WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
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NORTH-HOLLAND MATHEMATKS STUDIES
97
Wave Phenomena: ModernTheory and Applications Edited by
C. ROGERS University of Waterloo Canada and
T. BRYANT MOODIE University ofAlberta Canada
1984 NORTH-HOLLAND -AMSTERDAM
NEW YORK
OXFORD
@ElsevierSciencePublishers B.V., 1984 All rights reserved. N o part of rhispublication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocoping, recording or otherwise, without the prior permission of the copyrighr o wner.-
ISBN: 0444875867
Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 IOOOBZ AMSTERDAM T H E NETHERLANDS Soledisiibutors f o r the U . S . A .and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A.
Library of Congress Calaloging in Publicalion Data
Main entry under title: Wave phenomena.
(North-Holland mathematics studies ; 97) Papers presented at an international meeting held at the University of Toronto, June 20-24, 1983 under the auspices of the Canadian Applied Mathematics Society. Includes bibliographical references. 1. Wave-motion, Theory of--Addresses, essays, lectures. I. Rogers, C. 11. Moodie, T. Bryant (Thomas Bryant), 111. Canadian Applied Mathematics Society. 1944IV. Series. 84-14142 QA927.W375 1984 531’ .1133 ISBN 0-444-87586-7
.
PRINTED IN T H E NETHERLANDS
PREFACE
T h i s volume c o n t a i n s t h i r t y f i v e c o n t r i b u t i o n s t o t h e i n t e r n a t i o n a l m e e t i n g Wave Phenomena: Modern Theory and A p p l i c a t i o n s h e l d a t t h e U n i v e r s i t y o f T o r o n t o June 20 - June 24, 1983 under t h e auspices of t h e Canadian A p p l i e d Mathematics S o c i e t y and s u p p o r t e d by t h e N a t u r a l Sciences and E n g i n e e r i n g Research Council of Canada. The B r i t i s h Council k i n d l y s u p p l i e d a d d i t i o n a l t r a v e l s u p p o r t . The purpose o f t h e m e e t i n g was t o b r i n g t o g e t h e r r e s e a r c h e r s o f i n t e r n a t i o n a l s t a t u r e i n v a r i o u s aspects o f wave t h e o r y , b o t h l i n e a r and n o n l i n e a r . The c o n t e n t s o f t h i s volume bear w i t n e s s t o t h e d i v e r s i t y o f p h y s i c a l phenomena embraced by t h e subject. The e d i t o r s w i s h t o express t h e i r deep g r a t i t u d e t o a l l c o n t r i b u t o r s whose a r t i c l e s made p o s s i b l e t h e p r e s e n t proceedings. E s p e c i a l thanks a r e due t o R.A. Ross f o r o r g a n i s a t i o n a l s u p p o r t and t o Helen Warren f o r h e r e x c e l l e n t e d i t o r i a l assistance.
vi
LIST OF CONTENTS
V
Papers
1
EQUATIONS OF EVOLUTION AND WAVES A1 an Jeffrey RATIONAL FUNCTION FREQUENCY EXTRAPOLATION I N ULTRASONIC TOMOGRAPHY F. S t e n g e r , M.J.
B e r g g r e n , S.A.
J o h n s o n and C.H.
NONLINEAR T R A V E L L I N G WAVES I N F L U I D S P.L.
19
Wilcox 35
Sachdev
SECOND ORDER WAVE INTERACTION W I T H LARGE STRUCTURES
49
M. Rahman ACCELERATION WAVES I N I S O T R O P I C CONSTRAINED THERMOELASTIC MATERIALS B.D.
71
Reddy
A LAGRANGIAN VIEW OF WAVE-MEAN FLOW INTERACTION
83
R. G r i m s h a w A THEORY OF DYNAMICAL FERROELECTRICITY
99
P e t e r J. Chen SOLITARY WAVE SOLUTIONS OF GENERALISED KORTEWEG-DE V R I E S EQUATIONS F.A.
GEOMETRICAL CRYSTAL ACOUSTICS R.S.D.
113
Thomas and H. C o h e n
RIEMANN I N V A R I A N T S A.M.
105
Howes
123
Grundland
VARIATIONAL FORMULATION OF THE SINGULAR SURFACE PROPAGATION I N NON-
193
S I M P L E E L A S T I C MATERIALS
Jacek T u r s k i FORCED INTEGRABLE SYSTEMS
163
D a v i d J. K a u p INHOMOGENEOUS PLANE WAVES I N INCOMPRESSIBLE E L A S T I C MATERIALS M i c h a e l Hayes
175
List of Contents
CAN ACOUSTIC FOCUSING GENERATE TURBULENT SPOTS?
vii
7 93
G r e g o r y A. K r i e g s m a n n a n d E d w a r d L. R e i s s NONLINEAR WAVES I N THE PELLET FUSION PROCESS
V.J. E r v i n , W.F.
199
Ames a n d E . Adams
ENTROPY PRINCIPLE, SYMMETRIC HYPERBOLIC SYSTEMS AND SHOCK WAVES
21 1
Tommaso R u g g e r i DIFFERENTIAL AND DISCRETE SPECTRAL PROBLEMS AND THEIR INVERSES P.J.
221
Caudrey
WAVEFRONT FIELDS I N THE SCATTERING OF TRANSIENT ELASTIC WAVES BY A
233
SURFACE-BREAKING CRACK J u l i u s M i k l o w i t z and Constantine Chazapis WIENER-HOPF ANALYSIS:
AN APPRAISAL AND A NON-STANDARD APPLICATION
253
H a r o l d L e v i ne STRESS NAVE PROPAGATION I N DISCRETE RANDOM SOLIDS
267
Martin Ostoja-Starzewski DO L I O U V I L L E ' S SOLUTIONS POSSESS TOPOLOGICAL CHARGE?
279
George L e i b b r a n d t DISLOCATIONS I N THE DYNAMIC ANALYSIS OF CRACKS
293
L o u i s M. B r o c k A BACKLUND TRANSFORMATION FOR A NONLINEAR TELEGRAPH EQUATION
299
B r i a n S e y m o u r a n d E r i c Varley ON LINEAR DISPERSIVE WAVES WITH D I S S I P A T I O N
307
Francesco Mainardi L I M I T I N G GASDYNAMIC THEORY AND I T S APPLICATIONS W.H.
SCATTERING OF ACOUSTIC WAVES T.S.
3 19
H u i a n d H.J. Van R o e s s e l
BY IMPEDANCE SURFACES
329
A n g e l 1 and R.E. K l e i n m a n
INTERACTING NONLINEAR WAVES I N A NEURAL CONTINUUM MODEL:
337
ASSOCIATIVE MEMORY AND PATTERN RECOGNITION M a r c u s S. Cohen SOLITON DYNAMIC STATES I N THE SINE-GORDON SYSTEM
peter Leth C h r i s t i a n s e n
349
viii
List of Contents
AN ASYMPTOTIC A N A L Y S I S O F ACOUSTIC SCATTERING B Y NEARLY R I G I D
361
OR SOFT TARGETS G r e g o r y A. K r i e g s m a n n , A n d r e w N o r r i s and E d w a r d L. R e i s s SYMMETRY REDUCTION FOR NONLINEAR WAVE EQUATIONS I N RIEMANNIAN AND PSEUDO-
373
RIEMANNIAN SPACES A.M.
G r u n d l a n d , J . H a r n a d and P. W i n t e r n i t z
INTEGRAL REPRESENTATIONS FOR SOLUTIONS TO SOME D I F F E R E N T I A L EQUATIONS
391
THAT A R I S E I N WAVE THEORY R o b e r t F. M i l l a r NONLINEAR WAVE PHENOMENA I N THE B R A I N
409
R o b e r t M. M i u r a SOLITON SOLUTIONS TO ZAKHAROV-SHABAT SYSTEMS BY THE REDUCTION METHOD
423
J . H a r n a d , Y . S a i n t - A u b i n and S. S h n i d e r M U L T I P L E SCATTERING BY SH WAVES BY RANDOMLY D I S T R I B U T E D D I S S I M I L A R
433
SCATTERE RS M a t t h e w F. M c C a r t h y and M i c h a e l M. C a r r o l l AN INTEGRAL EQUATION A R I S I N G I N D I F F R A C T I O N THEORY R.A.
Ross
453
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
EQUATIONS OF EVOLUTION AND WAVES
Alan J e f f r e y Department o f E n g i n e e r i n g Mathematics U n i v e r s i t y o f Newcastle upon Tyne U.K.
INTRODUCTION The purpose o f t h i s paper i s t o d i s c u s s some o f t h e many d i f f e r e n t e v o l u t i o n e q u a t i o n s which a r i s e i n c o n n e c t i o n w i t h wave m o t i o n , some d i r e c t l y and o t h e r s as a s y m p t o t i c a p p r o x i m a t i o n s . Wave m o t i o n i t s e l f may be regarded as t h e m a n i f e s t a t i o n o f some f o r m o f i n h e r e n t i n s t a b i l i t y i n a process, so perhaps i t i s t o be expected t h a t t h e e q u a t i o n s which govern i t s many d i f f e r e n t aspects b e l o n g t o no c l e a r l y d e f i n e d c l a s s . The e v o l u t i o n e q u a t i o n s d i s c u s s e d h e r e by way o f example emphasise t h i s v a r i e t y o f t y p e , w h i l e a t t h e same t i m e showing something o f t h e d i v e r s i t y of b e h a v i o u r w h i c h can a r i s e i n c o n n e c t i o n w i t h wave p r o p a g a t i o n . We s h a l l , f o r example, see a way i n which e q u a t i o n s o f d i f f e r e n t c o m p l e x i t y can d e s c r i b e t h e b e h a v i o u r o f t h e same i n i t i a l v a l u e problem, and how t h i s can cause a s o l u t i o n t o t h e s i m p l e r f o r m o f e q u a t i o n (degenerate s o l u t i o n ) t o " f o r g e t " some o f t h e i n i t i a l d a t a t h a t gave r i s e t o i t . Then, i n t h e h y p e r b o l i c case, t h e b e h a v i o u r o f an a c c e l e r a t i o n t y p e o f wave w i l l be seen t o i n v o l v e an o r d i n a r y d i f f e r e n t i a l e q u a t i o n o f t h e B e r n o u l l i t y p e . Depending on t h e c o e f f i c i e n t s o f t h i s e q u a t i o n , so t h e i n t e n s i t y o f t h e a c c e l e r a t i o n wave may, o r may n o t , become i n f i n i t e c a u s i n g t h e c o r r e s p o n d i n g s o l u t i o n o f t h e h y p e r b o l i c system t o produce a shock ( d i s c o n t i n u o u s ) s o l u t i o n . More extreme s t i l l i s t h e b e h a v i o u r o f a s i m p l e q u a s i l i n e a r h y p e r b o l i c system i n which i t w i l l be seen t h e s o l u t i o n i t s e l f becomes unbounded a f t e r a f i n i t e t i m e (blow-up t i m e ) d e s p i t e t h e i n i t i a l d a t a b e i n g bounded and C1. F o r e q u a t i o n s of t h i s t y p e no g e n e r a l i s a t i o n o f t h e s o l u t i o n i s p o s s i b l e which w i l l e n a b l e i t t o be extended beyond t h e blow-up t i m e . I m p l i c i t i n some o f t h e s e examples, and i n o t h e r s t h a t w i l l a l s o be discussed, i s t h e i d e a t h a t t h e s t r u c t u r e o f t h e s o l u t i o n t o an i n i t i a l v a l u e problem o f t e n s i m p l i f i e s a f t e r a s u i t a b l y l o n g t i m e . By o n l y s t u d y i n g t h e s o l u t i o n a f t e r a s u f f i c i e n t l a p s e o f t i m e we a r e , i n e f f e c t , c o n s i d e r i n g i t s f o r m a t l o c a t i o n s f a r removed f r o m t h e i n i t i a l l i n e . A c c o r d i n g l y , i f t h e s o l u t i o n i s c o n s i d e r e d t o d e f i n e a f i e l d , t h i s t y p e of s o l u t i o n may be termed a f a r f i e l d f o r t h e problem. Far f i e l d s i n v o l v e the asymptotic behaviour o f t h e s o l u t i o n t o the o r i g i n a l problem i n some a p p r o p r i a t e sense. As an example o f t h i s l a s t process we c o n s i d e r l o n g waves i n s h a l l o w w a t e r i n t h e cases when t h e f l u i d has a c o n s t a n t d e n s i t y , and when i t i s s t a b l y and c o n t i n u o u s l y s t r a t i f i e d . The a n a l y s i s f o r t h e s e d i f f e r e n t s i t u a t i o n s e s t a b l i s h e s t h a t t h e Korteweg-de V r i e s (KdV) e q u a t i o n governs t h e wave m o t i o n i n each case, though w i t h d i f f e r e n t c o n s t a n t c o e f f i c i e n t s . Thus t h e s o l i t o n s o l u t i o n s which a r e so f a m i l i a r i n t h e c o n s t a n t d e n s i t y case o c c u r a l s o i n a c o n t i n u o u s l y
1
A. Jeffrey
2 stratified fluid.
Rather than d i s c u s s t h e w e l l documented case o f KdV s o l i t o n s , we d e s c r i b e i n s t e a d a c o m p a r a t i v e l y new f o r m o f s o l i t o n c a l l e d t h e l o o p s o l i t o n which a r i s e s i n t h e case o f a v e r y f l e x i b l e rod. These l o o p s o l i t o n s have t h e t r u e s o l i t o n p r o p e r t y t h a t despite being s o l u t i o n s t o a nonlinear equation they preserve t h e i r i d e n t i t y e x a c t l y a f t e r i n t e r a c t i o n w i t h one a n o t h e r . The l o o p s o l i t o n s a l s o p r o v i d e a simple p h y s i c a l e x p l a n a t i o n o f t h e phase change t h a t o c c u r s as a r e s u l t o f such interaction. F i n a l l y , an example i s g i v e n o f a s t o c h a s t i c o r d i n a r y d i f f e r e n t i a l e q u a t i o n w h i c h c o u l d a r i s e i n wave p r o p a g a t i o n t h r o u g h a medium c o n t a i n i n g randomly d i s t r i b u t e d inhomogeneities. Although t h e random c o e f f i c i e n t i n t h e e q u a t i o n i s d e s c r i b e d by a g e n e r a l i s e d random t e l e g r a p h process, which i s a s t a t i o n a r y random process, t h e moments o f t h e s o l u t i o n depend on t h e t i m e , showing t h e s o l u t i o n i t s e l f does n o t e v o l v e as a s t o c h a s t i c process which i s s t a t i o n a r y i n t i m e . DEGENERATE SOLUTIONS AND FAR FIELDS The degeneracy o f s o l u t i o n s t h a t a r i s e s when c o n s i d e r i n g f a r f i e l d s i n wave p r o p a g a t i o n problems i s w e l l i l l u s t r a t e d by a s i m p l e l i n e a r example i n v o l v i n g t h e telegraph equation
i n which a , b and c a r e c o n s t a n t s . Here, when a , b > 0, t h e l a s t two terms r e p r e s e n t d i s s i p a t i v e e f f e c t s i n t h e system. I n terms of t h e wavenumber k and frequency
the function
= RefA e x p ( i ( k x
u(x.t)
w i l l be a s o l u t i o n o f (2.1) when
W,
-
wt))}
(2.2)
and k s a t i s f y t h e d i s p e r s i o n r e l a t i o n
w
T h i s shows t h e s o l u t i o n u s u a l l y e x h i b i t s t h e e f f e c t s o f b o t h d i s s i p a t i o n and d i s p e r s i o n , f o r (2.2) t h e n becomes
iI
i
u ( x , t ) = Re A e x p ( - a t / 2 ) exp i k x 2
& (4c2 k2
t
46
-
a2)?’))>.(2.4)
L e t us t a k e as t h e i n i t i a l c o n d i t i o n s f o r (2.1)
Then, u s i n g (2.41, a f t e r some c a l c u l a t i o n i t f o l l o w s by use o f t h e F o u r i e r i n t e g r a l t h a t t h e s o l u t i o n o f (2.1) s u b j e c t t o (2.5) i s u(x,t) = t
1
1 c exp(- a t / 2 )
Q(X - c t ) i
rtct &
exp(- at/E)[o(x + c t )
t
~ ( s )
x-ct
I([ a2-4b F] % ( c 2 t 2
-
(~-s)~)’]ds
3
Equations of Evolution and Waves
+ 1 c exp(- a t / 2 ) x
(c2t2 - (x
-
s)~)']~s
,
where I denotes t h e m o d i f i e d Bessel f u n c t i o n o f t h e f i r s t k i n d . The e n t i r e s o l u t i o n ( 2 . 6 ) i s seen t o decay l i k e e x p ( - a t / 2 ) as shown i n (2.4). However, t h e f i r s t group o f terms r e p r e s e n t s a damped s o l u t i o n o f t h e o r d i n a r y wave e q u a t i o n , w h i l e t h e o t h e r two terms r e p r e s e n t a decaying r e s i d u a l s o l u t i o n determined by t h e domain o f dependence [ x - c t , x + c t l on t h e i n i t i a l l i n e of t h e p o i n t ( x , t ) i n question. I f t h e i n i t i a l d a t a has compact s u p p o r t e x p r e s s i o n (2.6) s i m p l i f i e s , as may b e seen by c o n s i d e r i n g t h e s i t u a t i o n i n which o ( x ) i s a r e c t a n g u l a r p u l s e and ~ ( x =) c o n s t . o v e r t h e p u l s e b u t i s z e r o e l s e w h e w . The r e s u l t i s i l l u s t r a t e d i n F i g . 1 which shows t h e r e s o l u t i o n o f t h e i n i t i a l p u l s e i n t o two damped wedge-shaped s o l u t i o n s o f t h e o r d i n a r y wave e q u a t i o n moving t o t h e r i g h t and l e f t , l i n k e d by t h e decaying r e s i d u a l s o l u t i o n .
The p o s i t i o n and shape o f these p u l s e s s u i t a b l y l o n g a f t e r t h e s t a r t o f t h e d i s t u r b a n c e r e p r e s e n t s t h e s a l i e n t f e a t u r e o f t h e s o l u t i o n a t such a t i m e . We now show h e u r i s t i c a l l y how t h i s i n f o r m a t i o n may be determined d i r e c t l y f r o m (2.1) w i t h o u t t h e complication o f the r e s i d u a l s o l u t i o n being involved. I f t h e s o l u t i o n t o (2.1) i s considered t o d e f i n e a f i e l d , then i t i s appropriate t o c a l l t h e s o l u t i o n s describing these pulses t h e f a r - f i e l d s o f (2.1).
As t h e damped wedge-shaped s o l u t i o n s i n (2.6) p r e s e r v e t h e i r shape as t h e y propagate t h e y must be f r e e f r o m d i s p e r s i o n . From (2.4) t h i s can be seen t o be t h e case i f a2 = 4b when, i n c i d e n t a l l y , t h e t a i l i n ( 2 . 6 ) vanishes. S e t t i n g a2 = 4b i n ( 2 . 1 ) t h e n enables i t t o be w r i t t e n i n e i t h e r o f t h e two forms ( ' t c a x + -aa
]
( g - c a au x + l1a u ] = ~ ,
(2.7)
or
L e t t h e functions
u")
s a t i s f y t h e f o l l o w i n g two f i r s t o r d e r e q u a t i o n s
and (2.10) Then u = u(') a r e a u t o m a t i c a l l y s o l u t i o n s o f (2.1) i n which a' = 4b , though t h e y a r e degenerate s o l u t i o n s s i n c e ( 2 . 1 ) i s a second o r d e r e q u a t i o n whereas t h e u(*) a r e s o l u t i o n s o f f i r s t o r d e r e q u a t i o n s . The i n t e r p r e t a t i o n o f u ( * ) p r e s e n t s no problem, because (2.9) and (2.10) show t h e u(') a r e propagated along t h e C(*) c h a r a c t e r i s t i c s o f ( Z . l ) , respectively, given by (2.11) The
u(+)
s o l u t i o n o f (2.1) propagates t o t h e r i g h t and t h e
u(-)
solution t o
4
A. Jeffrey
4u
Figure 1 the l e f t . If or
v epresents e i t h e r u(+) or u ( - ) and t denotes the time along a C (+I C(-! characteristic, i t follows t h a t (2.9) and (2.10) are each o f the form
5
Equations of Evolution and Waves
*+ dt
1 a v = 0 2
-
along
.
characteristics
C")
(2.12)
S o l v i n g (2.12) a l o n g a c h a r a c t e r i s t i c g i v e s v ( t ) = v(0) exp(- at/2)
,
(2.13)
where v(0) i s t h e v a l u e o f v a t t h e p o i n t a t which t h e c h a r a c t e r i s t i c c u t s t h e i n i t i a l l i n e . T h i s i s t h e r e q u i r e d s o l u t i o n , f o r i t shows t h a t t h e ' n i t i a l shape o f v = u(') i s p r e s e r v e d as i t propagates a l o n g t h e r e s p e c t i v e C!)' The f u n c t i o n s cha a c t e r i s t i c s , though t h e s o l u t i o n s decays l i k e e x p ( - a t / 2 ) a r e a s y m p t o t i c s o l u t i o n s t o ( 2 . 1 ) , and t h e r e d u c t i o n from two t o one o f t h e u(' o r d e r o f the equation governing t h e f u l l s o l u t i o n t o t h e order o f the equation g o v e r n i n g t h e a s y m p t o t i c s o l u t i o n i s t y p i c a l o f such s o l u t i o n s . I n a sense we may say t h e f a r f i e l d s have " f o r g o t t e n " some o f t h e i n i t i a l c o n d i t i o n s which gave r i s e t o them.
.
7
I n passing, we remark t h a t a s p e c i a l case o f ( 2 . 1 ) a r i s e s when a = KO. and K i s l a r g e . For s e t t i n g E = 1/K , e q u a t i o n ( 2 . 1 ) becomes
,b
= Kg
(2.14) ax T h i s may be analysed r i g o r o u s l y i n t h e f o r m o f a s i n g u l a r l y p e r t u r b e d i n i t i a l v a l u e problem f o r which e r r o r e s t i m a t e s and an e x i s t e n c e theorem can be e s t a b l i s h e d . By c o n s t r u c t i n g a formal a s y m p t o t i c expansion i n terms o f E and u s i n g a c o n t r a c t i o n mapping p r i n c i p l e f o r o p e r a t o r s i n a Banach space a s o l u t i o n can be shown t o e x i s t c l o s e t o t h e a s y m p t o t i c expansion f o r s u i t a b l y s m a l l E . T h i s e q u a t i o n a r i s e s i n overdamped v i b r a t i o n problems [ll,i n t h e p r o p a g a t i o n of r a d i a t i o n t h r o u g h a h i g h l y a b s o r b i n g medium and a v a r i e t y o f s i t u a t i o n s i n continuum mechanics [2,31.
BREAKDOWN OF S O L U T I O N S U n l i k e t h e s o l u t i o n t o t h e t e l e g r a p h e q u a t i o n s which e x i s t s f o r a l l t > 0 , t h e s o l u t i o n t o n o n l i n e a r e v o l u t i o n e q u a t i o n s need n o t n e c e s s a r i l y e x i s t f o r a r b i t r a r i l y large t The s t a n d a r d example b y which t h i s may be i l l u s t r a t e d [2,41 i s t h e C1 s o l u t i o n o f
.
au % + f ( u ) jy= 0 , at
(3.1)
subject t o the i n i t i a l condition u(x,O) = @ ( X )
, --
<
x <
-
(3.2)
We may w r i t e ( 3 . 1 ) as du = 0 dt along t h e c h a r a c t e r i s t i c s
C
g i v e n by
c
:
;;
---
(3.3)
f(u)
t
(3.4)
showing u = c o n s t . a l o n g t h e s e c h a r a c t e r i s t i c s , which a r e m e r e l y s t r a i g h t l i n e s . I n t e g r a t i n g (3.4) t o f i n d t h e c h a r a c t e r i s t i c t h r o u g h t h e p o i n t ( 5 , 0 ) and u s i n g (3.2) we f i n d t h e c h a r a c t e r i s t i c s C have t h e f o r m x = 5 + t f (cJ(5)) Y where 5 i s now t o be regarded as a parameter. As o f t h e s t r a i g h t l i n e c h a r a c t e r i s t i c t h r o u g h ( 5 , 0) u i s g i v e n i m p l i c i t l y by
(3.5) u = ~ ( 5 )a t every p o i n t i n (3.5), i t f o l l o w s t h a t
A. Jeffrey
6
-
u = @ (x
.
t f(u))
(3.6)
D i f f e r e n t i a t i o n o f (3.6) then shows - f(u) a # (x - t f ( u ) ) 1 + t f ' ( u ) o t (x - t f(u)) '
au =
at
(3.7)
and
Consequently, a t any t i m e
t
f o r which
1 + t f'(u) ol (x - t f(u)) = 0 y (3.9) these d e r i v a t i v e s become i n f i n i t e and u ceases t o be a C1 f u n c t i o n . This i s t h e s o - c a l l e d g r a d i e n t catastrophe, and f o r any given i n i t i a l data t h e l e a s t time t c f o r which (3.9) i s t r u e i s t h e maximum time f o r which (3.1) possesses a ~1 s o l u t i o n . The relevance o f t h i s t o more general wave propagation problems may be seen by considering a special s o l u t i o n t o an i n i t i a l value problem f o r the homogeneous quasi 1i n e a r h y p e r b o l i c system
-aU+ at
A(U)
au
,
0
(3.10)
,...,
un i n which U i s an n x 1 vector w i t h elements u1 , u2 u2 ,..., un) an n x n m a t r i x w i t h c o e f f i c i e n t s a i j = a i j ( u 1
.
and A(U) Here, by
is
A(n) o f A are r e a l and h y p e r b o l i c we mean t h e eigenvalues ~ ( l , ) A(2),..., t h e r e i s a f u l l s e t o f l i n e a r l y independent eigenvectors. Systems o f t h i s type are o f frequent occurrence i n p h y s i c a l problems C4,5,61. Let us consider a s o l u t i o n U o f (3.10) i n which n - 1 elements a r e functions o n l y o f t h e remaining element, say u1 , so we w r i t e U = U(u1) This i s , i n f a c t , j u s t a generalised simple wave s o l u t i o n . S u b s t i t u t i o n i n t o (2.9) then gives
.
aul
aul
[TI + =A(;) with
I
(3.11)
the u n i t m a t r i x .
Non-trivial solutions f o r
dfi/dul
12 so t h a t i f
A
I
+aula xA ( U-) I
i s an eigenvalue o f aul
-/ at There w i l l be t h e r e w i l l be
n n
w i l l o n l y e x i s t when = 0
,
A(;) au
2 ax =
r e a l eigenvalues different solutions
(3.12)
-x(u,)
A(’) u(i) 1
. ,...
(3.13)
Y A(n) satisfying
of
A
, so
i t follows
(3.14) s u b j e c t t o some gixen i n i t i a l c o n d i t i o n . Once s o l u t i o n s ~ ( 1 )= ~ ( u ( 1 ) ) f o l l o w from (2.10). 1
ul(j)
has been found t h e
n
7
Equations of Evolution and Waves
Each o f t h e n e q u a t i o n s (3.14) i s o f t h e f o r m ( 3 . 1 ) , s o a g r a d i e n t c a t a s t r o p h e Each s o l u t i o n may a r i s e i n any one o f t h e s e n d i f f e r e n t s o l u t i o n s U(i) U ( i ) r e p r e s e n t s a p o s s i b l e f a r f i e l d . However, i f t h e g r a d i e n t c a t a s t r o p h e occurs i n such a f a r f i e l d , i t s t i m e o f o c c u r r e n c e p l a c e s an upper bound on t h e t i m e o f e x i s t e n c e f o r t h a t f a r f i e l d as a C1 f u n c t i o n .
.
When system (3.10) i s i n d i v e r g e n c e form, and t h u s r e p r e s e n t s a system o f c o n s e r v a t i o n laws, t h e g r a d i e n t c a t a s t r o p h e corresponds t o t h e development o f a shock wave f r o m a smooth s o l u t i o n . Shock waves a r e non-unique d i s c o n t i n u o u s s o l u t i o n s whose jumps across t h e d i s c o n t i n u i t y s u r f a c e and whose speed o f p r o p a g a t i o n s a t i s f y a system of n o n l i n e a r a l g e b r a i c e q u a t i o n s . F o r a d i s c u s s i o n o f t h e i r p r o p e r t i e s and t h e way i n which a u n i q u e p h y s i c a l s o l u t i o n may be s e l e c t e d f r o m amongst t h e m a t h e m a t i c a l l y p o s s i b l e b u t n o n p h y s i c a l s o l u t i o n s we r e f e r t o t h e f o l l o w i n g r e f e r e n c e s [4,5,6,71. I t i s i n t e r e s t i n g t o n o t i c e t h a t even i n problems i n which t h e f i e l d e q u a t i o n s a r e l i n e a r a g r a d i e n t c a t a s t r o p h e and t h e e n s u i n g shock waves can r e s u l t i f t h e c o n s t i t u t i v e e q u a t i o n s c o n s i d e r e d a r e n o n l i n e a r . T h i s i s t h e case, f o r example, f o r waves i n a f e r r o m a g n e t i c medium, f o r a l t h o u g h t h e f i e l d e q u a t i o n s a r e t h e Maxwell e q u a t i o n s t h e c o n s t i t u t i v e e q u a t i o n s a r e n o n l i n e a r and o f t h e f o r m [81.
PROPAGATION OF ACCELERATION WAVES
-
BERNOULLI EQUATION
I n s t e a d o f c o n s i d e r i n g t h e b e h a v i o u r o f t h e complete s o l u t i o n , e i t h e r t o t h e f u l l e q u a t i o n s g o v e r n i n g some wave p r o p a g a t i o n phenomenon o r t o t h e i r a s y m p t o t i c form, i t o f t e n s u f f i c e s t o c o n s i d e r o n l y t h e b e h a v i o u r o f some a t t r i b u t e o f t h e s o l u t i o n . The s i m p l e s t case i s when a L i p s c h i t z d i s c o n t i n u i t y i n a f i e l d v a r i a b l e governed by h y p e r b o l i c e q u a t i o n s o c c u r s across some i n i t i a l s u r f a c e . We may t h e n ask how i t s i n t e n s i t y as c h a r a c t e r i s e d by t h e jump i n t h e d e r i v a t i v e across t h e s u r f a c e v a r i e s as t h e d i s c o n t i n u i t y propagates. I n t h r e e dimensions t h i s i n v o l v e s t h e geometry o f c h a r a c t e r i s t i c s u r f a c e s and r a y s and we r e f e r t o t h e f o l l o w i n g r e f e r e n c e s f o r d e t a i l s o f t h e a n a l y s i s I f t h e v a r i a t i o n o f t h e i n t e n s i t y i s considered along a r a y o f the [9,10,111. system t h i s may be shown t o be governed by an e v o l u t i o n e q u a t i o n which i s an o r d i n a r y d i f f e r e n t i a l equation o f t h e B e r n o u l l i type w i t h a f u n c t i o n o f the a r c l e n g t h a l o n g t h e r a y as t h e e v o l u t i o n parameter. An a n a l y s i s o f t h e p r o p e r t i e s o f a B e r n o u l l i e q u a t i o n w i t h v a r i a b l e c o e f f i c i e n t s s u b j e c t t o c o n d i t i o n s which u s u a l l y a p p l y i n p r a c t i c e p r o v i d e s theorems f o r t h e e s t i m a t i o n o f t h e b e h a v i o u r o f t h e s o l u t i o n C9,121. O f p a r t i c u l a r i n t e r e s t i s t h e case i n which t h e i n t e n s i t y becomes unbounded a f t e r a f i n i t e e l a p s e d t i m e t c , c a l l e d t h e escape t i m e f o r t h e s o l u t i o n . T h i s i s u s u a l l y t h e t i m e a t which a d i s c o n t i n u o u s s o l u t i o n ( s h o c k ) forms. To i l l u s t r a t e t h i s s i t u a t i o n we c o n s i d e r a s i m p l e p r o b l e m i n v o l v i n g t h e d e t e r m i n a t i o n o f t h e t i m e o f b r e a k i n g o f w a t e r waves i n a channel o f a r b i t r a r i l y v a r y i n g d e p t h and w i d t h . I f h ( x ) i s t h e a r b i t r a r i l y v a r y i n g depth below t h e e q u i l i b r i u m l e v e l , q ( x , t ) i s t h e s u r f a c e e l e v a t i o n above t h e e q u i l i b r i u m l e v e l and W(x) i s t h e s l o w l y v a r y i n g w i d t h , t h e s h a l l o w w a t e r e q u a t i o n s t a k e t h e f o r m
-au+ at
and
u -au + ax
qan= 0 . ax
A. Jeffrey
8
a
an
+
ax Cu(n + h ) l
-a+t
U(V
+
h)
(4.2)
Then, supposing a disturbance advances i n t o water a t r e s t , t h e r e i s a wavefront across which an/ax i s discontinuous. Denote t h e jump i n a d a x across t h e wavefront by a ( x ) , then since the water ahead of t h e wavefront i s a t r e s t a ( x ) i s simply t h e value o f an/ax immediately behind t h e wavefront. A s t r a i g h t forward a n a l y s i s C131 shows t h a t a ( x ) , which i s t h e i n t e n s i t y o f t h i s a c c e l e r a t i o n - t y p e wave, s a t i s f i e s t h e B e r n o u l l i equation (4.3) I n t e g r a t i o n o f (4.3) gives
where
a.
, Wo
= a(0)
= W(0)
and
A wave o f e l e v a t i o n corresponds t o a. < 0 and a wave o f depression t o a. > 0 The wave w i l l be s a i d t o break i f f o r some x = xc the water surface behind the wavefront becomes v e r t i c a l , so t h a t a(x,) =
-.
.
We conclude from (4.4) t h a t : ( a ) A wave o f e l e v a t i o n always breaks i n water o f f i n i t e depth i f I ( x ) i s such t h a t 1 + a0 I ( x c ) = 0 , and xc > 0 i s f i n i t e . I f t h e depth o f the water shelves t o zero a t x = II , say, so t h a t h ( i ) = 0 , a wave o f e l e v a t i o n propagating towards the shore w i l l break before reaching t h e shore l i n e i f l a g [ > l / I ( a ) , and a t the s h o r t l i n e i f ]a01 < l/I(.t) (b) A wave o f depression can o n l y break i f t h e depth o f the water shelves t o zero, and then o n l y a t t h e shore l i n e provided I ( e ) <
-.
.
As an example o f the more extreme behaviour t h a t i s p o s s i b l e w i t h s o l u t i o n s o f h y p e r b o l i c equations we c i t e the f o l l o w i n g i n i t i a l value problem C41. q u a s i l i n e a r h y p e r b o l i c system
-aU + at
A - aU =
0
ax
The
(4.6)
with ’-
‘9 ’ u
u2
=
,A
=
cosh 2u2
0
cosh u2
0
-
sinh
on t h e i n t e r v a l
x
E
C-h
,h l
, u ~ ( x 0, ) with
= 0
h > 0
.
u2
(4.7)
cosh 2u2
s i n h 2u2 0 ,u3, i s required t o satisfy the i n i t i a l conditions u ~ ( x 0) , = x/h
s i n h 2u2’
J
,
u 3 ( x , 0) = -x/h
I t s solution i s
(4.8)
9
Equations of Evolution and Waves
(4.9)
and t h e s e f u n c t i o n s a r e seen t o become i n f i n i t e when example o f t h e "blow-up" o f a s o l u t i o n .
t = h
, so
we have an
Since b o t h t h e d e r i v a t i v e s and t h e s o l u t i o n i t s e l f become i n f i n i t e a t t h e blow-up t i m e t = h , t h e s o l u t i o n o f t h i s i n i t i a l v a l u e problem cannot be extended beyond t h i s t i m e . The s i t u a t i o n here i s i n c o n t r a s t w i t h t h e o r d i t l a r y g r a d i e n t c a t a s t r o p h e where o n l y t h e d e r i v a t i v e s become i n f i n i t e . T h i s u s u a l l y corresponds t o shock f o r m a t i o n , w i t h t h e c l a s s o f weak s o l u t i o n s t o which shocks belong p r o v i d i n g a g e n e r a l i s a t i o n o f C1 s o l u t i o n s t o a c l a s s c o n t a i n i n g discontinuous solutions. We remark t h a t t h e blow-up o f t h e s o l u t i o n i n t h e example d i s c u s s e d h e r e i s n o t due t o i n t e r s e c t i o n o f c h a r a c t e r i s t i c s , s i n c e t h e t h r e e f a m i l i e s o f c h a r a c t e r i s t i c s belonging t o (4.7) are j u s t f a m i l i e s o f p a r a l l e l s t r a i g h t l i n e s w i t h g r a d i e n t s -1 , 0 and 1 , r e s p e c t i v e l y . I n f a c t t h e system i s an example o f a c o m p l e t e l y e x c e p t i o n a l system o f e q u a t i o n s 141. The s t u d y o f t h e p r o p a g a t i o n o f L i p s c h i t z d i s c o n t i n u i t i e s p r o v i d e s o n l y l o c a l r e s u l t s on t h e r e g u l a r i t y o r breakdown o f s o l u t i o n s . To o b t a i n g l o b a l r e s u l t s d i f f e r e n t approaches a r e necessary. We s h a l l n o t d i s c u s s t h i s m a t t e r here, b u t r e f e r i n s t e a d t o t h e work o f L e v i n e and Payne 1141, and t o t h a t o f Glassey [151, as examples o f what may be achieved, and a l s o f o r f u r t h e r references. ASYMPTOTIC SOLUTION TO A NONLINEAR SCALAR EQUATION The g e n e r a l i s e d s i m p l e waves c o n s i d e r e d i n Sec. 3 were a s p e c i a l f o r m o f s o l u t i o n t o a f i r s t o r d e r q u a s i l i n e a r h y p e r b o l i c system. The d u r a t i o n o f t h e i r e x i s t e n c e was l i m i t e d by t h e t i m e a t which t h e s o l u t i o n t o t h e g o v e r n i n g e v o l u t i o n e q u a t i o n au -a +t x(u) jauy = 0 1 ceased t o be a C f u n c t i o n . A d i f f e r e n t s i t u a t i o n may a r i s e i f an a s y m p t o t i c s o l u t i o n i s sought i n o t h e r circumstances. To i l l u s t r a t e i d e a s we s h a l l c o n s i d e r t h e b e h a v i o u r o f w a t e r waves governed by t h e Boussinesq e q u a t i o n [2,161
(5. Here u ( x , t ) i s a one-dimensional wave f i e l d , c determines t h e magnitude o f t h e d i s p e r s i v e e f f e c t .
i s t h e phase speed and
p
We t a k e as o u r s t a r t i n g p o i n t t h e f a c t t h a t a s t u d y o f t h e a s y m p t o t i c b e h a v i o u r o f l o n g waves f o r t h e l i n e a r i s e d f o r m o f t h i s e q u a t i o n i n w h i c h t h e r i g h t - h a n d s i d e o f (5.1) i s n e g l e c t e d shows t h a t s i m i l a r i t y o f a s y m p t o t i c b e h a v i o u r h o l d s f o r a coordinate transformation s a t i s f y i n g
10
A. Jeffrey
z =
- ct = const. ( 3 c ~ / 2 ) l / t1/3 ~
(5.2)
T h i s suggests t h a t i n t h e f u l l n o n l i n e a r f o r m o f t h e Boussinesq e q u a t i o n ( 5 . 1 ) we adopt t h e s c a l e t r a n s f o r m a t i o n 5 = E ~ ( x/ -~ A t ) , T = ~ 3 1 % where A i s an as y e t a r b i t r a r y speed a t which t h e new c o o r d i n a t e system i s moving. We then f i nd ( 5 . 1 ) becomes
Making t h e f o r m a l a s y m p t o t i c expansion
(5.4) and s u b s t i t u t i n g t h i s i n t o (5.3) b r i n g s us t o a system o f p e r t u r b a t i o n e q u a t i o n s The f i r s t two a r e 2 O(E
2
1:
(A
2
- c
2
12 -
(5.5)
aE
and
Setting x = i velocity c . o r d e r terms on x = c , so t h e ( 5 . 6 ) t h a t u1
c causes t h e c o o r d i n a t e frame ( E ; , T ) t o move w i t h t h e phase T h i s s a t i s f i e s e q u a t i o n (5.5) a u t o m a t i c a l l y and makes t h e second t h e l e f t hand s i d e o f ( 5 . 6 ) v a n i s h . Making t h e s p e c i f i c c h o i c e wave moves t o t h e r i g h t , we t h e n f i n d a f t e r an i n t e g r a t i o n o f s a t i s f i e s t h e equation
3+ i3, u [F] 2c 1 at aT
3
a c3
(5.7)
As would be expected i n t h e l o n g wavelength case, t h i s i s s i m p l y t h e Korteweg-de V r i e s (KdV) e q u a t i o n , and i t i s known f r o m s t u d i e s o f i t s s o l u t i o n s t h a t t h e y remain a t l e a s t C1 f u n c t i o n s . The s o l i t o n s o l u t i o n s o f ( 5 . 7 ) a r e , i n f a c t , C f u n c t i o n s . There i s t h u s no g r a d i e n t c a t a s t r o p h e w i t h t h i s e q u a t i o n because t h e s t e e p e n i n g produced by t h e h y p e r b o l i c p a r t o f t h e e q u a t i o n c o m p r i s i n g t h e f i r s t two terms i s balanced and smoothed by t h e t h i r d t e r m which i s d i s p e r s i v e i n nature. The KdV e q u a t i o n ( 5 . 7 ) i s n o t expressed i n terms o f t h e o r i g i n a l v a r i a b l e s x and t , b u t t h e r e d u c t i o n o f o r d e r a s s o c i a t e d w i t h a s y m p t o t i c r e s u l t s i s a g a i n apparent f o r i t i s f i r s t o r d e r i n t i m e ( u n i - d i r e c t i o n a l waves), whereas t h e Boussinesq e q u a t i o n (5.1) which gave r i s e t o i t was second o r d e r i n t i m e ( b i d i r e c t i o n a l waves). The change o f c o o r d i n a t e s t o g and T used above, though
11
Equations of Evolution and Waves
suggested by somewhat d i f f e r e n t r e a s o n i n g , l e d T a n i u t i and Wei C171 t o develop t h e i r r e d u c t i v e p e r t u r b a t i o n method f o r g e n e r a l n o n l i n e a r h y p e r b o l i c systems C6,16,181. I t can b e shown t h a t t h e KdV e q u a t i o n a l s o governs l o n g waves on t h e s u r f a c e o f a s t a b l y and c o n t i n u o u s l y s t r a t i f i e d f l u i d . T h i s i s a s i t u a t i o n i n which t h r e e t i m e s c a l e s a r i s e and we now i n d i c a t e t h e e s s e n t i a l f e a t u r e s o f t h e approach necessary t o e s t a b l i s h t h i s r e s u l t C191.
The g o v e r n i n g e q u a t i o n s a r e
q =0 ,
div
-
*+ aT
39. + a-r
(5.8)
N
cisgrad p = 0 ,
(5.9)
- F , -p t h e pressure,
(q.grad)q + 1 grad p =
-
P
(5.10)
where T i s t h e t i m e , p t h e f l u i d d e n s i t y , q the fluid v e l o c i t y and t h e e x t e r n a l f o r c e a c t i n g on t h e f l u i d ( g r a v i t y ) . The axes a r e t a k e n as shown i n F i g . 2 w i t h 9 h a v i n g components U i n t h e X - d i r e c t i o n and V i n t h e Y-direction.
Figure 2 The f l u i d i s assumed t o b e s t a b l y and c o n t i n u o u s l y s t r a t i f i e d w i t h d i m e n s i o n l e s s d e n s i t y 7 ( Y ) , e q u i l i b r i u m depth h and a t y p i c a l l o n g wavelength L , w h i l e t h e wave m o t i o n which i s assumed t o be c r e a t e d by a d i s t u r b a n c e a t X = 0 has dimensionless amplitude E w i t h respect t o L
.
We i n t r o d u c e t h e d i m e n s i o n l e s s v a r i a b l e s X
u =
U/fi
= x/L
,y
with
U = E L / T ~, v = V / v
=
Y/h , t =
TIT
0
with
=
Eh/TO
(5.11)
w i t h T~ t h e p e r i o d o f a t y p i c a l o s c i l l a t i o n . T a k i n g a r e f e r e n c e d e n s i t y p o (average d e n s i t y ) and t h e c o r r e s p o n d i n g r e f e r e n c e p r e s s u r e po = pogh we have i n the disturbed state
- = p o ( 7 ( y ) + Ep*(X p
I
y
I
TI)
(5.12)
12
A. Jeffrey
w i t h p*(X , Y , T ) the dimensionless d e n s i t y p e r t u r b a t i o n . approximation then a l l o w s us t o w r i t e
-
p = P0(P(Y)
EP*U
+
where P(Y) i s t h e e q u i l i b r i u m pressure and pressure p e r t u r b a t i o n .
Y
The usual h y d r o s t a t i c
Y)) ,
*
(5.13)
,Y )
p (X
t h e dimensionless
I n terms o f these q u a n t i t i e s t h e dimensionless form o f equations (5.8) t o (5.10) becomes
au+av=o ay ax ap at+ EU % +
6(f(Y) +
EP)
I
v f'(y)
[s t
(5.14)
' +
E
*1
,
= 0
ay
(5.15) (5.16)
ax EU
av -+ ax
EV
ayl
where 6 = (h/L)' and f , p and p correspond t o a f t e r t h e change t o dimensionless v a r i a b l e s .
+ 2P ay
T,
p*
0 ,
t p =
and
(5.17)
p*, respectively,
The usual boundary c o n d i t i o n s a r e a p p l i e d a t t h e f r e e s u r f a c e r e q u i r i n g i t t o be a surface o f constant pressure w i t h a v e l o c i t y equal t o t h a t o f t h e f l u i d .
If w
i s a t y p i c a l f i e l d v a r i a b l e we have
w =W(X,Y,t;E,G) , so we seek a d i r e c t asymptotic expansion i n t h e form
w = woo +
E Wl0
+ 6 wol
...
+
(5.18)
which w i l l y i e l d s e c u l a r terms i n v o l v i n g w j 0 and wol . I n o r d e r t o o b t a i n a u n i f o r m l y v a l i d expansion procedure, i n which we may consider simultaneously the l i m i t s E + 0 , 6 + 0 , we d e f i n e two slow time v a r i a b l e s t' =
E t
and
,
t* = s t
t , tt
and seek an expansion i n v o l v i n g t h e t h r e e time scales w = woo(x I Y I t
3
tt
3
t*) + EWl0(X
.Y
+ 6WO1(X,Y, t , t t , t* ) +
3
t
and
t
*
(5.19)
, tt , t* 1
... .
(5.20)
This then y i e l d s t h e t y p i c a l formal r e s u l t (5.21) The u n i f o r m l y v a l i d l e a d i n g term WOO i s then found by r e q u i r i n g t h e vanishing o f t h e s e c u l a r term i n (Ew10 + 6 ~ 0 1 ) i n t h e f i r s t o r d e r approximation. The outcome o f t h i s approach i s t h e KdV equation
i!
-_
E
aF a'F - 0 2AF - 62C ac at3 -
,
(5.22)
13
Equations of Evolution and Waves
w i t h 5 = U t - x and U , A and C c o n s t a n t s . To t h i s o r d e r o f a p p r o x i m a t i o n t h e non-dimensional h e i g h t o f t h e f r e e s u r f a c e i s t h e n Given b y y
=
1 + EF ( ~ , t ) .
(5.23)
T h i s same r e s u l t c o u l d have been o b t a i n e d by a number o f d i f f e r e n t approaches C181, b u t t h i s one shows c l e a r l y t h e r o l e s p l a y e d by E and 6 and t h e way i n which d i f f e r e n t t i m e s c a l e s e n t e r i n a n a t u r a l manner. B e f o r e t u r n i n g t o a n o t h e r t o p i c , i t i s a p p r o p r i a t e a t t h i s p o i n t t h a t some remarks s h o u l d be o f f e r e d about s o l i t o n s . L e t us c o n s i d e r t h e e q u a t i o n
I t was shown by Konno e t a l . 1201 t h a t t h i s e q u a t i o n i s r e l a t e d t o t h e t r a n s v e r s e o s c i l l a t i o n o f a v e r y e l a s t i c beam and t o t h e m o t i o n o f a s t r e t c h e d rope, w i t h t t h e time, y t h e displacement measured normal t o t h e undeformed a x i s of t h e r o p e taken t o be t h e x - a x i s , and s t h e a r c l e n g t h measured a l o n g t h e r o p e . I t was a l s o showed t h a t a t r a v e l l i n g wave s o l u t i o n i n t h e f o r m o f a l o o p i s p o s s i b l e and t h i s was c a l l e d t h e l o o p s o l i t o n .
By g e n e r a l i s i n g t h e AKNS scheme f o r s o l v i n g i n v e r s e s c a t t e r i n g problems C16,211 Konno and J e f f r e y C221 s o l v e d t h e general n l o o p s o l i t o n problem. T h e s o l u t i o n i l l u s t r a t e s s o l i t o n i n t e r a c t i o n v e r y c l e a r l y , and i n t h i s case p r o v i d e s a s i m p l e p h y s i c a l e x p l a n a t i o n f o r t h e phase change t h a t a r i s e s a f t e r i n t e r a c t i o n .
The s m a l l e r a l o o p s o l i t o n i s t h e f a s t e r i t t r a v e l s , and F i g . 3 shows t h e o v e r t a k i n g o f a l a r g e slow l o o p s o l i t o n by a s m a l l e r f a s t e r one, The phase d e l a y i s seen t o be caused m a i n l y by t h e t i m e taken f o r t h e s m a l l s o l i t o n t o t r a v e l around t h e l o o p o f t h e l a r g e r one. The phase s h i f t i t s e l f i s shown i n F i g . 4 i n which t h e s o l i d l i n e s a r e t h e l o c i o f t h e two l o o p s o l i t o n s and t h e c h a i n d o t t e d l i n e i s t h e l o c u s t h e s m a l l e r l o o p s o l i t o n would have f o l l o w e d were i t n o t f o r t h e i n t e r a c t i o n . The l o c u s d u r i n g t h e i n t e r a c t i o n has been o m i t t e d .
STOCHASTIC EVOLUTION EQUATIONS N o t a l l p h y s i c a l processes a r e p u r e l y d e t e r m i n i s t i c , and wave m o t i o n t h r o u g h a random medium i s o f t e n o f c o n s i d e r a b l e importance. T h i s happens, f o r example, when s m a l l i n h o m o g e n e i t i e s a r e d i s t r i b u t e d i n a random f a s h i o n t h r o u g h o u t a medium t h r o u g h w h i c h a wave i s t o pass. The s t u d y o f such problems i s w e l l d e s c r i b e d by Hersh [231 and by Papanicolaou and K e l l e r C241, and i s a l s o d i s c u s s e d i n 1161. The i l l u s t r a t i v e example chosen here i n v o l v e s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n w i t h a c o e f f i c i e n t w h i c h i s a g e n e r a l i s a t i o n o f t h e random t e l e g r a p h process i n t r o d u c e d by K e n r i c k 1251. The o r d i n a r y random t e l e g r a p h process i s a f u n c t i o n which may t a k e t h e values 1 w i t h equal p r o b a b i l i t y , and i s s u b j e c t t o f u r t h e r assumptions r e g a r d i n g t h e manner i n w h i c h i t s w i t c h e s f r o m one v a l u e t o t h e o t h e r . The g e n e r a l i s a t i o n here i n v o l v e s a l l o w i n g t h e f u n c t i o n t o assume two a r b i t r a r y values w i t h a r b i t r a r i l y assigned p r o b a b i l i t i e s , s u b j e c t o n l y t o t h e i r sum b e i n g u n i t y .
A. Jeffrey
14
-
"2
-
v2
t,
c
t 2 c t3
v,
.=<
Figure 3
Figure 4
v,
15
Equations of Evolution and Waves
S p e c i f i c a l l y , we c o n s i d e r a random f u n c t i o n [al
, defined
a(t)
on
0
with probability
p1
with probability
p2 = 1
al
with probability
1
a2
with probability
2bp26t + o ( 6 t ) ,
a(t) =
-
p1
5
by
t <
(6.1)
.
Ia2 Also, i f
a ( t ) = al
a(t
-
, then
6t) =
1
-
2bp26t + o ( 6 t ) (6.2)
.
I t follows from t h i s d e f i n i t i o n t h a t w i t h a s i m i l a r c o n d i t i o n i f a ( t ) = a2 t h e c o n s t a n t b has a p h y s i c a l i n t e r p r e t a t i o n t h r o u g h t h e f a c t t h a t t h e average number o f changes o f v a l u e p e r u n i t t i m e i s 4bp1p2 . The mean v a l u e
a(t)
of
a
is
S.
a(t) > =
and t h e a u t o c o r r e l a t i o n f u n c t i o n o f < a(t
a
+
T)
and < a ( t + T ) a ( t ) s t a t i o n a r y random process.
As
+ p2a2 = const.,
= p a
a(t)
>
is
+ p1p2(al
a(t) > =
(6.3)
a r e independent o f
- a2) 2 e - 2 b l ~ l t
t h e process
(6.4) a(t)
is a
F o r o u r s t o c h a s t i c e v o l u t i o n e q u a t i o n we c o n s i d e r t h e s i m p l e s t p o s s i b l e v a r i a b l e c o e f f ic i e n t e q u a t i o n d t = -a(t)y
,
(6.5)
i n which a ( t ) i s t h e g e n e r a l i s e d random t e l e g r a p h process (6.1) and subject t o the i n i t i a l condition y(0) = 0
y
is
.
(6.6)
The s o l u t i o n y ( t ) w i l l be random so i t s p r o p e r t i e s must be d e s c r i b e d s t a t i s t i c a l l y . Rather t h a n w r i t i n g down t h e form o f s o l u t i o n t o (6.5) and ( 6 . 6 ) which i s
and t h e n s e e k i n g t o f i n d t h e p r o p e r t i e s o f y ( t ) f r o m those o f a ( t ) t o be e a s i e r t o i n f e r t h e s t a t i s t i c a l p r o p e r t i e s o f t h e f u n c t i o n y ( t ) d i f f e r e n t i a l e q u a t i o n i t s a t i s f i e s and t h e p r o p e r t i e s o f a ( t ) .
, it
proves from t h e
The d e t a i l s o f t h e a n a l y s i s w i l l be g i v e n elsewhere, and i t w i l l s u f f i c e h e r e m e r e l y t o say t h a t t h e y i n v o l v e d e r i v i n g and s o l v i n g a p a r t i a l d i f f e r e n t i a l e q u a t i o n f o r h ( y , t ) d e f i n e d by PrIy
5
y(t)
I t can be shown t h a t t h e moments
5
y + dyl
< yn >
of
=
h(y
y(t)
,t)dy
.
have t h e f o r m
16
where
A. Jeffrey
r\n)
,
rhn) > 0
r ( n ) = b + 21( a l 1, 2
with
+ a2) n
J b2 + b ( a l f a 2 - 2 a ) n + l ( a 4
1
- a 2 ) 2 n2
.
(6.9)
The a u t o c o r r e l a t i o n f u n c t i o n < y ( t + T ) y ( t ) > may be s i m i l a r l y expressed and a l s o i n v o l v e s t h e t i m e t . The presence o f t i n b o t h t h e moments < yn > and t h e a u t o c o r r e l a t i o n f u n c t i o n show t h e s o l u t i o n y ( t ) e v o l v i n g f r o m (6.5) i s n o t a s t a t i o n a r y random f u n c t i o n , d e s p i t e t h e f a c t t h e random c o e f f i c i e n t a ( t ) i s such a f u n c t i o n .
If,f o r example, wave p r o p a g a t i o n was t o be governed b y an e q u a t i o n such as ( 6 . 5 ) a l o n g a c h a r a c t e r i s t i c , then t h e f a r f i e l d would be s t o c h a s t i c i n n a t u r e w i t h i t s moments g i v e n by ( 6 . 8 ) . I n l i n e a r wave p r o p a g a t i o n t h e f a r f i e l d c o u l d comprise a d e t e r m i n i s t i c p a r t and a s t o c h a s t i c p a r t , each o f which c o u l d be examined s e p a r a t e l y . The problem becomes f a r more c o m p l i c a t e d when n o n l i n e a r wave p r o p a g a t i o n i s i n v o l v e d . REFERENCES L11 Smith, D.R. and Palmer, J.T., On t h e b e h a v i o u r o f t h e s o l u t i o n o f t h e t e l e g r a p h i s t ’ s e q u a t i o n f o r l a r g e a b s o r p t i o n , Arch. Rat. Mech. Anal. 39 (1970), 146-157. L2 1 Whitham, G.B.,
L i n e a r and N o n l i n e a r Waves ( W i l e y , New York, 1974).
C31 Geel, R., S i n g u l a r P e r t u r b a t i o n s o f H y p e r b o l i c Type, Thesis, Mathematisch Centrum, Amsterdam (1978). C41 J e f f r e y , A., Q u a s i l i n e a r H y p e r b o l i c Systems and Waves, Research N o t e s i n Mathematics 5 (Pitman P u b l i s h i n g , London, 1976).
151 Smoller, J . , Shock Waves and R e a c t i o n - D i f f u s i o n Equations ( S p r i n g e r , B e r l i n , 1982).
C61 T a n i u t i , 1983)
.
T. and N i s h i h a r a , K. (Ed. A. J e f f r e y ) (Pitman P u b l i s h i n g , London,
171 L i u , T.P., A d m i s s i b l e S o l u t i o n s o f H y p e r b o l i c Conservation Laws, Amer. Math. SOC. Memoir, No. 240, Providence ( 1 9 8 1 ) . I 8 1 Katayev, I.G., 191 Chen, P.J.,
E l e c t r o m a g n e t i c Shock Waves ( I l i f f e , London, 1966).
S e l e c t e d Topics i n Wave P r o p a g a t i o n ( N o o r d h o f f , Leyden, 1976).
C l O l Achenbach, J.D., Gautesen, A.K. and McMaken, H., Ray Methods f o r Waves i n E l a s t i c S o l i d s , Monographs and S t u d i e s i n Mathematics 14 (Pitman P u b l i s h i n g , London, 1982). C l l l E n g e l b r e c h t , J . , N o n l i n e a r Wave Processes o f Deformation i n S o l i d s , Monographs and S t u d i e s i n Mathematics 16 (Pitman P u b l i s h i n g , London, 1983).
Equations of Evolution and Waves
17
C121 Menon, V . V . , Sharma, V.D. and J e f f r e y , A., On t h e g e n e r a l b e h a v i o u r o f a c c e l e r a t i o n waves, Appl. A n a l y s i s , 1983 ( i n p r e s s ) . E l 3 1 J e f f r e y , A. and Mvungi, J., On t h e b r e a k i n g o f w a t e r waves i n a channel o f a r b i t r a r i l y v a r y i n g depth and w i d t h , J. Appl. Math. & Phys. (ZAMP), 31 (1980) 758-761. 1141 L e v i n e , H.A. and Payne, L.E., On n o n e x i s t e n c e o f g l o b a l s o l u t i o n s t o some a b s t r a c t Cauchy problems o f s t a n d a r d and non-standard t y p e s , Rend. Matematica 8 (1975) 413-428. C151 Glassey, R., Blow-up theorems f o r n o n l i n e a r wave e q u a t i o n s , Math. Z e i t 132 (1973) 183-203. C161 J e f f r e y , A. and Kawahara, T., A s y m p t o t i c Methods i n N o n l i n e a r Wave Theory, A p p l i c a b l e Mathematics S e r i e s (Pitman P u b l i s h i n g , London, 1982). Reductive p e r t u r b a t i o n method i n n o n l i n e a r wave 1171 T a n i u t i , T. and Wei, C.C., p r o p a g a t i o n , J. Phys. SOC. Japan 24 (1968) 941-946. 1187 J e f f r e y , A. and K a k u t a n i , T . , Weak n o n l i n e a r d i s p e r s i v e waves: a d i s c u s s i o n c e n t e r e d around t h e Korteweg-de V r i e s e q u a t i o n , S I A M Rev. 14 (1972) 582-643. 1191 J e f f r e y , A., ( i n press).
The KdV e q u a t i o n i n s t r a t i f i e d f l o w , App. A n a l y s i s , 1983
1201 Konno, K . , Ichikawa, Y.H. and Wadati, M., A l o o p s o l i t o n p r o p a g a t i n g a l o n g a s t r e t c h e d rope, 3. Phys. SOC. Japan 50 (1981) 1025-1026. 1211 A b l o w i t z , M.J., Kaup, D.J., Newell, A.C. and Segur, H., N o n l i n e a r e v o l u t i o n e q u a t i o n s o f p h y s i c a l s i g n i f i c a n c e , Phys. Rev. L e t t . 31 (1973) 125-127. 1221 Konno, K. and J e f f r e y , A., Someremarkable p r o p e r t i e s o f two l o o p s o l i t o n s o l u t i o n s , J. Phys. SOC. Japan 52 (1983) 1-3. 1231 Hersh, R., Random e v o l u t i o n s : J. Math. 4 (1974) 443-477.
A s u r v e y o f r e s u l t s and problems, Rocky M t .
1241 Papanicolaou, G. and K e l l e r , J.B., S t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s w i t h a p p l i c a t i o n s t o random harmonic o s c i l l a t o r s and wave p r o p a g a t i o n i n random media, S I A M J. Appl. Math. 21 (1971) 287-305. The a n a l y s i s of i r r e g u l a r m o t i o n s w i t h a p p l i c a t i o n s t o t h e energy f r e q u e n c y spectrum o f s t a t i c and o f t e l e g r a p h s i g n a l s , P h i l . Mag. Ser. 7, 7 (1929) 176-196.
[251 K e n r i c k , G.W.,
This Page Intentionally Left Blank
Wave Phenomena: Modern Theory and Applicatlons C. Rogers and T.E. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
19
RATIONAL FUNCTION FREQUENCY EXTRAPOLATION I N ULTRASONIC TOMOGRAPHY F. S t e n g e r ' Department o f Mat hemat ics U n i v e r s i t y o f Utah M. J. Berggren' y 3
Department o f B i o e n g i n e e r i n g U n i v e r s i t y o f Utah S.A.
Johnsonzy3
Department o f B i o e n g i n e e r i ng U n i v e r s i t y o f Utah C.H. ~ i l c o x " ~ Depa r tmen t o f Math emat ics U n i v e r s i t y o f Utah I n t h i s paper we d e s c r i b e a procedure f o r s o l v i n g t h e i n v e r s e s c a t t e r i n g problem i n u l t r a s o n i c imaging. Although t h e d e r i v a t i o n o f t h e method i s based on t h e Helmholtz e q u a t i o n model, O'U
(11
+ kz(l + f)u = 0
i t i s a p p l i c a b l e t o any model f o r w h i c h t h e s p a c i a l sound p r e s s u r e u = u ( r , k ) s a t i s f i e s an a s y m p t o t i c e q u a l i t y o f the form
where k i s p r o p o r t i o n a l t o t h e frequency, P denotes t h e r a y p a t h a l o n g which t h e p r e s s u r e wave t r a v e l s from t h e source p o i n t Fs t o t h e d e t e c t o r p o i n t and u i s a p o s i t i v e c o n s t a n t . The method i s Dased on p r e d i c t i n g +(-) = IpF(?)ds v i a a r a t i o n a l f u n c t i o n procedure, u s i n g s e v e r a l v a l u e s + ( k l ) , $ ( k 2 ) ... , $(kZm+l) . The method i s i l l u s t r a t e d f o r t h e case o f t h e Rytov a p p r o x i m a t i o n t o ( l ) , i n w h i c h case t h e p a t h s P a r e s t r a i g h t l i n e s . A p e r t u r b a t i o n method f o r c o r r e c t i n g f o r curved r a y paths i s a l s o d e s c r i b e d . The a l g o r i t h m can a l s o be m o d i f i e d t o image m a t e r i a1 s w i t h more c o m p l i c a t e d frequency dependent a t t e n u a t i o n . Examples o f images r e c o n s t r u c t e d f r o m computer s i m u l a t e d d a t a w i t h and w i t h o u t Gaussian a d d i t i v e n o i s e a r e g i v e n . The b e n e f i c i a l e f f e c t o f a n o i s e t o l e r a n t f i r s t norm d a t a f i t t i n g a l g o r i t h m i n i m p r o v i n g image q u a l i t y i s shown.
rd
F. Stenger et al.
20 1.
INTRODUCTION AND SUMMARY
T t - ir2thods d e r i v e d i n t h i s paper a r e based on t h e H e l m h o l t z d i f f e r e n t i a l e q u a t i o n model
v 2u + k 2 ( 1 + f ) u = O where u denotes 3 dimensions, and
t k i
(1.1) i s t h e Laplacian operator i n
s D a c i a l Cound pressure, v2 2
f(F) =
co C2(
r)
1
.
I n Eq. ( 1 . 2 ) , = (x,y,z) , co i s t h e speed o f sound i n t h e r e g i o n ( u s u a l l y a l i q u i d ) s u r r o u n d i n g t h e body B , k = W / C O , where w = 27 x ( f r e q u e n c y ) and c ( F ) i s t h e "speed" o f sound a t t h e p o i n t F i n B We assume t h a t B i s a subset o f a volume V , and t h a t sound sources and d e t e c t o r s a r e l o c a t e d on a V I We have d i s p l a y e d t h e word "speed" i n quotes above, s i n c e i n t h e boundary o f V t h e d e r i v a t i o n o f Eq. ( 1 . 1 ) c ( F ) i s indeed t h e speed o f sound a t t h e p o i n t F , whereas, i n r e a l l i f e , m a t e r i a l s have a t t e n u a t i o n , and t h e r e f o r e I m f > 0 In t h i s paper we d e r i v e
.
.
.
V = V1 u B
F i g u r e 1.1:
a numerical a l g o r i t h m f o r r e c o n s t r u c t i n g t h e f u n c t i o n
F in
V
, based
=
m
on t h e model ( 1 . 1 ) .
The f a c t t h a t t h e f u n c t i o n c(F) t u r n s o u t t o be complex, as w e l l as o t h e r c r i t i c i s m s o f ( 1 . 1 ) have r a i s e d q u e s t i o n s w i t h regards t o t h e v a l i d i t y o f t h e model ( l . l ) , and t h i s has l e d s e v e r a l a u t h o r s [1,41 as w e l l as some o f us C31 t o d e r i v e o t h e r models. However, under e i t h e r p l a n e wave o r p o i n t source e x c i t a t i o n , a l l o f t h e s e newly d e r i v e d models i n c l u d i n g ( 1 . 1 ) have t h e common f e a t u r e t h a t
1
rd =
log 5
1
F ds + o ( 1 )
,k
+
-
(1.4)
1
where uk(F) i s t h e s p a c i a l sound p r e s s u r e and 1 i s a p a t h i n V a l o n g w h i c h t h e p r e s s u r e wave t r a v e l s f r o m i t s source p o i n t rs t o t h e d e t e c t o r p o i n t Fd While o t h e r i n v e r s i o n t e c h n i q u e s a r e based on t h e p a r t i c u l a r model, t h e method o f
.
21
Frequency Extrapolation in Ultrasonic Tomography
t h i s paper a p p l i e s t o any such model.
r
i n p r a c t i c e , and t h e y a r e Although f and Uk a r e n o t Cm f u n c t i o n s o f t h e r e f o r e d i f f i c u l t t o compute, we show t h a t t h e f u n c t i o n
-
$(k)
=
1 l o g uk(F) ik
(1.5)
7
rS
i s a v e r y smooth and s l o w l y v a r y i n g f u n c t i o n o f k t h e r e a l l i n e , where ko > 0 Indeed, on Cko,ml the relation
.
$(k) =
1
F ds
+
O(k-&)
on an i n t e r v a l Ckg,-l of function $(k) satisfies
, the
,k
-f
-,
(1.6)
L where 6 i s a p o s i t i v e c o n s t a n t depending o n l y on cannot g i v e a complete p r o o f , we e x p e c t t h a t $ ( k ) sector S(ko,e) = t k
E
C : Ikl
2
.
f I n a d d i t i o n , a l t h o u g h we i s a n a l y t i c and bounded i n a
ko , [ a r g k l s 8 1
(1.7)
and where k o and e a r e p o s i t i v e c o n s t a n t s . R e c e n t l y d i s c o v e r e d r e s u l t s by a p p r o x i m a t i o n t h e o r y C81 have shown t h a t such a f u n c t i o n can b e v e r y a c c u r a t e l y approximated on Ckg,-l by a r a t i o n a l f u n c t i o n o f k . Thus i f $ ( k ) i s known o n l y on a s u b i n t e r v a l o f [ko,-l , we can e x p e c t t h e " p - a l g o r i t h m " ( s e e e.g. [ 1 2 ] ) t o y i e l d a v e r y a c c u r a t e a p p r o x i m a t i o n t o $(-) = IL F ds
.
To t h i s end, we have s i m u l a t e d some t e s t s f o r t h e case o f t h e Rytov a p p r o x i m a t i o n t o t h e s o l u t i o n o f (1.1), t a k i n g - b I F-F, I 2
f(F) i s a p l a n e wave. I n and when u! e v a l u a t e t h e Rytov a p p r o x i m a t i o n , l i n e s L p a r a l l e l t o t h e incoming
(1.8) t h i s case i t i s p o s s i b l e C6,71 t o e x p l i c i t l y R y t C u k ( r ) l and t h e paths L a r e s t r a i g h t f i e l d uf By computing = e
.
a t 9 equi-spaced p o i n t s i n t h e range o f f r e q u e n c i e s between 1 4 megahertz, t h e p a l g o r i t h m enables us t o a c c u r a t e l y compute
megahertz and
( 1. l o ) T h i s s i m p l e a l g o r i t h m works w e l l 4 i f t h e numbers $ ( k ) can be computed a c c u r a t e I n p r a c t i c e , we t o 4 o r more s i g n i f i c a n t f i g u r e s , b u t breaks down a t 3.. e x p e c t t h a t we w i l l be a b l e t o measure $ ( k ) a c c u r a t e t o 3 s i g n i f i c a n t f i g u r e s . To t h i s end, we have i n t r o d u c e d an "a1-adjustment" t o o u r a l g o r i t h m , which i n e f f e c t computes t h e l i m i t of s e v e r a l r a t i o n a l f u n c t i o n i n t e r p o l a t i o n s o f $ ( k ) , and t h e n e v a l u a t e s t h e a1 minimum o f t h e s e l i m i t s . T h i s has t h e e f f e c t o f under-emphasizing t h e l a r g e r e r r o r s o f t h e s e l i m i t s , and i f t h e $ ( k ) a r e accurate t o 3 a c c u r a t e t o 3 s i g n i f i c a n t f i g u r e s , we a r e a b l e t o compute $(-) s ig n i f i c a n t f i g u r e s
.
22
F. Stenger e t a / .
I n o r d e r t o r e c o n s t r u c t an image from t h e $(-) , we use s t a n d a r d X-ray CT a l g o r i t h m s [21 which assume t h a t t h e p a t h s L a r e s t r a i g h t l i n e s L While t h i s a p p r o x i m a t i o n i s u s u a l l y n o t u n d u l y p e s s i m i s t i c f o r t h e case o f sound waves i n t i s s u e , i t i s n o t Exact. To t h i s end, we have i n t r o d u c e d a n u m e r i c a l scheme f o r c o r r e c t i n g t h e F which we c o n s t r u c t e d under t h e assumption t h a t t h e paths a r e s t r a i g h t l i n e s . T h i s method i s based on a d i s c r e t i z a t i o n o f t h e e q u a t i o n s r e c e n t l y d e r i v e d i n C71. W h i l e we have i n c l u d e d t h i s a l g o r i t h m f o r completeness, we have n o t y e t t e s t e d i t i n p r a c t i c e .
.
I n t h e f i n a l s e c t i o n o f t h i s paper we i l l u s t r a t e a r e c o n s t r u c t i o n v i a t h e p-algorithm o f the function (1.11) u s i n g s i m u l a t e d d a t a o b t a i n e d v i a t h e Rytov a p p r o x i m a t i o n . We a l s o i l l u s t r a t e t h e r e s u l t when 0.1% n o i s e was added t o t h e Rytov data, and f i n a l l y , we i l l u s t r a t e t h e r e s u l t when o u r a l - a d j u s t m e n t a l g o r i t h m i s a p p l i e d t o t h e p - a l g o r i t h m r e s u l t s computed u s i n g t h e n o i s y data. I t i s p o s s i b l e t h a t we c o u l d have o b t a i n e d b e t t e r r e s u l t s b y f i r s t a p p l y i n g t h e 21 smoothing a l g o r i t h m t o t h e d a t a , and then a p p l y i n g t h e r a t i o n a l e x t r a p o l a t i o n a l g o r i t h m . We s h a l l i n v e s t i g a t e t h i s procedure i n t h e f u t u r e . I n o u r model ( 1 . 1 ) we have assumed t h a t f i s independent o f w . We c o u l d s t i l l r e c o v e r f i f i t s f u n c t i o n a l f o r m were known e x p l i c i t l y , such as, i f
by s u i t a b l e normalization o f
f ( k , ? ) = e(F)k6 + l o g u k ( F ) , e.g.,
B(r) ,
(1.12)
by a l t e r i n g (1.5) t o ( 1 .13)
T h i s , t h e n would enable us t o r e c o v e r
$(-)
Jor(r)ds .
=
(1.14)
L Next, we c o u l d r e c o v e r
~ ( r by )
sampling
$(k)
= k'l$(k)
- $(-)I (1.15)
2.
INTEGRAL EQUATION FORMULATION OF THE HELMHOLTZ EQUATION
By means o f Green's theorem, one can d e r i v e t h e i n t e g r a l e q u a t i o n
Uk(')
= uF(r)
which i s e q u i v a l e n t t o ( . l )
+
k2
fliv
Gk(F-F')f(?')u(?')dV(F') ,
, where ikr e Gk(F) = 4nr '
23
Frequency Extrapolation in Ultrasonic Tomography
and where u t ( r ) denotes t h e i n p u t s p a c i a l p r e s s u r e wave. u s u a l l y has p o i n t sources o f t h e f o r m ikl u p ) = i n which
rs
I n p r a c t i c e one
r-rsI (2.3)
417I F-Fs I
denotes a source p o i n t , a l t h o u g h t h e p l a n e wave f o r m
i s sometimes a l s o used, where
E 3.
ASYMPTOTIC ESTIMATE OF
= (kx,ky,kz)
;
I[
(2.5)
l o g uk
L e t uk0 be d e f i n e d by e i t h e r Eqs. (2.3) o r 2 . 4 ) . o p t i c s t h e n p r e d i c t s t h a t ( s e e e.g. 13 p. 1341)
The t h e o r y o f g e o m e t r i c
L denotes t h e r a y p a t h emanating where i n t h e case o f u[- d e f i n e d by ( 2 . 3 ) , from t o t h e p o i n t rd , w h i l e i n t h e case o f u l d e f i n e d by ( 2 . 4 ) , L s t a r t s o u t i n t h e d i r e c t i o n o f k a t some p o i n t on t h e boundary o f V and ends up a t rd
rs
.
L e t us now assume t h a t f E L i p (V) a p p l i c a t i o n s we must always havA‘)o 2 o u r a l g o r i t h m , any p o s i t i v e v a l u e o f ( 3 . 1 ) , w i t h Fd r e p l a c e d by r i n t o we f i n d t h a t t h e o(1) t e r m i n ( 3 . 1 )
.
, f o r some u > 0 We b e l i e v e t h a t i n 1/2 , a l t h o u g h f o r purposes o f d e r i v i n g u w i l l s u f f i c e . Then, upon s u b s t i t u t i n g (2.1) and assuming L t o be any C1 p a t h , can be r e p l a c e d b y O(k-‘”) That i s ,
.
T h i s r e s u l t may be e s t a b l i s h e d v i a a procedure s i m i l a r t o t h a t a p p l i e d t o t h e e q u a t i o n s (2.2) and (1.7) o f C101, w i t h m 2 0 , n 2 0 and m + n = u
.
4.
ANALYTIC PROPERTIES OF THE SOLUTION
I n t h i s s e c t i o n we s t u d y t h e a n a l y t i c p r o p e r t i e s o f t h e s o l u t i o n Uk o f (2.1) as a f u n c t i o n o f k , i n o r d e r t o show t h a t we can p e r f o r m a c c u r a t e a p p r o x i m a t i o n d e f i n e d i n ( 1 . 5 ) on Cko,ml . o f the function
+
0 We s h a l l show t h a t w i t h Uk d e f i n e d by ( 2 . 3 ) , t h e s o l u t i o n uk(F) o f ( 2 . 1 ) i s The p o l e s o f t h i s f u n c t i o n o c c u r a t t h e a r a t i o o f two e n t i r e f u n c t i o n s o f k p o i n t s k = k, f o r w h i c h t h e homogeneous e q u a t i o n c o r r e s p o n d i n g t o (2.1) has s o l u t i o n s . We t h e n show t h a t t h e s e numbers km cannot be r e a l .
.
24
F. Stenger et al.
L e t V be a bounded r e g i o n i n IR3 , and l e t f E Lip(,)(V) , such t h a t I m f > 0 a.e. on V , and such t h a t f 0 on R3 - V Hence
.
V
2
2 Uk(r) + k [l+f(r)lUk(')
v 2 u k ( r ) + k2u k ( r )
Im k
Now Wilcox 1113 has shown t h a t i f o f (4.2) must s a t i s f y
11%-
= 0
1'
iku
0
2
= 0
,r
,IR3
-
V
on IR3
-
V , t h e n any s o l u t i o n
,r
,R
dS(r) + 0
v
E
(4.1
.
+
1
(4.2) u
(4.3)
m
r=R and
j
I u I 2 dS(r)
+
c(0
2 c
<
-)
,R
-+ m
.
(4.4)
r=R By Green's theorem, we have
j
j
uv2udV(F) +
r=
1
I v u I 2 dV(F) =
r2R
i
that i s ,
-k2
j
(l+f)(u['dV(r)
+
k2
1
( v u (2 dV(F) =
rsR
r=
k
(Im f ) l u \ ' d V ( ? )
i s real.
V
(4.5)
I
i
-!$dS(F) .
(4.6)
u=R
Then, by t a k i n g i m a g i n a r y p a r t s o f ( 4 . 6 ) ,
u au d S ( r )
I
= Im
dS(r)
r=R
r=R =
1
Im
u [ g
-
ikuldS(?) + I m ( - i k )
r=R
I
r=R
By a p p l y i n g S c h w a r t z ' s i n e q u a l i t y t o t h e f i r s t i n t e g r a l on t h e i g h t hand s i d e , l e t t i n g R = , u s i n g (4.3) and ( 4 . 4 ) , and r e c a l l i n g o u r assumption t h a t I m f > 0 a.e. on V , we g e t -+
0 < k2
[ (Im f ) I u I 2dV(?)
= -kc 2 0
V T h i s c o n t r a d i c t i o n proves LEMMA 4.1:
If
Im f > 0
a.e.
on
V
,
then t h e equation
cannot have any n o n t r i v i a l s o l u t i o n s f o r
k
real.
.
(4.8)
25
Frequency Extrapolation in Ultrasonic Tomography
Indeed, a t t e m p t s a t e x t e n d i n g Lemma 4.1 t o complex v a l u e s o f we s e t
1
k
suggest t h a t i f
f ( F ) d V ( r ) = peie (4.10)
V
IVI
dV(r)
=
,
V where
o ,o
<
e
- ~ / 8< 8 '
<
0
tan 20'
=
-
>
and i f
8'
i n t h e range
,p/lV
<
i s d e f i n e d by
+ v e tp
~1
71
-
(4.12)
cos
t h e n t h e e q u a t i o n ( 4 . 9 ) has no n o n t r i v i a l s o l u t i o n s i f -8'
(4.11)
< 1/2
8'
I kle"
k
satisfies
.
(4.13)
Although we cannot p r o v e t h i s , we e x p e c t t h a t t h e f o l l o w i n g assumption i s p r o b a b l y v a l i d i f t h e e q u a t i o n ( 4 . 1 ) has no s o l u t i o n s Uk f o r which t h e r a y p a t h s are trapped i n V : ASSUMPTION 4.2: If Im f 2 0 then t h e r e e x i s t s a sector S(ko,e)
=
V
in
tk
E
and i f (4.10) and (4.11) a r e s a t i s f i e d
C: Ik
where ko and e a r e p o s i t i v e numbers s o l u t i o n uk t h e n k ,d S(ko,e)
.
LEMMA 4.3:
The s o l u t i o n
u
2
ko
,
larg kl
5
(4.14)
el
such t h a t whenever (4.9) has a n o n t r i v i a l
o f Eq. ( 2 . ) may be r e p r e s e n t e d i n t h e f o r m (4.15)
where f o r each f i x e d
r ,X
and
Y
are e n t i r e functions o f
PROOF: We c a r r y o u t t h e p r o o f o n l y f o r t h e case o f u i p r o o f i n t h e case o f ( 2 . 4 ) i s s i m i l a r , and we o m i t i t .
k
.
d e f i n e d by ( 2 . 3 ) .
The
I n s t e a d o f ( 2 . 1 ) we c o n s i d e r t h e e q u a t i o n
(4.16)
We know f r o m t h e t h e o r y o f Fredholm i n t e g r a l e q u a t i o n s (see e.g. C41) t h a t f o r , t h e s o l u t i o n u o f Eq. (4.16) i s a r a t i o e v e r y f i x e d a E C and F = x,y,z) o f two e n t i r e f u n c t i o n s o f k , and hence o f k , i . e . ,
h
F. Stenger et a/.
26
u(k,a,?)
= X(k,a,F) Y ( k ,ay:)
(4.17)
where X and Y i n (4.17) a r e e n t i r e f u n c t i o n s o f k . I n s p e c t i o n o f t h e p r o o f o f t h i s r e s u l t f o r t h e case o f (4.12) shows t h a t f o r f i x e d k and and Y i n (4.17) a r e a l s o e n t i r e f u n c t i o n s o f a Hence f o r f i x e d , X(k,k,r) and Y(k,k,F) a r e e n t i r e f u n c t i o n s o f i( .
r
r
,
X
LEMMA 4 . 4 : I f f E L i p a ( V ) , if u! i s d e f i n e d by e i t h e r Eqs. (2.3) o r (2.4), and i f Assumption 4.2 h o l d s , t h e n t h e f u n c t i o n $ d e f i n e d by ( 1 . 5 ) i s a n a l y t i c and bounded i n t h e s e c t o r S(ko,e) , where i t s a t i s f i e s t h e r e l a t i o n
+(k)
=
I
Jl+f ds
,k
+ O(k-'/*)
-f
-
in
S(k0,8)
.
(4.18)
1 PROOF: By Assumption 4.2 t h e r e a r e no n o n t r i v i a l s o l u t i o n s Uk o f Eq. ( 4 . 9 ) , w i t h k i n S(k0.e) . Hence t h e f u n c t i o n U(k,r) i n Eq.(4.15) does n o t v a n i s h i f k E S(ko,e) . Hence $ ( k ) i s a n a l y t i c i n S(k0,e) Eq. (4.18) t h e r e f o r e f o l l o w s f r o m Eq. ( 3 . 2 ) .
.
5.
RATIONAL APPROXIMATION PREDICTION
I n t h i s s e c t i o n we s h a l l show t h a t we can " p r e d i c t " +(a)
=
J
Jl+f
ds
(5.1)
L using the Thiele algorithm f o r constructing a r a t i o n a l f u n c t i o n which i n t e r p o l a t e s We f i r s t prove $(k)
.
LEMMA 5.1: L e t $(k) be d e f i n e d as i n Lemma 4.2. Corresponding t o e v e r y i n t e g e r N > 0 , there e x i s t s a r a t i o n a l f u n c t i o n N c0k + c,kN-' + + cN r (k) = N kN + dlkN-' + + dN
... ...
such t h a t
(5.3) where c and C a r e p o s i t i v e c o n s t a n t s independent o f N and where d e f i n e d as i n Lemma 4.1. Indeed, f o r any E > 0 , we may t a k e
ko
where
.
u
i s d e f i n e d as i n (3.2) and
8 as i n (4.10).
Then
C = C(E)
is
PROOF: I n C91 a r a t i o n a l f u n c t i o n o f z s i m i l a r t o ( 5 . 2 ) was c o n s t r u c t e d , f o r a p p r o x i m a t i n g a f u n c t i o n $ ( z ) on -1 < z 5 1 , where $ was a n a l y t i c on t h e u n i t d i s c . A s i m p l e v a r i a t i o n o f t h e aFgument i n C91 y i e l d s (5.3) and (5.4). The i n e q u a l i t y ( 5 . 3 ) t e l l s us t h a t r a t i o n a l f u n c t i o n s t o a c o n s i d e r a b l y b e t t e r j o b o f a p p r o x i m a t i n g + ( k ) t h a n p o l y n o m i a l s i n l / k . For example, i f r$(k)-$(m)l ck-'/' , c f 0 , t h e n no m a t t e r how t h e c o n s t a n t s bo,bl, b, are
-
...,
27
Frequency Extrapolation in Ultrasonic Tomography
selected, N
max k0$y where
C'
.
/ $ ( k ) - j =O 1 bjk-JI
C'
> = Nu/2
i s a p o s i t i v e c o n s t a n t which depends n e i t h e r on t h e
(5.5) bj
n o r on
N
.
I n p r a c t i c e , t h e c o n s t a n t u i s n o t known a p r i o r i . N e v e r t h e l e s s by t h e l i n e a r methods o f [8,91, ( 5 . 3 ) i s s t i l l s a t i s f i e d f o r some p o s i t i v e c o n s t a n t s c and C We n o t e a l s o , f r o m ( 5 . 1 ) ,
( 5 . 2 ) and ( 5 . 3 ) , t h a t
T h i s f a c t , and t h e f a c t t h a t Lemma 5.1 t e l l s us t h a t $ ( k ) i s " v e r y n e a r l y " a r a t i o n a l f u n c t i o n , suggests t h a t t h e T h i e l e r a t i o n a l f u n c t i o n a l g o r i t h m s h o u l d p r o v i d e an e f f e c t i v e procedure f o r d e t e r m i n i n g co . Given t h e d a t a 2N
, $(kj)l
Ikj
(5.7) j =O
where t h e k j a r e d i s t i n c t and r e a l , t h e T h i e l e A l g o r i t h m f o r c o n s t r u c t i n g an r N ( k ) of t h e f o r m o f Eq. ( 5 . 2 ) i s d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n s :
,j
= $(kj)
P:
i
i-z+ki+j - k j
p. = p. J j+l
i-1 Pj+l
'
i-1
-
P j
= 0,l
,...,2N
j = O,l,
...,ZN-1
j = O,l,
..., 2N-i
i = 2,3,
..., 2N
Then t h e r a t i o n a l f u n c t i o n r N ( k ) o f t h e form ( 5 . 2 ) which i n t e r p o l a t e s p o i n t s k j may be computed v i a i t s c o n t i n u e d f r a c t i o n r e p r e s e n t a t i o n
0 rN(k) = Po
k - k o
a t the
(5.9)
1 k-kl PO
$
+ 20 P o - Po +...+
-
k2N-l 2N-2 Po - 0 0 2N
I n p a r t i c u l a r , we have
co 6.
.k'
ZN = Po
(5.10)
IMPROVEMENT OF N O I S Y DATA
has been t e s t e d on t h e f u n c t i o n ( 1 . 9 ) , The a l g o r i t h m ( 5 . 8 ) f o r p r e d i c t i n g $(-) under t h e assumption t h a t we can compute $ ( k ) i n t h e range o f f r e q u e n c i e s between 1 and 4 megahertz. Indeed, t a k i n g N as l o w as 3 and e v a l u a t i n g $ ( k ) a c c u r a t e t o 7 decimals we can compute t h e i n t e g r a l i n (1.10) t o 6 decimals accuracy. Our t e s t r u n s have shown however, t h a t w h i l e (5.8) s t i l l
28
F. Stenger et a/.
y i e l d s r e l i a b l e r e s u l t s when t h e numbers + ( k j ) a r e rounded t o 4 decimals accuracy, i t f a i l s when t h e y a r e rounded t o 3 We have found t h a t t h i s problem can be remedied by t a k i n g t h e ( u s u a l ) e l minimum o f these. That i s i f
.
a r e M such p r e d i c t i o n s o b t a i n e d by a p p l y i n g ( 5 . 8 ) t o M d i f f e r e n t s e t s o f d a t a o f t h e f o r m ( 5 . 7 ) , t h e e1 minimum o f t h e c r ) i s t h e number a which minim i zes
I n t h i s way, we a r e a b l e t o use n o i s y d a t a o f t h e form (5.7) which i s a c c u r a t e t o o n l y 3 decimals t o p r e d i c t $(-) = a a c c u r a t e t o 3 decimals. I n t h e case when t h e
c, 0
minaS(a)=rnin(m)S(co(m)) .
a r e r e a l , i t i s e a s i l y seen t h a t
Hence, even f o r general complex cbm), i t i s c o n v e n i e n t t o r e p l a c e t h e e x a c t s o l u t i o n a o f (6.1) by t h e approximation, a C( t i g , where C( and g a r e r e a l , and where
7.
CORRECTION FOR RAY REFRACTION
Our e x p e r i e n c e has shown t h a t t h e r a y paths 1 o f sound waves i n t i s s u e a r e We may t h u s use t h e e q u a t i o n d e r i v e d i n usually very n e a r l y s t r a i g h t l i n e s L [ 7 1 t o d e r i v e a c o r r e c t i o n on t h e F , where P i s c o n s t r u c t e d under t h e assumption t h a t t h e p a t h s 1 a r e s t r a i g h t l i n e s L . To t h i s end, i f r S =( O , O , O ) , r d = (a,o,o)
.
FL
=
1 L
that is,
F ds =
je F(x,E+(x),E$(x))
/1
+ E
2
+'(x)
2
+ E
2
$'(x)
2
dx
,
(7.1)
0
L i s the path L = {(x,y,z):
y = €$(X)
, z=
€$(XI
, 0 =< x =< !zl ,
(7.2)
and F
=
M
(7.3)
.
I n t h e n o t a t i o n o f 171, we w r i t e F = l + ~ h ,
(7.4)
i.e. E h = m - l . I n u l t r a s o n i c tomography I c ( r ) - COI / l c o l 2 1/10 assuming t h a t I h l =< 1 , we may assume t h e r e l a t i o n
(7.5)
,
so t h a t If1 I E ~ =< 1/10 .
=<
1 / 5 ; hence
29
Frequency Extrapolation in Ultrasonic Tomography
In order t o reconstruct
F v i a s t r a i g h t l i n e methods, we would l i k e t o know
re 7.6) To t h s end, t h e fol owing e q u a t i o n s a r e d e r i v e d i n [ 7 1 .Q
e = I
E*
I $ h y + $hz + 7 1 4"
+7 1 $ ' * I dx + O ( E ~ ),
(7.7)
0
where 7.8)
which a r e solved, s u b j e c t t o t h e boundary c o n d i t i o n s $ ( O ) = $ ( a ) = $ ( O ) = $(E) = 0
.
(7.10)
We now a p p l y t h e s e r e s u l t s t o o u r case. Assuming t h a t t h e paths a r e s t r a i g h t l i n e s , we can r e c o n s t r u c t an F1 =
’I
P , such
that
= P(x,O,O)
.
(7.11)
Fo dx ; Fo = F(x,O,O)
.
(7.12)
Po
dx ;
Po
0
We would l i k e t o know
Fo
, such
1
that
L
FL
=
0 Then we have FL = F l - e
(7.13)
where, upon i g n o r i n g terms o f o r d e r s3 , and t a k i n g E = 1 ( i m p l y i n g a c o r r e s p o n d i n g r e d u c t i o n i n t h e magnitudes o f t h e c o r r e s p o n d i n g q u a n t i t i e s ) , Fo = where by ( 7 . 8 ) ,
Po
2 2 1 + 7 C$'(X) + $ ' ( x ) I
,
(7.14)
( 7 . 9 ) and (7.10)
(7.15)
L e t us now d e r i v e a n u m e r i c a l scheme f o r making t h e c o r r e c t i o n ( 7 . 1 4 ) . To t h i s end we assume t h a t F i s known on a 3-dimensional p i x e l a r r a y w i t h s i d e h , whose c e n t e r s a r e l o c a t e d a t t h e p o i n t s
F. Stenger et al.
30
(7.16) We t h e n use t h e a p p r o x i m a t i o n s
F!.1Jk
'ifjk=
j:i
1,
aF ay
:- ( x , , y
-F.
i
ij,k+l
i' k
1
-F
.
i,J+1 ,k
-
Fi , j - 1 ,k
2k
-
- Fij,k-l 2k
$ hryjk
F (t,y-,z ) d t "Y J k
X.
) =
-z F (t,yj,Zk)dt
i-1 + h m=2 1
- 1 =
7 hrfjk + h
'iij +k $ h qjk :Aijk
i - 1 -z 1 Fmjk
(7.17)
+
12. hFfjk
= 'ijk
m=2
R
j
Rx 0
(t-t,)
Pz
I h 1 [ ( m - T 1) h - k x l r k j k (t,y.,z ) d t = J k 'x m = l
:Djk
t o get 1 [ ( A i j k - Cjk) 2 Fijk = eijk - 7
t
( B j j k - D j k )21
.
(7.18)
A f t e r making such an "x-sweep" f o r each j and k , we then make a s i m i l a r "y-sweep", and r e p e a t , u n t i l t h e changes i n t h e numbers Fijk become n e g l i g i b l e . 8.
AN EXAMPLE
I n t h i s s e c t i o n we i l l u s t r a t e t h e a p p l i c a t i o n o f t h e f o r m u l a s developed i n S e c t i o n s 5-6 o f t h e paper v i a a s i m u l a t i o n c a r r i e d o u t f o r t h e case o f t h e Rytov R y t . [ u ( F ) l i s o f t e n made f o r t h e s o l u t i o n model. The Rytov a p p r o x i m a t i o n u(F) t o ( l . l ) , where
and where
w k (F)
= - -
8.2)
Our s i m u l a t i o n was c a r r i e d o u t f o r t h e case o f
4 f(F) =
1
j=l
-b[F-F.l 2 e
J
8.3)
31
Frequency Extrapolation in Ultrasonic Tomography
s i n c e i t i s p o s s i b l e t o e x p l i c i t l y e v a l u a t e W when f has t h e f o r m (8.3) C101. I n ( 8 . 3 ) t h e p o i n t s F j a r e l o c a t e d a t r j = (O,Yj,O) , where i n u n i t s o f m i l l ? The number b waschosen meters, y1 = 3.75 , y2 = 0 , y3 = -2.5 , y4 = -4.375 so t h a t each Gaussian
.
-b
e
1 r-rJ. 1 L
had a f u l l w i d t h a t h a l f maximum equal t o 1 mm, i . e . , about 2 wavelengths a t a frequency o f 2.5 MH (megahertz), and sound speed C, = 1500 m/sec.
A l l r e c o n s t r u c t i o n s a r e based on t h e f o r m u l a lim k-t-
2
Wk
where L i s a s t r a i g h t l i n e p a t h [ l o ] , and where t h e numbers (8.4) w e r e o b t a i n e d by using t h e algorithms (5.8)-(5.10), by t a k i n g 9 values o f k corresponding t o 9 equi-spaced f r e q u e n c i e s i n t h e range o f 1 t o 4 MH
.
F i g u r e 8.1 i s a p i c t u r e o f t h e e x a c t o b j e c t d i g i t i z e d t o a each p i x e l h a v i n g dimension .3125 mm x .3125 mm.
41
x
41 p i x e l a r r a y ,
The p i c t u r e s i n F i g u r e s 8.2, 8.3 and 8.4 (as w e l l as F i g u r e 8.1) were o b t a i n e d from reconstructions c a r r i e d o u t i n t h e Ultrasound Laboratory a t the U n i v e r s i t y o f Utah, and based on t h e methods o f t h i s paper. The p i x e l a r r a y i n each dimension i s t h e same as i n F i g u r e 8.1. The p r o j e c t i o n d a t a was o b t a i n e d f r o m 3 0 v i e w s o v e r 180 degrees. The l i n e o f d e t e c t i o n was p o s i t i o n e d a t 8 cm f r o m the center o f rotation. The p i c t u r e i n F i g u r e 8.2 was o b t a i n e d by a p p l y i n g t h e a l g o r i t h m s ( 5 . 7 ) - ( 5 . 1 0 ) accurate t o 7 decimals. I t deviates from the t o d a t a i k j , 2W,,.(F)/(ikj)jg J j=l e x a c t o b j e c t a t each end, s i n c e t h e r e t h e v a l u e o f f ( F ) i n ( 8 . 3 ) was v e r y s m a l l , so t h a t t h e r e s u l t i n g s i g n i f i c a n t f i g u r e accuracy i n t h e d a t a was l e s s t h a n 4 p l a c e s . The r a t i o n a l f u n c t i o n a l g o r i t h m was t h e r e f o r e l e s s a c c u r a t e a t t h e s e ends. The p i c t u r e i n F i g u r e 8.3 i s a r e c o n s t r u c t i o n o b t a i n e d as i n F i g u r e 8.2, b u t b y a d d i n g random n o i s e w i t h normal d i s t r i b u t i o n and s t a n d a r d d e v i a t i o n o f 0.1% t o t h e s i m u l a t e d data. F i g u r e 8.4 i s a p i c t u r e o f t h e r e s u l t o f t h e r e c o n s t r u c t i o n o f t h e s i m u l a t e d d a t a f r o m 10 s e t s o f 9 f r e q u e n c i e s e q u a l l y spaced i n t h e range f r o m 1 t o 4 MH. Random n o i s e w i t h normal d i s t r i b u t i o n and s t a n d a r d d e v i a t i o n o f 0.1% was a g a i n added t o t h e d a t a as f o r t h e d a t a o f F i g u r e 8.3, b u t h e r e t h e f i r s t norm procedure o f Eqs. (6.2) was used on t h e p r e d i c t i o n s c6): t o o b t a i n more accurate predictions.
F. Stenger et a/.
32
FIGURE 8.1:
D i g i t i z e d Image o f T e s t O b j e c t
The t e s t o b j e c t c o n s i s t e d o f f o u r Gaussian d i s t r i b u t i o n s as shown w i t h i n t h e l o w e r box. The b r i g h t e n e d l i n e through t h e c e n t e r o f t h e o b j e c t s i n d i c a t e t h e l o c a t i o n f r o m which t h e p r o f i l e ( p l o t t e d i n t h e upper box) was taken. The s e p a r a t i o n s o f t h e Gaussian o b j e c t s a r e 3.75, 2.5, and 1.875 mm, and t h e f u l l w i d t h a t h a l f maximum i s 1.0 mm f o r each o f t h e o b j e c t s .
FIGURE 8.2:
R e c o n s t r u c t i o n o f Test O b j e c t f r o m R a t i o n a l F u n c t i o n P r e d i c t i o n i n t h e Absence o f Noise.
Frequency Extrapolation in Ultrasonic Tomography
33
The r e c o n s t r u c t e d o b j e c t i s almost o f t h e same q u a l i t y as t h e o r i g i n a l t e s t o b j e c t . S c a t t e r i n g d a t a were generated by a p p l i c a t i o n o f t h e Rytov a p p r o x i m a t i o n s c a t t e r i n g i n t e g r a l . P r o j e c t i o n d a t a were made f r o m t h e r a t i o n a l f u n c t i o n p r e d i c t i o n u s i n g n i n e e q u a l l y spaced f r e q u e n c i e s f r o m 1 t o 4 MHz. R e c o n s t r u c t i o n f r o m t h e e x t r a p o l a t e d d a t a was done u s i n g t h e ART a l g o r i t h m a l o n g s t r a i g h t l i n e s .
FIGURE 8.3:
R e c o n s t r u c t i o n o f T e s t O b j e c t i n t h e Presence o f Random Noise W i t h o u t Data C o n d i t i o n .
Random n o i s e w i t h a normal d i s t r i b u t i o n o f mean z e r o and a s t a n d a r d d e v i a t i o n o f 0.1% was added t o t h e s i m u l a t e d d a t a a t each frequency. The r a t i o n a l f u n c t i o n e x t r a p o l a t i o n was t h e n o b t a i n e d from t h i s n o i s y d a t a and, t h e r e c o n s t r u c t i o n was c a r r i e d o u t i n t h e same manner as f o r F i g . 8.2.
FIGURE 8 . 4 : R e c o n s t r u c t i o n o f T e s t O b j e c t i n t h e Presence o f Noise Using t h e Data C o n d i t i o n i n g A l g o r i t h m o f Equation ( 6 . 2 ) .
F. Stenger et al.
34
S c a t t e r i n g d a t a was s i m u l a t e d f o r 10 d i f f e r e n t s e t s o f 9 u n i f o r m l y spaced f r e q u e n c i e s f r o m 1-4 MHx and random n o i s e w i t h a mean o f z e r o and a s t a n d a r d d e v i a t i o n o f 0.1% was added t o a l l of t h e d a t a . The r a t i o n a l f u n c t i o n e x t r a p o l a t i o n was a p p l i e d t o each o f t h e t e n s e t s of f r e q u e n c i e s , and then t h e f i r s t norm procedure o f Eq. ( 6 . 2 ) was a p p l i e d t o t h e e x t r a p o l a t i o n s . F i n a l l y , t h e r e c o n s t r u c t i o n a l g o r i t h m was a p p l i e d t o t h e averaged d a t a i n t h e same manner as t h e p r e v i o u s two f i g u r e s . REFERENCES 1.
Bates, R.H.T., Boerner, W.M. and Dunlop, G.R., An extended Rytov Approximat i o n and I t s S i g n i f i c a n c e f o r Remote Sensing and I n v e r s e S c a t t e r i n g , O p t i c s Communications 4 (1976) 421-423. 2. Brooks, R.A. and Chiro, G.D., P r i n c i p l e s o f Computer A s s i s t e d Tomography i n Radiographic and R a d i o i s o t r o p i c Imaging, Phys. Med. B i o l . 21 (1976) 689-732. 3. Felsen, L.B. and M a r c u r i t y , N.M., R a d i a t i o n and S c a t t e r i n g o f Waves ( P r e n t i c e H a l l , 1973). 4. Fredholm, I . , Oeuvres Compl6tes (Malmo, 1 9 5 5 ) . 5 . Johnson, S.A., Stenger, F., Wilcox, C., B a l l , J. and Berggren, M., Wave Equations and I n v e r s e S o l u t i o n s f o r S o f t Tissue, U n i v e r s i t y o f Utah, Department o f B i o e n g i n e e r i n g M a n u s c r i p t ( 1 9 8 1 ) . 6. M u l l e r , R.K., D i f f r a c t i o n Tomography I : The Wave Equation, U l t r a s o n i c Imaging 2 (1980) 213-222. 7. Norton, S.J. and L i n z e r , M., C o r r e c t i n g f o r Ray R e f r a c t i o n i n V e l o c i t y and A t t e n u a t i o n Tomography: A P e r t u r b a t i o n Approach, U l t r a s o n i c Imaging 4 (1982) 201 -233. 8. Stenger, F., Optimal R a t i o n a l Approximation w i t h Preassigned Poles, i n manus c r ip t 9. Stenger, F., Optimum Convergence o f Minimum Norm Approximations i n Hp , Numer. Math. 29 (1978) 345-362. 10. Stenger, F., A s y m p t o t i c I n v e r s i o n Based on Using More t h a n One Frequency, A c o u s t i c a l Imaging 11 (1981) t o appear. 11. Wilcox, C.H., A G e n e r a l i z a t i o n o f Theorems o f R e l l i c h and Atkinson, Proc. h e r . Math. SOC. 7 (1956) 271-276. 12. Wynn, P., Continuous P r e d i c t i o n A l g o r i t h m s , E s t r a d a da C o l c o l o 9 (1972) 197-278.
.
'Supported by U.S.
Army Research C o n t r a c t Number DAAG-29-77-G-0139.
2Supported by N a t i o n a l I n s t i t u t e o f H e a l t h Grant Number 1 R01 CA 29728. 3Supported by t h e American Cancer S o c i e t y Grant Number PDT-110 B. 4 I n o u r s i m u l a t i o n we a r e a b l e t o d i s t i n g u i s h two o b j e c t s o f t h e form (1.8) separated by 2 wave1 engths.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. M w d l e (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
35
NONLINEAR TRAVELLING WAVES I N FLUIDS P.L.
Sachdev
Courant I n s t i t u t e o f Mathematical Sciences New York U n i v e r s i t y New York, New York 10012 U.S.A. and Department o f A p p l i e d Mathematics I n d i a n I n s t i t u t e o f Science Bangalore, I n d i a 560012 I t i s shown w i t h t h e h e l p o f s e v e r a l examples f r o m f l u i d mechanics t h a t systems o f q u a s i - l i n e a r inhomogeneous p a r t i a l d i f f e r e n t i a l equations admit exact t r a v e l 1 i n g o r decaying s o l u t i o n s . These s o l u t i o n s r e p r e s e n t f l o w s w i t h shocks o r w i t h f r e e s u r f a c e s , o r c o u l d be s i m p l y p e r i o d i c i n t h e phase f u n c t i o n i n an i n f i n i t e domain. F u r t h e r p o s s i b i l i t i e s and d i f f i c u l t i e s a s s o c i a t e d w i t h t h e s o l u t i o n s o f such systems a r e b r i e f l y i n d i c a t e d . As u s u a l , boundary c o n d i t i o n s i n each case d i c t a t e t h e f i n a l form o f t h e s o l u t i o n .
INTRODUCTION The s i m p l e s t example o f t r a v e l l i n g waves i s p r o v i d e d by t h e wave e q u a t i o n , Utt
-
2 c uxx = 0 ,
(1 . I )
which has t h e g e n e r a l s o l u t i o n i n t h e f o r m u = f ( x - c t ) + g(x + c t ) , d e s c r i b i n g a s u p e r p o s i t i o n of two waves t r a v e l l i n g i n o p p o s i t e d i r e c t i o n s w i t h c o n s t a n t speed c . T h i s e q u a t i o n i s h y p e r b o l i c so t h a t t h e harmonic s o l u t i o n w i t h amplitude A u = A ei ( k x has a l i n e a r d i s p e r s i o n r e l a t i o n , d i s p e r s i v e type i s
w = ?
ck
-
wt)
.
(1.2)
An example o f e q u a t i o n s o f t h e
t h e s o - c a l l e d beam e q u a t i o n . It has s o l u t i o n of t h e f o r m ( 1 . 2 ) if t h e frequency o and wave number k a r e r e l a t e d by t h e n o n l i n e a r r e l a t i o n
so t h a t we have a g a i n two waves p r o p a g a t i n g i n o p p o s i t e d i r e c t i o n s . The above examples r e l a t e t o l i n e a r e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s . I n t h e p r e s e n t paper, we i l l u s t r a t e , w i t h t h e h e l p o f s e v e r a l examples f r o m f l u i d mechanics, t r a v e T l i n g wave s o l u t i o n s o f n o n - l i n e a r systems o f p a r t i a l d i f f e r e n t i a l equations ( PDEs) which a r e quasi1 i n e a r and inhomogeneous. They d e s c r i b e b o t h h y p e r b o l i c and d i s p e r s i v e waves. The d i s p e r s i v e c h a r a c t e r may s p r i n g f r o m t h e e q u a t i o n s themselves o r f r o m t h e boundary c o n d i t i o n s . The n o n l i n e a r h y p e r b o l i c e q u a t i o n s
P. L. Sachdev
36
naturally allow solutions with shocks. The importance of t h i s c l a s s of t r a v e l l i n g solutions has been l u c i d l y brought out by Barenblatt and Zeldovich C11 who characterise them as intermediate asymptotes, "which do not merely represent s p e c i f i c examples b u t describe t h e intermediate asymptotic behavior of solutions of wider c l a s s e s of i n i t i a l , boundary and mixed problems, t h a t i s , they describe t h e behavior of these s o l u t i o n s away from t h e boundaries of t h e regions of independent variables o r a l t e r n a t i v e l y , in t h e region where i n a sense t h e solution i s no longer dependent on the d e t a i l s of the i n i t i a l and/or boundary conditions b u t the system i s s t i l l f a r from being in a s t a t e of equilibrium". They have a l s o shown t h a t the t r a v e l l i n g waves a r e , i n a sense, more general than the s i m i l a r i t y s o l u t i o n s of t h e 'second k i n d ' , the l a t t e r being included in the former. The asymptotic nature of these special solutions has been c l e a r l y analysed f o r some nonlinear parabolic equations ( s e e , f o r example, P e l e t i e r [21) b u t s i m i l a r analysis f o r t h e systems under study here s t i l l poses a formidable task. The examples we s t u d y i n t h e present review include the uniform shock produced by a spherical o r cylindrical piston i n t o a homogeneous medium, simple self-propagating o r decaying waves on sheared flows, quasi-simple acoustic gravity waves i n an isothermal atmosphere, and t r a v e l l i n g waves in a s e l f g r a v i t a t i n g medium.
2.
SHOCKS OF UNIFORM STRENGTH
Courant and Friedrichs [31 studied propagation of spherical and cylindrical shocks moving w i t h constant velocity i n t o a uniform medium. They referred t o solutions depending only on the r a t i o of distance t o time as quasi-simple waves since they share several f e a t u r e s with simple waves f o r homogeneous systems, though d i f f e r i n 9 s i g n i f i c a n t l y in several o t h e r ways. The equations of gasdynamics, in spherical symmetry, w i t h usual notation, ( c y l i n d r i c a l l y symmetric case may be t r e a t e d s i m i l a r l y ) a r e
& (p
p-y)
=
0
.
I f the shocks a r e produced by a uniformly expanding piston and a r e themselves uniform i n s t r e n g t h , t h e flow behind t h e shock has uniform entropy and t h e l a s t equation ( 2 . 3 ) i s replaced by t h e i s e n t r o p i c r e l a t i o n , p = kp-Y , where k i s a constant. Then physical considerations imply t h a t p , p and u a r e functions of f only, so t h a t they remain constant on the rays f = constant. Thus, we a r r i v e a t t h e following form of t h e solution of t h e inhomogeneous system of quasi-linear hyperbolic equations ( 2 . 1 ) - ( 2 . 3 ) , t h e inhomogeneity depending both on the dependent variable u as well as the independent variable r :
u
= Tl
-1
U(n) , c =
Tl
-1
C(n) , p =
with c2 = 22 P
, square of sound speed ,
P(n)
,
(2.4)
37
Nonlinear Travelling Waves in Fluids
where
q =
t
and
D
=
(1
-
U)
2
-
C
2
.
(2.6)
I t i s p o s s i b l e t o reduce t h e above system t o t h e phase p l a n e ( C , U) , p r o v i d e d t h e s o l u t i o n nowhere crosses t h e c r i t i c a l l i n e 0 = 0 . We have t h e n
The phase diagram has been d i s c u s s e d q u i t e t h o r o u g h l y by Courant and F r i e d r i c h s and has a l s o been c o r r o b o r a t e d b y o t h e r s , f o r example, i n t h e c o n t e x t o f t h e t h e o r y o f l a s e r s . The phase p i c t u r e p r o v i d e s s o l u t i o n s o f s e v e r a l p h y s i c a l problems as d e t a i l e d by Courant and F r i e d r i c h s . Here we b r i e f l y r e v i e w t h e p i s t o n problem mentioned above. L e t t h e v e l o c i t y a t t h e p i s t o n be g i v e n by 5 = 50 , where 6 = = Q-1 , such t h a t a c c o r d i n g t o (2.4), U(,) = 1 a t t h e f p i s t o n . Such a p i s t o n m o t i o n l e a d s t o a shock moving w i t h a c o n s t a n t speed, say, I: , whose v a l u e i s t o be determined. The shock c o n s e r v a t i o n laws g i v e , i n t h e s i m i l a r i t y form, (1 - L12)cg = ( 1 - Us) - ir 2(1 - U S ) 2 = ( 1 - Ll2 )(I - u s ) 2 + U S ( 1 - U s ) ,
(2.8a ,b) where and Us and Since c > C,
p2 =
C denote values o f U and ,'one can e a s i l y see t h a t
Y + l
C
i m m e d i a t e l y b e h i n d t h e shock.
f o r an expanding f o r w a r d f a c i n g shock f r o n t . We choose a p o i n t (Us , Cs) on t h e shock e l l i p s e ( 2 . 8 a ) s a t i s f y i n g t h e i n e q u a l i t i e s ( 2 . 9 ) , f o r a g i v e n shock v e l o c i t y c and Co , and i n t e g r a t e ( 2 . 7 ) u n t i l U = 1 , i d e n t i f y i n g t h e p i s t o n . To g e t t h e c o r r e s p o n d i n g ( n , U) r e l a t i o n , we have q - l = 5 = c , and U = Us a t t h e shock and i n t e g r a t e ( q , U) e q u a t i o n f o r d e c r e a s i n g v a l u e o f 6 o r increasing value o f q u n t i l t h e i n t e g r a l curve h i t s t h e l i n e U = 1 w i t h o u t Thus, t o each v a l u e o f I: crossing t h e c r i t i c a l l i n e s D = 0 o r I l - U I = I C I a t t h e shock corresponds a c e r t a i n v a l u e o f t h e p i s t o n ( p a r t i c l e ) v e l o c i t y c a t t h e p i s t o n s u r f a c e . The f u n c t i o n a l r e l a t i o n between z and 5 , n u m e r i c a l r e s u l t s show, i s monotone.
.
The p r e s e n t example, h i s t o r i c a l l y , i s one o f t h e f i r s t d e s c r i b i n g q u a s i - s i m p l e wave s o l u t i o n s o f an inhomogeneous system o f h y p e r b o l i c e q u a t i o n s i n two i n d e pendent v a r i a b l e s . From t h e p h y s i c a l p o i n t of view, t h i s i s n o t a v e r y i n t e r e s t i n g example, s i n c e i t r e q u i r e s c o n t i n u o u s f e e d i n g o f t h e energy by t h e p i s t o n t o m a i n t a i n a shock o f c o n s t a n t s t r e n g t h . The e a r l i e s t s i m i l a r a n a l y s i s f o r a homogeneous system i s due t o Earnshaw 141
38
P. L. Sachdev
who sought s o l u t i o n s o f t h e f o r m P = P(q) u = u(q) where q = q ( x , t ) 9
f o r one-dimensional i s e n t r o p i c unsteady f l o w s and a r r i v e d a t t h e (now) f a m i l i a r s i m p l e waves. T h i s s t u d y was extended t o n-dimensions by S c h i n d l e r 151. The s p e c i a l case of two-dimensional d y n a m i c a l l y - s i m i l a r f l o w s was c o n s i d e r e d by Mackie C61, and reviewed i n d e t a i l by L e v i n e [ 7 1 w i t h s p e c i a l r e f e r e n c e t o t h e S o v i e t approach t o t h e t o p i c . 3. S I M P L E W A V E S ON SHEAR FLOWS T h i s s e c t i o n d e a l s w i t h e x a c t s o l u t i o n s d e s c r i b i n g s i m p l e waves on i n v i s c i d shear f l o w s i n s h a l l o w channels, e i t h e r s e l f - p r o p a g a t i n g i n t h e h o r i z o n t a l d i r e c t i o n o r decaying w i t h t i m e , Freeman C81, Sachdev 191. I n t h e h y d r a u l i c a p p r o x i m a t i o n , t h e equations o f motion are ut + uuX + vuY + ghx = 0 ,
(3.1)
=o,
u x + vY
(3.2)
where ( u , v ) a r e v e l o c i t y components i n t h e x and y d i r e c t i o n s measured a l o n g and p e r p e n d i c u l a r t o u n i f o r m bottom. The h y d r a u l i c a p p r o x i m a t i o n r e p l a c e s t h e p r e s s u r e t e r m i n ( 3 . 1 ) by t h e u n i f o r m g r a v i t a t i o n a l p r e s s u r e gph , where p i s t h e d e n s i t y of w a t e r , g a c c e l e r a t i o n due t o g r a v i t y and h , t h e depth o f t h e w a t e r i n t h e channel. The boundary c o n d i t i o n s r e q u i r e
v
+
ut + uux on
A t the bottom
,
+ uh,
= ht
qhX = 0
,
(3.3)
y = h .
v = O
on
The s p e c i a l s o l u t i o n s sought assume t h a t
y = O .
u
and
h
(3.4) a r e c o n s t a n t on l i n e s
(3.5)
= c(h(x,t))
( C f . Sec. 2) so t h a t u = u(x
-
ct ,y)
,
(3.6)
.
h = h(x - c t ) By i n t r o d u c i n g v a r i a b l e s W,
equations (3.1)
-
= ~ / ( ~ h 5/ 5~ =~ x) - c t
,h
= h(C)
(3.2) and t h e boundary c o n d i t i o n s (3.3)
-
c , ( u - c ) -au + ah
wl=O
(3.4) become
a + g = o , 1 ay
( u - c ) - au +w ah
w1 = u
-
(3.7)
3
on
g = 0 , on y = h
y = O .
,
(3.10) (3.11)
39
Nonlinear Travetfing Waves in Fluids
Eliminating
* ah
f r o m ( 3 . 8 ) and ( 3 . 9 ) , we may w r i t e (3.12)
which can be i n t e g r a t e d t o g i v e (3.13)
rg
The f i r s t o f boundary c o n d i t i o n s (3.10) now becomes
0
(u
-
d Y 2 = l , c)
(3.14)
w h i l e ( 3 . 1 1 ) has a l r e a d y been imposed i n w r i t i n g (3.13). function
I n t r o d u c i n g a new
(3.15)
2 i n t o ( 3 . 8 ) and u s i n g (3.13),
0 we g e t an e q u a t i o n f o r
I : (3.16)
The boundary c o n d i t i o n s on
I become
I(h I n terms o f t h e n o r m a l i s e d depth
,h )
= 1
Y =
f , (3.16)
, I ( h , 0)
= 0
.
becomes
h - a2 1 + ( I - y ) Ta21 = ayah ay w i t h boundary c o n d i t i o n s (3.18) 1( h , 1) = 1 , I ( h , 0 ) = 0 . The i n t r i c a t e n o n l i n e a r h y p e r b o l i c e q u a t i o n ( 3 . 1 7 ) would have t o be s o l v e d s u b j e c t t o ( 3 . 1 8 ) and a p r e s c r i b e d boundary c o n d i t i o n I = I ( y ) on h = ho , a supernumer'ary s e t o f boundary c o n d i t i o n s r e q u i r e d t o o b t a i n t h e unknown f u n c t i o n c ( h ) . However, t h e s p e c i a l s e l f - s i m i l a r s o l u t i o n s I = I ( Y ) can be o b t a i n e d by imposing t h e c o n d i t i o n c = 2a(gh)1/2 , where a i s a c o n s t a n t . E q u a t i o n ( 3 . 1 7 ) becomes a n o n l i n e a r o r d i n a r y d i f f e r e n t i a l e q u a t i o n (I
-
dI Y) -d21 - - +- dY dy2
dI 2 - d[ -I - dY dY
d I 1/2,
a
and r e q u i r e s s e v e r a l t r a n s f o r m a t i o n s t o become t r a c t a b l e . t h a t (3.19) assumes t h e f o r m dJ ' JJ' - = dJ The s u b s t i t u t i o n
2 N = J' + 1
2(1 + J ' )
2
(3.19)
(3)
- 2 a ( l + J')312
Put
+ (1 +
enables us t o f i n d t h e i n t e g r a l
J = I-Y
J')
.
so
(3.20)
40
P. L. Sachdev
(3.21) where
a
and
b
are roots o f 1 ,2 - aX + 2
dJ = dY
Since
2 N
-
1
, we
= 0
(3.22)
have t h e s o l u t i o n
dN -
N3 where
C1
i s a constant. 4a
’
(3.23)
F i n a l l y , ( 3 . 2 3 ) can be w r i t t e n as 2
-
1
-
2(a2
u
1)
B
2a ( - 1 7’ 2 a 2 - 1 Y= 2 4a - 1 2(a2 - 1 ) B( 7 ’ 2 2 a - 1 2 a - 1
’
2a
- a-l a -1 )
-
(3.24)
~
where B denotes t h e i n c o m p l e t e Beta f u n c t i o n . T h i s s o l u t i o n s a t i s f i e s t h e boundary c o n d i t i o n s (3.18) a t t h e b o t t o m and t h e s u r f a c e . To v e r i f y t h a t , we note t h a t N2 = -d=I dY
gh (u - c)
(3.25)
2
so t h a t u = (2a
-
N - l ) ( g h ) 1/ 2
W r i t i n g U = u/(gh)1’2 and n o t i n g ( 3 . 1 8 ) , o f J = I - Y , we see t h a t U=l/a and
U=2a
at at
(3.21),
.
(3.26)
( 3 . 2 2 ) and, t h e d e f i n i t i o n
Y = O ,
Y = l .
Thus, i t i s c l e a r t h a t (3.24) s a t i s f i e s these boundary c o n d i t i o n s . t i o n speed i s
(3.27) (3.28) The propaga-
(3.29) We n o t e t h a t f o r c/(gh)1/2 , etc. point
.
c
-
a > 1 , > 2a > umax , b a r d e n o t i n g t h e n o r m a l i s e d v a r i a b l e s Thus u z c anywhere i n t h e range and t h e r e i s no c r i t i c a l
The above s o l u t i o n r e f e r s t o s e l f - p r o p a g a t i n g waves i n an i n f i n i t e l y l o n g channel bounded below by a p l a n e b o t t o m and above by a f r e e s u r f a c e . The wave speed depends on t h e l o c a l h e i g h t h , which i s i t s e l f a r b i t r a r y . Now we g i v e a n o t h e r c l a s s o f s o l u t i o n s L91 o f t h e above problem w h i c h decay e x p l i c i t l y w i t h t i m e and f o r w h i c h t h e shape o f t h e f r e e s u r f a c e i s determined, i n c o n t r a s t t o above wherein h remains g e n e r a l . I n t h e p r e s e n t case i t i s c o n v e n i e n t t o change t h e b a s i c system so t h a t i n terms o f t h e new v a r i a b l e s
41
Nonlinear Travelling Waves in Fluids
,w
z =
(here H
replaces
h
{v - z(Ht
=
=
uHx)}/H
-t
,
(3.30)
of the e a r l i e r p a r t ) we have Ht
-t
0
,
(3.31)
Hx = 0
,
(3.32)
z = l .
(3.33)
uHx + H ( u x + wz)
+ uux + wuz + Ut
=
with boundary conditions on
w = O
z = O and
Here the acceleration due t o gravity i s denoted by
9.
We seek d i r e c t l y s i m i l a r i t y solutions H
=
tal h ( E ) ,
u
=
tbl f ( c ,
rl)
(3.34)
c = x t
, l l = z .
A simple c a l c u l a t i o n shows t h a t
al=-2a,bl=-a,c1=-1 , n l = a - l ,
(3.35)
where a i s a parameter. A s e t o f transformations and s t e p s s i m i l a r t o those c a r r i e d out by Freeman lead t o t h e solution of t h e form f = ( 2 ; ) - 1ELI
,g
=
,5
(2a)-lG
(2;)(th)”2
=
(3.36)
where 1 u+2(;
2G( 1 -4a)N1 -3 2(; 1 -2a)N1-1 ’
2Cr( 1-4a)N2-3
2Cr( 1 -4a)N1 -3
1-4a)N2-3 B( 2a( - 1-2a)N2-1
2Cr(l-n) N2 1 1 1 N,L N, I
r l =
B(
-
I
2a( 1-2a)N2-1
(3.37)
1)
Y
2Cr( 1 -2a)N1 -1
provided 1 + 2a; N1 lJ:1--
(; 1 -2a)N1 -1
’0
I
(3.38)
1 + 2a;N2 v z l - -2 a( - 1-2a)N2-1
> o .
I t was shown by t h e author C91 t h a t the s o l u t i o n s a r e without c r i t i c a l l e v e l s i f
-2 1 2.25 < a <-, a 1 - a
0
1
5 c1 < -3 .
(3.39)
P. L. Sachdev
42
I t can be e a s i l y checked t h a t t h e f r e e surface i n t h e p r e s e n t case i s e x p l i c i t l y g i v e n by H = x 2 and t h e s o l u t i o n has some o f t h e f e a t u r e s found by Longuet4a t H i g g i n s [ l o 1 ,[11] f o r f r e e s u r f a c e f l o w s i n t h e absence o f t h e e f f e c t s o f g r a v i t y . They a r e decaying and l o c a l i n n a t u r e .
In a r e l a t e d paper B l y t h e e t a l . [121 have found n e u t r a l l y s t a b l e d i s t u r b a n c e s which n e i t h e r grow n o r decay i n a m p l i t u d e a l t h o u g h , l i k e s i m p l e waves i n an unsheared f l o w , t h e y can break. T h e i r s p e c i a l e x p l i c i t s o l u t i o n d e s c r i b e s a c e n t e r e d expansion wave which i s caused by a d i s c o n t i n u i t y a t x = 0 a t t = 0 P a r t i c l e paths e t c . a r e d e l i n e a t e d and compared w i t h those i n an unsheared f l o w . 4.
.
ACOUSTIC GRAVITY WAVES
I n c o n t r a s t t o p r e v i o u s s e c t i o n wherein t h e f l o w was r e s t r i c t e d t o a f i n i t e channel bounded above b y a f r e e s u r f a c e and below by a h o r i z o n t a l bottom, we c o n s i d e r now t r a v e l l i n g waves i n an i n f i n i t e atmosphere e x t e n d i n g above t h e ground, Seshadri and Sachdev C131. The atmosphere i s t a k e n t o be i s o t h e r m a l and t h e governing system o f PDEs i s q u a s i l i n e a r and inhomogeneous, which, f o r t h e i s o t h e r m a l atmosphere, has t h e RHS's depending on t h e dependent v a r i a b l e s o n l y . The b a s i c e q u a t i o n s f o r an i n v i s c i d c o m p r e s s i b l e f l o w under g r a v i t y a r e 0
P t + UPx -k W P Z + PUX + PWZ = Ut
+ uux
-k
wuz +
Wt
+
uwx
-k
wwz
Pt + UPx + WP,
p1 1
+0
,
px = 0 , p,
+
,
-g
YPWZ = 0
+ YPU,
Here t h e f l o w v a r i a b l e s have t h e u s u a l n o t a t i o n and t o gravity.
g
*
i s t h e a c c e l e r a t i o n due
We f i r s t i n t r o d u c e t h e non-dimensional v a r i a b l e s
where H i s t h e v e r t i c a l s c a l e h e i g h t , and "0" r e f e r s t o c o n s t a n t r e f e r e n c e c o n d i t i o n s . We seek f o r t h e t r a n s f o r m e d system s o l u t i o n s o f t h e f o r m
U = U($)
,W
A s i m p l e argument shows t h a t
= $
W($) , P
= P(4)
,R
has t o be l i n e a r i n
=
R($) X, Z
. and
(4.3) T ,
, + , =x- t - - Tz , x1 l2 where x1 , x2 are a r b i t r a r y c o n s t a n t s . s o l v i n g f o r t h e d e r i v a t i v e s , we have
S u b s t i t u t i n g ( 4 . 3 ) i n t h e PDEs and
(4.4)
43
Nonlinear Travelling Waves in Fluids
= - E[ ( 1
W
Q
(1-P E2 ) - - W- IE ,P
- - )P R
D
2 R x1
(4.5)
l2
n
(anP)+
=
EL [WE +
(1
2 (QnR)$ =
E
[WE
2
- El1 ,
(4.6)
R
,
P Wn P +E - ( 1 - -) R - ( y - 1) R ’ x2
(4.7)
where
E
= (-1
u
w
x1
x2
+-+-)
,
D = E2 ( E2 - $ n ) ,
I t seems q u i t e n a t u r a l t o i n t r o d u c e t h e v a r i a b l e square, and w r i t e combining ( 4 . 6 ) and ( 4 . 7 ) ,
When ( 4 . 8 ) i s s u i t a b l y m a n i p u l a t e d w i t h (4.4) integral
E ~"(y-') Using (4.8) and (4.9),
x
c o n s t = -B
=
-
P
K =
, the
"sound speed"
( 4 . 5 ) , we o b t a i n an i n t e r m e d i a t e
,
say, B >
o
.
(4.9)'
we can o b t a i n a s i n g l e e q u a t i o n i n t h e w - K p h a s e plane:
{ ( y - l ) K CWKn + XK-1 B 2
1 Y 4 ] ) 4 K- -
.
(4.10)
The s i n g u l a r p o i n t s o f t h i s e q u a t i o n a r e
B~ p l a y s an i m p o r t a n t The two p o i n t s c o i n c i d e when K = 1 . The parameter 6 yn r o l e i n t h e d i s c u s s i o n o f t h e s o l u t i o n o f (4.10). A thorough phase p l a n e a n a l y s i s has been c a r r i e d o u t b y Seshadri and Sachdev 1131, r u l i n g o u t t h e e x i s t e n c e o f shock waves i n t h i s c l a s s o f e q u a t i o n s , a c o n c l u s i o n w h i c h perhaps c o u l d be a r r i v e d a t by n o t i n g t h e d i s p e r s i v e n a t u r e o f t h e e q u a t i o n s . Indeed t h e o n l y p h y s i c a l l y r e a l i s t i c s o l u t i o n corresponds t o X2 + m and 7
$ hlY
< 1
, when
the singular point
CL
i s a c e n t e r and
6 a saddle p o i n t .
P.L. Sachdev
44
The bounding c l o s e d c u r v e , i n t h i s case, c u t s t h e singul,. l i n e K = Kg at the B s i n g u l a r p o i n t (see F i g u r e 1 ) . Thus a f a m i l y o f p e r i o d i c s o l u t i o n s e x i s t s These s o l u t i o n s r e p r e s e n t f o r d i f f e r e n t values o f t h e parameter B 2 / ~ 2 y < 1 h o r i z o n t a l l y p r o p a g a t i n g g e r j 3 d i c waves, w i t h an a r b i t r a r y speed 11 . The p r e v i o u s s t u d y o f t h i s k i n d i s due t o D r a z i n E l 4 1 who a t t e m p t e d n o n l i n e a r p e r i o d i c s o l u t i o n s f o r a s l i g h t l y s t r a t i f i e d atmosphere i r l t h e Boussinesq a p p r o x i m a t i o n .
.
5. FINITE AMPLITUDE WAVES I N A SELF-GRAVITATIONAL MEDIUM We r e v e r t t o one dimension and c o n s i d e r waves of f i n i t e a m p l i t u d e i n a s e l f g r a v i t a t i n g medium C151. The b a s i c s t a t e i s a homogeneous s e l f - g r a v i t a t i n g medium w i t h d e n s i t y p o , u n i f o r m l y r o t a t i n g around t h e x - a x i s . The system i s i n e q u i l i b r i u m i f t h e a n g u l a r v e l o c i t y of r o t a t i o n R i s g i v e n by
n=-.
(5.1)
By choosing a s u i t a b l e frame o f r e f e r e n c e and c o n s i d e r i n g l o n g i t u d i n a l waves t r a v e l l i n g i n t h e x - d i r e c t i o n , t h e C o r i o l i s f o r c e vanishes and t h e system g o v e r n i n g t h e waves i s Pt
= 0
+ (PU),
3
(5.2)
(5.3) (5.4)
(5.5) where, besides o t h e r s t a n d a r d n o t a t i o n , J, r e p r e s e n t s t h e p e r t u r b a t i o n g r a v i t a t i o n a l p o t e n t i a l and t h e medium i s t a k e n t o be b a r o t r o p i c . We l o o k f o r t r a v e l l i n g wave s o l u t i o n s depending on X = x - U t o n l y . T h i s a n s a t z l e a d s t o t h e ODES
L
9= d X2
where
w = u
-
U and
h =
& !! . P
4aG(p - po)
,
(5.8)
We c o n s i d e r t h e i s o t h e r m a l case f o r which
(5.9) t h e p o l y t r o p i c case g i v e s s i m i l a r r e s u l t s . variables
By i n t r o d u c i n g t h e non-dimensional
(5.10) and i n t e g r a t i n g t h e system (5.6)
-
( 5 . 8 ) , we have P*W*
=
-1
3
(5.11)
(5.12)
45
Nonlinear Travelling Waves in Fluids
(5.13)
$ * = P * - l ,
47TGPOA2
X
where d o t denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o x, = T,@=, h, = En a*, and h i s t h e wave l e n g t h . E l i m i n a t i n g w, w i t h t h e h e l p o f "'(5.11), we o b t a i n $* f r o m (5.12) and s u b s t i t u t i n g i n (5.13) we g e t an e q u a t i o n f o r p* ,
.
d
1 .
1
5 P * dp, I P *
(-
3+ P+
dh* a -11 dP *
= - ( p * - 1)
where a = (-)a. i s t h e i n v e r s e o f Mach number square. be i n t e g r a t e d l t o g i v e
(5.14)
,
E q u a t i o n (5.14) can
(5.15) 1
where and
1
w,
f ( p * ; a ) = - - -+ p* 2P*
C
-
(5.16)
a kn P *
i s a c o n s t a n t such t h a t C-f(1 -A2;a)=C-f(l
(5.17)
-A1;al)=O.
Equation ( 5 . 1 7 ) f o l l o w s f r o m t h e assumption t h a t , a t t h e maximuththe d e n s i t y i s 1 + A1 and a t t h e minimum i t i s 1 - A2 , t h e p o i n t s where p, = 0 E q u a t i o n (5.17) t h u s takes ( 5 . 1 5 ) i n t o
.
p* =
and, besides, g i v e s t h e v a l u e o f 1 1
-
A2
+
as 1 A1
1
2 ( 1 - A2)'
a =
' 2 ( 1 + A1)'
(5.19)
1 + A1
A1 + A2 - Qn 1 - A2 The p o s i t i v i t y o f t h e LHS o f ( 5 . 1 8 ) l e a d s t o t h e c o n d i t i o n o f e x i s t e n c e o f t h e s o l u t i o n as BCf(1 - A2 ; a ) - f(p,;
a)l
2
0 for 1 - A
2 -<
p*
s 1 +A1 ,
t h e e q u a l i t y i n (5.20) h o l d i n g a t t h e end p o i n t s o f t h e i n t e r v a l .
(5.20)
Since (5.21)
t h e c o n d i t i o n (5.20)
implies
f(,,,;
a)
-
f(1
-
A2 ; a )
<
0 f o r 'I
-
A2 < P* < 1
= 0 for p* = 1
From'the d e f i n i t i o n o f
f(p,;a)
i n ( 5 . 1 6 ) , we have
-
+
A1
A2 o r 1 + A1
.
(5.22)
46
P. L. Sachdev
(5.23) g i v i n g a minimum a t
px =
1
.
Thus, we have
(5.24)
The f a c t o r p*
< 1
,
(1
-
1 -)
i n (5.23) i s p o s i t i v e f o r
p* >
1
and n e g a t i v e f o r
p*
i m p l y i n g t h a t (5.24) i s s a t i s f i e d f o r t h e e n t i r e range of
p*
if a >
1 7 . P*
This i n t u r n requires t h a t
(5.25) T h i s i s t h e necessary and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of p e r i o d i c waves i n an i s o t h e r m a l s e l f - g r a v i t a t i n g medium. W i t h t h e d e f i n i t i o n (5.19) o f a , (5.25) d e l i n e a t e s t h e r e g i o n i n t h e A1 - A 2 p l a n e f o r which such s o l u t i o n s , we have from ( 5 . 2 5 ) t h a t t h e s o l u t i o n e x i s t . Furthermore s i n c e a = i n the region o f existence s a t i s f i e s t h e c o n d i t i o n
(F)' - U< 1
- A 2 < 1 ,
aO t h a t i s , t h e p r o p a g a t i o n speed o f t h e n o n l i n e a r p e r i o d i c wave i n a s e l f g r a v i t a t i n g medium i s always subsonic, and t h e l a r g e r A2 , t h e s m a l l e r t h e Yueh [151 has shown t h a t t h e same r e s u l t s h o l d f o r t h e maximum v a l u e o f U/ao p o l y t r o p i c case.
.
L o c a t i n g t h e o r i g i n a t t h e peak where wave, a c c o r d i n g t o ( 5 . 1 5 ) as
p+ =
1 + A1 , we have t h e p r o f i l e o f t h e
= -1+
5
2
X,
5
1
.
(5.26)
The second p a r t o f ( 5 . 2 6 ) f o l l o w s f r o m t h e assumption t h a t t h e non-dimensional wave l e n g t h o f t h e p r o f i l e i s 1 , and t h e l e n g t h o f t h e p r o f i l e f r o m t h e peak t o the valley i s The n o n l i n e a r d i s p e r s i o n r e l a t i o n f o r t h e p e r i o d i c s o l u t i o n 1 i s o b t a i n e d by p u t t i n g X, = when p * = 1 - A2 i n t h e f i r s t p a r t o f ( 5 . 2 6 ) :
k..
+F= 1
r+Al 1-A2
1-
f ( l - A2 ; a ) -
f(S ;a)
ds
(5.27)
47
Nonlinear Travelling Waves in Fluids
o r by t h e d e f i n i t i o n o f
B
, (5.28)
Here
xJ
=
a.
i s t h e Jean's length.
The appearance o f A2 , t h e a m p l i t u d e parameter, i n t h e d i s p e r s i o n r e l a t i o n (5.29) c l e a r l y b r i n g s o u t t h e n o n l i n e a r c h a r a c t e r o f t h e d i s p e r s i o n r e l a t i o n . Thus f o r each p a i r (A2 , A T ) o f t h e a d m i s s i b l e domain i n t h e (A1 , A2) p l a n e , we g e t a0 2 ~1 f r o m (5.19) and t h e r e f o r e t h e wave speed f r o m = a ; t h e wavelength i s Instead o f g i v e n by (5.29) w h i l e t h e wave p r o f i l e i s d e s c r i b e d by (5.26). t a k i n g (A1 , A2) as t h e two independent parameters o f t h e f a m i l y o f s o l u t i o n s , we may p r o f i t a b l y choose A2 and t o b r i n g o u t t h e p h y s i c s o f t h e problem
(r)
hJ
more c l e a r l y . F i g u r e 6 o f Yueh shows t h a t t h e r e a r e t h r e e d i s t i n c t r e g i o n s i n t h e a m p l i t u d e wavelength p l a n e : ( 1 ) a r e g i o n i n which a shock wave i s expected t o form, ( 2 ) t h e r e g i o n i n which t h e n e u t r a l ( p e r i o d i c ) waves e x i s t , and ( 3 ) a r e g i o n where t h e r e i s g r a v i t a t i o n a l i n s t a b i l i t y . I n t h e t h i r d case, i t i s e x p e c t e d t h a t t h e u n s t a b l e waves w i l l e v e n t u a l l y e v o l v e t o a n e u t r a l wave w i t h f i n i t e a m p l i t u d e . Yueh has f u r t h e r shown t h a t no s o l i t a r y wave can be found i n a homogeneous s e l f - g r a v i t a t i n g medium. A s i m i l a r s t u d y i s c a r r i e d o u t f o r a homogeneous plasma f o r which t h e g o v e r n i n g system o f PDEs i s s i m i l a r b u t t h e r e s u l t s come o u t t o be q u i t e d i f f e r e n t . 6.
CONCLUSIONS
I n t h i s r e v i e w we have a t t e m p t e d t o demonstrate, w i t h t h e h e l p o f s p e c i f i c examples, t h e e x i s t e n c e of e x a c t p r o g r e s s i v e o r t r a v e l l i n g wave s o l u t i o n s o f systems o f q u a s i l i n e a r inhomogeneous PDEs. The s o l u t i o n s may have d i s c o n t i n u i t i e s such as shocks, t h e y may r e p r e s e n t waves w i t h i n f i x e d o r f r e e b o u n d a r i e s o r may be p e r i o d i c p r o p a g a t i n g i n an i n f i n i t e domain. These examples would show t h a t s i m i l a r s o l u t i o n s may b e found f o r o t h e r systems, p a r t i c u l a r l y i n s e v e r a l ( s p a t i a l ) dimensions i n areas such as plasma p h y s i c s o r r o t a t i n g f l u i d s . The e x i s t e n c e o f i n t e r m e d i a t e i n t e g r a l s i s a fortunate circumstance w h i c h e n a b l e s t h e d i s c u s s i o n t o be reduced t o t h e f a m i l i a r phase p l a n e . F o r o t h e r problems i n more t h a n one dimension, which a w a i t s o l u t i o n , such i n t e r mediate i n t e g r a l s may n o t e x i s t and t h e d i s c u s s i o n would have t o be c a r r i e d o u t i n phase space o f t h r e e o r more dimensions. T h i s would pose mathematical c o m p l e x i t i e s s i n c e no s t a n d a r d d i s c u s s i o n o f t h e phase space seems a v a i l a b l e . The o t h e r aspect t h a t remains t o be t r e a t e d i s t h e a s y m p t o t i c s t a b i l i t y o f such s o l u t i o n s . We g i v e some f u r t h e r s o l u t i o n s o f t h e t y p e c o n s i d e r e d h e r e f o r w a t e r wave problems i n C161 and C171.
REFERENCES
111 B a r e n b l a t t , G.I. and Z e l ' d o v i c h , Ya.B.,
S e l f - s i m i l a r s o l u t i o n s as i n t e r -
48
P. L. Sachdev
mediate asymptotes, Ann. Rev. F l u i d Mech. 4 (1972) 285-312. 121 P e l e t i e r , L.A., A s y m p t o t i c b e h a v i o r o f s o l u t i o n s o f t h e porous media e q u a t i o n , S I A M J . Appl. Math. 21 (1971) 542-551. [ 3 1 Courant, R. and F r i e d r i c h s , K . O . , Supersonic Flow and Shock Waves, I n t e r s c i e n c e , New York (1948) 424-429. C41 Earnshaw, S., On t h e mathematical t h e o r y o f sound, Trans. Royal. SOC. London 150 (1860) 133-148. [ 5 ] S c h i n d l e r , G.M., Simple waves i n m u l t i d i m e n s i o n a l gas f l o w , S I A M J . Appl. Math. 19 (1970) 390-407. 161 Mackie, A.G., Two-dimensional q u a s i - s t a t i o n a r y f l o w s i n gas dynamics, Proc. Camb. P h i l . SOC. 64 (1968) 1009-1108. C71 Levine, L.E., Two-dimensional unsteady s e l f - s i m i l a r f l o w s i n gas dynamics, ZAMM 52 (1972) 441-460. 181 Freeman, N.C., Simple waves on shear f l o w : S i m i l a r i t y s o l u t i o n s , J . F l u i d Flech. 56 (1972) 257-263. 191 Sachdev, P.L., Exact, s e l f - s i m i l a r , time-dependent f r e e s u r f a c e f l o w s under g r a v i t y , J . F l u i d Mech. 96 (1980) 797-802. C l O l Longuet-Higgins, M.S., A c l a s s o f e x a c t , time-dependent, f r e e s u r f a c e f l o w s , J . F l u i d Mech. 55 (1972) 529-543. 1111 Longuet-Higgins, M.S., S e l f - s i m i l a r , time-dependent f l o w s w i t h a f r e e surface, J . F l u i d Mech. 73 (1976) 603-620. C121 B l y t h e , P.A., Kazakia, Y. and V a r l e y , E., The i n t e r a c t i o n o f l a r g e a m p l i t u d e s h a l l o w - w a t e r waves w i t h an ambient shear f l o w : N o n - c r i t i c a l f l o w s , J . F l u i d Mech. 56 (1972) 241-255. C131 Seshadri, V . S . and Sachdev, P.L., Q u a s i - s i m p l e wave s o l u t i o n s f o r a c o u s t i c g r a v i t y waves, Phys. o f F l u i d s 20 (1977) 888-894. C141 D r a z i n , P.G., N o n l i n e a r i n t e r n a l g r a v i t y waves i n a s l i g h t l y s t r a t i f i e d atmosphere, J . F l u i d Mech. 36 (1969) 433-446. C151 Yueh, T.Y., N o n l i n e a r waves i n a s e l f - g r a v i t a t i n g medium, S t u d i e s i n Appl . Math. 65 (1981) 1-36. E l 6 1 Sachdev, P.L. and Singh, M.C., E x a c t p e r i o d i c s o l u t i o n s o f l o n g wave e q u a t i o n s f o r a r o t a t i n g f l u i d , J . Appl. Mech., t o appear. C171 Sachdev, P.L. and Singh, M.C., Exact time-dependent f l o w s f o r nons h a l l o w w a t e r e q u a t i o n s (1983), t o be p u b l i s h e d . Appendix
The phase
y = 2 .
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
49
SECOND ORDER WAVE INTERACTION WITH LARGE STRUCTURES M. Rahman Department of Applied Mathematics Technical University of Nova Scotia Halifax, Nova Scotia B3J 2 x 4 Canada
The present study is mainly concerned with the investigation of second order wave forces on offshore structures. This paper extends Lighthill’s method for deep water waves to the waves in intermediate depth and shallow water cases and clearly demonstrates that all nonlinear boundary conditions including the radiation condition are satisfied by the scattered velocity potential. An exact expression for linear velocity potential applicable for shallow water waves has been used in these calculations. These results have been verified with those obtained by direct perturbation technique reported recently by Rahman and Heaps (1983) and with the experimental data. These comparisons show excellent agreement. INTRODUCTION The estimation of hydrodynamic wave interaction with large offshore structures has received considerable mathematical and engineering interest due to many practical applications to ocean engineering. Recently, the United States Commission on Marine Science and Engineering Resources have predicted that underwater structures would be built in connection with oil and gas recovery and production. The dynamic characteristics of these strueures could be determined provided an accurate prediction of the wave-induced forces on these structures is available. The mathematical form of the Morison equation (1950) has been applied to predict these forces for the last thirty years. However, certain difficulties in using it in the design and construction of offshore structures have been reported in the literature. Based upon the classical work on linear diffraction theory for water waves by Havelock (1940), MacCamy and Fuchs (1954) developed a linear diffraction theory for wave forces on large cylindrical piles immersed in the ocean. This study was also found to have limited applications because the nature of the water waves is inherently nonlinear. In a recent paper by Chakrabarti (1975), the nonlinear theory has been attempted for the case of a vertical cylinder and difficulties were found in satisfyinq the free surface boundarv condition and the radiation condition. The to this objection method of Raman and Venkatanarasaiah (1976), is not subject but is extremely complicated. The recent solution of Rahman (1981)has satisfied the free surface boundary conditions only at the surface of the cylinder. Hogben et a1 (1977) and Lighthill (1979) have published comprehensive review articles on the recent developments in the subject with a critical evaluation of both the methods and the results concerning the waves and hydrodynamic loading on offshore structures.
M. Rahman
50
Hunt and W i l l i a m s (1982) have r e c e n t l y d e m o n s t r a t e d a second o r d e r t h e o r y t o e v a l u a t e t h e wave l o a d i n g s on a c y l i n d e r . Another second o r d e r t h e o r y a l m o s t i d e n t i c a l t o t h a t o f Hunt and W i l l i a m s w a s r e p o r t e d by Rahman and Heaps (1983) by using formal p e r t u r b a t i o n technique. Both t h e s e t h e o r i e s seem t o produce identical results. However, d i f f i c u l t i e s were found i n e v a l u a t i n g t h e F o u r i e r B e s s e l i n t e g r a l a r i s i n g i n t h e q u a d r a t i c f o r c e s because of t h e h i g h l y complicated s i n g u l a r behaviour of t h e i n t e g r a n d . Motivated by t h e above d i s c u s s i o n , Rahman and C h a k r a v a r t t y (1981) have r e c e n t l y r e p o r t e d a t h e o r e t i c a l i n v e s t i g a t i o n o f t h e n o n l i n e a r wave l o a d i n g on o f f s h o r e s t r u c t u r e s extending t h e L i g h t h i l l ' s deep w a t e r t h e o r y t o t h e shallow w a t e r waves. These c a l c u l a t i o n s w e r e o n l y performed f o r t h e s m a l l kb v a l u e s where k is t h e i n c i d e n t wave number and b i s t h e c y l i n d e r r a d i u s . The p r e s e n t i n v e s t i g a t i o n i s mainly concerned w i t h t h e e x a c t second o r d e r c a l c u l a t i o n u s i n g t h e e x a c t e x p r e s s i o n f o r t h e l i n e a r v e l o c i t y p o t e n t i a l of d i f f r a c t i o n t h e o r y . These r e s u l t s have been v e r i f i e d w i t h t h o s e o b t a i n e d by I t is interestRahman and Heaps (1983) and compared w i t h t h e e x p e r i m e n t a l d a t a . i n g t o n o t e t h a t b o t h t e c h n i q u e s l e a d t o i d e n t i c a l r e s u l t s a n d comparison shows e x c e l l e n t aqreement. The a d v a n t a g e o f t h i s new t h e o r y i s t h a t t h e second o r d e r e f f e c t s of t h e Morison e q u a t i o n can be i n c o r p o r a t e d from t h e knowledge of t h e lirr?ar v e l o c i t y p o t e n t i a l a l o n e w i t h o u t c a l c u l a t i n g t h e q u a d r a t i c wave p o t e n t i a l as i t i s necessary i n t h e formal p e r t u r b a t i o n technique.
MATHEMATICAL FORMULATION We c o n s i d e r a l a r g e r i g i d v e r t i c a l c y l i n d e r o f r a d i u s b which is a c t e d upon by a t r a i n o f t w o d i m e n s i o n a l p e r i o d i c , p r o g r e s s i v e waves p r o p a g a t i n g on t h e I t i s assumed t h a t t h e f l u i d i s s u r f a c e o f f l u i d of a r b i t r a r y u n i f o r m d e p t h h . i n c o m p r e s s i b l e , i n v i s c i d , and t h e motion i s i r r o t a t i o n a l i n t h e r e g i o n bounded The by t h e f r e e s u r f a c e , r i g i d bottom boundary and t h e s u r f a c e o f t h e c y l i n d e r . governing d i f f e r e n t i a l e q u a t i o n s and t h e r e l e v a n t boundary c o n d i t i o n s may be found e l s e w h e r e i n c l u d i n g Rahman and Heaps ( 1 9 8 3 ) . A s d e m o n s t r a t e d by Rahman and C h a k r a v a r t t y ( 1 9 8 1 ) , t h e second o r d e r c o r r e c t i o n i n t h e l i n e a r f r e e s u r f a c e c o n d i t i o n may b e o b t a i n e d a s
where
Q, i s t h e v e l o c i t y p o t e n t i a l z is the v e r t i c a l axis
t is the t i m e
g i s t h e ' a c c e l e r a t i o n due t o g r a v i t y , and
( V @ i 2 is t h e f l u i d speed squared. Using t h e p e r t u r b a t i o n series (see L i g h t h i l l
(1979)), the velocity potential,
4 , and t h e f r e e s u r f a c e e l e v a t i o n , n , may b e e x p r e s s e d as
51
Second Order Wave Interaction with Large Structures
where qQ a r e t h e ordinary s o l u t i o n s s a t i s f y i n g t h e l i n e a r i z e d f r e e surface c o n d i t i o n s and I$ 4 , nq r e p r e s e n t t h e second o r d e r c o r r e c t i o n o f t h e o r d e r of t h e s q u a r e of t h e d i s t u r b a n c e s (and h i g h e r o r d e r c o r r e c t i o n s a r e n e g l e c t e d ) . Now u s i n q t h e T a y l o r s e r i e s e x p a n s i o n of @ a b o u t z = 0 , $ can b e e x p r e s s e d as
+ higher o r d e r terms The d i f f e r e n t q u a n t i t i e s i n t h e f r e e s u r f a c e c o n d i t i o n (1) may s i m i l a r l y b e o b t a i n e d and s u b s t i t u t i n g t h e s e v a l u e s i n t o (l), we o b t a i n t h e e q u a t i o n s f o r t h e l i n e a r p o t e n t i a l , $9., and t h e q u a d r a t i c p o t e n t i a l , + q , i n t h e form
a2+,
adl,
y + g - - = o
at
(5)
az
where
E q u a t i o n (6), a l t h o u g h d e r i v e d a t t h e e x a c t f r e e s u r f a c e , c a n b e a p p l i e d a t t h e u n d i s t u r b e d f r e e s u r f a c e z = 0 b e c a u s e b o t h s i d e s a r e q u a d r a t i c so t h a t e r r o r i n r e l o c a t i n g them a t t h e f r e e s u r f a c e i s o f s t i l l h i g h e r o r d e r .
PHYSICAL SOLUTION OF L I N E A R DIFFRACTION THEORY The l i n e a r d i f f r a c t i o n v e l o c i t y p o t e n t i a l , @ & , f o r a n incoming wave t r a i n i n t h e p r e s e n c e of a v e r t i c a l c y l i n d e r f o r d e e p w a t e r waves w a s g i v e n by Havelock ( 1 9 4 0 ) and h i s t h e o r y h a s been e x t e n d e d by MacCamy and Fuchs ( 1 9 5 4 ) f o r s h a l l o w w a t e r waves and t h e v e l o c i t y p o t e n t i a l i n t h i s c a s e may b e e x p r e s s e d i n t h e complex form
where Re stands f o r real p a r t ,
a
0
= 1
,
a
=
2(-i)n
,
i =
fi ( n
> 01,
and
i n which r i s t h e r a d i a l d i s t a n c e and a i s t h e a m p l i t u d e of t h e wave. g i v e n by t h e d i s p e r s i o n r e l a t i o n
Here w i s
52
M. Rahman
o2
gk t a n h kh
=
(11)
and H F ) i s t h e n t h o r d e r Hankel f u n c t i o n o f second k i n d , d e f i n e d by Hij2)(kr) = J ( k r )
-
i y (kr)
(12)
i n which J n ( k r ) and y n ( k r ) a r e B e s s e l f u n c t i o n s of f i r s t and second k i n d , respectively. Here a prime d e n o t e s t h e d i f f e r e n t i a t i o n .
($,
The s o l u t i o n $ f i n ( 8 ) i s i n complex form, a r e a l s o l u t i o n of which i s g i v e n by + $); where 6; i s t h e complex c o n j u g a t e o f $, may b e e x p r e s s e d i n the form io t
P.
where w
=
a w cosh k ( z + h ) a n An(kr) e s i n h kh n=-m
c o s nO
2k
i s d e f i n e d as
w
o
if n
-o
if n
= \
1
The f u n c t i o n an and An are d e f i n e d by ( 9 ) and 1101, r e s p e c t i v e l y f o r p o s i t i v e v a l u e s o f n , and by t h e r e l a t i o n s
a
-n
= n
*
* A-n
= An
*
*
f o r n e g a t i v e v a l u e s of n where a n and An are t h e complex c o n j u g a t e s of an and An, respectively. The summation i n ( 1 3 ) i n c l u d e s b o t h n = 0, and n = 0- i n o r d e r t o ( k r ) and e-iwt A. ( k r ) i n c l u d e t e r m s l i k e elwt A.
.
E q u a t i o n ( 9 ) may b e r e d e f i n e d as
a
n
= 6
n
e
-nni/2
where
60 = 1
n = O
6
n # O
= 2
WAVE FORCES FORMULATIONS
I n t h i s s e c t i o n w e w i l l b e mainly concerned w i t h t h e f o r m u l a t i o n o f wave f o r c e s on o f f s h o r e s t r u c t u r e s . Making r e f e r e n c e t o t h e work o f Rahman and C h a k r a v a r t t y (19811, t h e t o t a l f o r c e s may b e o b t a i n e d i n t h e form
where
Second Order Wave Interaction with Large Structures
53
FL i s t h e l i n e a r f o r c e
F
d
i s t h e second o r d e r dynamic f o r c e
F,
i s t h e second o r d e r w a t e r l i n e f o r c e , and
F
is the quadratic force
q
The a c c u r a c y o f t h e wave f o r c e s c a n b e g r e a t l y i n c r e a s e d by a d d i n g t o t h e l i n e a r f o r c e , F Q , t h e second o r d e r c o n t r i b u t i o n g i v e n b y t h e term (Fa + F, + Fq) which i s t h e key t o t h e u n d e r s t a n d i n g o f t h e i r r o t a t i o n a l flow t e r m i n t h e Morison equation.
In f o r m u l a t i n q t h e t o t a l f o r c e s w e have n e g l e c t e d t h e c o n t r i b u t i o n due t o v i s c o u s d r a g on t h e a s s u m p t i o n t h a t t h i s f o r c e may b e n e g l e c t e d when t h e r e l a t i v e dimension of t h e s t r u c t u r e i s l a r q e . Denoting t h e l i n e a r f o r c e by FQ which i s e q u a l t o t h e r e s u l t a n t of t h e t r a n s i e n t p r e s s u r e s of - P -
a%
at
w e can w r i t e
where s i s t h e body s u r f a c e and nx i s t h e d i r e c t i o n c o s i n e between t h e outward normal and t h e d i r e c t i o n of t h e f o r c e FQ. Here p i s t h e d e n s i t y o f t h e f l u i d . The second o r d e r f o r c e a r i s e s from t h e p r e s s u r e d i s t r i b u t i o n s d e r i v e d from t h e l i n e a r p o t e n t i a l , 61. These p r e s s u r e d i s t r i b u t i o n s have second o r d e r terms which e x e r t a l o a d i n g on a s t r u c t u r a l component. A c t u a l l y two f o r c e s c o n s t i t u t e One c o r r e s p o n d s t o t h e dynamic f o r c e , Fd, which t h i s second o r d e r c o n t r i b u t i o n . is t h e r e s u l t a n t o f t h e dynamic p r e s s u r e ,
o v e r t h e s u r f a c e o f t h e body. representation
Thus, t h e f o r c e , Fd, h a s t h e f o l l o w i n g i n t e g r a l
I t s h o u l d b e p o i n t e d o u t t h a t t h e p o t e n t i a l i n c l u d e s t h e r e s p o n s e s o f t h e body t o b o t h f l u c t u a t i n g m o t i o n s and t h e e x t e n s i o n a l m o t i o n s due t o waves.
Another f o r c e p r o v i d e s a second o r d e r c o n t r i b u t i o n t o t h e wave l o a d i n g on I t is s t r u c t u r e s whenever t h e body p e n e t r a t e s t h e f r e e s u r f a c e of t h e l i q u i d . d i r e c t l y a s s o c i a t e d w i t h t h e t r a n s i e n t and h y d r o - s t a t i c p r e s s u r e s of t h e i r r o t a t i o n a l f l o w s . According t o L i q h t h i l l (1979),t h i s a d d i t i o n a l second o r d e r h o r i z o n t a l x-component o f t h e f o r c e , F, a c t i n g a t t h e w a t e r l i n e w (where t h e s t r u c t u r a l component i n t e r s e c t s t h e f r e e s u r f a c e o f t h e f l u i d ) i s g i v e n by
54
M. Rahman
where t h e i n t e g r a n d i s t h e h o r i z o n t a l l y r e s o l v e d f o r c e p e r u n i t l e n g t h a c t i n g a t I t is t h e w a t e r l i n e and t h e i n t e q r a l g i v e s i t s r e s u l t a n t i n t h e x - d i r e c t i o n . i n t e r e s t i n g t o note t h a t t h i s equation is equivalent t o
a s p r e s e n t e d i n E q u a t i o n ( 3 0 ) o f Rahman and Heaps ( 1 9 8 3 ) .
SIMPLE APPROACH TO CALCULATING F 4 The s i m p l e approach t o c a l c u l a t i n g Fq c o u l d b e summarized a s f o l l o w s : W e w i l l assume t h e f o l l o w i n g form o f t h e l i n e a r p o t e n t i a l , $, and t h e q u a d r a t i c : potential,
%
$J
$
9" = R e [ @a. e
4
= R e [ @ e 4
iwt
I
2iwt
(22)
1
(23)
I n s e r t i n g t h e s e v a l u e s i n E q u a t i o n ( 6 ) , w e o b t a i n a f t e r some r e d u c t i o n
where 2 K = % =
9
4k t a n h kh
.
(25)
With t h e p o t e n t i a l due t o a u n i t t r a n s l a t i o n o s c i l l a t i o n o f t h e body i n t h e form Re t $ e2iwtl, it f o l l o w s t h a t on t h e s u r f a c e s of t h e body,
(26-ab)
where n
i s t h e x-component of a u n i t inward normal t o t h e s u r f a c e s .
w h i l e $, l i k e aq, s a t i s f i e s t h e r a d i a t i o n c o n d i t i o n at i n f i n i t y . us t o apply Green's e q u a t i o n
JJ [O V 2 $
v
q
-
$0 2 @
1 dV
9
=
J
suz=0
[Oq
2-
Furthermore,
This then l e a d s
a@
$
anq1 d s
where t h e s u r f a c e i n t e g r a l i s t a k e n o v e r t h e b o u n d a r i e s of t h e f l u i d ( s and z ' = 0 ) and n is t h e normal outward from t h e f l u i d . S i n c e b o t h 0 and Q s a t i s f y L a p l a c e ' s e q u a t i o n , t h e l e f t hand s i d e of v a n i s h e s so t h a t 7281 g i v e s
(28)
Second Order Wave Interaction with Large Structures
55
A f t e r p e r f o r m i n g t h e i n d i c a t e d i n t e g r a t i o n i n ( 2 9 ) and a p p l y i n g t h e c o n d i t i o n s
( 2 6 ) and ( 2 7 ) , i t f o l l o w s from ( 2 9 ) t h a t
ao
I
I@
$ [ + - K @ l d x d y = 4
z=O
q
s
n
ds
.
(30)
x
Thus, t h e q u a d r a t i c f o r c e , F c a n be w r i t t e n a s q
= Re
I Oq
[-2piw e 2 i w t
nx d s l
.
s
I n v o k i n g ( 3 0 ) and ( 2 4 ) , i t f o l l o w s t h a t
= Re
F
[(-2iwp) e
2iwt
CJ
ao
a$
1 z=O
z=o
az -
K 3 '
I dx dy
(33)
(34)
= Re
[-
I
PW
(VOQ)
2
1
+ 2
2
(tanh
2 kh-1) ( k $ L )
dx dyl
(35)
z=O where W = (-)a$
az
i s t h e v e r t i c a l v e l o c i t y on t h e f r e e s u r f a c e 2=0
a s s o c i a t e d w i t h t h e u n i t o s c i l l a t i o n o f t h e body. I n o r d e r t o determine t h e v e r t i c a l v e l o c i t y , W , it is necessary t o f i n d t h e s o l u t i o n f o r t h e p o t e n t i a l , $. A s s t a t e d e a r l i e r t h e f u n c t i o n , $, s a t i s f i e s t h e L a p l a c e ' s e q u a t i o n , t h e c o n d i t i o n (26ab) on t h e c y l i n d e r r = b , t h e f r e e s u r f a c e The s o l u t i o n , $, c a n c o n d i t i o n on z = 0 and t h e r a d i a t i o n c o n d i t i o n a t i n f i n i t y . be e x p r e s s e d i n t h e form
M. Rahman
56
where (P) denotes the Cauchy principal value of the integral and K1 is the modified Bessel function of order one. It is noted that the integral in (36) represents the standing wave part of the motion and the nonintegral part is the radiation component of the motion. The r derivative of the integral on r = b is a simple Fourier integral which can easily be evaluated as 1 - e2kz, and the r-derivative of the radiation component on r = b is simply 2ekz so that the potential I$ satisfies the boundary condition
Some simple algebraic calculation gives the real integral form of W
From the asymptotic nature of the Bessel function for small Kr and Kb, it turnsout that K (Kru) 1 b2 lim [UK, Kr+O Kb4
b2
Thus the integral in (37) is asymptotically equal to K - , and the nonintegral term is approximately equal to - 2 K b 2 / r when both Kr andrKb tend to zero. This
leads to the final asymptotic value of W as
.
Kb2
w----Ose
In order to estimate the effective range b we can define B by the equation B Re
J
2
Wdrl
= -
Kb
b
(39) i
r < B with which (39) holds,
B
en (r) ” cos 0
(40)
on the other hand, it follows from (37) that m
J b
K (Kbu)
m
2 Wdr = [ J
{-
0 2
+: I
2H(2) du
u K Ki(Kbu) l+u
0
(Kb)
1 cos e
K H ~ ’~( ~) b )
.
In evaluating the r-integral we have used the Bessel relation that
We next use the asymptotic result for small K b as
(41)
Second Order Wave Interaction with Large Structures
57
t o o b t a i n t h e a s y m p t o t i c r e p r e s e n t a t i o n f o r ( 4 1 ) i n t h e form
where
y is t h e E u l e r ' s constant. This r e s u l t a g r e e s w i t h ( 4 0 ) provided t h a t B = 1 (2e-Y)
K =
1 e-' 2
-
C k t a n h kh
(k t a n h k h )
-1
(45)
where C =
2
e-'
= 0.2807
a numerical c o n s t a n t
.
T h i s asymptc _ i c b e h a v i o u r i s of p r a c t i c a l i m p o r t a n c e t o second o r d e r i r r c a t i o n a l f l o w l o a d i n g s f o r v e r y s m a l l kb. I n t h e f o l l o w i n g however, w e are g o i n g t o p r e s e n t a n e x a c t second o r d e r s o l u t i o n f o r a r b i t r a r y kb.
EXACT CALCULATIONS FOR SECOND ORDER FORCES For a d i f f r a c t e d wave whose l i n e a r p o t e n t i a l i s of t h e form (13), t h e t o t a l h o r i z o n t a l f o r c e , F e , c a n b e o b t a i n e d from e q u a t i o n (18). T h e r e f o r e ,
2n FE =
a+,
0
1
1
8=0
z=-h
(-p
at
)
dzl
r=b
( - c o s 8) bd 8
.
A f t e r i n s e r t i n g t h e e x p r e s s i o n f o r $p, from (13), and p e r f o r m i n g t h e i n d i c a t e d i n t e g r a t i o n , F i t u r n s o u t t o b e t h e e x p r e s s i o n which h a s been o b t a i n e d by many r e s e a r c h e r s i n c l u d i n g MacCamy and Fuchs 1 1 9 5 4 ) . To c a l c u l a t e t h e second o r d e r c o n t r i b u t i o n s , F d , Fg, and Fq, t h e c a l c u l a t i o n s are summarized a s f o l l o w s . To e v a l u a t e t h e dynamic f o r c e , Fd, t h e f o r m u l a ( 1 9 ) may be r e w r i t t e n as
I n v o k i n g t h e e x p r e s s i o n f o r $ k from ( 1 3 ) and c a r r y i n g o u t t h e z - i n t e g r a t i o n of t h e e x p r e s s i o n (V$ )2, w e o b t a i n P,
M. Rahman
58
+
cos
I
(m-n)
2
+ D
m
-
B 11
[ c o s (m+n)O
n
+
cos ( m - n ) ~ ]
(48)
where t h e Wronskian p r o p e r t y of B e s s e l f u n c t i o n s g i v e s
- 2i
-iD,
Am(kb) =
= R
(nkb)
ffm (2
)'
(49)
e
(kb)
i n which
I n view of t h e s u b s e q u e n t 8 - i n t e g r a t i o n d e s c r i b e d i n e q u a t i o n ( 4 7 ) , w e r e q u i r e o n l y t h e c o e f f i c i e n t of cos 8 i n t h e d o u b l e summation i n (481 and t h i s y i e l d s t o the following
2
-
4a 9
c
T2k2b2 9,=0
[(l
-
2 kh s i n h 2kh
Q x [ E ~ (-1)
+
R ( R + 1) k 2 b2 ('
[c9,cos
2wt -
where
.iy'R+1 -
EL = (J
"i+l yi)'Ta.
sQ
+
2 kh s i n h 2kh )
sin 2wtIl
(52)
Second Order Wave Interaction with Large Structures
The B e s s e l f u n c t i o n s arguments b e i n g K b .
59
Then i n t e g r a t i n g w i t h r e s p e c t t o 8 ,
t h e t o t a l dynamic f o r c e , F d , may b e e x p r e s s e d as
k
.
- (-1) Cc, c o s 2 w t - s2 s i n 2wt)I
x [E,
(57)
The r e s u l t ( 5 7 ) i s t h e sum of t h e s t e a d y - s t a t e and o s c i l l a t o r y p a r t s which c a n be w r i t t e n as
Fss =
2
m
21)4a 1 nbk2
2
Fos = - %!% d
nbk2 x
[(1 -
8=0 "
c
, 2kh s i n h 2kh )
('
2kh , s i n h 2kh
(-1)9" { (1 -
k=O
Ic,
a. (9"+1) +
+
+
2kh s i n h 2kh
R(R+l) k2,,2
('
+
(58)
)I
2 kh s i n h 2kh
c o s 2 w t - S~ s i n 2 w t J
(59)
.
(60)
such t h a t ss
Fd = Fd
+
0s
Fd
T o c a l c u l a t e t h e w a t e r l i n e f o r c e , Fu, t h e f o r m u l a ( 2 0 ) may b e r e w r i t t e n a s
z=b Inserting t h e expression f o r
4,
i n t o (61), t h e t e r m i n (-1a$ at
written as
==o
may b e
r=b
+
Bm
-
O n ) 1 x [cos (m+n)O + cos (m-n) 8 1
.
(62)
I n view of t h e s u b s e q u e n t 8 - i n t e g r a t i o n g i v e n i n (61), we r e q u i r e o n l y t h e c o e f f i c i e n t of cos 8 term i n t h e d o u b l e summation i n ( 6 2 ) and t h i s y i e l d s t o t h e following
60
M. Rahman
where E ,,
CL, and SL a r e as p r e v i o u s l y d e f i n e d .
Then i n t e g r a t i n g w i t h r e s p e c t t o 0 , t h e t o t a l w a t e r l i n e f o r c e , F ,,
may b e
o b t a i n e d as 2
-
c
F w = - - 4pga nbk2
[EL
Q
+ (-1) ice
cos w t -
sQ s i n
.
2wt11
(64)
&=O
T h i s r e s u l t may b e e x p r e s s e d a s t h e s t e a d y - s t a t e and o s c i l l a t o r y p a r t s a s follows : ss
0s
Fw = Fo
Fw
+
where
2
-
F,”” =
%
-
nbk
il=O
II
C
(-1)
{C,
cos 2ot
.
- SQ s i n 2wt)
(67)
Now combining t h e s t e a d y - s t a t e and o s c i l l a t o r y p a r t s o f Fd and Fo, we o b t a i n
2
Fos = -
m Z
2pga
x
!2 (-1)
[(3
-
sinh
2kh 2kh )
L=O
Tbk2
[cL c o s
+
k(L+l)
____ k2b2
2wt
- sR s i n
(’
+
2kh s i n h 2kh
.
2wtl
11
(69)
I t i s i n t e r e s t i n g t o n o t e t h a t t h e s e two r e s u l t s p e r f e c t l y a g r e e w i t h t h o s e
o b t a i n e d by d i r e c t p e r t u r b a t i o n t e c h n i q u e , To obtain t h e quadratic f o r c e , F
g
2n
m
F
q
1
= Re {
I
(-rdr)
b
pW
[(V@L)
2
1 +7
r e p o r t e d by Rahman and Heaps ( 1 9 8 3 ) .
t h e formula ( 3 5 ) i s r e w r i t t e n as f o l l o w s :
(tanh’ kh
2
- 1) ( k @ L )1
where t h e r e a l p a r t of W i s g i v e n by
K = - 4wL g
d0 z=o
0
= 4k t a n h k h , a s d e f i n e d i n ( 2 5 )
.
(70)
Second Order Wave interaction with Large Structures Invoking t h e expression f o r $%, ( C $ y ) 2 2
2
( v + y ) z = O=
2
a+L
a@l. + (--) +
2
m
a 9k c o t h kh
(-1
lz=O
aZ
m
I: I: 6 m=O n=O
2
a 9k c o t h kh 4
2 a 9k t a n h kh 4
c a n be c a l c u l a t e d and i s o b t a i n e d a s
2
[(x’ rao a@9.
61
m
,
~Q; ~O; [ c o s ((m+n)
m
z
6m6n Q,
~1,
m=O n=O
m
mn
22
+ B~ + O~
[ c o s ((m+n) P 2 +
B~ + B~
k r
m
z
I: m=O n=O
6 6 9 o [ c o s ((m+n) m n -m -n
+
am +
bn - 2 o t )
where
i n which Qm(kr) Qm(kr)=
(74) iJ:(kb)
P,(kr)
= J
m
+ y:(kb)
(kr) y'(kb)
m
as p r e v i o u s l y d e f i n e d i n ( 5 1 ) .
- J;(kb)
y (kr)
m
(75)
Here a prime d e n o t e s t h e d i f f e r e n t i a t i o n .
S i m i l a r l y t h e e x p r e s s i o n f o r ( k + e ) 2 i s q i v e n by
M. Rahman
62
I n view of t h e c o s 0 t e r m p r e s e n t i n t h e W e x p r e s s i o n , i n t h e 8 - i n t e g r a t i o n i n ( 7 ) we o n l y need t o c o l l e c t t h e c o e f f i c i e n t of c o s 8 term i n t h e d o u b l e summation of ( 7 2 ) and ( 7 6 ) . Thus, w e o b t a i n , a f t e r some r e d u c t i o n ,
may b e Then i n t e g r a t i n g w i t h r e s p e c t t o 8 , t h e t o t a l q u a d r a t i c f o r c e , F g’ e x p r e s s e d i n i n t e g r a l form
- -
m
F
q
= - 2pgka
+
2
-
f,
rdr
1 [3 t a n h kh OR QQ+l R=O
2 L (P+1) ( r - 1) OR QR+,]l k r
[ J m0
.9
[El - (-1)
K (Kur) du 1 { - uK'(Kub) I,+n lfu 1
J1(Kr)
. c o t h kh 1 2 Q i Qi+l
f
{CL cos 2ot
J;(Kb)
2
Ji
f
- Sl s i n 2 w t ) J
y l ( K r ) y;(Kb)
(Kb) + y ;
2
(78) (Kb)
Here t h e t e r m s a s s o c i a t e d w i t h EQ i s t h e s t e a d y - s t a t e p a r t and t h e t e r m s a s s o c i a t e d w i t h c o s 2 w t and s i n 2 w t c a n be r e c o g n i z e d t o b e t h e o s c i l l a t o r y p a r t of t h e F q' F o r t h e c o n v e n i e n c e of t h e p r e s e n t a t i o n a n d comparison w i t h t h e e x p e r i m e n t a l d a t a , t h e f o r c e e x p r e s s i o n s a r e made d i m e n s i o n l e s s and t h e i r d i m e n s i o n l e s s forms a r e g i v e n below
- Fd pgD3
2 kh ~ T ~ ~ ( D k/ L0 ) ~
R x [ E -~ (-1) {c,
a. (9"+1) k2b2
(1
c o s 2 w t - sa. s i n 2 w t 3 1
+
s i n2kh h 2kh )
I
Second Order Wave Interaction with Large Structures
63
Hence, t h e d i m e n s i o n l e s s form o f t h e t o t a l f o r c e may b e e x p r e s s e d a s
DISCUSSION OF RESULTS An e x a c t second o r d e r d i f f r a c t i o n t h e o r y h a s been d e v e l o p e d i n t h i s paper. L i g h t h i l l ‘ s t h e o r y f o r d e e p w a t e r waves h a s been e x t e n d e d t o t h e s h a l l o w water waves u s i n q an e x a c t e x p r e s s i o n € o r t h e v e l o c i t y p o t e n t i a l i n t h e c a s e of a s u r f a c e piercing circular cylinder. The t h e o r y h a s b e e n t e s t e d by comparison w i t h t h e r e s u l t s o f t h e a v a i l a b l e e x p e r i m e n t a l d a t a . The maximum h o r i z o n t a l f o r c e was e x p r e s s e d a s t h e a v e r a g e of t h e a b s o l u t e v a l u e s of t h e maximum p o s i t i v e f o r c e and t h e maximum n e g a t i v e f o r c e . The p r e s e n t t h e o r y d i f f e r s from t h e p r e v i o u s one ( s e e Rahman and Heaps (1982)) i n t h a t t h e q u a d r a t i c f o r c e , Fq i n t h e p r e s e n t p a p e r h a s been o b t a i n e d u s i n g G r e e n ’ s f u n c t i o n f o r m u l a t i o n w i t h t h e p r e v i o u s knowledge of t h e l i n e a r v e l o c i t y p o t e n t i a l , alone; thus avoiding t h e t ed i o u s algebr ai c c a l c u l a t i o n of t h e q u a d r a t i c v e l o c i t y p o t e n t i a l , which i s a b s o l u t e l y n e c e s s a r y i n t h e f o r m a l p e r t u r b a t i o n t e c h n i q u e u s e d i n o u r p r e v i o u s work. I t h a s been found t h r o u g h t h i s a n a l y s i s t h a t both t h e t h e o r i e s l e a d t o t h e i d e n t i c a l r e s u l t s except t h a t t h e i n t e g r a l r e p r e s e n t a t i o n o f t h e q u a d r a t i c f o r c e w a s t u r n e d o u t t o be i n s l i g h t l y d i f f e r e n t form. T h i s d i f f e r e n c e , however, h a s b e e n a t t r i b u t e d t o two d i f f e r e n t t h e o r e t i c a l developments. I n any c a s e , t h e f i n a l r e s u l t s w e r e found t o b e a l m o s t I n t h i s p a p e r t h e d r i f t f o r c e h a s b e e n d e f i n e d a s t h e sum o f s t e a d y identical. s t a t e p a r t o f t h e dynamic FZS, w a t e r l i n e F F and q u a d r a t i c f o r c e F F b e c a u s e of t h e absence o f t h e s t e a d y state p a r t i n t h e q u a d r a t i c f o r c e e x p r e s s i o n i n t h e p r e v i o u s t h e o r y (1983), t h e d r i f t f o r c e i n t h a t c a s e w a s o b t a i n e d a s t h e s t e a d y s t a t e p a r t of t h e dynamic and w a t e r l i n e f o r c e s and i s g i v e n by t h e e q u a t i o n ( 6 8 ) . F i g u r e s 1-3 show a comparison of t h e p r e d i c t e d r e s u l t s o b t a i n e d by t h e p r e s e n t t h e o r y and the p r e v i o u s t h e o r y w i t h t h e e x p e r i m e n t a l d a t a c o l l e c t e d by I t may b e n o t e d t h a t t h e p r e s e n t t h e N a t i o n a l Research C o u n c i l o f Canada, O t t a w a . theory l e a d s t o t h e b e t t e r c o r r e l a t i o n with t h e experimental d a t a . I n Figure 4 , comparison w a s made between t h e r e s u l t s o f t h e s e two t h e o r i e s w i t h e x p e r i m e n t a l d a t a o f C h a k r a b a r t i ( 1 9 7 5 ) . Again t h e p r e s e n t t h e o r y a p p e a r s t o c o r r e l a t e w e l l with t h e experimental d a t a .
M. Rahman
64
Experiment *** L i n e a r Theory Second-Order Theory of Rahman and Heaps (1983) ----P r e s e n t Second Order Theory 0 0 0
m R tr a \
F i g u r e 1 Comparison of L i n e a r and Second Order Wave F o r c e s w i t h E x p e r i m e n t a l Data of Moyridge and Jamieson ( 1 9 7 6 )
65
Second Order Wave interaction with Large Structures Experiment * * * Linear Theory __ Second-Order Theory of Rahman and Heaps ( 1 9 8 3 ) P r e s e n t Second Order Theory 000
Figure 2
-----
Comparison of Linear and Second Order Wave Forces with Experimental Data o f Mogridge and Jamieson (1976)
66
M. Rahrnan Experiment * * * L i n e a r Theory Second-Order Theory o f Rahman and Heaps (1983) ----P r e s e n t Second Order Theory 000 ~
0
0.02
0.04
0.06
n/L
Figure 3
Comparison of L i n e a r and Second Order Wave F o r c e s
with E x p e r i m e n t a l D a t a of Mogridge and Jamieson (1976)
Second Order Wave Interaction with Large Structures
67 n u
3 Y
in d
0 rl
in
0
0
Fig. 4
Comparison of Linear and Second-Order Wave Forces with Experimental Data of Chakrabarti (1975)
M. Rahman
68
we have u s e d S i m p s o n ' s i n t e g r a t i o n scheme i n e v a l u a t i n g t h e d o u b l e i n t e g r a l i n t h e q u a d r a t i c f o r c e g i v e n by ( 8 2 ) . T h i s i n t e g r a t i o n i s performed i n c o n j u n c t i o n w i t h IMSL l i b r a r y t o e v a l u a t e t h e B e s s e l f u n c t i o n s of d i f f e r e n t o r d e r s arising in the integrand. The u - i n t e g r a l is v e r y s i m p l e b e c a u s e t h e i n t e g r a n d is a f u n c t i o n o f m o d i f i e d B e s s e l f u n c t i o n o f f i r s t o r d e r and of s e c o n d k i n d . Due t o i t s e x p o n e n t i a l l y decaying n a t u r e , t h e v a l u e o f t h e i n t e g r a l q u i c k l y converges t o a finite limit. However, t h e r - i n t e g r a l i s e x t r e m e l y d i f f i c u l t b e c a u s e o f t h e c o m p l i c a t e d c o m b i n a t i o n of d i f f e r e n t o r d e r s o f B e s s e l f u n c t i o n s o f f i r s t and second k i n d . I t h a s been found t h r o u g h o u r n u m e r i c a l a n a l y s i s t h a t t h e r - i n t e g r a n d i s h i g h l y o s c i l l a t o r y . The i n t e g r a t i o n h a s been c a r r i e d o u t from t h e s u r f a c e of t h e c y l i n d e r t o a l a r g e d i s t a n c e B. Extensive numerical tests i n d i c a t e t h a t beyond t h e l i m i t i n g B t h e c o n t r i b u t i o n o f t h e i n t e g r a l i s insignificant. Fortunately, t h e s i n g u l a r i t y a r i s i n g i n t h e u-integrand i n t h i s p a p e r h a s been removed,while i n t h e p r e v i o u s r e p o r t t h e Cauchy p r i n c i p a l v a l u e a t the singular point of t h e integrand w a s considered. The s o l u t i o n method f o r t h e d o u b l e i n t e g r a l s was programmed i n F o r t r a n V f o r t h e CDC Cyber 170 d i g i t a l computer a t t h e T e c h n i c a l U n i v e r s i t y o f Nova S c o t i a . A Simpson's i n t e g r a t i o n scheme h a s b e e n f o r m u l a t e d and was u s e d t o i n t e g r a t e w i t h very f i n e grid s i z e .
ACKNOWLEDGEMENTS The a u t h o r i s g r a t e f u l t o I m p e r i a l O i l o f Canada L i m i t e d and t o t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada f o r t h e i r f i n a n c i a l s u p p o r t l e a d i n g t o t h i s p a p e r . Thanks are also due t o M r . P a u l C o l b o u r n e , a summer research s t u d e n t , f o r h i s s k i l l e d c o m p u t a t i o n .
REFERENCES Morison, J. R . , O ' B r i e n , M . R . , J o h n s o n , J. W. and S c h a a f , S . A., "The F o r c e s E x e r t e d by S u r f a c e Waves on P i l e s " , P e t r o l e u m T r a n s a c t i o n s , A I M E , 1 8 9 , 1950, 149-154. Havelock, T. H . , " P r e s s u r e o f Water Waves on a F i x e d O b s t a c l e " , P r o c . Roy. SOC., A175, 1940, 409-421. MacCamy, R . C . and Fuchs, R . A., "Wave F o r c e s o n P i l e s , A D i f f r a c t i o n Theory", Beach E r o s i o n Board T e c h n i c a l Memo N o . 6 9 , 1954. C h a k r a b a r t i , S . K . , "Second-Order Wave F o r c e on Large V e r t i c a l C y l i n d e r " , J o u r n a l o f t h e Waterways, Harbours and C o a s t a l E n g i n e e r i n g D i v i s i o n , A X E , V O l . 101, No. WW3, 1975, 311-317. Raman, H . and V e n k a t a n a r a s a i a h , P., " F o r c e s d u e t o N o n l i n e a r Waves on V e r t i c a l C y l i n d e r s " , J o u r n a l o f t h e Waterways, Harbours and Coastal E n g i n e e r i n g D i v i s i o n , A X E , V o l . 1 0 2 , NO. WW3, 1976, 301-316. Rahman, M . , " N o n l i n e a r Wave Loads o n L a r g e C i r c u l a r C y l i n d e r s : A P e r t u r b a t i o n T e c h n i q u e " , Ad. Water R e s o u r c e s , 4 , 1981, 9-19. Hogben, N . , M i l l e r , B . L . , S e a r l e , J. W . and Ward, G . , " E s t i m a t i o n o f F l u i d Loading on O f f s h o r e S t r u c t u r e s " , N a t i o n a l M a r i t i m e I n s t i t u t e Report NO. 11, 1977, 1-60. L i g h t h i l l M. J . , "Waves and Hydrodynamic Loading", P r o c . Second I n t e r n a t i o n a l C o n f e r e n c e on t h e Behaviour o f O f f s h o r e S t r u c t u r e s , 1, 1 9 7 9 , 1-40.
Second Order Wave Interaction with Large Structures
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[91
Hunt, J. N. and Williams, A. N., "Nonlinear Diffraction of Stokes’ Water Waves by a Circular Cylinder for Arbitrary Uniform Depth", J. de Mechanique Theo. et Appl. 1(3), 1982, 429-449
[lo]
Rahman, M. and Heaps, H. s . , "Wave Forces on Offshore Structures: Nonlinear Wave Diffraction by Large Cylinders", J. Physical Oceanography, 1983 (in press).
[ll]
Rahman, M. and Chakravartty, I. C., "Hydrodynamic Loading Calculations for Offshore Structures", SIAM J. of Applied Mathematics, 41, 1981, 445-458.
[12]
Mogridge, G. R. and Jamieson, W. W., "Wave Loads on Circular Cylinders: A Design Method", Hydraulics Laboratory, National Research Council, Canada, MH-111, December, 1976, 34 pp.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.E. Moodie (eds.) 0 Elsevier Science PublishersB.V. (North-Holland), 1984
71
ACCELERATION WAVES I N ISOTROPIC CONSTRAINED THERMOELASTIC MATERIALS B.D. Reddy Department o f A p p l i e d Mathematics and Department o f C i v i l E n g i n e e r i n g U n i v e r s i t y of Cape Town Rondebosch 7700 South A f r i c a 1.
INTRODUCTION
I n a r e c e n t paper ( 1 ) p r o p a g a t i o n c o n d i t i o n s have been d e r i v e d f o r a c c e l e r a t i o n waves i n constrained h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s . A t t e n t i o n has been c o n f i n e d t o t h e cases o f homothermal and homentropic waves; f o r b o t h t h e p r o p a g a t i o n c o n d i t i o n s a r e analogues of t h e c l a s s i c a l F r e s n e l -Hadamard c o n d i t i o n . Growth e q u a t i o n s have a l s o been d e r i v e d f o r p l a n e waves p r o p a g a t i n g i n t o homogeneously deformed m a t e r i a l which i s a t r e s t ahead o f t h e w a v e f r o n t . I n t h e above-mentioned work, no assumption has been made r e g a r d i n g t h e symmetry o f t h e m a t e r i a l o r t h e d i r e c t i o n o f t h e wave normal, a l t h o u g h t h e l a t t e r may be r e s t r i c t e d when c e r t a i n c l a s s e s o f c o n s t r a i n t s a r e p r e s e n t ( ( I ) , e q u a t i o n (6.4)). I t would be i n t e r e s t i n g , though, t o examine t h e e f f e c t s o f m a t e r i a l symmetry on t h e r e s u l t s f o r t h e g e n e r a l case. I n t h e p r e s e n t c o n t r i b u t i o n we s p e c i a l i s e t h e r e s u l t s i n ( 1 ) t o t h e case i n which t h e m a t e r i a l i s isotropic, and re-examine t h e c o n d i t i o n s f o r p r o p a g a t i o n of a c c e l e r a t i o n waves i n t h e l i g h t o f t h e i s o t r o p y o f t h e m a t e r i a l . As i n t h e p r e v i o u s work, t h e c o n s t r a i n t s a r e a r b i t r a r y . I n S e c t i o n 2 b a s i c r e s u l t s f r o m ( 1 ) a r e summarised and t h e n s p e c i a l i s e d f o r i s o t r o p i c m a t e r i a l s . We t a k e f u l l advantage o f t h e i n h e r e n t s i m p l i c i t y o f r e s u l t s when t h e orthonormal bases used a r e t h e t r i a d s o f p r o p e r v e c t o r s o f t h e r i g h t - and l e f t Cauchy d e f o r m a t i o n t e n s o r s . R e s t r i c t i o n s imposed by t h e c o n s t r a i n t e q u a t i o n s on t h e p r o p a g a t i o n o f a c c e l e r a t i o n waves a r e examined i n S e c t i o n 3; here we p a r t i c u l a r i s e o u r s t u d y t o t h r e e s p e c i a l cases f o r which t h e c o n s t r a i n t e q u a t i o n s t a k e s i m p l e forms, and f o r which t h e a c c e l e r a t i o n wave i s a principaZ wave. We show f o r t h e s e t h r e e cases t h a t t h e c o n s t r a i n t s a r e ZinearZy dependent ( i n a sense t o be d e f i n e d ) , and f u r t h e r m o r e , t h a t a c c e l e r a t i o n waves a r e aZways trrmsuerse ( a n d a l s o homothema2 i f a d d i t i o n a l c o n d i t i o n s a r e s a t i s f i e d ) . These o b s e r v a t i o n s a l s o h o l d f o r a c c e l e r a t i o n waves propagating i n material which i s i n a s t a t e o f uniform d i l a t a t i o n . I n S e c t i o n 4 t h e p r o p a g a t i o n c o n d i t i o n s o b t a i n e d i n ( 1 ) a r e summarised, and we subsequently r e c o n s i d e r t h e s e c o n d i t i o n s i n t h e c o n t e x t o f i s o t r o p i c m a t e r i a l s , f o c u s s i n g a t t e n t i o n on t h e t h r e e s p e c i a l cases mentioned above. Analogues a r e established o f t h e r e s u l t s f o r unconstrained e l a s t i c m a t e r i a l s given e a r l i e r by Chen ( 2 ) f o r h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s and b y T r u e s d e l l ( 3 ) f o r t h e p u r e l y mechanical t h e o r y . N o t a t i o n . I n g e n e r a l , s c a l a r s a r e denoted by l o w e r - c a s e Greek l e t t e r s , v e c t o r s by lower-case Roman l e t t e r s and t e n s o r s o f o r d e r t w o and h i g h e r b y upper-case Roman l e t t e r s . E x c e p t i o n s a r e X f o r p o s i t i o n v e c t o r o f a p a r t i c l e , U f o r wave speed, and, ai f o r p r i n c i p a l s t r e t c h e s . The v e c t o r space o f t e n s o r s o f
72
B.D. Reddy
o r d e r p i s denoted b y EP ; we w r i t e E1 E E and Eo 5 R . Very o f t e n we s h a l l come across e x p r e s s i o n s o f t h e f o r m C a i q i B p i , and so we m o d i f y t h e r e p e a t e d i n d e x appears on one side usual summation c o n v e n t i o n as f o l l o w s : i f only of an e q u a t i o n , o r i f i t appears i n a group o f q u a n t i t i e s which do n o t form p a r t o f an e q u a t i o n , t h e n s m u t i o n is impZied o v e r t h e r e p e a t e d i n d e x . However, i f a r e p e a t e d i n d e x appears on b o t h s i d e s o f an e q u a t i o n , no s m u t i o n is i m p l i e d ; f o r example, t h e e q u a t i o n S i = g i b j ( c j d j ) r e p r e s e n t s Si = higi ,;bj ( cj dJ . ) , i = 1 ,... 3 .
a
2.
CONSTRAINED ISOTROPIC THERMOELASTIC MATERIALS
We t a k e as s t a r t i n g p o i n t t h e c o n s t i t u t i v e e q u a t i o n s f o r an u n c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l i n t h e a l t e r n a t i v e forms = paQ0/aF
,Q
=
- a$o/ae
(2.1)
= E ~ ( F , Q ), S = paEo/aF
,e
=
aEo/ae
(2.3)
,S
$ = $O(F,e)
or E
Here F = Grad x ( X , t ) i s t h e d e f o r m a t i o n g r a d i e n t , J, i s t h e s p e c i f i c f r e e t h e s p e c i f i c entropy, e the energy, S E E 2 ( V ) t h e P i o l a - K i r c h h o f f s t r e s s , temperature, E t h e s p e c i f i c i n t e r n a l energy and p t h e d e n s i t y i n t h e r e f e r e n c e c o n f i g u r a t i o n . We denote by V t h e subset o f R3 o c c u p i e d by t h e body i n t h e r e f e r e n c e c o n f i g u r a t i o n . The r e f e r e n t i a l h e a t f l u x v e c t o r l i s denoted by q , and i n ( 2 . 2 ) and (2.4) g E E(V) i s t h e t e m p e r a t u r e g r a d i e n t ae/aX , X being t h e p o s i t i o n vector o f p a r t i c l e s i n t h e reference configuration. For i s o t r o p i c t h e r m o e l a s t i c m a t e r i a l s t h e f u n c t i o n $0 i s a s c a l a r - v a l u e d i s t h e l i s t of p r i n c i p a l f u n c t i o n o f T A and e , where TA(A = 1,2,3) i n v a r i a n t s o f t h e r i g h t Cauchy-Green t e n s o r C = FTF , w h i l e qo i s a f u n c t i o n o f TA,e and t h e i n v a r i a n t s l g l , lFgl and lCgl . S i m i l a r c o n s i d e r a t i o n s a p p l y t o t h e f u n c t i o n EO i n ( 2 . 3 ) , t h e r o l e s o f temperature and e n t r o p y b e i n g interchanged. We t a k e t h e a l t e r n a t i v e b u t e q u i v a l e n t view t h a t t h e l i s t T A may be r e p l a c e d by t h e t h r e e principuZ s t r e t c h e s a i E R(V) as independent v a r i a b l e s , so t h a t t h e c o n s t i t u t i v e e q u a t i o n s now become
or E
= Eo(ai,n)
,S
= P(aEo/aai)qi
q = qo(ai
n
191
B pi
IFgl
,e
Icsl)
= aEo/an
.
, (2.6)
I n ( 2 . 5 ) and ( 2 . 6 ) $0 and EO a r e symetYic f u n c t i o n s o f ai , w h i l e p i and qi are, r e s p e c t i v e l y , t h e orthonormal t r i a d s o f p r o p e r v e c t o r s o f C and o f
73
Acceleration Waves in Therrnoelastic Materials
T
B=FF
.
Constraints a r e d e f i n e d v i a t h e f u n c t i o n s $' : E2 and t h e c o r r e s p o n d i n g c o n s t r a i n t e q u a t i o n s a r e = 0
$,(F,e) zB(F,e).g
,
CY
1 ,...,N
=
,6
= 0
x
= 1
R
+
R
and
,
z6 : E2 x R + E , (2.7)
,...,M .
(2.8)
On t h e assumption t h a t t h e c o n t r i b u t i o n o f t h e c o n s t r a i n t s t o S and ~1 do n o t c o n t r i b u t e t o t h e e n t r o p y p r o d u c t i o n , i t can be shown ( 1 ) t h a t t h e c o n s t i t u t i v e equations f o r thermoelastic m a t e r i a l s subject t o t h e constraints (2.7 - 8) t a k e t h e f o r m jl =
T(F,e,xCY) = jlo(F,e)
s
paji/aF ,
q
=
= q(F,e,g,rB)
,
-$aa
q =
XCY$"(F,e) ,
+
4' = aJl/axa
= qo(F,@,s)
+
(2.9)
,
u6zB(F.e) ,
(2.10) (2.11)
where jlO(F,e) i s d e f i n e d i n ( 2 . 1 ) 1 and A,, yB E R ( V ) are a r b i t r a r y f i e l d s o f s c a l a r m u l t i p l i e r s . Here, and h e n c e f o r t h , summation i s imp i e d o v e r r e p e a t e d Greek in d i ces . The c o n j u g a t e r e p r e s e n t a t i o n o f ( 2 . 9 - 11) i s a c h i e v e d b y d e f n i n g t h e augmented i n t e r n a l energy f u n c t i o n E
= ;(~,q.h~)=
ji
t
ne
, e , a;/an
,
$'
=
(2.12)
;
t h e n we f i n d t h a t S = pai/aF
q = q(F,v,g,yB,ACY)
.
a;/aA,
,
(2.13) (2.14)
I n g e n e r a l , c o n s t r a i n t s have some degree o f d i r e c t i o n a l i t y s p e c i f i e d b y one o r more v e c t o r f i e l d s . F o r example, w h i l e t h e c o n s t r a i n t $(R,B) = det F - f ( a ) which corresponds t o temperature-dependent c o m p r e s s i b i l i t y , e x h i b i t s no such directionality, the constraint
$(F,@) = IFel2 - g(a) c o r r e s p o n d i n g t o temperature-dependent e x t e n s i b i l i t y possesses t h e d i r e c t i o n a l dependence s p e c i f i e d by t h e u n i t v e c t o r f i e l d e ( X ) . Such v e c t o r s a r e n o t u s u a l l y s p e c i f i e d e x p l i c i t l y i n t h e l i s t o f arguments o f c o n s t r a i n t s , presumably because as a r u l e t h e y a r e independent o f t h e d e f o r m a t i o n . However, i f we a r e t o c o n s i d e r t h e consequences o f material isotropy, i t i s e s s e n t i a l t h a t we make c l e a r t h e dependence o f 4 on v e c t o r f i e l d s er by w r i t i n g $"(F,e,er)
..., .
= 0
,
(2.15)
where r = 1, re O f course, n o t a l l o f t h e er need be p r e s e n t i n each c o n s t r a i n t . The c o n s t r a i n t s (2.8) a f f e c t t h e c o n d i t i o n s f o r growth o f a c c e l e r a t i o n waves b u t t h e p r o p a g a t i o n c o n d i t i o n s a r e u n a f f e c t e d by them. S i n c e we a r e concerned c h i e f l y w i t h p r o p a g a t i o n c o n d i t i o n s , we do n o t c o n s i d e r (2.8) f u r t h e r .
74
B.D. Reddy
F o r i s o t r o p i c m a t e r i a l s $' i n (2.15) a r e i s o t r o p i c f u n c t i o n s o f F , e and e, ; a c c o r d i n g l y , i f we i n v o k e a r e p r e s e n t a t i o n theorem ((5), ( 6 ) ) f o r s c a l a r - v a l u e d i s o t r o p i c f u n c t i o n s we f i n d t h a t $
where
u
-a
= $
I
.
A = 1 ,. .3, r,A = 1 ,...,re
= 0
,
(2.16)
; here
, h,
f, = Fe,
and
, )h,
8 , f,
= Ce,
,, ,h,
f,
,
= h
= f .f
(2.17)
r . ha
(2.18)
.
a r e symmetric m a t r i c e s whose elements a r e t h e i n v a r i a n t s i n v o l v i n g er We assume t h a t l e r l = 1 and t h a t t h e s c a l a r s ( e e ) a r e independent o f t h e d e f o r m a t i o n , so t h a t t h e y do n o t appear i n t h e o f arguments o f i n (2.16). As i n t h e case o f (2.5) we r e p l a c e dependence on TA i n (2.16) by symmetric dependence on a i , so t h a t (2.16) becomes
7is.t
"U
$ (ai
, e , ,f
,h),
= 0
.
(2.19)
Using t h e i d e n t i t i e s aai/aF ah
k
where
rA / a F
r
=
= Fh
af,,/aF
+ k,
er e,
= k,
Bf,
,
er pi
= qi
= f,
+ f,
b er
er eA + f, 8
e,
8
hA + f, 8 h,
, (2.20)
,
r '
t o g e t h e r w i t h e q u a t i o n s (2.19 - 14), (2.5 - 6 ) and (2.16), t h e c o n s t i t u t i v e e q u a t i o n s f o r i s o t r o p i c c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l s become
-
JI = $(ai P -1
s
-
, e , xu
= (a$/aai)qi
,)h,,
f, 8
pi + k
+
~r
-JIO(ai
=
,e)
(aT/af,,)(f,
ere
e,
8
, e , f,, ,)h,
+ hccTu(ai + fA
Q
,
e r ) + (aT/ah,,)(k,
(2.21) 8
eA)
+f,sh,+f,sh,),
(2.22)
- aT/ae
(2.23)
17 =
or E
p
-1
s
= JI +
ne ,
E
€(ai , n . ~ ~ , f ,,~,'A)
=
= (a€/aai)qi
B pi + (a;/af,,)(f,
(ac/ahr,)(k,
B e,
e
+ k,
,
= a2an
+ fr
er e, =
a;/axu
'
(2.24)
8
e,
+ f,
8
h,
+ f, B h,)
.
8
e,)
+
(2.25)
, (2.26)
I n e q u a t i o n s (2.21) and (2.24) t h e f u n c t i o n s and 'E a r e assumed symmetrised w i t h r e s p e c t t o f r A and h r a ; t h a t i s , t h e i r dependence on f,A and ,h, is Representations f o r the heat f l u x vector v i a ( f r A + f A r ) and ( h r 4 + h A r ) have been o m i t t e d s i n c e t h i s v e c t o r f i e l d i s n o t r e l e v a n t t o o u r subsequent discussion.
.
3.
CONSEQUENCES OF THE CONSTRAINT EQUATIONS
We examine here necessary c o n d i t i o n s f o r t h e e x i s t e n c e o f a c c e l e r a t i o n waves, based on t h e c o n s t r a i n t e q u a t i o n s (2.7 - 8 ) . The s t a n d a r d r e s u l t s
75
Acceleration Waves in Thermoelastic Materials
I F 1 = 0 , 1il
=
0
,
1
[FI = U- s B n , [Sin = - PUS (3.1) w i l l be needed l a t e r ; here, s = 1x1 i s t h e a m p l i t u d e and n and U a r e t h e d i r e c t i o n and speed of p r o p a g a t i o n o f t h e wave r e l a t i v e t o t h e reference c o n f i g u r a t i o n . A s s o c i a t e d r e s u l t s i n v o l v i n g jumps i n t h e temperature, t h e e n t r o p y and i n t h e i r f i r s t d e r i v a t i v e s a r e
o
lei = I g I = on ,
[il= -
UO
, 1ni
,
=
o
[On1 = Hn ,
I:[
=
-
UH
,
(3.2)
where Q and H a r e , r e s p e c t i v e l y , t h e thermal and e n t r o p i c a m p l i t u d e s . a c c e l e r a t i o n wave i s homothema2 i f o = 0 and homentropic i f H = 0 .
An
When no assumption i s made a b o u t t h e degree o f m a t e r i a l symmetry, t h e c o n d i t i o n [ G a l = 0 implies t h a t (1) -a 2-a c . s = - u w e
(3.3) . For homothema2 waves ( 3 . 3 ) i m p l i e s where Fa = ( a r / a F ) n and I;"= t h a t a t most two c o n s t r a i n t s o f t y p e (2.7) may b e p e r m i t t e d i n o r d e r f o r a r e ZinearZy a c c e l e r a t i o n waves t o propagate, assuming t h a t t h e v e c t o r s independent. F o r c o n s t r a i n t s w h i c h a r e such t h a t a r / a F = V a B wa , t h a t i s when aza/aF i s a s i m p l e t e n s o r , t h e c o n d i t i o n wa.n = 0 i s a l s o s u f f i c i e n t f o r (3.3) t o h o l d w i t h o = 0, . Then any number o f c o n s t r a i n t s of t h i s t y p e a r e permitted provided t h a t Wa E I n } l .
aT/ae
ca
When c o n s t r a i n t s o f t h e f o r m (2.8) a r e p r e s e n t , b y e v a l u a t i n g t h e jump i n t h i s e q u a t i o n we f i n d t h a t o z B. n = O ,
(3.4)
which i s s a t i s f i e d i d e n t i c a l l y f o r homothermal waves, F n r non-hornothernlal \.caves ( 3 . 4 ) imposes a l i m i t a t i o n on t h e p e r m i s s i b l e wave normals, w h i l e ( 3 . 3 ) , when r e w r i t t e n i n t h e form
-N c .s
=
- u2w-N o , m'.s ;Nmu
=
=o N ;
o , - Nu;;
(5
= 1 ,...N-1
, (3.5)
I
shows t h a t a t w s t t h r e e l i n e a r l y independent c o n s t r a i n t s a r e p e r m i t t e d . However, when mu = (v' 8 w')n f o r some v' , w' E E ( V ) any number of such c o n s t r a i n t s a r e p e r m i t t e d i f wa E I n } '
.
An a l l - e m b r a c i n g s e t o f necessary c o n d i t i o n s f o r t h e e x i s t e n c e o f a c c e l e r a t i o n waves, which a p p l i e s a l s o t o t h e cases when dim{? , a = 1, ...,N } < N i s dim{?}
2
f o r homothema2 waves
3
otherwise
2
F o r isotropic m a t e r i a l s t h e c o n d i t i o n (a7/aai)qi.s)(pi.n)
+ 2(ap/ah,,)
.
[Gal = 0
t o g e t h e r w i t h (2.20)
2(a~/afrA)(eA.n)(fr.s) 2-0 C(eA.n)(kr.s) + (h,.n)(f,.s)l = -U w CI
imply t h a t
+
.
.
(3.7)
and ,h, As i t stands, e q u a t i o n Here we have made use of t h e symmetry o f A,f (3.7) i s t o o g e n e r a l i n n a t u r e f o r any u s e f u l r e s u l t s t o be deduced f r o m i t .
76
B.D. Reddy
However, t h e w a r e a number o f n o n - t r i v i a l s p e c i a l cases which m e r i t a t t e n t i o n , and f o r which ( 3 . 7 ) s i m p l i f i e s c o n s i d e r a b l y . We s h a l l a c c o r d i n g l y devote a t t e n t i o n t o t h e s e s p e c i a l cases, w h i c h concern f i r s t l y principal! waves p r o p a g a t i n g i n a c o n s t r a i n e d m a t e r i a l f o r which ( i ) = p ( a i , 0 ) : we r e f e r t o these as i s o t r o p i c c o n s t r a i n t s ; ( i i ) t h e v e c t o r s e r a r e c o i n c i d e n t w i t h p r o p e r v e c t o r s p i ; and ( i i i ) t h e v e c t o r s er l i e i n t h e subspace (n-1' A f o u r t h s p e c i a l case concerns waves p r o p a g a t i n g i n a m a t e r i a l which i s i n a s t a t e o f pure d i l a t i o n . I n what f o l l o w s , we say t h a t a s e t o f c o n s t r a i n t s i s ZinearZy independent ( r e s p . dependent) if the s e t is l i n e a r l y independent (resp. dependent).
.
{c"}
P r i n c i p a l waves, i s o t r o p i c c o n s t r a i n t s
Case ( i ) :
To be d e f i n i t e , suppose t h a t t h e wave normal = ( a p / a a l ) q 1 and ( 3 . 7 ) becomes
n
c"
(aT/aal)ql.s
=
-
U'P~
coincides w i t h
.
p1 ; t h e n
(3.8)
The c o n s t r a i n t s a r e l i n e a r l y dependent ( i n f a c t , c o l l i n e a r ) , and t h e r e i s consequently no r e s t r i c t i o n on t h e number w h i c h may be p e r m i t t e d i n o r d e r f o r a c c e l e r a t i o n waves t o propagate. A number o f o b s e r v a t i o n s a r e summarised i n t h e f o l 1owi ng THEOREM 1. -a
(ai,e)
I n an i s o t r o p i c t h e m a e l a s t i c material w i t h i s o t r o p i c constraints
= O
I
( a ) a l l homothemal waves are transverse; (hl if w"" = 0 ( i . e . none of th8 constraint functions 4" depends on temperature), then a12 waves are transtrerse; (ci if W" = 0 for onZy a , then an acceZeration waue is homothermal! and transverse; ( d ) i f W" z 0 f o r a l l a and (iBaa"/aa, - ;"aTB/aal) f 0 .for a t l e a s t one p k r ($",@j of c o n s t r a i n t s , t h & z m ' a c c e l e r a t i o ~wave is-both transverse and homothermal.
some
i s t r i v i a l . To p r o v e ( e l , suppose t h a t . Then (3.8) g i v e s ( a T a / a a l ) q l . s = 0 and so q1 .s= 0 and t h e wave i s t r a n s v e r s e . Thus ( a a B / a a l ) q l . s = 0 = -U21;'o , i n d i c a t i n g t h a t t h e wave i s homothermal. To prove I d ) , we have o n l y t o observe -5 -a t h a t ( W a $ / a a l - o"aaT5/aa1!~1.5 = 0 ; i f t h e t e r m i n b r a c k e t s i s non-zero f o r a t least-one p a i r o f c o n s t r a i n t s , t h e n t h e wave i s t r a n s v e r s e . B u t t h i s i m p l i e s = 0 f o r a l l ~1 . Hence t h e wave, i f i t e x i s t s , i s homothermal. t h a t U2,"o PROOF.
F
The p r o o f o f l a ) and
f'bi
O and o"@ z 0 f o r some a ,
5
We a l s o r e c o r d h e r e f o r completeness some r e s u l t s f o r generalised transverse waves, these b e i n g d e f i n e d as waves which a r e b o t h homothermal and homentropic. P r o p e r t i e s o f these waves have been d i s c u s s e d i n some d e t a i l by Chadwick and C u r r i e ( 7 ) , (8) i n t h e c o n t e x t o f u n c o n s t r a i n e d t h e r m o e l a s t i c media. When c o n s t r a i n t s are p r e s e n t t h e i d e n t i t y = -ajl/ae and e q u a t i o n s (3.1-2) y i e l d -U 2H = s.(a';/Fae)n
+ U'o
-
p-'o""[ial
(3.9)
. For and so a g e n e r a l i s e d t r a n s v e r s e wave s a t i s f i e s ps.(a2;/aFae)n = G"[i,l t h e s p e c i a l case b e i n g d i s c u s s e d n = p1 and a*q/aFae = (a2$/aaiae)qi Q) p i , so t h a t general i s e d t r a n s v e r s e waves i n u n c o n s t r a i n e d m a t e r i a l s a r e t r a n s v e r s e .
77
Acceleration Waves in Thermoelastic Materials
Furthermore, i f a22 l" = 0 , an a c c e l e r a t i o n wave i s a g a i n n e c e s s a r i l y t r a n s v e r s e i n b o t h t h e o r d i n a r y and g e n e r a l i s e d senses. Case ( i i ) :
P r i n c i p a l waves,
e,
= p,
We assume h e r e o f c o u r s e t h a t t h e r e a r e a t most t h r e e v e c t o r s ep . An e l e m e n t a r y a p p l i c a t i o n o f t h e p o l a r decomposition theorem l e a d s t o t h e r e p r e s e n t a t i o n s f r = aSqr and hi- = afpr The c o n s t r a i n t e q u a t i o n (2.7) c l e a r l y reduces t o a f u n c t i o n of t h e p r i n c i p a l s t r e t c h e s and t e m p e r a t u r e o n l y , w i t h t h e r e s u l t t h a t a l l o f t h e c o n c l u s i o n s reached i n case ( i ) h o l d here.
.
Case ( i i i ) :
P r i n c i p a l waves,
en
E
.
In}'
F o r t h i s case i t i s easy t o show t h a t hi- E I n } ' ; hence ( 3 . 7 ) reduces t o ( 3 . 8 ) and once again t h e r e s u l t s d e s c r i b e d i n Theorem 1 h o l d . Case ( i v ) :
State o f pure d i l a t a t i o n
Here a1 = a2 = a3 = a say, F = a1 and e l e m e n t a r y c a l c u l a t i o n s show t h a t f r A = a2erA , ,h, = a4erA where erA = gA.e, , and so $a reduces t o a f u n c t i o n o f a and e o n l y , w i t h -a (3.10) c = (ar/aa)n . Thus t h e c o n s t r a i n t s a r e c o l l i n e a r . O f course any orthonormal t r i a d p i i n t h e r e f e r e n c e c o n f i g u r a t i o n i s an e i g e n b a s i s , and f u r t h e r m o r e p i = q i . Suppose we choose p i s o t h a t p1 = n ; t h e n ( 3 . 3 ) becomes ( a p / a a ) q l . s =-U21ao which i s q u a l i t a t i v e l y t h e same as (3.8). C l e a r l y , then, Theorem 1 h o l d s h e r e as well. 4.
PROPAGATION CONDITIONS FOR I S O T R O P I C MATERIALS
We d e r i v e i n t h i s s e c t i o n p r o p a g a t i o n c o n d i t i o n s f o r homothermal and homoentropic waves i n i s o t r o p i c c o n s t r a i n e d m a t e r i a l s . We have shown i n ( 1 ) t h a t t h e concepts o f a d e f i n i t e c o n d u c t o r and a non-conductor a r e e a s i l y extended t o c o n s t r a i n e d m a t e r i a l ; s i n c e t h e thermal c o n d u c t i v i t y t e n s o r K = a i / a g = aqo/ag i s d e f i n e d t o be one f o r which = 0 , we have, as i n t h e case o f u n c o n s t r a i n e d m a t e r i a l s (1 1
6
u.(sym K ) u > 0
-q = O -
Vu
E r, o = 0
E
,
H = O .
(4.1
1
T h a t i s , a c c e l e r a t i o n waves i n a d e f i n i t e c o n d u c t o r (non c o n d u c t o r ) a r e homothermal ( h o m o e n t r o p i c ) . The f i r s t - o r d e r moduli
1,1 E
4 E (V)
-A = p a 2 j l / a F a F
f o r constrained m a t e r i a l s are defined by
,A
and t h e c o r r e s p o n d i n g a c o u s t i c t e n s o r s
f o r a l l vectors
u
HomokhemaZ waves. condition i s (1)
tju = , A{(.)}
iiru
8
A
2"
= pa daFaF
0,6
2
E
E (V)
n l n , Qu = A{u
d e n o t i n g t h e map
E2
8
(4.2) a r e d e f i n e d by
nln
(4.3) E2
.
When t h e c o n s t r a i n t s a r e ZinearZy independent t h e p r o p a g a t i o n p u 2 s = PUS
(4.4)
B.D.
78
-
where- P i s t h e projection operator and da defined by
Reddy
c'
I -
, c" being defined e a r l i e r
B
C".';j B = 6" B , 2, E span cC"} . (4.5) t h e s e t of spurious proper vectors [81 ; We c a l l E, E Is : 6s E span {?I} f o r s E E, , U = 0 . For N c o n s t r a i n t s there e x i s t a t o t a l of (3-N) proper vectors which have associated w i i h them non-zero proper numbers. These proper vectors form a basis of {span C C " } } ~ ; furthermore, s i n c e 0 i s real a n d symmetric the proper numbers a r e real and non-zero. We have seen in Section 3 t h a t t h e c o n s t r a i n t s a r e c o l l i n e a r f o r a number of special cases; on the o t h e r hand, ( 4 . 3 ) was derived i n ( 1 ) on the assumption t h a t t h e constraints are l i n e a r l y independent. We therefore derive here t h e propagation condition f o r materials with c o n s t r a i n t s which a r e c o l l i n e a r . To be d e f i n i t e , suppose t h a t T' = GaE where 7' a r e known s c a l a r s and E a mit vector. The preliminary form of t h e propagation condition, obtained from ( 3 . 1 ) 4 , is ( P U2 I - 0 ) s = -E"[iOll = -UvaCi'lz . (4.6) Y
c . s 7 0 ; fur-thermore, from equation UvaCXal = ?.Qs . The propagation
Since we are dealing with homothermal waves ( 4 . 5 ) we have 3 = c , and so we find t h a t condition thus becomes
iJu2 s
c
(I B c,ts (4.7) which i s just t h e propagation condition f o r a material with a s i n g l e c o n s t r a i n t . The difference here, though, i s t h a t whereas t h e quantity .Cil i s a b l e t o b e found f o r a singly constrained m a t e r i a l , t h e q u a n t i t i e s [ A 1 are indeterminate when there are a number o f c o l l i n e a r c o n s t r a i n t s (see also 'the remarks by Whi tworth ( 9 ) ) . =
I n t h e case of isotropic materials t h e tensor
The term (a$aa.)qi Ipi by Chadwick and bgden (10) t o t h e bases pi and qi term. The contribution of we find eventually t h a t p-'R
=
q i.
Ip
k
+. ,
A
i s found from
i s comiaZ with and so t h e techniques employed f o r t h e derivation of non-zero components r e l a t i v e f o r tensors l i k e E may be applied d i r e c t l y t o t h i s t h e remaining terms t o a r e readily evaluated and
[Taia;/aai-akaT/aak a 2i. - a 2k
(
I
+ 2 q i B e r B ($af,,qi
+ 2ajr/ah,,f, + 2aT/ahrAqi +
B e,
+ a;/ah,,
B pi B
L
f,
(aif,
B
B pi + a i q i
pi + aiqi 2 B eA + q i B h,)] B
e,]
w h, B q i B e, + 2f, B e, BX(aT/af,,)
2 ( k r B e,
+
f,
Ih a )
&(a;/ah,,)
(4.9)
79
Acceleration Waves in Thermoelastic Materials
where
&: R & =
-f
E2
qi
i s t h e tensor valued d i f f e r e n t i a l operator. pi a/aai + 2{f,
I
I
eA
a/af,,
+ (kn
I
e, + f,
I
h,)a/ahaA1.(4.10)
I t i s now a straightforward matter t o obtain t h e acoustic tensor corresponding
t o i s o t r o p i c m a t e r i a l s ; from (4.3)1 and (4.9) we have
N
Y
Here, 7 E f.n f o r any vector f , $,! 5 a$/aai and I v j ) E 4(u. I v . + u . I v . ) 1 J J J represents the symmetric p a r t of t h e simple tensor u i 8 v j . Once again we focus a t t e n t i o n on t h e special cases outlined in Section 3 , r e s t r i c t i n g our discussions t o homothermal waves in t h e f i r s t instance. For Cases ( i ) , ( i i ) and ( i i i ) t h e acoustic t e n s o r reduces t o -14 = o i q i @ q i P Qi = $ , 1 1 6 i l + ( 1 - S i l ) ( a i G , i - a l T , l ) ( / ( a i - a:)) N
c
and t h e vector
(4.12)
in ( 4 . 7 ) i s given by
-c =
.
(aT/aa,)q,
(4.13)
From ( 4 . 7 ) we therefore have THEOREM 2 .
In m i s o t r o p i c heat-conducting e l a s t i c body i n which t h e nomaZ t o
an acceleration wave coincides with t h e proper vector pi , E d for which t h e a -a c o n s t r a i n t s are such t h a t f i ) @ = $ ( a i , e ) ; o r f i i ) $a = $ a ( a j , e , e r ) w i t h e r = pr ; or (iii) ga = p ( a i , e , e r ) w i t h e r E { n P , there e x i s t two homothermal a c e e l e m t i o n waves. Both waves arc transverse, w i t h amplitudes q 2 , 43 and corresponding wave speeds Q$ , Q3k , provided t h a t 0 i s positive definite.
The above applies t o t h e case when the s e t of principal d i r e c t i o n s i s d i s t i n c t , t h a t i s , a1 a2 a3 a1 , when n = p1 and a1 f a2 z a3 any vector s E I n ) L i s an amplitude, while f o r a s t a t e of d i l a t i o n any vector orthogonal t o n i s again an amplitude. For these l a t t e r two cases t h e components of t h e a c o u s t i c tensor a r e found by proceeding t o the l i m i t i n (4.12) (see equivalent calculations in (10))and are e a s i l y shown t o be f
f
-
-1Qi = $ , l , 6 i l
P
+
for
(1
al
and
-1-
4,
-
= $,,16il
+
-
6il)(ai$.i f
a2 = a3
4(1 - 6 i l ) a -
1
-
2 alG,,)/(ai
-
2 all
(4.14)
’
(T,i
Y
+
a$,ii
-
(4.15)
80 for
B. D. Reddy
al = a2 = a3
=
a
or
al = ai
, i = 2
3 .
or
Hornentropic waves When d e a l i n g w i t h homentropic waves i t i s p r e f e r a b l e t o develop t h e t h e o r y u s i n g t h e form (2.24) - (2.26) o f t h e c o n s t i t u t i v e equations. The c o n d i t i o n e q u i v a l e n t t o (3.3) i s t h e n
&/axU)*l where
c"
= (a;/aF)n
.
*
o
=
CU.s
u ( a 2 g / a A a a A B ci,1 .
=
(4.16)
B u t we have shown ( 1 ) t h a t = w*a^, w /U ,.
a 2-E / a x U a x
(4.17)
B
where
( R e l a t i o n s h i p s between e n t r o p i c q u a n t i t i e s l i k e U' and t h e i r thermal c o u n t e r p a r t s ;;" are g i v e n i n d e t a i l i n ( 1 ) ) . The c o n s t r a i n t c o n d i t i o n t h e r e f o r e becomes, f o r t h e case o f N c o n s t r a i n t s , GN.s =
m" =
^-1
!J
^aAB
u w w Ci,I
,2.5
iNc" - w"cN , u
=
= 0
, (4.19)
1, ..., N - 1
.
We see from Hence, and h e n c e f o r t h , i n d i c e s a range o v e r 1 t o (N - 1 ) ( 4 . 1 9 ) ~ t h a t a maximum o f t h r e e c o n s t r a i n t s may be p e r m i t t e d i n o r d e r t h a t homentropic a c c e l e r a t i o n waves may propagate, assuming t h a t t h e v e c t o r s mu a r e l i n e a r l y independent. The p r o p a g a t i o n c o n d i t i o n f o r homentropic waves i s d e r i v e d u s i n g arguments s i m i l a r t o t h o s e used i n t h e d e r i v a t i o n o f ( 4 . 4 ) f o r homothermal waves, and i s 2
The t e n s o r
a*
pU
s = PQ*s
.
(4.20)
i s related t o the acoustic tensor
Q (eqn. (4.3)) by (4.21)
and and
i s t h e p r o j e c t i o n o p e r a t o r I - m B iu where i n t h e sense t h a t
mu
i s d e f i n e d i n (4.19)
ia i s r e c i p r o c a l t o 6"
mu.;,
= A',
, iT
c span{m* U 1
.
(4.22)
.
I n t h e e v e n t o f t h e r e b e i n g o n l y one c o n s t r a i n t , f' reduces t o I For a t o t a l o f N independent c o n s t r a i n t s t h e symmetry o f $* i m p l i e s t h e e x i s t e n c e o f (44) r e a l non-zero p r o p e r number pU2 w i t h c o r r e s p o n d i n g m u t u a l l y o r t h o g o n a l p r o p e r v e c t o r s f o r m i n g a b a s i s o f (span{m'})'
.
We t u r n now t o isotropic m a t e r i a l s , and t o t h e c o n s t i t u t i v e e q u a t i o n s i n t h e f o r m ( 2 . 2 4 ) - ( 2 . 2 6 ) . We know f r o m ( 3 . 8 ) t h a t f o r t h e s p e c i a l cases c o n s i d e r e d i n S e c t i o n 3, t h e c o n s t r a i n t s a r e c o l l i n e a r whether o r n o t t h e wave i s homent r o p i c . I t i s t h u s once again necessary t o r e d e r i v e t h e p r o p a g a t i o n c o n d i t i o n on t h e assumption t h a t t h e c o n s t r a i n t s a r e c o l l i n e a r . Suppose t h e n t h a t C' = vaC w i t h = 1 ; t h e p r e l i m i n a r y form o f the propagation condition i s e a s i l y shown t o be
81
Acceleration Waves in Thermoelastic Materials
2
(pU
- Q)
I
^ ^
-
s =
(4.23)
,
a = N
On t h e o t h e r hand (4.16) y i e l d s , w i t h
.
Ucv"Cia1
,
w b c i 1 = U-lj;N.S/,N
B
(4.24)
so t h a t *CIA *N-o ^N 2 . s = v c.s = v w (c.s)/w A
or
(LN2 where
u = 1,
^N"
... , N -
1
^u^N
w
)(c.s) = 0 ,
i,
4.25
~
.
^oAN
Thus w v' = w w o r c . s = 0 ; i f t h e f o r m e r c o n d i t i o n h o l d s , we o b t a i n f r o m (4.23) and (4.24) t h e c o n d i t i o n pU
2
s
*
(4.26
= Q*s
^N2 where Q* = Q - ( u / ( w ) ) c B c . Then, as i n t h e case o f homothermal waves t h e p r o p a g a t i o n c o n d i t i o n reduces t o t h a t f o r a s i n g l y - c o n s t r a i n e d m a t e r i a l . c.s = 0 , on t h e o t h e r hand, we f i n d f r o m (4.23) t h a t A
~
(pU2I where
b
:
E
-f
Ic}'
- 64)s
0
=
If
(4.27)
I- c
i s the projection operator
8
c
.
By f o l l o w i n g t h e s t e p s t a k e n i n e q u a t i o n s ( 4 . 9 ) t h r o u g h (4.12) for i s o t r o p i c and ?, , t h i s t i m e m a t e r i a l s we may deduce s i m i l a r r e s u l t s f o r t h e t e n s o r i n v o l v i n g t h e augmented i n t e r n a l energy f u n c t i o n . Then f o r t h e s p e c i a l cases t r e a t e d e a r l i e r i n t h e c o n t e x t o f homothermal waves, s i m i l a r r e s u l t s may a g a i n be deduced f a r homentropic waves. I t is i n s t r u c t i v e , though, t o proceed b y a s l i g h t l y d i f f e r e n t r o u t e and t o use t h e r e s u l t d e r i v e d i n C l l , t h a t =
- U-lf;i
c =
C" - ;-IN
Fi
Q
(4.28)
and
where
-u = pa2?/ae2
and
8
N
i=
(4.29)
pa2T/aFae
.
(4.30)
The c o r r e s p o n d i n g a c o u s t i c t e n s o r s a r e r e l a t e d by Q =
Q
- U-lMn
8
Fin
.
(4.31)
I n t h e case o f a p r i n c i p a l wave p r o p a g a t i n g i n an i s o t r o p i c m a t e r i a l w i t h c e r t a i n o r w i t h e,.n = 0 , we f i n d f r o m ( 4 . 1 2 ) , i s o t r o p i c c o n s t r a i n t s , w i t h e r = p, (4.29) and (4.30) t h a t
Q = Qi qi
8
qi
,
=
G;" q1
(4.32)
where
Qi and
=
iji - --1 ( a 2 $/aalae) 2 6il
= aTa/aal
- i-'Za(a2T/aalae)
(4.33)
.
(4.34)
B.D. Reddy
82
^ - l ^ u ^ B = --1-a-B From t h e i d e n t i t y - p w w p w w (C11, eqn. ( 6 . 3 0 ) ) we a l s o deduce t h a t has t h e same s p e c t r a l r e s p r e s e n t a t i o n as Q , w i t h
a*
67 = Gi
+
6il
- -N u(W
-2
{aTN/aal
- --1-N u w a 2-a/aalae}
2
(4.35)
We thus have, u s i n g e q u a t i o n ( 4 . 2 5 ) onwards, 3. In an i s o t r o p i c heat-conducting e l a s t i c body in which t h e normal t o an acceleration wave coincides with ?he proper vector -pl ,-and f o r which t h e = $(ai,e) ; or ( i i ) = $‘(ai,e,er) with constraints are such t h a t (i) er = Pr ; or ( i i i l $ 01 = -a $ (aj,e,er) with e r E I n } * , there e x i s t three hornentropic acceleration waves, two of which are transverse w i t h F p Z i t u d e s 42 , q3 and corresponding wave speeds $ ; , Q? , provided that Q i s p o s i t i v e . d e f i n i t e . The t h i r d acceZeration wave i s longitudinal, w i t h wave speed
THEOREM
The d i s c u s s i o n f o l l o w i n g Theorem 2 has an o b v i o u s analogue here, t h e d e t a i l s o f which we o m i t . We a l s o observe, from (3.7) s u i t a b l y s p e c i a l i s e d o r i n d e e d f r o m Theorem 1, t h a t t h e t r a n s v e r s e waves d e s c r i b e d i n Theorem 3 a r e b o t h homentropic and homothermal. ACKNOWLEDGEMENT T h i s work was f i n a n c i a l l y s u p p o r t e d b y t h e Council f o r S c i e n t i f i c and I n d u s t r i a l Research o f South A f r i c a . REFERENCES
Reddy, B.D, The p r o p a g a t i o n and growth o f a c c e l e r a t i o n waves i n c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l s , J. E l a s t . ( t o appear). 2. Chen, P.J., Thermodynamic i n f l u e n c e s on t h e p r o p a g a t i o n and growth o f a c c e l e r a t i o n waves i n e l a s t i c m a t e r i a l s , Arch. Rat, Flech. Anal. 31 (1968) 228-254. 3. T r u e s d e l l , C., General and e x a c t t h e o r y o f waves i n f i n i t e e l a s t i c s t r a i n , Arch. Rat. Mech. Anal. 8 (1961) 263-296. R e p r i n t e d i n Problems o f N o n l i n e a r E l a s t i c i t y . Gordon and Breach (New York) 1965. 4. Chadwick, P. and Seet, L . T . C . , Second o r d e r t h e r m o e l a s t i c i t y f o r i s o t r o p i c and t r a n s v e r s e l y i s o t r o p i c m a t e r i a l s . I n Trends i n E l a s t i c i t y and T h e r m o e l a s t i c i t y (Nowacki A n n i v e r s a r y Vol . ) W o l t e r s - N o o r d h o f f (Groningen) (1971) 29-57. 5. Wang, C.-C., A new r e p r e s e n t a t i o n theorem f o r i s o t r o p i c f u n c t i o n s , Arch. Rat. Flech. Anal. 36 (1970) 166-19Z. 6. Chadwick, P. and C u r r i e , P.K., The p r o p a g a t i o n and growth o f a c c e l e r a t i o n waves i n h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s , Arch. Rat. Mech. A n a l . 49 (1972) 137-158. 7. Chadwick, P. and C u r r i e , P.K., On t h e p r o p a g a t i o n o f g e n e r a l i s e d t r a n s v e r s e waves i n h e a t - c o n d u c t i n g e l a s t i c m a t e r i a l s , J. E l a s t . 4 (1974) 301-315. 8. S c o t t , N.H., A c c e l e r a t i o n waves i n c o n s t r a i n e d t h e r m o e l a s t i c m a t e r i a l s , Arch. Rat. Mech. A n a l . 58 (1975) 57-75. 9. Whitworth, A.M., Simple waves i n c o n s t r a i n e d e l a s t i c m a t e r i a l s , Q u a r t . J. Mech. Appl. Math. 35 (1982) 461-484. 10. Chadwick, P. and Ogden, R.W., A theorem o f t e n s o r c a l c u l u s and i t s a p p l i c a t i c n t o i s o t r o p i c e l a s t i c i t y , Arch. Rat. Mech. Anal. 44 (1971) 54-68. 1.
’See Chadwick and Seet ( 4 ) f o r a d e f i n i t i o n .
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
83
A LAGRANGIAN VIEW OF WAVE-MEAN FLOW INTERACTION R. Grimshaw
Department of Mathematics U n i v e r s i t y of Melbourne P a r k v i l l e , V i c t o r i a 3052 Australia
It i s shown how t h e wave a c t i o n e q u a t i o n i s r e a d i l y o b t a i n e d from a Lagrangian Formulation of t h e e q u a t i o n s of motion. The c o u n t e r p a r t t o t h e wave a c t i o n e q u a t i o n i s t h e mean-flow e q u a t i o n which c a n a l s o be simply d e r i v e d i n a Lagrangian Eormulation; i n t h i s e q u a t i o n t h e wave f o r c i n g c a n b e expressed e i t h e r by a r a d i a t i o n s t r e s s t e n s o r o r by t h e wave pseudomomentum. A f t e r d e v e l o p i n g t h e t h e o r y i n a g e n e r a l framework t h e r e s u l t s a r e a p p l i e d t o a compressLble f l u i d . In p a r t i c u l a r i t is shown t h a t , u s i n g s p h e r i c a l p o l a r coo r d i n a t e s , t h e wave a c t i o n e q u a t i o n and t h e mean-flow e q u a t i o n t a k e Eorms u s e f u l f o r a p p l i c a t i o n s f n s t r a t o s p h e r i c meteorology. F i n a l l y t h e t h e o r y i s a p p l i e d t o a s e t of e q u a t i o n s which d e s c r i b e t h e i n t e r a c t i o n of b a r o t r o p i c t o p o g r a p h i c , @-plane Rossby waves w i t h a mean Elow.
91.
INTRODUCTION
The i n t e r a c t i o n between waves and mean flows is a t o p i c w i t h r e l e v a n c e t o many d i E f e r e n t b r a n c h e s of p h y s i c s . I n r e c e n t y e a r s c o n s i d e r a b l e i n t e r e s t in t h i s s u b j e c t h a s developed w i t h i n t h e g e o p h y s i c a l f l u i d dynamics community. Here, as e l s e w h e r e , t h e b e n e f i t s of d e s c r i b i n g t h e L n t e r a c t i o n w i t h i n a L a g r a n g i a n framework a r e now g e t t i n g i n c r e a s i n g r e c o g n i t i o n . Thus, on t h e one hand, t h e r e s p o n s e of t h e waves t o t h e mean flow i s b e s t d e s c r i b e d by t h e wave a c t i o n e q u a t i o n , which i s a n out-growth from t h e a c t i o n p r i n c i p l e s of c l a s s i c a l mechanics. In t h e f l u i d dynamics c o n t e x t t h e c e n t r a l r o l e of t h e wave a c t i o n e q u a t i o n w a s f i r s t recognized i n t h e p i o n e e r i n g work of Whitham (1965, 1970) and B r e t h e r t o n and G a r r e t t (1968). On t h e o t h e r hand, t h e r e s p o n s e of t h e mean f l o w t o t h e waves i s d e s c r i b e d w i t h i n t h e c o n t e x t of a g e n e r a l i z e d Lagrangian-mean Eormulation, i n which t h e a c c e l e r a t i o n of t h e Lagrangian-mean f l o w due t o t h e waves i s d e s c r i b e d e i t h e r by t h e r a d i a t i o n s t r e s s t e n s o r , O K by t h e pseudomomentum. The p r e l i m i n a r y i d e a s were developed by D e w a r (1970) and B r e t h e r t o n (1971) and r e c e i v e d t h e i r f u l f i l m e n t i n t h e g e n e r a l i z e d Lagrangian-mean t h e o r y of A n d r e w s and McIntyre (1978a,b). A g e n e r a l review of b o t h wave a c t i o n and wavemean Elow i n t e r a c t i o n i s c o n t a i n e d in a r e c e n t review by Grimshaw (1984).
In t h i s a r t i c l e w e s h a l l f i r s t g i v e a b r i e f a c c o u n t of t h e g e n e r a l t h e o r y based o n a L a g r a n g i a n f o r m u l a t i o n of t h e e q u a t i o n s of motion in o r d e r t o s e t t h e s c e n e f o r t h e developments of subsequent s e c t i o n s . A more comprehensive a c c o u n t of t h e g e n e r a l t h e o r y i s d e s c r i b e d by Grimshaw (1984). Then i n 92 we s h a l l d e s c r l b e t h e i m p l e m e n t a t i o n of t h e g e n e r a l t h e o r y f o r a c o m p r e s s i b l e f l u i d u s i n g t h e g e n e r a l i z e d Lagrangian-mean t h e o r y of Andrews and P k I n t y r e (1978a,b). In 93 t h i s t h e o i y i s r e c a s t i n s p h e r i c a l p o l a r c o - o r d i n a t e s and we d e m o n s t r a t e how t h e wave a c t i o n e q u a t i o n and t h e Lagrangian-mean flow e q u a t i o n s a r e r e l a t e d t o t h e Eliassen-Palm r e l a t i o n s and t h e Charney-Drazin theorems of s t r a t o s p h e r i c meteorology. Then i n 94 w e d e s c r i b e how t h e g e n e r a l t h e o r y c a n b e used t o s t u d y b a r o t r o p i c t o p o g r a p h i c Rossby waves on a p-plane.
84
R. Grimshaw
To d e s c r i b e t h e g e n e r a l t h e o r y we s h a l l suppose t h a t t h e p h y s i c a l system is s p e c i f i e d by t h e vector-valued f i e l d Q ( t , x ) where t i s t h e time and xi ( i = 1, 2 , 3) a r e s p a t i a l c o - o r d i n a t e s . i I n t h e a b s e n c e of d i s s i p a t i o n t h e p h y s i c a l system obeys a v a r i a t t o n a l p r i n c i p l e w i t h Lagrangian d e n s i t y Then t h e e q u a t i o n s of motion a r e L(c$~, Qxi, Q; t , x i ) .
where t h e g e n e r a l i z e d f o r c e Q r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . Our d e r i v a t i o n of t h e wave a c t i o n e q u a t i o n w i l l b e based o n t h e i d e a s of Whitham (1965, 1970) b u t w i l l f o l l o w t h e development by Hayes (1970) more c l o s e l y . Thus, w e suppose t h a t @ ( t , x i ; 8 ) depends o n t h e ensemble parameter 8 such t h a t
e+
Q ( t , xi;
2n)
Q ( t , xi;
=
el,
(1.2)
and t h e n d e f i n e t h e a v e r a g i n g o p e r a t o r
For s i m p l i c i t y w e s h a l l d e n o t e t h e mean f i e l d <$> by 3 . The a v e r a g i n g o p e r a t o r commutes w i t h d i f f e r e n t i a t i o n and h a s o t h e r s i m p l e and obvious p r o p e r t i e s (Andrews e n e x t d e f i n e t h e wave p e r t u r b a t i o n o r d i s t u r b a n c e f i e l d andAMcIntyre, 1978a,b). W by Q where
(1.4)
C$=?+C$
We s h a l l assume t h a t Q i s O(a) where a i s a measure of wave a m p l i t u d e , a l t h o u g h t h e r e i s a t p r e s e n t no r e s t r i c t i o n on t h e magnitude of a vis-a-vis t h a t oE I$,nor on t h e i r r e l a t i v e s c a l e s w i t h r e s p e c t t o t and xi. I f (1.1) is m u l t i p l i e d by $ e ( i . e . a$/ae) and t h e r e s u l t a v e r a g e d , i t may b e shown t h a t
(1.5a)
(1.5b)
where
=
@i
aL < Q e -->, wx
(1.5~)
i
a =< ieQ >.
(1.5d)
ai
Equation (1.5a) is t h e wave a c t i o n e q u a t i o n , * i s t h e wave a c t i o n d e n s i t y , is t h e wave a c t i o n f l u x I n d 8 is a t e r m r e p r e s e n t i n g t h e e f f e c t s of d i s s i p a t i o n . B o . t h k and 6, are O(a ) wave q u a n t i t i e s , and a r e t h e a p p r o p r i a t e g e n e r a l measures of wave a c t i v i t y . A p p l i c a t i o n s of ( 1 . 5 a ) depend on d e l i n e a t i o n of t h e f a m i l y C$(t,x i ; 0 ) . One c h o i c e which fs commonly used i s t o i d e n t i f y 8 a s a phase s h i f t , s o t h a t t h e a v e r a g i n g o p e r a t o r c a n b e i n t e r p r e t e d as a phase a v e r a g e . Thus w e p u t ,
Q a nd
.
A
=
$ ( t , xi;
s ( t , xi)
w = -s
- el,
(1.6a) (1.6b)
t'
Ki
= sxi'
85
A Lagrangian View of Wave-Mean Flow Interaction Here s ( t , x i ) is t h e phase, 4 i s p e r i o d i c i n s , w i s t h e wave frequency and i s t h e wavenumber. It f o l l o w s t h a t
Ki
a
(1.7)
i
and t h e wave a c t i o n e q u a t i o n (1.5a) t a k e s t h e form o b t a i n e d by Whitham (1970) i n t h e absence o f d i s s i p a t i o n . Another c h o i c e i s t o i d e n t i f y 9 w i t h a c o - o r d i n a t e so t h a t t h e a v e r a g i n g o p e r a t o r i s a c o - o r d i n a t e a v e r a g e . It i s remarkable t h a t t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) is f o r m a l l y e x a c t , v a l i d w i t h o u t any a p p a r e n t r e s t r i c t i o n o n wave a m p l i t u d e o r w i t h o u t any assumption t h a t t h e mean f i e l d i s s l o w l y v a r y i n g w i t h r e s p e c t t o t h e waves. There i s , of c o u r s e , a n a s y m p t o t i c e r r o r i n c u r r e d in t h e i d e n t i f i c a t i o n of t h e f a m i l y ( 1 . 2 ) (Hayes, 1970). Turning next t o t h e mean flow e q u a t i o n w e f i r s t d e f i n e t h e " u n d i s t u r b e d " Lagr a n g i a n by
Note t h g t a l t h o u g h Lo does n o t depend e x p l i c i t l y o n t h e wave f i e l d $, t h e mean f i e l d s 4 g e n e r a l l y c o n t a i n O(a2) wave-induced components. Next w e p u t
L~ i s a n o ( a 2 ) wave p r o p e r t y , and i t s e x p l i c i t dependence o n t , x i l i n c l u d e s i t s dependence on t h e mean f i e l d s 0. I f we now m u l t i p l y (1.1) by $. and a v e r a g e J t h e r e s u l t , we c a n show t h a t
aT. aT . -&! +&
=
-<
ax1 > +
,
i
j
(1.10a)
j
where
(1.10b)
>
T.. = J1
- n,>bji'
(1.1Oc)
is t h e pseudomomentum, and -T.i t h e c o r r e s p o n d i n g f l u x . The s i g n s are H e r e -T a g r e e w i t h h i s t o r i c a l conventl!on (Andrews and M c l n t y r e , 1978b). A chosen similar e q u a t i o n can b e o b t a i n e d f o r t h e pseudoenergy by m u l t i p l y i n g ( 1 . 1 ) by 0, and a v e r a g i n g t h e r e s u l t (Grimshaw, 1984). Note t h a t when 9 is i d e n t i f i e d w i t h t h e wave a c t i o n d e n s i t y A (1.5b) r e d u c e s t o -Tjo, and t h e t h e c o - o r d i n a t e -x j’ wave a c t i o n f l u x 6 ( 1 . 5 ~ ) r e d u c e s t o -Tji; a l s o e q u a t i o n (1.10a) t h e n r e d u c e s t o t h e wave a c t i o n e q u a t i o n ( 1 . 5 a ) .
2:
To make f u r t h e r p r o g r e s s w e now assume t h a t - t h e mean f i e l d s v e l o c i t y ii and a vector-valued mean f i e l d h , where dh - + A dt
a; ij
i
h--=O,
ax
j
7
c o n s i s t oE a mean
(1.11)
86
R. Grimshaw
x
where u r e p r e s e n t s t h e e f f e c t s of d i s s i p a t i o n . In a p p l i c a t i o n s w i l l incorporate t h e e f f e c t s of mean d e n s i t y , mean f l u i d d e p t h e t c . (compare Garrett (1968) and Note t h a t i n t h e absence of d i s s i p a t i o n A i s a mean q u a n t i t y Dewar-(1970)). ( A = 1); however i t i s u s e f u l i n a p p l i c a t i o n s t o l e t h have a f l u c t u a t i n g , d i s s i p a t i v e component. We n e x t d e f i n e
which i s thg: time d e r i v a t i v e f o l l o w i n g t h e meanlmotion, and assume t h a t L1 only t h r o u g h i t s dependence on d4ddt; f u r t h e r a p a r t from t h e depends on dependence L1 o! Ci through d $ / d t any o t h e r e x p l i c i t denpendence o f L1 o n These h y p o t h e s e s ii i s b i l i n e a r i n ui and t h e d i s t u r b a n c e v a r i a b l e s Q and $x
+
OF
i
.
a r i s e from t h e f a c t t h a t t h e f u l l Lagrangtan i s u s u a l l y a t most q u a d r a t i c i n t h e v e l o c i t y f i e l d . The mean flow e q u a t i o n c a n now b e o b t a i n e d by a v e r a g i n g (1.1). The d e t a i l s of t h e d e r i v a t i o n a r e d e s c r i b e d by Grimshaw (1984), and t h e r e s u l t i s
a aLO aLoE ( Y ) +a F- b_ aui j j a;,
+ R
where
ij
=
~
a R i jax j
-T
ij
<
~
aL axi
+;.T
10
J
.? aLO iLOsij)
A
>e + -
ah
iJ
(1.13a)
,
i j
aL 1 h-->
(1.13b)
?A
Here R i j is t h e r a d i a t i o n stress t e n s o r , a l t h o u g h n o t e t h a t t h e terminology i s not u n i v e r s a l . A l s o e d e n o t e s t h e e x p l i c i t d e r i v a t i v e of L w i t h r e s p e c t t o xi when t h e d i s t u r b a n c e f i e l d s Q and t h e mean v a r i a b l e s Ci, A a r e a l l held c o n s t a n t . Equation (1.13a) c a n b e recognized as t h e mean momentum e q u a t on and d e m o n s t r a t e s t h e manner in which t h e r a d i a t i o n s t r e s s t e n s o r , a n O(a ) wave q u a n t i t y , i s r e s p o n s i b l e f o r t h e a c c e l e r a t i o n of t h e mean flow. An a l t e r n a t i v e t o (1.13a) i n v o l v i n g t h e pseudomomentum i n s t e a d of t h e r a d i a t i o n stress t e n s o r c a n b e The r e s u l t is o b t a i n e d from ( 1 . 1 3 a , b ) by u s i n g (1.lOa).
!
a (-=aLO
at
i
)
+ aa x( -u j j
aLo ) hi
>e
+
+ - < -aA ax. J
+
. h aL a;> iJ
$x Q>.
+ax a no> i
(1.14)
i
T h i s i s t h e form p r e f e r r e d by Andrews and McIntyre (1978a). s e c t i o n by n o t i n g t h e a l t e r n a t i v e e x p r e s s i o n f o r Rij,
We c o n c l u d e t h i s
(1.15)
a7
A Lagrangian View of Wave-Mean Flow Interaction
Here (i3Ll/a$x2)d d e n o t e s t h e d e r i v a t i v e w i t h r e s p e c t t o constant.
J
S i m l l a r l y I t c a n b e shown t h a t
8,
ix
4 J
when d $ / d t i s h e l d
=
(1.16a)
Uik+4+
(1.16b)
where
92.
GENERALIZED LAGRANGIAN-MEAN
FORMULATION
The equati-ons of motion f o r a c o m p r e s s i b l e f l u i d a r e (2.la) (2.lb)
(2.lc) (2.ld)
where
Here x: i s t h e Eul r i a n c o - o r d i n a t e such t h a t a f l u i d p r t i c l e a t x i h a s The n o t a t i o n i s s t a n d a r d ; t h u s Qi i s t h e c o n s t a n t a n g u l a r v e l o c i t y v e l o c i t y ui. of t h e frame of r e f e r e n c e , @ ( x i ) i s t h e p o t e n t i a l f o r g r a v i t a t i o n a l and c e n t r i f u g a l f o r c e s , p(p, S ) i s t h e thermodynamic p r e s s u r e , p i s t h e d e n s i t y and S i s t h e e n t r o p y . The terms Xi and h r e p r e s e n t t h e e f f e c t s of d i s s i p a t i v e and d i a b a t ic e f f e c t s r e s p e c t i v e l y
.
The a p p r o p r i a t e L a g r a n g i a n f o r m u l a t i o n of t h e s e e q u a t i o n s i s t h e g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n of Andrews and McIntyre (1978a), which c o n t a i n s a comprehensive a c c o u n t of t h i s t h e o r y . Here we s h a l l g i v e only a b r i e f o u t l i n e ( s e e a l s o McIntyre (1977, 1980), Grimshaw (1979) o r Dunkerton (1980)). L e t xi b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s and l e t c i ( t , xo) b e t h e p a r t i c l e displacements defined s o t h a t
Then, f o r any g i v e n ui t h e r e i s a u n i q u e " r e f e r e n c e " v e l o c i t y i l ( t , x . ) s u c h J t h e p o i n t x i moves w i t h t h a t when t h e p o i n t xi moves w i t h v e l o c i t y The material time d e r i v a t i v e ( $ . l a ) i s t h e n g i v e n by v e l o c i t y ui.
The g e n e r a l i z e d Lagrangian-mean f o r m u l a t i o n i s now o b t a i n e d by l e t t i n g mean v e l o c i t y , and r e q u i r i n g t h a t
= 0.
ui
be t h e
(2.4)
Thus (2.3) a g r e e s w i t h ( 1 . 1 2 ) . The r e a d e r should n o t e t h a t o u r n o t a t i o n d i f f e r s i n one s i g n i f i c a n t r e s p e c t from Andrews and McIntyre (1978a,b); h e r e ii d e n o t e s
R. Grimshaw
88
zt
t h e Lagrangian-mean v e l o c i t y , r a t h e r t h a n used by Andrews and McIntyre (1978a,b), who u s e t h e s i n g l e o v e r b a r t o d e n o t e E u l e r i a n means. No c o n f u s i o n should arise as E u l e r i a n means w i l l n o t b e d i s c u s s e d i n t h i s a r t i c l e .
-
Next w e d e f i n e a mean d e n s i t y p s o t h a t
au g+p - = i ax
0.
(2.5)
i
It i s a n immediate consequence of ( 2 . l b ) and ( 2 . 5 ) t h a t
PJ where
(2.6b)
= P,
J = det[
axi ~
ax.
1.
(2.6b)
J
We l e t K i j b e t h e i , j
-
c o - f a c t o r of J , and s o
Note t h a t U i j is t h e d e r i v a t i v e of 3 w i t h r e s p e c t t o B x i / a x . and aU /axj = 0 . J ij The momentum e q u a t i o n ( 2 . l a ) now becomes
-
dui p-+ dt
-
2 p ~ Q u ijk j k
ui = ui
(2.8a)
.) = pXI,
iJ
j
-
where
.
@ a + p- -a-axi ;+-(pK ax d5,
+dt
(2.8b)
The g e n e r a l i z e d L a g r a n g i a n m e a n e q u a t i o n s a r e t h u s ( 2 . 8 a ) , (2.5) mean f i e l d s a r e Gi, p and S and t h e wave p e r t u r b a t i o n f i e l d i s Lagrangian is
and ( 2 . 1 ~ ) . The A suitable
ci.
- 1 p{.j uiui
+
E
B
X’U
ijk i j k
-
@(xi)
-
E(p, S)}.
(2.9)
-
Here w e r e c a l l t h a t p i s d e f i n e d in terms of p by ( 2 . b a ) , and E ( p , S) is t h e i n t e r n a l energy p e r u n i t mass, so t h a t
(2.10)
where T is t h e temperature. V a r i a t i o n s i n x; ( o r e q u i v a l e n t l y i n ti) g i v e (2.8a) w i t h Q i = pXi. I n o r d e r t o k e e p the-correspondence w i t h t h e g e n e r a l t h e o r y o f $1 a s c l o s e as p o s s i b l e w e d e f i n e Qs = pT s o t h a t v a r i a t i o n s w i t h r e s p e c t t o S-can b e d e f i n e d . The v e c t o r A of $1 i s i d e n t i f i e d as t h e 2-vector w i t h components p and S , and w e n o t e t h a t ( 2 . 1 ~ ) and (2.5) h a v e t h e r e q u i r e d form (1.11).
89
A Lagrangian View of WaveMean Flow interaction The wave a c t i o n e u a t i o n c a n now b e o b t a i n e d from t h e g e n e r a l t h e o r y of 51, u s i n g ( 1 . 5 a , b , c , d ) and q l . l 6 a , b ) .
where
,. ac.,
.4= < p -
ae (2.11c)
8 =< -. p ( ac.,
ae Xi
as T>. +-
(2.11d)
ae
ThLs a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978b) a l t h o u g h t h e d i s s i p a t i v e term& h a s been w r i t t e n i n a d i f f e r e n t form h e r e . F o r l i n e a r i z e d wave motion, i t i s u s e f u l t o i n t r o d u c e t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n
%+
=
2 O(a ) .
(2.12)
It may t h e n b e shown t h a t
(2.13)
Note t h a t t h e second term h e r e i s i d e n t i c a l l y non-divergent and c a n b e o m i t t e d from (2.11a). To d e r i v e t h e mean flow e q u a t i o n s we f i r s t i d e n t i f y LO from (1.8).
Thus
where L i s g i v e n by ( 2 . 9 ) . The mean f l o w e q u a t i o n i s t h e n o b t a i n e d from ( 1 . 1 3 a , b ) , o r more d i r e c t l y by a v e r a g i n g ( 2 . 8 a ) . The r e s u l t i s
(2.15a) where
and
-
-
P = P(P,
S),
(2.15b) (2.15~)
It c a n b e v e r i f i e d t h a t Rij i s t h e same r a d i a t i o n stress t e n s o r d e f i n e d by An a l t e r n a t i v e form of (2.15a) i n v o l v i n g t h e pseudomomentum -Ti0 i n s t e a d (1.14). of t h e r a d i a t i o n stress t e n s o r c a n b e o b t a i n e d u s i n g t h e p r o c e d u r e d e s c r i b e d i n $1 and a v e r a g i n g ( s e e ( s e e (1.14), o r more d i r e c t l y by m u l t i p l y i n g ( 2 . l a ) by ax;/ax j Andrews and McIntyre (1978a)). The r e s u l t i s
90
R. Grimshaw
;a = + aTio - a t + - aa ~ (. U- j T i o ) Tjo 2 ax, J +
where
lt =
a + E> +m- < P
-
P<
'j
2 (; 2 + d5
ax' axi
> + <; T^ as >, ax,
(2.16a)
d5
EjklQ$,)>'
and
(2.16b)
(2.16~)
T h i s a g r e e s w i t h t h e r e s u l t of Andrews and McIntyre (1978a) a l t h o u g h t h e d i s s i p a t t v e terms on t h e right-hand s i d e of (2.16a) have been w r i t t e n i n a d i f f e r e n t form h e r e .
s3.
GENERALIZED ELIASSEN-PALM AND CHARNEY-DRAZIN THEOREMS
One of t h e most promising a r e a s f o r a p p l i c a t i o n of t h e g e n e r a l r e s u l t s of $2 i s s t r a t o s p h e r i c meteorology. With t h e a v e r a g i n g o p e r a t o r i n t e r p r e t e d as a z o n a l a v e r a g e , t h e c o u n t e r p a r t s of t h e wave a c t i o n e q u a t i o n (2.11a) and t h e mean f l o w e q u a t i o n (2.15a) ( o r (2.16a)) are u s u a l l y c a l l e d Eliassen-Palm and Charney-Drazin theorems r e s p e c t i v e l y , a f t e r t h e p i o n e e r i n g work of E l i a s s e n and Palm (1961) and Charney and D r a z i n (1981) ( f o r r e c e n t reviews see Andrews and & I n t y r e (1978c), McIntyre (1980), Dunkerton (1980) and Uryu (1980)). To o b t a i n t h e s e theorems i n t h e i r most g e n e r a l form w e must f i r s t e x p r e s s t h e e q u a t i o n s of motion i n s p h e r i c a l A s i m p l e and c o n v e n i e n t way of d o i n g t h i s i s t o i n t r o d u c e polar co-ordinates. s p h e r i c a l p o l a r c o - o r d i n a t e s i n t o t h e Lagrangian (2.9); t h i s i s t h e approach used by R r e t h e r t o n (1982) f o r small-amplitude waves. The c o r r e s p o n d i n g r e s u l t s i n c y l i n d r i c a l p o l a r c o - o r d i n a t e s are d e s c r i b e d by Grimshaw (1984). Thus w e l e t t h e c o - o r d i n a t e s x i of 51 b e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , h a n d v, where r i s t h e r a d i u s , h t h e c o - l a t i t u d e and v t h e l o n g i t u d e . These a r e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , whose E u l e r i a n c o u n t e r p a r t s x a r e I. r ' , A' and v ' . The p a r t i c l e d i s p l a c e m e n t s are t h e n d e f i n e d by
The v e l o c i t y components Ln t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s a r e u
where
= _d _r ' dt
,
d h' dv' u2 = r' , u3 = r' sin h' dt ' dt
(3.2a)
(3.2b)
Here Ui are t h e mean v e l o c i t y components in t h e L a g r a n g i a n c o - o r d i n a t e d i r e c t i o n s , which must b e c a r e f u l l y d i s t i n g u i s h e d from t h e E u l e r i a n c o - o r d i n a t e d i r e c t i o n s . The Lagrangian i s a g a i n g i v e n by ( Z a g ) , where, f o r s i m p l i c i t y , we s h a l l suppose t h a t t h e a x i s of r o t a t i o n i s i n t h e N-S d i r e c t i o n . Thus t h e L a g r a n g i a n i s
91
A Lagrangian View of Wave-Mean Flow Interaction
- 1
r2 s i n A PI- u u + 9 r ' s i n 2 i i
r f 2 s i n A ' PJ
where
=
u3 - ~(x;) - ~ ( p , s)},
i.
r2 s i n A
(3.3a) (3.3b)
and J i s a g a i n d e f i n e d by (2.6b) b u t now xi and xi a r e t h e s p h e r i c a l p o l a r coo r d i n a t e s . V a r i a t i o n s i n Si now g i v e t h e e q u a t i o n s o € motion.
-p { dul ~
dt
L
L
u2 - I u 3 - 2Q s i n h' u3 - -T K r
+ aQ ar
}
(3.4a)
(3.4b)
(3.4c) The c o u n t e r p a r t of ( 2 . 5 ) i s
w h i l e S a g a i n s a t i s f i e s (2.1.2). The r e s u l t i s
The wave a c t i o n e q u a t i o n c a n now b e o b t a i n e d from ( 1 . 5 a , b , c , d ) .
-a + - ( -a
at
where
=
-
art r2 s i n ~ p < u
+ r2
ae
1
sin h
;=
ui
axi
hi
+
rf
*i )
=
$ u2 + r'
l p ~
<,I2
+
s i n h'
(3.6a)
I
s i n A’
a v r >, s i n 2 A’ -
ae
ac ae pKji>,
ae
u
3
> (3.6b)
(3.6~)
92
R. Grimshaw
b a nd
h
1
=
=
-
r 2 sin Ap
ae
i
as +-T>.
(3.6d)
ae
h ' = 1; h = r, h ' = r ' ; h = r s i n A, h ' = r ' s i n 1 2 2 3 3
A’.
(3.6e)
Here w e remind t h e r e a d e r t h a t xi a r e t h e s p h e r i c a l p o l a r c o - o r d i n a t e s r , A and v. For l i n e a r i z e d wave motion w e i n t r o d u c e t h e E u l e r i a n p r e s s u r e - p e r t u r b a t i o n p' by ( 2 . 1 2 ) . It may t h e n b e shown t h a t
The second term i s i d e n t i c a l l y t h e b a s i c f l o w is z o n a l ( i - e . i n t e r p r e t e d as a z o n a l a v e r a g e e q u a t i o n (3.6a) r e d u c e s t o t h e Andrews and McIntyre ( 1 9 7 8 ~ ) .
non-divergent and c a n be o m i t t e d from (3.6a). When and z2 a r e O(a2)) and t h e a v e r a g i n g o p e r a t o r is ( i . e . 8 i s i d e n t i f i e d w i t h -v) t h e wave a c t i o n g e n e r a l i z e d Eliassen-Palm r e l a t i o n d e r i v e d by Also -(r2 s i n ?-,)-!& i s t h e a n g u l a r pseudomomentum.
The mean flow e q u a t i o n s c a n now b e o b t a i n e d from (1.13a) ( o r ( 1 . 1 4 ) ) ( a f t e r a l l o w i n g f o r t h e p r e s e n c e of t h e g e o m e t r i c a l f a c t o r s h i i n L ) , o r more d i r e c t l y by a v e r a g i n g ( 3 . 4 a , b , c ) . The e s s e n t i a l s t e p in t h i s l a t t e r approach is
(3.8a)
where
R
ij
=
6
.<,I2
iJ
sin A' PJ
-
r 2 sin A
p"> (3.8b)
*
-
Here p is p ( p , 9) and R i j i s t h e r a d i a t i o n stress t e n s o r ( i . e . t h e a n a l o g u e o f ( 2 . 1 5 ~ ) in s p h e r i c a l p o l a r c o - o r d i n a t e s ) . For i n s t a n c e , t h e z o n a l mean flow e q u a t i o n is
-
1
aR
A= p
A' X3>,
(3.9c)
r2 s i n A a x j where
M =
A’ u3
+ GrI2
s i n 2 XI>.
(3.9b)
Here M is t h e z o n a l mean s p e c i f i c a n g u l a r momentum; € o r t h e r e l a t i o n s h i p between M and t h e z o n a l mean f l o w , see Dunkerton (1980). When t h e a v e r a g i n g o p e r a t o r is i n t e r p r e t e d as a z o n a l a v e r a g e ( i . e . 0 i s i d e n t i f i e d w i t h -v) t h e o f f - d i a g o n a l components of R v j are i d e n t i c a l t o Bq; simply compare (3.8b) w i t h ( 3 . 6 ~ ) . Note t h a t t h e d i a g o n a l components of R v j do not now a p p e a r in ( 3 . 9 ~ ) . With t h e s e i n t e r p r e t a t i o n s , and u s i n g ( 3 . 6 a ) , e q u a t i o n ( 3 . 9 ~ )becomes
A Lagrangian View of Wave-Mean Flow Interaction
dM dt
--
dt
(
r2L -?*--+ pr
x3>.
93
(3.10)
pr2 s i n A
sin A
With t h e f u r t h e r r e s t r i c t i o n t o l i n e a r i z e d waves on a z o n a l b a s i c flow (3.10) r e d u c e s t o a g e n e r a l i z e d Charney-Drazin theorem ( s e e t h e s i m i l a r r e s u l t s o b t a i n e d by Andrews and McIntyre ( 1 9 7 8 ~ )and B r e t h e r t o n ( 1 9 8 2 ) ) . It f o l l o w s t h a t M w i l l change o n l y i n r e s p o n s e t o wave t r a n s i e n c e r e p r e s e n t e d by t h e term i n v o l v i n g k i n (3.10), o r due t o d i s s i p a t i v e e f f e c t s r e p r e s e n t e d by t h e terms i n v o l v i n g 3 and X3.
S4.
TOPOGRAPHIC, @-PLANE ROSSBY WAVES
The r e s u l t s of $2 and 53 have been o b t a i n e d w i t h o u t any r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e s c a l e of t h e waves, o r w i t h o u t making any of t h e a p p r o x i m a t l o n s such a s t h e h y d r o s t a t i c a p p r o x i m a t i o n , o r t h e q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n , o f t e n used i n g e o p h y s i c a l f l u i d dynamics. These v a r i o u s a p p r o x i m a t i o n s c a n b e used i n p o s t e r i o r i i n t h e e v a l u a t t o n of t h e wave a c t i o n d e n s i t y Jk ((2.11b) o r ( 3 . 6 b ) ) , t h e wave a c t i o n f l u x B 2 ( ( 2 . 1 1 ~ ) o r ( 3 . 6 c ) ) , e t c . However, i t i s i n s t r u c t i v e and sometimes more u s e f u l t o d e r i v e a wave a c t i o n e q u a t i o n and a mean flow e q u a t i o n d i r e c t l y from a n approximate s e t of e q u a t i o n s . In t h i s r e s p e c t t h e g e n e r a l t h e o r y of § l p l a y s a c o - o r d i n a t i n g r o l e . To i l l u s t r a t e t h i s a s p e c t w e s h a l l d e r i v e t h e wave a c t i o n e q u a t i o n and t h e mean f l o w e q u a t i o n s € o r a homogeneous f l u i d u s i n g t h e h y d r o s t a t i c and @-plane a p p r o x i m a t i o n s . In thLs c o n t e x t t h e wave p e r t u r b a t i o n s a r e g e n e r i c a l l y c a l l e d Rossby waves, and d e r i v e t h e i r wave-like c h a r a c t e r from a combination of t h e t o p o g r a p h i c s l o p e , t h e @ - e f f e c t and t h e s h e a r of t h e mean flow. Many a u t h o r s have s t u d i e d t h e i n t e r a c t i o n of Rossby waves w i t h mean flows ( s e e Dickinson (1978) f o r a review). From t h e p o i n t of view adopted i n t h l s a r t i c l e t h e p a p e r s by Grimshaw ( 1 9 7 7 ) , Rhines and Holland (1979), Young and Rhines (1980) and Huthnance (1981) a r e p a r t i c u l a r l y r e l e v a n t , a l t h o u g h n e a r l y a l l t h e r e s u l t s o b t a i n e d by t h e s e a u t h o r s a r e r e s t r i c t e d t o s m a l l a m p l i t u d e waves, o r s u b j e c t t o some o t h e r r e s t r i c t i o n c o n c e r n i n g t h e s t r u c t u r e of t h e mean flow. The r e s u l t s t o b e o b t a i n e d below c o n t a i n no r e s t r i c t i o n on wave a m p l i t u d e , o r on t h e scale of t h e waves v i s a - v i s t h e mean f l o w , and hence c a n b e r e g a r d e d as a g e n e r a l i z a t i o n of t h e c o r r e s p o n d i n g r e s u l t s of t h e s e a u t h o r s . The E u l e r i a n e q u a t i o n s of motion f o r a homogeneous f l u i d i n t h e h y d r o s t a t i c a p p r o x i m a t i o n are
(4. l a )
(4.lb)
(4.1~) where
H = h(x', y ' )
f = f and
0
+
@1x '
-d= = + au , , + v a Ya ' dt
+
+
5,
(4.ld)
p2y1,
(4.le)
a
(4.1f)
94
R. Grimshaw
Here ( x ' , y ' ) a r e E u l e r i a n h o r i z o n t a l c o - o r d i n a t e s such t h a t a f l u i d p a r t i c l e a t (x’ y ' ) h a s h o r i z o n t a l v e l o c i t y components (u, v ) , w h i l e is t h e f r e e s u r € a c e d i s p l a c e m e n t , and h ( x ' , y ' ) i s t h e e q u i l i b r i u m d e p t h of t h e f l u i d . The terms F and G r e p r e s e n t t h e combined e f f e c t s of wind s t r e s s f o r c i n g and d i s s i p a t i o n . One immediate and useful consequence of ( 4 . 1 a , b and c ) i s t h e p o t e n t i a l v o r t l c i t y equation
<
where
dd tx = E ,
(4.2a)
HX = q + f ,
(4.2b)
(4.2~) and
(4.2d)
To u s e t h e g e n e r a l t h e o r y of $1 w e mst f i r s t r e c a s t t h e s e e q u a t i o n s u s i n g t h e g e n e r a l i z e d Lagrangian-mean Eormulation. The development i s similar t o t h a t d e s c r i b e d i n $2 f o r a g e n e r a l c o m p r e s s i b l e f l u i d . Thus, w e l e t x and y b e g e n e r a l i z e d Lagrangian c o - o r d i n a t e s , and l e t 5 and 7) b e t h e p a r t i c l e d i s p l a c e m e n t s d e f i n e d so t h a t x ' = x + E .
=
y ' = y + q ,
= 0,
and
(4.3a) (4.3b)
(4.3c)
Here i, a r e t h e LagrangLan-mean v e l o c i t i e s . We l e t J b e t h e J a c o b i a n of t h e t r a n s f o r m a t t o n from (x, y ) t o (x’, y ' ) w i t h c o - f a c t o r s K i j ( i . e . (2.6b) and ( 2 . 7 ) r e s t r i c t e d t o h o r i z o n t a l co-ordinates). Then ( 4 . 1 ~ ) becomes
(4.4a) where
-
-
Note t h a t H i s mean q u a n t i t y .
H = JH.
(4.4b)
The e q u a t i o n s o f motion ( 4 . l a , b ) become
Here we are t e m p o r a r i l y u s i n g i n d e x n o t i o n where u1 = u, u2 = v e t c . a b l e Lagrangian i s
A suit-
95
A Lagrangian View of Wave-Mean Flow Interaction
2
where
u u
i i
+
f(k
-
+
e) u }
x
, i i
+
e = f x’ 0-
p,x'
2
g(hC
+
i
,
I
21
f
< 2 )J -
gH6,
2
(4.6a)
(4.6b)
p2y' j . I
Here i,, j and k, a r e u n i t v e c t o r s i n the_ x , y and z - d i r e c t i o n s r e s p e c t i v e l y . V a r i a t i o n s i n Ei g i v e (4.5) w i t h Q i = HFi; v a r i a t i o n s in g i v e (4.4b). W e are now i n a p o s i t i o n t o a p p l y t h e g e n e r a l t h e o r y of $1, n o t i n g t h a t (4.4a) h a s t h e r e q u i r e d Eorm ( 1 . 1 1 ) .
<
The wave a c t i o n e q u a t i o n i s g i v e n by ( 1 . 5 a , b , c , d ) , (4.7a)
- xi
+
ae
1
(k
x
e)i>,
(4.7b)
I
(4.7c)
8
=
- ac,
~
ae
F
i
>.
(4.7d)
An i n t e r e s t i n g f e a t u r e of (4.7b) i s t h e manner t n which t h e p-plane e f f e c t , h e r e r e p r e s e n t e d by t h e p a r a m e t e r s pi, m o d i f i e s t h e e x p r e s s i o n f o r t h e wave a c t i o n d e n s i t y (2.11b) o b t a i n e d i n $2. T h i s is more a p p a r e n t from t h e a l t e r n a t i v e expression t o (4.7b).
(4.8a) where
-
€ = €
0
+ p x + p y . 2 1
(4.8b)
F u r t h e r , i t c a n b e shown t h a t ( 4 . 8 a ) a g r e e s w i t h (3.6b) when t h e a p p r o p r i a t e p-plane, h y d r o s t a t i c and " t r a d i t i o n a l " a p p r o x i m a t i o n s ( i . e . n e g l e c t of t h e h o r i z o n t a l component of t h e e a r t h ' s r o t a t i o n ) a r e made i n ( 3 . 6 b ) . For l i n e a r i z e d wave motion t h e l a s t term i n k ( 4 . 8 a ) c a n b e n e g l e c t e d a n d m ? i s g i v e n by
where C ' i s t h e E u l e r i a n p r e s s u r e p e r t u r b a t i o n (compare ( 2 . 1 2 ) ) and t h e second term i n (4.9) i s i d e n t i c a l l y non-divergent, and hence c a n b e o m i t t e d from ( 4 . 7 a ) .
96
R. Grimshaw
The mean flow e q u a t i o n s a r e o b t a i n e d from (1.13a) o r (1.14) a f t e r a l l o w i n g f o r t h e e x p l i c i t dependence o n xi c o n t a i n e d i n t h e term i n v o l v i n g e i n ( 4 . 6 a ) , as a consequence of whtch t h e Lagrangian L i s n o t b i l i n e a r i n ui and The f o r m e r i n v o l v e s t h e r a d i a t i o n stress t e n s o r and i s r e a d i l y o b t a i n e d by a v e r a g i n g ( 4 . 5 ) . However, i n t h i s c o n t e x t , i t i s not so u s e f u l as s i g n i f i c a n t wave-forcing terms a r e c o n t a i n e d i n e ( s e e (1.13a)) which a r i s e from t h e e x p l i c i t dependence of f and h on xi. The a l t e r n a t i v e form (1.14) i s p r e f e r r e d , and i s most simply o b t a i n e d by m u l t t p l y i n g ( 4 . 5 ) w i t h a x i / a x j and a v e r a g i n g . The r e s u l t i s
ci.
where
(4.10b)
(4.10d)
and
A comparison of (4.10a) w i t h (1.14) o r (2.16a) s u g g e s t s Lhat Pi(4.10b) i n (4.10a) p l a y s t h e r o l e of pseudomomentum, a l t h o u g h w e n o t e t h a t HPi d i f f e r s from t h e pseudomomentum - T i o ( l . l O b ) , where h e r e (4.11) The d i f f e r e n c e arises due t o t h e afore-mentioned p r e s e n c e of L.
e,
i n t h e Lagrangian
In many a p p l i c a t i o n s t h e mean-flow e q u a t i o n (4.10a) i s dominated by t h e nearg e o s t r o p h i c b a l a n c e between t h e mean C o r i o l i s f o r c e and t h e mean p r e s s u r e g r a d i e n t . In t h i s case a more u s e f u l form of t h e mean flow e q u a t i o n i s o b t a i n e d W e find that by a v e r a g i n g ( 4 . 2 a ) . -
g =E,
(4.12a)
where
(4.12b)
and
(4.12~)
Note t h a t (4.12b and c ) d e m o n s t r a t e t h a t (4.12a) i s a l s o a n immediate consequence of (4.10a). The s i m p l i c i t y of (4.12a) i s s e l f - e v i d e n t ; t h i s e q u a t i o n i s a
97
A Lagrangian View of Wave-Mean Flow Interaction
g e n e r a l i z a t i o n of similar r e s u l t s which h a v e a p p e a r e d i n t h e l i t e r a t u r e , u s u a l l y f o r l i n e a r i z e d wave m o t i o n ( s e e Rhines a n d Holland ( 1 9 7 9 ) , o r Huthnance ( 1 9 C l ) ) . In p a r t i c u l a r , f o r t h g s e s i t u a t i o n s where t h e dominant component o f is T/n and t h e d i s s i p a t i v e term E i s small ( e . g . where t h e waves a r e s l o w l y v a r y i n g r e l a t i v e t o t h e mean f l o w ) , e q u a t i o n (4.12a) s u c c i n c t l y d e m o n s t r a t e s t h a t t h e mean f l o w c l o s e l y follows t h e geostrophic contours.
X
REFERENCES Andrews, D.G. and M c I n t y r e , M.E., 1978a: "An e x a c t t h e o r y of n o n l i n e a r waves on a Lagrangian-mean flow", J . F l u i d Mech., 8v , 609-646. Andrews, D.G. a n c M c I n t y r e , M.E., J . F l u i d Mech., 8 9 , 647-664.
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"On wave-action and i t s r e l a t i v e s " ,
Andrews, D.G. and M c I n t y r e , M.E., 1978c: " G e n e r a l i z e d E l t a s s e n - P a l m and Charney-Drazin theorems f o r waves o n a x i s y m m e t r i c mean f l o w s i n c o m p r e s s i b l e 175-185. a t m o s p h e r e s " , J. Atmos. S c i . , 3, B r e t h e r t o n , F.P., 1982: " C o n s e r v a t i o n of wave a c t i o n and a n g u l a r momentum i n a s p h e r i c a l atmosphere". P r e p r i n t . B r e t h e r t o n , F.P. and Garrett C . J . R . , 1968: "Wavetrains i n inhomogeneous moving media", Proc. R. SOC. London, S e r . A . TET, 539-554. Charney, J . G . and D r a z i n , P.G., 1961: " P r o p a g a t i o n o f p l a n e t a r y - s c a l e d i s t u r b a n c e s from t h e lower i n t o t h e u p p e r atmosphere", J . Geophys. R e s . , ii6, 83-109. D e w a r , R.L., 1970: " I n t e r a c t L o n between hydromagnetic waves and a timed e p e n d e n t , inhomogenous medium", Phys. F l u i d s , 3 3 , 2710-2720.
D i c k i n s o n , R . E . , 1978: "Rossby waves - l o n g p e r i o d o s c i l l a t i o n s o f a t m o s p h e r e s and o c e a n s " , Ann. Rev. F l u i d Mech. 159-198.
m,
Dunkerton, T . , 1980: "A L a g r a n g i a n mean t h e o r y of wave, mean-flow i n t e r a c t i o n w i t h a p p l i c a t i o n s t o n o n a c c e l e r a t i o n and i t s breakdown", Rev. 387-400. Geophys. Space Phys.,
m,
[LO] E l i a s s e n , A. and Palm, E., 1961: "On t h e t r a n s f e r of e n e r g y i n s t a t i o n a r y m o u n t a i n waves", Geofys. Publ. = ( 3 ) , 1-23. 1968: "On t h e i n t e r a c t i o n between i n t e r n a l g r a v i t y waves and a s h e a r flow", J. F l u i d Mech., 71, 497-512.
[ll] Garrett, C . J . R . ,
[ 1 2 ] Grimshaw, R . , 1977: "Non-linear a s p e c t s o f s h e l f waves and mean c u r r e n t s " , in "Waves on Water o f V a r i a b l e Depth", e d . D.G. P r o v i s and R. Radok; pp 206211; L e c t u r e Notes i n P h y s i c s , bG, S p r i n g e r and A u s t r a l i a n Academy o f S c i e n c e , pp 235. [ 1 3 ] Grimshaw, R., 1979: "Mean f l o w s induced by i n t e r n a l g r a v i t y wave p a c k e t s p r o p a g a t i n g i n a s h e a r flow", P h i l . T r a n s . R . SOC. London, S e r A 391417.
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[ 1 4 ] Grimshaw, R., 1984: "Wave a c t i o n and wave-mean f l o w i n t e r a c t i o n , w i t h a p p l i c a t i o n t o s t r a t i f i e d s h e a r f l o w s " , Ann. Rev. F l u i d Mech., 11-43.
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Hayes, W.D., 1970: " C o n s e r v a t i o n of a c t i o n and modal wave a c t i o n , "Froc. R. SOC. London., S e r A m(T, 187-208.
(161 Huthnance, J.M.,
1981: waves", J. F l u i d Mech.,
"On mass t r a n s p o r t s g e n e r a t e d by t i d e s and long
m, 367-387.
[17] McIntyre, M . E . , 1977: "Wave t r a n s p o r t in s t r a t i f i e d r o t a t i n g f l u i d s " i n "Problems o f S t e l l a r Convection"; pp 290-314, ed. E . A . S p i e g e l and J.P. S p r i n g e r , pp 363. Zahn: L e c t u r e Notes in P h y s i c s [ 1 8 ] McIntyre, M . E . , 1980: "An i n t r o d u c t i o n t o t h e g e n e r a l i z e d Lagrangian-mean d e s c r i p t i o n of wave, mean-flow a c c e l e r a t i o n " , Pageoph., 152-176.
m,
[19] Rhines, P.B. and H o l l a r d , W.R., 1979: "A t h e o r e t t c a l d i s c u s s i o n of eddyd r i v e n mean flows", Dyn. Amos. Oceans, 7 , 289-325. [20] U r y u , M . ,
Pageoph.,
1980:
" A c c e l e r a t i o n of mean z o n a l f l o w s by p l a n e t a r y waves",
m, 661-693.
[ 2 1 ] Whitham, G.B.,
1965: "A g e n e r a l approach to l i n e a r and n o n l i n e a r d i s p e r s i v e 273-283. waves u s i n g a Lagrangian", J. F l u i d Mech.,
[ 2 2 ] Whitham, G.B., F l u i d Mech., [ 2 3 ] Young, W.R.
m,
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"Two-timing,
a,373-395.
v a r i a t i o n a l p r i n c i p l e s and waves", J .
and R h i n e s , P.B., 1980: "Bossby wave a c t l o n , e n s t r o p h y and energy i n f o r c e d mean f l o w s " , Geophys. Astrophys. F l u i d Dynamics, 'fs, 39-52.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland),1984
99
A THEORY OF DYNAMICAL FERROELECTRICITY P e t e r J . Chen Sandia N a t i o n a l L a b o r a t o r i e s Albuquerque, New Mexico 87185 U.S.A.
I n t h i s p a p e r 1 s h a l l g i v e a dynamical t h e o r y o f f e r r o e l e c t r i c i t y . The t h e o r y t a k e s i n t o account t h e t r a n s i e n t e f f e c t s due t o domain s w i t c h i n g and d i p o l e dynamics. A s p e c i f i c representation i s derived f o r departures from the v i r g i n s t a t e which i s t a k e n t o be a n a t u r a l r e f e r e n c e s t a t e , and a l l o w s f o r changes i n m a t e r i a l symmetry due t o domain s w i t c h i n g and d i p o l e dynamics. INTRODUCTION F e r r o e l e c t r i c ceramics a r e f a i r l y complex m a t e r i a l s w h i c h e x h i b i t spontaneous p o l a r i z a t i o n i n t h e i r n a t u r a l s t a t e s below t h e i r C u r i e temperatures. The u n i t c e l l d i p o l e s o f t h e m a t e r i a l s may be e f f e c t i v e l y a l i g n e d v i a t h e a p p l i c a t i o n o f u n i - d i r e c t i o n a l e l e c t r i c f i e l d s o f s u f f i c i e n t magnitude and d u r a t i o n . These m a t e r i a l s a r e t h e n s a i d t o be p o l e d and t h e y e x h i b i t remanent p o l a r i z a t i o n . In t h e p o l e d s t a t e s f e r r o e l e c t r i c ceramics a r e e l e c t r o m e c h a n i c a l l y a c t i v e , and have f o u n d numerous a p p l i c a t i o n s such as t r a n s d u c e r s , sonars and f i l t e r s . These a p p l i c a t i o n s make use o f t h e r e v e r s i b l e responses o f t h e ceramics. However, f e r r o e l e c t r i c ceramics a l s o e x h i b i t i r r e v e r s i b l e responses depending on t h e n a t u r e o f t h e e x t e r n a l s t i m u l i such as t h e a p p l i c a t i o n o f l a r g e e l e c t r i c f i e l d s and/or l a r g e mechanical l o a d i n g s e i t h e r s t a t i c o r dynamic i n n a t u r e . I n t h e s e i n s t a n c e s , t h e remanent s t a t e s may a l t e r , t h e u n i t c e l l d i p o l e s may e x h i b i t dynamical e f f e c t s , and t h e m a t e r i a l s may undergo phase t r a n s i t i o n s . I n t h i s t a l k , I s h a l l d e s c r i b e how one may model t h e dynamical responses o f f e r r o e l e c t r i c ceramics.'
B A S I C EQUATIONS To b e g i n w i t h l e t us r e s t r i c t o u r a t t e n t i o n t o t h e case o f an i s o t h e r m a l t h e o r y and n e g l e c t t h e e f f e c t s o f phase t r a n s i t i o n s . L e t t h e v e c t o r denote t h e u n i t c e l l e l e c t r i c d i p o l e moment. We presume t h a t i t s magnitude and d i r e c t i o n depend on t h e mechanical s t r a i n S and t h e e x t e r n a l e l e c t r i c f i e l d k . We a l s o i n t r o d u c e t h e v e c t o r N, d e f i n e d by t h e r e l a t i o n 5
N = ( c -u . -j ) j- , (1) where J i s a u n i q u e u n i t v e c t o r such t h a t i f C l ~ * j _ # 0 t h e n C Q . ~must have maximum v a l u e , and where t h e summation i s c a r r i e d o u t o v e r each s u b - p a r t o f a f e r r o e l e c t r i c specimen. The d i r e c t i o n o f ,j denotes t h e d i r e c t i o n o f remanent p o l a r i z a t i o n , and C u - j . g i v e s t h e number o f u n i t c e l l d i p o l e s e f f e c t i v e l y a l i g n e d i n t h i s d i r e c t i o n . We g l s o presume t h a t t h e responses o f Q t o S and E_ can be p a r t i t i o n e d i n t o i t s i n s t a n t a n e o u s response pi and i t s t r a n s i e n t response ' J ~S O t h a t
100
P.J. Chen
The c o n s t i t u t i v e r e l a t i o n s f o r t h e s t r e s s T and t h e e l e c t r i c displacement D may, t h e r e f o r e , be g i v e n by
and we have t h e r a t e laws
The a p p r o p r i a t e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s a r e , o f course, t h e e q u a t i o n o f balance o f l i n e a r momentum and Gauss’ l a w i n t h e absence o f f r e e charge.
A S P E C I F I C REPRESENTATION The p r i n c i p a l d i f f i c u l t y i n t h e c o n s t r u c t i o n o f a s p e c i f i c r e p r e s e n t a t i o n o f t h e general t h e o r y i s t h e v e c t o r N_, e s p e c i a l l y s i n c e N v a r i e s b o t h s p a t i a l l y and t e m p o r a l l y . Hence, i t seems r e a s o n a b l e t h a t we c o u l d r e s t r i c t t h e p r o p e r t i e s o f i n some reasonable way and a d o p t t h e f o l l o w i n g assumptions:
N-
( i ) I t seems c o n v e n i e n t t o r e g a r d t h e s t a t e o f a v i r g i n specimen as t h e n a t u r a l r e f e r e n c e s t a t e and i n t h i s s t a t e t h e m a t e r i a l p r o p e r t i e s o f t h e specimen a r e isotropic. We assume ( i i ) As i s customary, we t a k e t h e p o l a r d i r e c t i o n as t h e X 3 - d i r e c t i o n . t h a t f o r a l l t h e p h y s i c a l processes t o which t h e specimen i s s u b j e c t e d t h e v e c t o r N i s always p a r a l l e l t o t h e X3-axis.
-
Assumption ( i i ) s t a t e s t h a t t h e cornpo-
Assumption ( i ) i s q u i t e s t r a i g h t f o r w a r d . nents o f N a r e g i v e n by I
N -” = (0,0,N3)
.
(5)
Very s i m p l y , o u r aim i s t o c o n s t r u c t a s p e c i f i c r e p r e s e n t a t i o n such t h a t i n a v i r g i n s t a t e t h e m a t e r i a l i s i s o t r o p i c e x h i b i t i n g no p i e z o e l e c t r i c e f f e c t , and t h a t whenever N3#0 t h e m a t e r i a l i s t r a n s v e r s e l y i s o t r o p i c w i t h r e s p e c t t o t h e Xja x i s e x h i b i t i n g p i e z o e l e c t r i c e f f e c t s which depend on Ng. We now i n t r o d u c e c e r t a i n s i m p l i f i c a t i o n s c o n c e r n i n g t h e r a t e laws ( 4 ) 1 .
We assume
t h a t t h e t r a n s i e n t response o f t h e d i p o l e moments t o t h e i n f l u e n c e s o f s t r a i n and e l e c t r i c f i e l d may be t a k e n i n t o a c c o u n t s e p a r a t e l y . Hence, i n s t e a d o f (4)1, we have
where we have i n t r o d u c e d t h e o b v i o u s s u g g e s t i v e n o t a t i o n f o r t h e s e p a r a t e dependences, and yt i s now t h e p a i r (ytS, ytE). We a l s o assume t h a t t h e t r a n s i e n t -
I
response o f domain s w i t c h i n g i s independent o f t h e t r a n s i e n t response o f t h e d i p o l e moments so t h a t ( 4 ) 2 r e s u l t s i n t h e r a t e l a w
A Theory of Dynarnical Ferroelectricity
101
f o r t h e e f f e c t i v e number o f a l i g n e d d i p o l e s N3 Since t h e c o n s t r u c t i o n o f t h e s p e c i f i c r e p r e s e n t a t i o n i n v o l v e s t h e expansion o f t h e c o n s t i t u t i v e r e l a t i o n s f o r t h e s t r e s s ( 3 ) 1 and t h e e l e c t r i c displacement ( 3 ) 2 w i t h t h e r e t e n t i o n o f t h e a p p r o p r i a t e terms and t h e c h o i c e o f s p e c i f i c e x p r e s s i o n s f o r t h e r a t e laws ( 6 ) and ( 7 ) , we s i m p l y l i s t t h e r e s u l t s , v i z . T i. J.
0.
1
I1J kR+a TiJ. kR3N3)SkE (C..
=
=
e.. N S. 1 ~ k 33 J k
+
(E
I
6.
+
+
( b ij kfdi j k3N3)”ts,
j . . N 1~ 153 3 t S . -J
I N ) E . + ( 6 6 . .+6..N )ptEj 153 3 J 1J 153 3
-
.+E..
1.J
+
’
k 13 . N3
(9)
t o g e t h e r w i t h t h e r a t e laws
fl3 I n f o r m u l a e ( 8 ) and ( 9 ) ,
+
(as(Sij)
-
CijkR
and
+
E
= ’bi jRm(N3)SRm
’
I = ’di jRm3(N3)SRm
’
“ E ( E ~ ) ) N=~ B s ( s i j )
-
-
+
BE(E3)
-
(14)
I a r e t h e i n s t a n t a n e o u s i s o t r o p i c e l a s t i c conT aijkk3 due t o domain s w i t c h i n g and a r e t r a n s v e r s e l y
s t a n t s and t h e i n s t a n t a n e o u s d i e l e c t r i c c o n s t a n t o f t h e v i r g i n specimen.
I
g i v e t h e t r a n s i e n t changes t o C . . ijkQ i s o t r o p i c . b . . LI t o g e t h e r w i t h t h e r a t e l a w ( 1 0 ) g i v e t h e t r a n s i e n t changes 1Jk tSk I t o Cijka. + aijTkR3N3 due t o t h e t r a n s i e n t response o f t h e d i p o l e moments, whereas
T d.. p t o g e t h e r w i t h t h e r a t e l a w ( 1 1 ) g i v e t h e t r a n s i e n t change t o aijkR3 i j k 3 tSk t o t h e t r a n s i e n t responses o f t h e d i p o l e moments and domain switching.BbijkR i s o t r o p i c when N3=0, o t h e r w i s e Obi jka and eijk3
due are
jka3 a r e t r a n s v e r s e l y i s o t f o p i c .
a r e t h e t r a n s i e n t p i e z o e l e t t r i c c o n s t z n t s due t o domain s w i t c h i n g , whereas
and j i J.31~t S t o g e t h e r w i t h t h e r a t e laws ( 1 2 ) and ( 1 3 ) g i v e t h e t r a n s i e n t -j due t o t h e t r a n s i e n t responses o f t h e d i p o l e moment and domain changes t o e . ijk3 gijk3 and 8~1 . .j k 3 a r e transversely s w i t c h i n g . A l l t h e t h r e e q u a n t i t i e s eijk3,
gijk3
isotropic.
I n a d d i t i o n , we r e q u i r e t h a t
-
102
P.J. Chen
T h i s c o n d i t i o n ensures t h e c o m p a t i b i l i t y of t h e p i e z o e l e c t r i c c o u p l i n g terms f o r a l l t i m e . h . . g i v e t h e a d d i t i o n a l s t r e s s due t o domain s w i t c h i n g and a r e t r a n s 315
I
v e r s e l y i s o t r o p i c . 6 t o g e t h e r w i t h t h e r a t e l a w ( 1 3 ) g i v e s t h e t r a n s i e n t change I t o E due t o t h e t r a n s i e n t response o f t h e d i p o l e moments o f t h e v i r g i n specimen. E~~~ a r e t h e t r a n s i e n t changes t o EI due t o domain s w i t c h i n g , whereas 6ij3 t o g e t h e r w i t h t h e r a t e l a w (13) g i v e t h e t r a n s i e n t changes t o
due t o t h e t r a n 153 s i e n t responses o f t h e d i p o l e moments and domain s w i t c h i n g . Both E~~~ and 6 . . 153 a r e t r a n s v e r s e l y i s o t r o p i c . F i n a l l y , ki3 g i v e t h e a d d i t i o n a l e l e c t r i c d i s p l a c e E..
ment due t o domain s w i t c h i n g . The r a t e l a w ( 1 4 ) c h a r a c t e r i z i n g t h e consequences o f domain s w i t c h i n g a l s o m e r i t s some comments. I t y i e l d s t h e s p e c i a l case
f o r domain s w i t c h i n g due t o s t r a i n alone, and t h e s p e c i a l case
-
-
f o r domain s w i t c h i n g due t o e l e c t r i c f i e l d a l o n e . F i r s t , domain s w i t c h i n g due t o s t r a i n depends n o n - l i n e a r l y on t h e s t r a i n . Whether t h i s dependence i s t o o general o r n o t i s open t o f u r t h e r i n v e s t i g a t i o n . However, we know t h a t i t depends a t l e a s t on Sii, because t h e i m p o s i t i o n o f h y d r o s t a t i c p r e s s u r e causes domain s w i t c h ing.* I n any case, we e x p e c t t h a t e x t e n s i v e domain s w i t c h i n g due t o s t r a i n o c c u r s o n l y i f t h e s t r a i n i s compressive.3 For compressive s t r a i n , we expect t h a t as and t h a t 16 /a ] i s a m o n o t o f i i c a l -
i s a monotonically increasing f u n c t i o n o f I S . . ] , l y decreasing f u n c t i o n o f I S . .
11
I
such t h a t sgn & / a s = sgn N 3 .
such t h a t ;gn
f i e l d depends o n l y on
expansive s t r a i n ,
3
/a i s p r s t t y much a c o n s t a n t f u n c t i o n o f
we expect t h a t as t e n d s t o z e r o and lSiiI
s sFor
11
2 s
/a = sgn Ng.
Second, domain s w i t c h i n g due t o e l e c t r i c s i t s E3-component. T h i s i s because o f t h e assumptions which
2.
we have adopted. We e x p e c t t h a t aE i s an even f u n c t i o n o f E3 and i n c r e a s e s monot o n i c a l l y w i t h l E 3 1 , and t h a t B / a - i s an odd f u n c t i o n o f E3 such t h a t sgn B /a = sgn
E3
E E
E E
and whose magnitude i n c r e a s e s m o n o t o n i c a l l y w i t h IE31.
There a r e a l s o c e r t a i n r e s t r i c t i o n s which must be imposed c o n c e r n i n g t h e r a t e laws (16) and ( 1 7 ) . Formula ( 1 6 ) h o l d s i f and o n l y i f
l N 3 1 > IBS 5
asI 5
and f o r m u l a ( 1 7 ) h o l d s i f and o n l y i f sgn N3 # sgn RE/aE
- -
or
sgn N3 = sgn 13ElaE and I N 3 \ < IRE/aE\
- -
- -
.
Otherwise, t h e r e i s no domain s w i t c h i n g . I t i s a s i m p l e m a t t e r t o i n f e r t h e i m p l i c a t i o n s o f these conditions concerning the r a t e law (14). The c o n s t i t u t i v e r e l a t i o n s ( 8 ) and ( 9 ) a r e a l s o g e n e r a l i z a t i o n s o f thos; which I f we appear i n t h e open l i t e r a t u r e c o n c e r n i n g p o l e d f e r r o e l e c t r i c m a t e r i a l s . suppress t h e e x p l i c i t dependences on N3 and assume t h a t t h e t i m e s c a l e i s such
A Theory of Dynamical Ferroelectricity
103
t h a t a l l t h e t r a n s i e n t processes have reached e q u i l i b r i u m , t h e n ( 8 ) and ( 9 ) may be condensed so t h a t t h e y a r e s i m i l a r t o t h o s e i n t h e open l i t e r a t u r e e x c e p t f o r t h e These terms account f o r t h e a d d i t i o n a l s t r e s s and e l e c t r i c terms h3ijN3 and ki3N3. displacement due t o domain s w i t c h i n g . ACKNOWLEDGMENT T h i s work was supported by t h e U.S. Department o f Energy under c o n t r a c t No. OE-AC04-76-DP00789 t o Sandia N a t i o n a l L a b o r a t o r i e s , a Department o f Energy f a c i l i t y FOOTNOTES 1. 2. 3.
4.
The r e s u l t s g i v e n h e r e a r e due t o Chen [l]. See, e.g., J a f f e , Cook and J a f f e 121, p. 180. Thus f a r we have n o t seen e x p e r i m e n t a l evidence o f e x t e n s i v e domain s w i t c h i n g due t o expansive s t r a i n . See, e.g., T i e r s t e n [3].
REFERENCES
[ l ] Chen, P . J., Three dimensional dynamic e l e c t r o m e c h a n i c a l c o n s t i t u t i v e r e l a t i o n s f o r f e r r o e l e c t r i c m a t e r i a l s , I n t . J. S o l i d s S t r u c t . 16, (1980) 1059-1067. [2] J a f f e , B., Cook, W . R. J r . , and J a f f e , H., P i e z o e l e c t r i c ceramics (Academic p r e s s , London-New York 1971 ) [3] T i e r s t e n , H.F., L i n e a r p i e z o e l e c t r i c p l a t e v i b r a t i o n s (Plenum Press, New York 1969).
.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
105
SOLITARY WAVE SOLUTIONS OF GENERALIZED KORTEWEG-DE VKIES EQUATIONS F. A. Howes
Department of Mathematics U n i v e r s i t y of C a l i f o r n i a a t Davis Davis, CA 95616 U.S.A.
We b e g i n w i t h a d i s c u s s i o n of t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of t h e s c a l a r Korteweg-deVries (KdV) and modified Korteweg-deVries e q u a t i o n s , i n c l u d i n g t h e d e r i v a t i o n of formulas f o r wave amplitude and speed when t h e waves move a g a i n s t a nonzero "background". This i s followed by a c o n s i d e r a t i o n of r e l a t e d q u e s t i o n s f o r system a n a l o g s of (KdV). Our approach i n v o l v e s t h e c o n s i s t e n t u s e of t h e asymptotic t h e o r y of second-order s i n g u l a r l y p e r t u r b e d boundary v a l u e problems, and so i t o f f e r s a convenient a l t e r n a t i v e t o more t r a d i t i o n a l a n a l y t i c a l and numerical methods.
SOLITARY WAVE SOLUTIONS OF THE KDV AND MODlFIED KDV EQUATIONS e q u a t i o n i n t h e form
Let us c o n s i d e r f i r s t t h e Korteweg-deVries
- + -2uux + E u = xxx Ut
0,
e
i s a s m a l l p o s i t i v e parameter and s u b s c r i p t s d e n o t e p a r t i a l d i f f e r e n -
tiation.
I n o r d e r t o s t u d y s o l u t i o n s of (KdV) which r e p r e s e n t permanent waves
where
moving t o t h e r i g h t w i t h speed c
-
u = u ( ~ , e )= u ( x , t , g ) .
2
0, we i n t r o d u c e t h e v a r i a b l e s 5
The new v a r i a b l e s a l l o w us t o w r i t e (KdV)
2
x
-
c t and
as t h e ordinary
d i f f e r e n t i a l equation 2
e u/'/ = ( c u ) '
-
2 (u / 2 ) f ,
' =
d/d5,
and a n i n t e g r a t i o n reduces t h i s t h i r d - o r d e r e q u a t i o n t o t h e second-order one
Here Urn i s t h e c o n s t a n t l i m i t i n g s t a t e of l i m u ( ~ ) ( ~ , E= ) 0, f o r k
IF1 (KdV) -4"
equation, N
variables
5
2
1.
u6,e)
l i m u(f,e)
t h a t is,
=
Um and
151 - b r n
(Note t h a t i f w e had s t a r t e d w i t h t h e unscaled
xt + uux + uxxx = 0,
z g(x-ct),
u,
w e would a l s o o b t a i n (1.1) by d e f i n i n g t h e
= T(x,t,g),
and proceeding a s above.)
Thus t h e s t u d y of permanent wave s o l u t i o n s of t h e Korteweg-deVries duces t o t h e s t u d y of t h e asymptotic b e h a v i o r ( a s s i n g u l a r l y p e r t u r b e d boundary v a l u e problem
g
-+
+
equation re-
0 ) of s o l u t i o n s of t h e
106
F.A. Howes
where
'p
I f cpu(Um) > 0 t h e n Urn is ob-
i s a smooth f u n c t i o n s a t i s f y i n g cp(U m ) = 0.
-
v i o u s l y a maximum p o i n t of t h e p o t e n t i a l energy f u n c t i o n a l @ ( u ) = mathematical t h e o r y of (1.2)
J:: based on phase-plane
(O'Malley (1976)),
cp(s)ds.
t e l l s us t h a t i f t h e r e i s a f i n i t e U-value u* such t h a t @(u+c)= ip(U ) m ’p(u*)
# 0, t h e n t h e problem (1.2) h a s a s o l u t i o n u
= u ( ~ , E as )
E -t
The
m
O+
except t h a t
analysis, =
0 and
satisfying
(1.3) lim
u(O,E)
+
=
u*.
E - t o
This s o l u t i o n c l e a r l y r e p r e s e n t s a s o l i t a r y wave of amplitude )u;k-U,I; wave of e l e v a t i o n ( d e p r e s s i o n ) i f u*
> Urn (u" < U,).
i t is a
We n o t e i n p a s s i n g t h a t i n
addition t o the solution satisfying (1.3), there are solutions u =
of (1.2)
w i t h "spikes" a t r e g u l a r l y spaced p o i n t s i n any f i n i t e i n t e r v a l (-L,L)
C (-m,m),
that is, lim c
-+ o+
G(5,s)
=
u,,
5
for
i n (-L,L),
except t h a t l i m +;(Tk,c) s + o
,
n=2,3,...
k=l,Z,...,n-l
= u",
f o r [k
(O'Malley (1976)).
=
L(Zk/n-l),
These f u n c t i o n s correspond t o p e r i -
odic c n o i d a l wave s o l u t i o n s o f e v o l u t i o n e q u a t i o n s l i k e (KdV),
a s discussed, f o r
example, by Karpman (1975). 2 Returning t o t h e s p e c i f i c problem (1.1) we s e e t h a t f o r ~ ( u )= c(u-Urn)- i ( u 2 -Urn),
- u;
cp(Um) = 0 and y U ( u ) = c
c o n s e q u e n t l y , 'pu(U,)
>
0 only i f Urn
<
c.
The poten-
- U ~s)o] ,@(u*) = 0, t i a l energy f u n c t i o n a l @ ( u ) i s e q u a l t o ~ ( U - U ~ ) ~ { U - U ~ - ~ ( C and f o r u* = 3c (1.3),
-
2U,.
This i m p l i e s t h e e x i s t e n c e of a s o l u t i o n of (1.1) s a t i s f y i n g
t h a t i s , a s o l i t a r y wave s o l u t i o n of t h e (KdV)
e l e v a t i o n of h e i g h t u*
-
e q u a t i o n i n t h e form of an
Urn = 3(c-Um), moving t o t h e r i g h t w i t h speed c > U r n .
It
i s i n s t r u c t i v e t o r e w r i t e t h i s amplitude-speed r e l a t i o n a s c = Urn thus, t h e l a r g e r t h e amplitude a d d i t i o n , (1.4) since
c
(1.4)
of t h e wave, t h e g r e a t e r i t s phase speed.
In
i m p l i e s t h a t t h e s o l i t a r y wave of amplitude -3Um i s s t a t i o n a r y ,
i s t h e n zero.
s t a t e (KdV)
A
+ A13 ;
C l e a r l y t h i s s o l i t a r y wave i s a s o l u t i o n of t h e s t e a d y -
equation u u x
+ E 2uxxx
= 0, which e x i s t s provided Urn
<
0.
I f Urn
2
t h e n t h e only time-independent s o l u t i o n of (KdV) s a t i s f y i n g t h e boundary condit i o n s i s t h e uniform s t a t e u
=
Urn.
0
107
Generalised Korteweg-De Vries Equations
A l l of t h e s e r e s u l t s a r e w e l l known; s e e , f o r example, Zabusky (1967) o r Karpman (1975) f o r t h e more t r a d i t i o n a l (but e q u i v a l e n t ) phase-plane a n a l y s i s .
The
novelty of our approach, we t h i n k , l i e s i n t h e e a s e with which r e s u l t s such a s (1.4) a r e obtained from t h e equation (1.1). We c l o s e t h i s s e c t i o n w i t h a d i s c u s s i o n of t h e modified Korteweg-deVries equation i n t h e form
u t The same v a r i a b l e s
7
-2+ u u
X
f e
2u
=
o.
(mKd V)
XXX
and u = u ( 5 , c ) allow us t o replace (mKdV) with t h e second-
o r d e r equation u" = C(U-Um)
-
4 ( u 3 - Urn)
cp(u),
(1.5)
2 t o which we apply t h e theory f o r (1.2)0 C l e a r l y 'p(Urn) = 0 and s i n c e cp (u) = c-u , Ll 2 we must r e q u i r e Um < c , i n order t h a t 'p (Urn) > 0. The corresponding energy func-
tional i s
1 2 2 @ ( u ) = E ( u - U ~ ) { (u-Um)
2 + 4Um(u-Um)+ 6(Um-c)3
.
(1.6)
Since t h e expression i n braces i s a q u a d r a t i c polynomial with p o s i t i v e discrimina n t , we s e e t h a t (mKdV) has s o l i t a r y wave s o l u t i o n s of e l e v a t i o n and depression 2 The amplitude A of a s o l i t a r y wave of moving t o t h e r i g h t with speed c > U r n . 2 e l e v a t i o n (compressive wave) i s u* Urn, where @(u*) = 0, and so Ac 4UrnAc 2 6(Um-c) = 0 from (1.6). Consequently t h e amplitude-speed r e l a t i o n f o r t h e com-
-
+
+
p r e s s i v e s o l i t a r y wave i s c = U,
2
1 + ;Ac
.
(Ac +&Urn)
On t h e o t h e r hand, t h e amplitude AR of a s o l i t a r y wave of depression ( r a r e f a c t i v e wave) i s Urn
-
u*, where @(u*) = 0, and s o now (1.6) implies t h a t A:
-
+
4UmAR
6(U?-c) = 0, t h a t i s ,
2 c = Urn I n g e n e r a l , t h e amplitudes Ac and A
1 + ;AR(% - 4Um)
R
.
a r e d i f f e r e n t , i f Urn f 0.
2
W e n o t e a l s o t h a t i f c = Urn then cp (U ) = 0, and t h e theory f o r problem (1.2) u m s t a t e d above does not apply. However, i f Urn < 0, then 'p (U,) > 0, and a modifiuu c a t i o n of t h e theory i n O'Malley (1976) ( s e e a l s o Howes (1978))allows us t o study 1 3 t h e problem (1.5) by a g a i n examining t h e f u n c t i o n a l @ ( u ) , equal t o =(u-Um) 2 { (u-U,) 4Um] f o r c = Urn. Thus t h e modified Korteweg-deVries equation has a
+
s o l i t a r y wave s o l u t i o n of e l e v a t i o n A = -4Um moving t o t h e r i g h t with speed 2 c = U2 = A 116. There is no q u a l i t a t i v e d i f f e r e n c e between t h i s wave and a wave m
f o r which 'pu(Um) > 0; however, t h e l a t t e r type of wave does converge t o Urn exponentially as
6
+
O',
while t h e former converges only a l g e b r a i c a l l y .
These r e s u l t s f o r (mKdV) a r e a l s o c l a s s i c a l (Zabusky (1967)); nonetheless, t h e i r
F.A. Howes
108
formulation i n terms of t h e s i n g u l a r l y perturbed problem (1.5) i s h e l p f u l , a s i n 2 t h e c a s e c = Urn. F i n a l l y l e t us mention t h a t t h e mathematical theory f o r the general problem (1.2) a l s o t e l l s u s when an e v o l u t i o n equation does n o t have solit a r y wave s o l u t i o n s . For example, consider another modified (KdV) equation 2- -2u u + e uxxx = 0. A s o l i t a r y wave u = u ( 5 , e ) = i ( x , t , ~ )moving t o the
-u
t
r i g h t w i t h speed c > 0 a g a i n s t a zero background (Urn = 0 ) must be a n o n t r i v i a l 2 3 s o l u t i o n of t h e equation 0 u" = cu + u / 3 c cp(u). However, u = 0 i s t h e only r e a l "U
s o l u t i o n of t h e equation
cp(s)ds
=
0, and so ( r e a l ) s o l i t a r y waves do not e x i s t .
JO
SOLITAHY WAVE SOLUTIONS OF SOME VECTOR ANALOGS OF THE KDV EQUATION
Several problems of c u r r e n t i n t e r e s t i n geophysics and p l a n e t a r y physics involve t h e e x i s t e n c e of permanent wave s o l u t i o n s of coupled d i s p e r s i v e equations l i k e
which govern the dynamics of s t r a t i f i e d f l u i d s (Benney (1966), Redekopp and Weidman (1978)).
Here
and
(y
p a r e assumed t o b e p o s i t i v e constants.
In order
t o f i n d s o l u t i o n s o f t h e system (2.1) which r e p r e s e n t permanent waves moving t o t h e r i g h t with speed c
5
-
2
0, we proceed a s i n t h e l a s t s e c t i o n and d e f i n e t h e v a r i -
-
-
These u l ( x , t , e ) and u2 = u 2 ( 5 , e ) E u 2 ( x , t , e ) . new v a r i a b l e s enable us t o r e p l a c e (2.1) with t h e system o f ordinary d i f f e r e n t i a l ables
x
c t , u1 = u1(5,e)
equations 2
-
IN
E u1 = (cu,)'
2
e u; = (cu,)'
for
' =
d/d{,
2
,
2
,
(U1/2+au2Ul)' (u2i2+pu1u2)'
and an i n t e g r a t i o n reduces t h e s e t h i r d - o r d e r equations t o t h e second-
order system e 2 u"1 = cul e 2u" = cu2
-
(u,.2 /2+W2U1) 2
(U2/2+pU1U2)
I
(2.2)
.
The constants of i n t e g r a t i o n a r e taken t o be zero, and s o we seek s o l u t i o n s of (2.2)
whose constant l i m i t i n g s t a t e i s (O,O),
(0,O)
for k
2
that is,
l i m (up),uF))(5,e)
=
151+m
0.
The problem (2.2) i s a s p e c i a l c a s e of t h e more g e n e r a l problem
where
u N
=
2=
(u,,u,)
((p
,(p ) i s a smooth vector-valued f u n c t i o n of t h e v e c t o r - v a r i a b l e 1 2 I n a n attempt t o mimic t h e s c a l a r theory f o r satisfying =
z(2) 2.
109
Generalised KortewegDe Vries Equations
problem (1.2), we now ask (following the theory of O'Donnell (1983)) t h a t , i n a d d i t i o n , t h e zero equilibrium s t a t e s a t i s f y t h e equation cp(u) = -
Y
2 in
t h e compon-
entwise sense t h a t CP 1( 0 , ~ ~ = )0
f o r a l l ul,
(2.4)
v2(u1,0) = 0,
and
u2 i n t h e region of i n t e r e s t .
The analog of t h e c o n d i t i o n t h a t
'P (0) > 0 i s then t h e condition t h a t ( 0 , ~ ~ > )0
,U1
cp
and
2,u2
(ul,O)
>
(2.5)
0,
These two f a i r l y r e s t r i c t i v e c o n d i t i o n s e s s e n t i a l l y allow
f o r a l l such u1,u2.
US
t o decouple t h e system (2.3) and apply t h e s c a l a r theory f o r problem ( 1 . 2 ) componentwise.
In o r d e r t o determine t h e e x i s t e n c e of s o l i t a r y wave s o l u t i o n s of
( 2 . 3 ) and t h e i r amplitudes, we d e f i n e t h e two p o t e n t i a l energy f u n c t i o n a l s @l(u1,u2)
=
-r
u1 (P1(s.u 2 Ids "0
and
u2
-Jo
m2(u1,u2) It follows from (2.4) and (2.5)
that u = N
(P2(U1,S)ds-
2 is
a maximum point of @
i'
and depending
on t h e n a t u r e of t h i s maximum, w e d i s t i n g u i s h two c a s e s . CASE I
S o l i t a r y Waves i n Each Component Centered a t D i f f e r e n t Points.
In t h i s
c a s e t h e s o l i t a r y waves i n each component evolve independently of each o t h e r (asymptotically, a s E @l(uJ,O) = @,(O,O)
-t
+
0 ).
I f t h e r e a r e f i n i t e values up, u f ~such t h a t 2 = 0 and cp 1(u-X,O) 1 f 0 , cp2(o,u;) f 0,
= 0, @ 2 (O,u*) 2 = @2( 0 , O )
then t h e theory (Howes (1983)) t e l l s u s t h a t t h e problem (2.3) has a s o l u t i o n (u1,u2) = (u1,u2)(5,e) a s
6
+ O+ s a t i s f y i n g
except t h a t
a t points
ti
CASE I1
S o l i t a r y Waves i n Each Component Centered a t t h e Same Point.
in
(-,a),
t1 # t2’ In t h i s
case we must t a k e i n t o account t h e coupling between components, s i n c e t h e waves " i n t e r a c t " through a term l i k e u1u2 i n equation (2.2).
uy, uy such (pl(u);*,uy)
#
I f there a r e f i n i t e values
t h a t @ (u**,u**) = @ ( 0 , O ) = 0, @ 2 ( u T * , u y ) = @,(O,O) = 0 and 1 1 2 1 0 , (p2(u;*,uy) # 0 , then t h e theory (Howes (1983),(1984)) t e l l s us
t h a t t h e problem (2.3) has a s o l u t i o n (u1,u2) = (u1,u2)(5,g) a s lim
s+o+ except t h a t
u.(<,~= ) 0,
=
--
<5 <
my
g
+
0'
satisfying
F.A. Howes
110
We note t h a t , i n g e n e r a l , t h e wave amplitudes u* and i t h e coupling of t h e c m p o n e n t s i n t h e c a s e of u’?.
uy
a r e d i f f e r e n t , owing t o
Let u s i l l u s t r a t e t h e s e r e s u l t s by looking f o r s o l i t a r y wave s o l u t i o n s of t h e s y s tem (2.1)
moving t o t h e r i g h t w i t h speed c = 1, t h a t i s , we c o n s i d e r t h e problem
I n any s u b i n t e r v a l of (-L,L) where respectively u l ) i s a s y m p t o t i c a l l y z e r o , t h e 2 ul-equation ( r e s p e c t i v e l y u2-equation) reduces t o t h e uncoupled e q u a t i o n E w“ = 2 w - w / 2 , which i s a s p e c i a l c a s e of e q u a t i o n (1.1). Consequently, t h e r e a r e s o l u t i o n s of ( 2 . 6 ) a s
6
+ ’0
r e p r e s e n t i n g two-component s o l i t a r y waves of ampli-
t u d e 3, moving t o t h e r i g h t w i t h u n i t speed, and l o c a l i z e d a t d i f f e r e n t <-values.
I n o t h e r words, we have s o l u t i o n s such t h a t
and lim
+
u2(t2,c) = 3,
e + o
t1 # 5 2 0
for locations
On t h e o t h e r hand, t h e r e a l s o e x i s t s o l i t a r y wave s o l u -
t i o n s , moving t o t h e r i g h t w i t h u n i t speed, which a r e l o c a l i z e d a t t h e same {-value,
say
5
= 0, w i t h t h e amplitude of t h e f i r s t component equal t o 3(301-1) t
(9018-1) and t h a t of t h e second component equal t o 3 ( 3 p - l ) / ( 9 ~ p - l ) , provided @
f 1/9.
--6
‘ 4 3 -ul
To s e e t h i s , n o t e t h a t t h e p a i r of a l g e b r a i c e q u a t i o n s @1(u1,u2) =
- 3m21
= 0,
rn2(Ul,U2) =
-6‘-u;[3
-
-u2 3pu ] = 0 h a s t h e n o n t r i v i a l 1
5,
5
p f and Olp # s o l u t i o n u** = 3(301-1)/(9&-1), u’;” = 3 ( 3 @ - 1 ) / ( 9 & - 1 ) , i f ci # 1 I n p a r t i c u l a r , f o r t h e symmetric c a s e cy = p (f t h e r e i s a s o l u t i o n of ( 2 . 6 )
5)
s q t i s f y i n g f o r i =1 , 2
and
Thus t h e coupling term m1u2 acts t o d e c r e a s e t h e amplitude of t h e wave i n each component from 3 t o 3/(3&1).
1 T.
111
Generalised Korteweg-De Vries Equations
CONCLUDING REMARKS
The reasoning employed i n t h e l a s t s e c t i o n a p p l i e s m u t a t i s mutandis t o t h e g e n e r a l p e r t u r b e d second-order boundary v a l u e problem
(3.1)
where
u, N
(3.1)
v,
and
N
U
a r e N-vectors, a s w e l l a s t o nonautonomous v e r s i o n s of
N
(Howes (1983), O'Donnell (1983)).
Korteweg-deVries
Thus we a r e a b l e t o t r e a t a g e n e r a l i z e d
system l i k e
and even a system w i t h s o u r c e o r r e a c t i o n terms such a s
ACKNaWLEDGMENTS The a u t h o r thanks U.C.
Davis f o r p r o v i d i n g a t r a v e l g r a n t which enabled h i s p a r t i -
c i p a t i o n i n t h i s t i m e l y and i n t e r e s t i n g meeting.
He a l s o thanks I d a Mae Zalac f o r
doing a s p l e n d i d j o b of typing. REFERENCES Benney, D. J., Long n o n - l i n e a r waves i n f l u i d f l o w s , J. Math. and Phys. 45 (1966) 52-62. Howes, F. A., B o u n d a r y - i n t e r i o r l a y e r i n t e r a c t i o n s i n n o n l i n e a r s i n g u l a r p e r t u r b a t i o n t h e o r y , Memoirs Amer. Math. SOC. no. 203 (1978). Howes, F. A , , Nonlinear d i s p e r s i v e systems: t h e o r y and examples, S t u d i e s i n Appl. Math. 69 (1983) 75-97. Howes, F. A., Asymptotic s t r u c t u r e s i n multicomponent f i e l d s , SIAM J. Math. Anal. (1984). Karpman, V. 1975).
I., Non-Linear Waves i n D i s p e r s i v e Media (Pergamon P r e s s , Oxford,
O'Donnell, M. A., Boundary and c o r n e r l a y e r b e h a v i o r i n s i n g u l a r l y p e r t u r b e d s e m i l i n e a r systems of boundary v a l u e problems, SIAM J. Math. Anal. (1983). O'Malley, R. E., Jr., Phase-plane s o l u t i o n s t o some s i n g u l a r p e r t u r b a t i o n problems, J. Math. Anal. Appl. 5 4 (1976) 449-466. Redekopp, L. G. and Weidman, P. D., S o l i t a r y Rossby waves i n z o n a l s h e a r flows and t h e i r i n t e r a c t i o n s , J. Atmospheric S c i . 35 (1978) 790-804. Zabusky, N. J., A s y n e r g e t i c approach t o problems of n o n l i n e a r d i s p e r s i v e wave p r o p a g a t i o n , i n : Ames, W. F. (ed.), Nonlinear P a r t i a l D i f f e r e n t i a l Equat i o n s (Academic P r e s s , New York, 1967).
This Page Intentionally Left Blank
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
113
GEOMETRICAL CRYSTAL ACOUSTICS R.S.D.
Thomas and H. Cohen
Departments of Applied Mathematics and Civil Engineering University of Manitoba Winnipeg, Manitoba Canada
Acceleration waves in an inhomogeneous and anisotropic two-dimensional elastic solid are studied using compatibility relations generalized from those of T. Y. Thomas (Plastic flow and fracture in solids, 1961). Ray directions and an integral expression for decay are obtained in the case of differentiable inhomogeneity of density and elastic coefficients and anisotropy. Decay is further given in the homogeneous case as an algebraic function of time.
1.
INTRODUCTION
Crystal acoustics is the name given by M.J.P. Musgrave, in his 1970 book of the same name [ll, to the study of the propagation of waves in elastic anisotropic solids. We use the term here in a somewhat broader sense to include inhomogeneity as well as anisotropy. The title is intended to reflect the analogy to the Friedlander treatise on geometrical acoustics 1 2 1 , which in particular studies such problems as wave fronts, rays, caustic formation, etc. Such topics are of current special interest t o solid state physicists who study heat waves in supercooled crystals C31. T.Y. Thomas, in his 1961 book C41, derived a formula for the decay at the front of a first-order or shock wave in a linear isotropic and homogeneous elastic
solid. Thomas’s result was generalized to the inhomogeneous case by Karal and Keller C5l and Keller C61.
Further generalizations for the case of acceleration
waves in nonlinear anisotropic and inhomogeneous elastic media were provided by Wright C71 and Bowen and Wang C81. The latter study was restricted to principal waves in orthotropic media. The analyses of C41 and L81 relied crucially on the use of the compatibility equations which govern the discontinuities that may exist at the wave front. Such compatibility relations are elaborated in the book by Thomas C41 and the article of Truesdell and Toupin 191.
They have recently been generalized by
Cohen and Wang [lo1 and Cohen and Thomas 1111 to include rays that are not orthogonal to the wave front, an extension which is necessary for application to anisotropic media.
114
R.S.D. Thomas and H. Cohen
I n this paper we study the problem of an acceleration wave in an inhomogeneous
and anisotropic two-dimensional elastic solid employing the compatibility relations of c111.
In particular we derive a formula, equation (3.15), for the
decay of such waves.
Its reduction to equation ( 4 . 3 ) provides simple general-
izations of the earlier shockwave results in the cases of isotropy and homogeneity.
I n the anisotropic and homogeneous case we obtain a result, equation
( 5 . 1 1 ) , generalizing a formula given by Cohen and Suh 1121 for acceleration waves
in the isotropic homogeneous case. In section two we recall from 1111 the derivatives and generalized compatibility In section three an integral
relations as they are needed in what follows.
expression for the decay of an acceleration wave is obtained without the assumptions of isotropy or homogeneity; only differentiable inhomogeneity of density and elastic coefficients is required. Section f o u r contains a couple of simple examples, one without and one with the additional specialization of isotropy.
Section five treats the other specialization, homogeneity, for full
seven-constant anisotropy, giving an algebraic expression for the decay of an acceleration wave over a time domain, the limit of which depends ray by ray on the direction o f propagation. 2.
DERIVATIVES AND COMPATIBILITY RELATIONS in a region Po
Let the smooth wave curve C(s,t)
co-ordinatized by Cartesians, be parametrized by is time and
s
is arc length when
, such that
t
S(s,t)
parametrized by
the surface S
,
xa(s,t)
and
t.
t = Of
, C(s,t)
field continuous in P
+
,
E3 , with
of
x
3
defines a surface
Along the moving curve C
or across
there hold compatibility relations in the sense of T.Y. Thoaas
C41 in discontinuities in the values of a tensor field on
T (so,to)
E2 ’ 1,2 , where t
=
Within a region P
t = 0.
Po = P n {(x1,x2,t):
called
of euclidean space
xa(s,t) , a
except across the singular surface
-
T (so,to) , denote the limits of
T
P. Let T be a tensor S.
Let
as its argument approaches
S(so,to> respectively from directions with t > to , t < to , and on the corresponding side of S; we assume these limits to exist, as their difference T+ - T-
, denoted
[TI
,
is the jump that is the wave.
Square brackets in
formulas will be reserved for this use throughout this section. Let w = (n ,n ) T be the unit normal to C in P 1 2 0 be unit tangent. Directional derivatives dT/dv and
T =
(tl,t2)T
dTfdr
,
=
(n2,-nl)T
defined for fixed
time, and the total derivative
LDtz = a atr + v dw d T + e dT dT along a curve in
S with tangent vector a
(2.1) =
~ ( n ~ , n ~ , O ) ~ + e ( n ~ , - n ~ ,(0,0,1) O)~+T
115
Geometrical Crystal Acoustics
will be required. x (s,t)
The co-ordinate
, let c(s,t)
v
is the speed of propagation of
be the continuous curvature of
orthogonal trajectory of the neighbouring curves
,
C
C(s,t) and
At
C.
, K be the
k(s,t)
be the
curvature of K. We recall the Frenet formulas as we shall use them, d7fd.r = CV , dv/dT
=
-
, dT/dT
=
- kv
dV/dv
=
kr
CT
, ,
v
and cite from 1111 total derivatives of
(2.2)
and
T
,
, D.r/dt = (ce+dv/dT)v
Dv/Dt = -(ce+dv/dT)T
,
(2.3)
and the relationship between the mixed curvilinear directional derivatives, dLT _ _ _ _ _ dLT =
dTdV
k dT -
dT dr
dv
dvdr
.
And we note that higher directional derivatives are not merely iterated but have expressions
The quantity fundamental to our considerations is the field on [DiT/Dvil , denoted
Ti , i
=
1,2,3.
of jumps
S
We shall represent partial derivatives in
the comma-subscript notation, where for example 3 , : a~ T/at ~ axb ~ axa
T
, a,b
=
1,2
.
The necessary compatibility relations, as derived in [111 from Hadamard’s lemma, are as follows when
[TI
and
T1
and all of their derivatives are set to zero
and the summation convention is used for repeated subscripts a,b,c, = 1 , 2 : Second Order:
,abl
IT
=
T2 nanb ,
=
- VT
na , [T,ttI = v
2
T2
.
(2.6)
Third Order:
+
= dT /dT (tanbnc natbnc + nanbtc)
2
-
cT (t t n + t n t + n t t ) f T3 nanbnc , 2 a b c a b c a b c
= cvT2 tatb
-
(2.7)
d(vT2)/d-r (tanb+natb) (2.8)
+
DT2 (--e
Dt
dT2 --vT dT
3
)n n a b ’
Fr0m’(2.6)~, we see how an acceleration wave will appear when displacement vector u.
T
is a
116
3.
R.S.D. Thomas and H. Cohen
PROPAGATION AND DECAY EQUATIONS
The equation of motion to be considered is of the form ’ab
’,ab
+
u
’ab,a
,b
=
Pu,tt
are matrix functions, and the scalar field
is the density of the medium. v2u 2 when are concerned with the propagation and decay of the’quantity [ul
5
cu
I
,a
E cu
1
,t
0 , which iinplies u
5
p
1
=
0
We
.
By taking jumps in (3.1) and applying the compatibility relations ( 2 . 6 ) we obtain 2 (n,nbAab-pv I)u2 = 0 , where I is the an eigenvalue problem for non-zero u
2. '
n n A Speeds of propagation vm , m = 1,2 a b ab’ (no summation convention on m) are then obtainable from the characteristic
two-by-two identity. Let
, the
equation for pv2 h
2
=
r
=
discriminant of which has the expression 2
T
C V (All-A2,)~I
+
T 2 IV (A12+A21)~I
.
(3.3)
We are concerned only with the case of distinct positive roots for which Let
h
denote the positive square root of 2pv
2
T
= v
(All+A22)v
(3.4)
The roots being distinct, the wave curves Cm(s,t) each with its own curvature
c m
will be distinct for
and orthogonal trajectories .,K
two-sided eigenvectors of the symmetric is a scalar and
> 0.
h2 ; then
.
f h
h
2
r
will be denoted
jm
t # 0
,
Corresponding
ern
, where jm
a unit vector.
E
m
To find a decay equation for the jump magnitudes jm , we differentiate equation (3.1) partially with respect to relations (2.6)2,
(C
dvm d-r
dv Dv m m C-30~ + 3 w v m I)jm 1 m m Dt ~
djmErn 2 +(3epv I - v 0 - er) _ _ m m d-r - ( r - pv 2I)u(m) = o , m m 3 where we have let 0
=
take jumps, and apply compatibility
(2.8), and (2.10), obtaining
v A--@-v
m m
,
t
(t n + n t )Aab
a b
a b
DjmEm + ( r - ~ P 2mVI) D t
ern (3.5)
117
Geometrical Crystal Acoustics
and
u(~) 3
u3
denote the value of
associated with
the product derivatives, multiplying through by
j E Expanding some of m m , which satisfies
E
m de /d-r = E~ DE /Dt = 0 , eliminates several terms of (3.5) but not that in m m m m djm/d-r , which makes (3.5) a partial differential equation. This term is
E~
eliminated by choosing e 2em pvm
e m
=
= ET
m
o
to satisfy Em
.
(3.7)
In 1111, it is shown that the integral curve of the velocity vector
a
so
determined is a bicharacteristic curve 1131. A s well as this elimination, this value of
e
the term in
allows one simplification of the coefficient of E
T
m
jm E~
and also of
0 dE /dT as follows. m
If (3.7) is differentiated in the direction
T
,
then the symmetry of
allows
@
the result to be written dv 2-pv dem
m
d-r
+2e
*v +2e m dT m m
The Frenet formulas (2.2) and (3.6)
1
T
d Em
P ~ = Z E O - + E
md-r
dT
(3.8)
md-rm
allow dO/d.r to be found, and (3.6)2 allows
simplification to
Progress
2cm ( r - A)
+
2nln2 Z(iill d - A 22) + (n22 - nl) 2 %(n12 d
dr
=
so
far allows (3.5) to be written
+A,,)
(3.9)
+
T d vm j mE m In 1n2 dT (All
Addition and subtraction of
- AZ2)
-
1 2 2 7 (n2 - nl)
n1n2d(A12+A21)/dv
d
(Al2
+ AZ1)
-C
~
.E
~
to the matrix factor of (3.9)
allows it to be resolved into (3.10)
However, the Frenet formulas ( 2 . 2 ) applied to the orthogonal trajectories Km
of
Cm
allow
dI /dv
-: :_- n n If the identity symmetry of
r
a b
to be found, and (3.6)1 __ dAab
dv
+
k 0 m
allows simplification to
.
= 0 is differentiated in the direction ET(r-pv2 m m allows the result to be written
(3.11)
v , the
of which the first term vanishes; hence (3.12)
118
R.S. D. Thomas and H. Cohen
by (3.11) and (3.7).
Dividing (3.8) by
and using (3.10) and (3.12)
pvij,
allows (3.9) to be written n
1
D En jLpv’
dv de m = c v - - m + 2(kmem ---) m m m dr dv
m Dt
The skew-symmetry of
-
A12
Application of (2.1) t o dv/dT
causes the last term of (3.13) to vanish. and comparison with (2.3)
T =
(3.13)
.
2
shows that
- kv (3.14) from the second term of (3.13) using (2.1) allows
Using (3.14) to eliminate k, (3.13) to be reduced to
which integrates to our first main result,
(3.15) To be any use, an equation like (3.15) needs calculable quantities.
The
curvature c is a feature of the particular wave-curve geometry, and we shall m return to it in Section 5, but the other quantities are calculable quite generally. By finding the unit eigenvector
E
m
explicitly and direct
substitution into (3.7), one finds that
,
2epv
=
.r’(All+A 22) v
f
=
[vT(All - A22)~IC~T(A11 - A22)v1
f f/h
(3.16)
where
+ I V ~ ( A ~ ~ + A ~ ~ T) (A12+A21)v1 VI~T
(3.17)
,
and the sign chosen on the right side corresponds to that chosen in (3.4) for the v
on the left side.
dem/dr in terms of
c m
The Frenet formulas (2.2) allow the calculation of and the derivatives of the
’ab.
In the remaining sections, wave modes will be treated entirely separately, and so the subscript m
will be dropped.
The sign
t
will continue to be used as
in (3.16). 4.
EXAMPLES
A s a first example, we consider an initial wave curve that is straight proceeding
through a half plane
Po
in which
the wave curve remains parallel;
or
p
c =
k
Aab
= 0.
or both vary in such a way that Let v = (-l,O)T
,r
=
(O,l)T, for a
119
Geometrical Crystal Acoustics
f r o n t b e g i n n i n g on t h e i s such t h a t
e # 0
x2
-
I f w e assume t h a t t h e medium
t = 0.
axis a t t i m e
i n i t i a l l y a l l along t h e f r o n t , then u n t i l
e
=
0
along
e a c h r a y we can u s e t h e i d e n t i t y de/d-r = e d(llne)/d.c
=
D(!Lne)/Dt - v d(llne)/dv
t o r e d u c e e q u a t i o n (3.15) t o ?
I
;
D lln jLepvJ = Dt which i s i n t e g r a b l e because
d Rn e
dxl Dx / D t = v
When e q u a t i o n ( 4 . 1 ) i s
along each ray.
1
integrated,
e # 0
v a l u e of
d e t e r m i n e s t h e o b l i q u e d i r e c t i o n i n which t h e wave moves, i t h a s no
e
c a n c e l s because i t a p p e a r s on b o t h s i d e s .
So, w h i l e t h e
e x p l i c i t e f f e c t o n decay;
1/2 v 2 j = v2(0) j ( 0 ) ~ p ( ~ ) v ( o ) / P v ~ (0)
where
,
(4.2)
i n d i c a t e s i n i t i a l v a l u e s , a c o n v e n t i o n we u s e throughout.
The second
example i s t h e p a s s a g e of a wave curve t h r o u g h an i s o t r o p i c r e g i o n example of one i n which
as a n Po I n t h i s c a s e ( 3 . 1 3 ) i s b e t t e r s i m p l i f i e d by
e :0 .
eliminating the last t e r m a s before, applying ( 2 . 1 ) , involving
e
and e l i m i n a t i n g a l l terms
t o obtain
which i n t e g r a t e s t o
1/2 2 2 v j = v (0) j ( 0 ) {p(O)v(O)/pv) exp
1
c/2dw
,
(4.3)
where t h e i n t e g r a l i s w i t h r e s p e c t t o a r c l e n g t h a l o n g t h e o r t h o g o n a l t r a j e c t o r i e s K
of t h e wave c u r v e s , which a r e t h e r a y s when
e
=
0.
Homogeneity, t h e t h i r d example, w i l l have a s e c t i o n t o i t s e l f .
5.
ANISOTROPIC HOMOGENEITY
W e c o n s i d e r t h e c o n s t a n c y of
speeds
v, the discriminant
and
p
,
h2
hab
of (3.1) i n
Po.
The d i s t i n c t wave
and t h e t a n g e n t i a l component
e
of t h e r a y
v e l o c i t y a r e s t i l l g i v e n r e s p e c t i v e l y by ( 3 . 4 ) , ( 3 . 3 ) , and ( 3 . 7 ) , b u t are a l l now f u n c t i o n s of
v
alone.
S e v e r a l d e r i v a t i v e s w i l l b e r e q u i r e d , a l l of which
c a n be o b t a i n e d by d i r e c t i o n a l d i f f e r e n t i a t i o n and a p p l i c a t i o n of t h e F r e n e t formulas.
From ( 3 . 3 ) u s i n g t h e symmetry of dh/dV = 2 k € / h
From (3.4) u s i n g t h e symmetry of
All
+
=
AZ2
- AZ2
- 2cf/h
and
.
h12
+
AZ1
come
(5.1)
and (5.1) come
dv/d-c = - c e df/d.r = c ( h 2 - k), where
dv/dv = k e S i m i l a r l y from (3.17)
,
, dh/dT
All
.
(5.2)
120
R.S.D. Thomas and H. Cahen
T + C V ~ ( A ~ ~ + A ~ ~ ) V I (.Al2 C T +A21)r1
+
2[rT(A12
+hZl)vI
.
2
Lastly, from ( 3 . 1 6 ) , (5.212, (5.1)2, and ( 3 . 4 ) comes de/d-r = cg ,
15.4)
where 2pvg
=
2
+
(2epv) /2pv2 T
-
T
2
2pv
(5.5)
3 k 2f /h + E/h 2
(All+A22)r
.
The initial consequence that we draw from the above is that the rays are straight lines; substitution of (5.2)
into (2.3)
from which it follows that
so
2 and
r,v,e
which is accordingly straight.
a
1 shows that Dv/bt
=
,
0
are all constant along a ray,
The second consequence presumes non-zero
curvature of the wave curve. LEMMA.
The radius of curvature of a smooth wave curve is a linear function of
time along a ray. Proof:
The proof proceeds by calculation of
Dc/Dt.
Application of the Frenet
formulas (2.2) to themselves gives 2
_d _v_ _ _ _ drdv
2
dk (-++)T
-
dvdr
d?
.
dc dv
Equation ( 2 . 4 ) on the other hand gives the value dc/dv
=
k2
+
c2
(k2+c2)r.
Accordingly,
- dk/dT
(5.6)
Comparison of (3.14) with (5.2)2 shows that
.
ec = vk We note that the existence of a radius of curvature requires c # 0 , since v # 0 , the conditions When
e # 0 , from ( 2 . 1 ) , Dc/Dt
The form of
e
function of nl
, given
=
&/at 2
c (v-g)
=
0 and
=
0 , (5.6), and (5.7),
In the latter case, e :0 =
,
(5.7) that,
0 are equivalent.
=
.
(5.8)
v , is a simple enough
is substituted for n2
has isolated zeros along a curve with straight so that k
k
by (3.16) as a function of
2 +d(l-nl)
if
e
so
as appropriate that it
c # 0 , unless it is identically zero.
the rays are all orthogonal trajectories but still
dk1d-c = 0 and s o from (5.6),
c2 = dc/dv = v-l Dc/Dt In consequence, the radius of curvature l/c holds because g = 0 by ( 5 . 4 ) .
. is linear in
t;
(5.9) in fact (5.8)
121
Geometrical Crystal Acoustics
It remains to consider a ray along which
along which
e
=
0 and which is surrounded by rays
e # 0 , corresponding to an isolated zero of
e
as a function of
Since the curvature of the wave curves along that ray are continuous with
nl. the curvatures along neighbouring rays, they must be governed by the same
linear law (5.8) though with
g
0 as above.
=
Equation (5.8) integrates to give
-- 1 -- 1 c(s,t) c(s,O)
+
(5.10)
’
(g-v)t
and the lemma is proved. This result corresponds to the expected shrinkage in the radius of curvature of a circular wave curve in the simple case with
e
=
g
=
0.
The second main result is easily deduced from the lemma and equations ( 3 . 1 5 ) , (5.4), and
h/Dt
,
= 0
v 2. 3 where
r
=
each ray.
l/c(s,O)
=
and
v 2 (o)j(0)r1 * v
=
v(O),
{r g
In the case of isotropy, g
the expected one cl21.
= g(0) =
t
to
0
.
[ g ( o ) -v(o)lt~-~ ~
(5.11)
since they are constant along
, and the expression (5.11) reduces
co
The negative exponent indicates a limited range of
validity in one time direction whenever The approach of
+
r/(v-g)
caustic, an envelope of rays.
g # v ;
t
must not reach r/(v-g).
indicates that the ray is approaching a The consequent concentration o f rays makes
measurements undependable. The attention of solid-state physicists, who seem to be the persons most likely to be concerned with such measurements, especially in materials of cubic symmetry
(Cll =
c22 ;
C16
=
cZ6
=
0) , has been called
to this formation of caustics elsewhere by the present authors C141.
In the case of a straight wave curve, neither the curvature nor the magnitude of the wave varies with time. or is not zero, c constant by (3.15).
From
c
=
0 , (5.7) gives k
=
0. Then as
is constant by (5.9) or (5.8) respectively, and vj
e
is is
R.S.D. Thomas and H. Cohen
122 REFERENCES
111 Musgrave, M.J.P., Crystal acoustics (Holden-Day, San Francisco, 1970). 121 Friedlander, F.G., Sound pulses (Cambridge University Press, Cambridge 1958).
131 Every, A.G., Ballistic phonons and the shape of the ray surface in cubic crystals, Phys. Rev. B 24 (1981) 3456-3467. C41
Thomas, T.Y., Plastic flow and fracture in solids (Academic Press, New York and London, 1961).
[51
Karal, F.C. and Keller, J.B., Elastic wave propagation in homogeneous and inhomogeneous media, J. Acoustical SOC. Amer. 31 (1959) 694-705.
C61 Keller, H . B . ,
Propagation of stress discontinuities in inhomogeneous
elastic media, SIAM Rev. 6 (1964) 356-382. C71
Wright, T.W., Acceleration waves in simple elastic materials, Arch, Rational Mech. Anal. 50 (1973) 237-277.
C81
Bowen, R.M. and Wang, C.-C.,
Acceleration waves in orthotropic elastic
materials. Arch. Rational Mech. Anal. 47 (1972) 149-170. 191 Truesdell, C. and Toupin, R.A., The classical field theories, in Fliigge, S. (ed.), Handbuch der Physik, V o l . III/l.
(Springer, Berlin-Gattingen-
Heidelberg-New York, 1965). ClOl
Cohen, H. and Wang, C.-C., Compatibility conditions for singular surfaces, Arch. Rational Mech. Anal. 80 (1982) 205-261.
1111 Cohen, H. and Thomas, R.S.D., Plane wave propagation and evolution for quasilinear hyperbolic systems, Utilitas Math. 24 (1983), to appear. El21
Cohen, H. and Suh, S.L., Wave propagation in elastic surfaces, Journ. Math. and Mech. 19 (1970) 1117-1129.
1131 Courant, R. and Hilbert, D., Methods of mathematical physics (Interscience, New York, 1953).
1141 Thomas, R.S.D. and Cohen, H., Caustics in geometrical crystal acoustics, Phys. Rev. B 28 (1983), to appear.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
RIEMA"
123
INVARIANTS
A . M . Grundland
McGill U n i v e r s i t y Department o f Mathematics and S t a t i s t i c s M o n t r e a l , Quebec Canada
T h i s p a p e r p r e s e n t s t h e g e n e r a l i z e d Riemann i n v a r i a n t s method of c o n s t r u c t i n g s o l u t i o n s f o r n o n l i n e a r and n o n e l l i p t i c s y s t e m of P . D . E . ' s . The approach adopted h e r e i s b a s e d on t h e method o f c h a r a c t e r i s t i c s . The main f e a t u r e o f t h i s method i s i n t r o d u c t i o n o f new dependent v a r i a b l e s ( c a l l e d Riemann i n v a r i a n t s ) which remain c o n s t a n t a l o n g a p p r o p r i a t e c h a r a c t e r i s t i c c u r v e s o f t h e b a s i c system o f e q u a t i o n s . T h i s method e n a b l e s us t o r e d u c e t h e number o f dependent v a r i a b l e s f o r some problems, so t h e s o l v i n g o f t h e b a s i c system o f e q u a t i o n s i s s i m p l i f i e d . We describe a procedurefor finding the existence conditions f o r s o l u t i o n s which have a p o s t u l a t e d form and p r e s e n t t h e i r p h y s i c a l interpretation. INTRODUCTION This paper has a methodological character;
i t concerns ways of c o n s t r u c t i n g
c e r t a i n classes of solutions of p a r t i a l d i f f e r e n t i a l equations.
I t c o n t a i n s an
a t t e m p t a t t h e d e s c r i p t i o n and a n a l y s i s [from t h e formal p o i n t o f view) o f t h e phenomena o f p r o p a g a t i o n and n o n l i n e a r s u p e r p o s i t i o n o f waves i n media.
We
assume t h a t t h e problem can be d e s c r i b e d by a n o n e l l i p t i c system of q u a s i l i n e a r
P.D.E.’s o f t h e f i r s t o r d e r as follows
where
s = 1 , ..., m 1-1 = 1,
. . . ,n
j = 1,. . . , L 1
x =
i s t h e number o f e q u a t i o n s , i s t h e number o f independent v a r i a b l e s , i s t h e number o f unknown f u n c t i o n s ( L E E; u = u ( x ) = ( U 1(x) ,...,u a. ( x ) )
,..., xn )
(X
The E u c l i d e a n s p a c e
m),
H.
E clRn - t h e s p a c e o f independent v a r i a b l e s - w i l l b e c a l l e d
a p h y s i c a l s p a c e , and t h e s p a c e hodograph s p a c e .
5 E
H
c R L - t h e s p a c e o f dependent v a r i a b l e s - a
Let us assume t h a t t h e c o e f f i c i e n t s
a ';
and
bS
o f t h e system
(1.1) are smooth f u n c t i o n s of t h e v a r i a b l e u d e f i n e d on an open s e t 0 moreover t h e i n i t i a l c o n d i t i o n s f o r t h e system ( 1 . 1 ) b e smooth, i . e .
c
H .
Let
A.M. Grundland
124
1 2 u(xo,x ,..,xn) = u (x) 0
E
1 n-1 C (R 1
We a r e l o o k i n g f o r s o l u t i o n s d e s c r i b i n g t h e p r o p a g a t i o n and n o n l i n e a r s u p e r p o s i t i o n The approach adopted i s based
of waves t h a t can b e r e a l i s e d i n t h e above s y s t e m s .
on a g e n e r a l i z e d method of c h a r a c t e r i s t i c s which o r i g i n a t e s i n t h e works of B. Riemann ( 1 8 6 9 ) .
H e was t h e f i r s t t o f o r m u l a t e t h e problem o f p r o p a g a t i o n and
s u p e r p o s i t i o n of s i m p l e waves ( l a t e r c a l l e d Riemann w a v e s ) .
Riemann c o n s i d e r e d a
h y p e r b o l i c s y s t e m o f homogeneous e q u a t i o n s w i t h two v a r i a b l e s
( d e p e n d e n t and i n -
dependent) and found t h e s o l u t i o n s c o r r e s p o n d i n g t o t h e s u p e r p o s i t i o n of s i m p l e waves i n t h e c a s e o f p a r a l l e l d i r e c t i o n o f p r o p a g a t i o n .
S i n c e then t h e problemof
Riemann waves and t h e i r s u p e r p o s i t i o n h a s been s t u d i e d by many a u t h o r s e . g .
R.
Mises ( 1 9 5 8 ) , J . L i g h t h i l l ( 1 9 6 8 ) , G . Whitham ( 1 9 7 4 ) , L e i b o w i t z and S e e b a s s ( 1 9 7 4 ) . I n t h e c u r r e n t l i t e r a t u r e one c a n f i n d many a t t e m p t s a t g e n e r a l i z i n g t h e methodof Riemann i n v a r i a n t s and i t s v a r i o u s a p p l i c a t i o n s ; a r e v i e w o f t h e s e problems can b e found i n t h e p a p e r s of T. T a n i u t i and A. J e f f r e y (19641, A . J e f f r e y ( 1 9 7 6 ) , Rozdestvenski and N .
yanienko ( 1 9 6 8 ) .
I n t h i s f i e l d t h e r e e x i s t s a Polish, theo-
r e t i c a l approach which e x t e n d s t h e method o f Riemann i n v a r i a n t s . w i t h t h e works of J . Bonder (19561, f o l l o w e d by M.
It o r i g i n a t e d
B u r n a t (1969-1974) and w a s
r e c e n t l y developed by 2 . P e r a d z y n s k i (1970-1981) and W. Zajaczkowski (1974-1980)
.
The r e s u l t s o b t a i n e d f o r the homogeneous s y s t e m were s o p r o m i s i n g t h a t i t seemed w o r t h w h i l e t o t r y t o e x t e n d t h e method and check i t s e f f e c t i v e n e s s f o r t h e c a s e of nonhomogeneous P.D.E. s. 1974-1983).
T h i s was done i n t h e a u t h o r ’ s p a p e r s (A.M.
Grundland
The a i m o f t h e p r e s e n t p a p e r i s t o g i v e a b r i e f o u t l i n e o f r e s u l t s
f o r q u a s i l i n e a r system o f P . D . E . ’ s .
The s t a r t i n g p o i n t f o r t h i s r e s e a r c h was an a l g e b r a i z a t i o n ( f o l l o w i n g t h e p a p e r s o f M. Burnat (1966),P e r a d z y n s k i ( 1 9 7 0 ) , A. Grundland ( 1 9 7 4 ) ) o f t h e system of e q u a t i o n s (1.11, which c a n b e w r i t t e n i n t h e form:
,uj E axP
DEFINITION.
A matrix
L3
u
{LJ: a . ” L 1 P l u
=
bs}
.
(1.2)
, s a t i s f y i n g t h e above c o n d i t i o n s a t a p o i n t uo
E
H,
w e s h a l l c a l l an i n t e g r a l e l e m e n t of t h e s y s t e m (1.1).
The m a t r i x
L =
l $l
i s a m a t r i x of t h e t a n g e n t mapping
by t h e f o r m u l a :
E 3 (6~’) The mapping
du(x)
-
( 6 ~ 1 ) E TUH
, where
du(x):
I auj 6u- = -
ax’
d e t e r m i n e s an e l e m e n t of t h e l i n e a r s p a c e
-
E
fixP
+ T
.
H
given
(1.3)
L ( E , T ~ ~ ,~ )which
125
Riernann Invariants
may be i d e n t i f i e d w i t h t h e t e n s o r p r o d u c t
E*
Tuff@E* (where
i s t h e dual t o
E)
Each element o f t h i s t e n s o r p r o d u c t i s a f i n i t e sum o f t h e s i m p l e t e n s o r s , i . e . L = y@X
ff
to
where
h
E*
E
a t the point
DEFINITION.
i s a c o v a r i a n t v e c t o r and
y
E
T
H
i s a vector tangent
u.
I n t e g r a l element
i s c a l l e d a simple element, i f
Lj
u
rankllL~11 = 1 ,
i . e . when t h e c o r r e s p o n d i n g t e n s o r i s t h e s i m p l e one.
L
I n o r d e r t o d e t e r m i n e a s i m p l e i n t e g r a l element
x
E
E*
(
s
=
we s h o u l d seek y
E
T
H
and
satisfying
1,..., m ,
,...,n ,
0
f o r homogeneous s y s t e m ,
j = 1,..., 1
p = 1
The n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r e x i s t e n c e o f a non-zero s o l u t i o n
y
Of
t h e equations (1.4) i s rankl/a?’A
1
< I.,
111
(if m
=
I. t h e n det(a?’X
)=O)
I
f o r homogeneous systems 11.5)
I ’ ,bS/I = r a n k ~ ~ a ~ p X( ipf~ m~ ,= 9. rank//aspA
then
yj = (a;’Au)-lbs)
f o r nonhomogeneous s y s t e m s . The r e l a t i o n s ( 1 . 4 ) and ( 1 . 5 ) are c a l l e d t h e wave r e l a t i o n and t h e d i s p e r s i o n r e l a t i o n ( F i s z d o n and Peradzynski 1976) r e s p e c t i v e l y . If t h e c o v e c t o r
X satisfies the
dispersion r e l a t i o n (1.5) then t h e r e e x i s t s a p o l a r i z a t i o n v e c t o r s a t i s f y i n g t h e wave r e l a t i o n ( 1 . 4 ) . y
and
A.
Hence
exists a vector a c t e r i s t i c vector
H
x
*
E
3
(u,X)
-+
y E TUH Thus t h e r e e x i s t s a r e l a t i o n between d i r e c t i o n s y ( u ,A)
E
T H.
A vector
y
f o r which t h e r e
such t h a t homogeneous system ( 1 . 4 ) h o l d s w i l l b e c a l l e d c h a r -
A
.
Simple e l e m e n t s of t h e s y s t e m ( 1 . 4 ) w i l l b e d e n o t e d by
f o r t h e homogeneous system ( t h a t i s when b ( u ) homogeneous s y s t e m (when
0 ) and by
b ( u ) f 0) r e s p e c t i v e l y .
yo
@
Xo
y@A
f o r t h e non-
Now w e w i l l show t h a t t h e s i m p l e
i n t e g r a l e l e m e n t s can b e used i n c o n s t r u c t i o n o f t h e s o l u t i o n o f t h e system ( 1 . 1 ) .
SIMPLb RIEMiWN WAVES. Let t h e mapping
u: U
+
ff
(U
c
E)
b e a s o l u t i o n o f t h e system ( 1 . 1 ) .
126
A.M. Grundland
DEFINITION.
o f t h e system ( 1 . 1 ) i s c a l l e d a simple wave ( f o r a
u
The s o l u t i o n
homogeneous system) or a s i m p l e s t a t e ( i n t h e c a s e o f a nonhomogeneous system) i f t h e t a n g e n t mapping
du(x)
i s a s i m p l e element a t each p o i n t
x0
E
D.
I n t h a t c a s e it f o l l o w s from t h e rank mapping theorem (Dieudonn6, 1960)
u(D)
that: R
-t-
f(R)
E
c
H
H
r.
i s a one-dimensional submanifold, i . e . some curve
r.
be a p a r a m e t r i z a t i o n o f
Then:
u(x)
=
f(R(x)),
Let
where
a J dfJ aR i s a s c a l a r f u n c t i o n C a l l e d a R i e m a n n i n v a r i a n t . Since -= - a xu dR a x u ’
R(x)
then for
t h e homogeneous system ( 1 . 1 ) t h e t a n g e n t v e c t o r :
i s a characteristic vector. y(R).
Since
lated t o
u(x)
satisfied.
be a c h a r a c t e r i s t i c c o v e c t o r r e l a t e d t o covectors proportional t o
dR(x)
Let u s assume t h a t t h e f o l l o w i n g e q u a l i t y h o l d s :
y(R).
= C(x)X(R(x)).
X(R)
Let
s a t i s f i e s eq. (l.l),
I f ranklla~’yj11
This is a typical
=
n
-
( i . e . g e n e r i c ) c a s e o f t h e system when we have more (m > n).
Now we show t h a t t h e r e e x i s t s a f u n c t i o n
ip
following equation holds:
R(x)
Indeed,
=
t h e above assumption i s a u t o m a t i c a l l y
1
e q u a t i o n s t h a n t h e independent v a r i a b l e s
o f a s i m p l e wave.
dR(x)
are re-
=
ip(O(x)),
of one v a r i a b l e such t h a t t h e
where
dO(x) = (1 + SAli
o(x)
x’)Avdxv,
=
Xu(R(x))xu thus
dO
A
i s t h e phase
so
dR = 0 ,
K R(x) = cp(O(x))
i n t h e neighbourhood o f t h e r e g u l a r p o i n t s where
dO(x)
doesn‘t
vanish. Let u s c o n s i d e r now a smooth c u r v e r : u = f ( R ) i n t h e hodograph s p a c e H w i t h df . S i n c e from (1.4) it f o l l o w s t h a t ranklla~’yj11 = q y(R) =
the tangent vector
k = 9. - q
then t h e r e e x i s t
ponding t o one v e c t o r (1.4) w i t h t h e same THEOREM 1.
y
y).
Let t h e c u r v e
given by O . D . E .
l i n e a r l y independent c o v e c t o r s
(i.e.
@Ai,
i = 1,
...,k
. . . , Xk
corres-
Then t h e f o l l o w i n g theorem h o l d s .
r
of the class
C1
be such t h a t t h e v e c t o r
(2.1) i s a characteristic vector.
independent c o v e c t o r s
A1,
a r e s o l u t i o n s of t h e equations
A A (R) := ( A l (R)
,. . . , A k ( R ) )
y(R)
Let a l s o a system o f l i n e a r l y be related t o the vector
y(R)
by t h e wave r e l a t i o n
(where q u a n t i t i e s
Er
a r e r e g a r d e d as v a r i a b l e s dependent on
x)
and l e t
cp(zA)
127
Riernann invariants
b e an a r b i t r a r y smooth f u n c t i o n of
can b e r e s o l v e d w i t h r e s p e c t t o v a r i a b l e s then the function
u(x)
I f the system
variables.
k
R (i.e. u = u ( x ) ,
and
u
R = R(x))
i s a s o l u t i o n o f t h e homogeneous system ( 1 . 1 ) .
This solution i s called, f o r
k = 1,
a p l a n a r s i m p l e Riemann wave and f o r
a n o n p l a n a r s i m p l e Riemann wave ( Z . Peradzynski 1 9 7 1 ) .
The s c a l a r f u n c t i o n
k z 1 R(x)
i s c a l l e d t h e Riemann i n v a r i a n t . Proof:
we o b t a i n
Differentiating (2.3)2
k
k dfj ulxu(x) = R,
dR
where
k 1 cp, s=1 ns
u(x)
~i =
yj.
s=l
k 1 - E s=l
dXsvxV cp,
* -
ns
’
dR
dXsvxV
dR #
o f t h e system ( 2 . 3 ) . the point
x
1 which i s j u s t t h e c o n d i t i o n o f l o c a l r e s o l v a b i l i t y
Then
U ~ , ~ U ( X )
i s a s i m p l e element o f t h e system ( 2 . 4 ) a t
o f t h e form k
du(x) = y
@
( Z
s=1
,A
su
)dxU
E
TUH
@
E*,
(2.5)
that is
where t h e q u a n t i t i e s
5’
$ 0
are c o n s i d e r e d as v a r i a b l e s depending on
b y v i r t u e o f t h e wave r e l a t i o n ( 2 . 2 ) , a f u n c t i o n o f t h e form
uJ
=
x.
fJ (R(s))
Thus is a
s o l u t i o n of t h e homogeneous e q u a t i o n s ( l . l ) , and t h e P f a f f i a n system ( 2 . 5 ) i s a c o m p l e t e l y i n t e g r a b l e one (Dieudonng, 1 9 6 0 ) .
Q.E.D.
THEOREM 2 . Let S b e a k = {A}-dimensional s u r f a c e i n a p h y s i c a l s p a c e E and 1 1 I, E C ( S , R ) b e a given f u n c t i o n . If t h e t r a n s v e r s a l i t y c o n d i t i o n i s s a t i s f i e d , i . e .
128
A.M. Grundland
//
XA(o ( x ) )
XES
I
a r e l i n e a r l y independent,
(2.6)
TXS
t h e n i n some neighbourhood
U
f o r which
S
3
u
E
1 C (U,H)
e x i s t s , a nonplanar
simple wave ( 2 . 3 ) s a t i s f i e s t h e boundary c o n d i t i o n :
= f(p)
Proof:
In some neighbourhood
xo
E
where
S,
R
(2.7)
i n t h e a p p r o p r i a t e map, t h e r e i s
A S = {xk = 0 , x - a r b i t r a r y }
Iu)
where
If t h e c o n d i t i o n ( 2 . 6 ) i s s a t i s f i e d , t h e n t h e m a t r i x Thus, we can change t h e dependence o f t h e f u n c t i o n
:= {AluIk}
IIAt(R)II AA
on
R
i s not degenerate
.
by l i n e a r t r a n s -
formation i n t h e f o l l o w i n g way A XB(R)
=
A AB
.
Then t h e c o n d i t i o n ( 2 . 7 ) h a s t h e form
P(X because R(.)
xk = 0 .
A
1
1 A k A A ipCAA(R)X ,...,A A(R)x 1 = cp(x 1 ,
=
Hence, i n such a chosen map we have
rp = p ,
so t h e f u n c t i o n
i s given i n t h e i m p l i c i t form A k R(x) = R(x , x )
In a p a r t i c u l a r c a s e when
= p(X
A
Q.E.D.
u (R(x))x’).
k = {A} = 1 t h e boundary c o n d i t i o n ( 2 . 7 ) can be g i v e n
on c e r t a i n c u r v e s . The s u r f a c e s o f constant R (i,e. t h e l e v e l o f t h e f u n c t i o n
REMARK.
a r e assumed t o be k-dimensional h y p e r p l a n e s .
The f u n c t i o n
hyperplane o f dimension (n-k) which i s a t t a c h e d a t t h e p o i n t forms xo
E
AA(p(xo)). S,
If t h e point
then t h e function
R
x
p(x,) xo
R(.))
determines t h e and d e f i n e d by
l i e s on s u c h a h y p e r p l a n e , p a s s i n g t h r o u g h
must be given b y
The t r a n s v e r s a l i t y c o n d i t i o n means t h a t t h e f a m i l y of k-dimensional h y p e r p l a n e s covers a c e r t a i n neighbourhood o f s u r f a c e s solution
R(.)
S.
Hence, i n t h i s neighbourhood, t h e
i s u n i q u e l y determined by t h e boundary c o n d i t i o n
RIS = p .
129
Riemann Invariants
Let us s t u d y now a p l a n a r simple Riemann wave (when k = 1 i n t h e e q u a t i o n ( 2 . 3 ) ) .
fT
Changing t h e dependent v a r i a b l e s
xu ( E ) : =
Au(R),
i n t h e e q u a t i o n ( 2 . 3 ) and d e n o t i n g
= (p-loR
we o b t a i n R(x)
=
XU(R(x))xu = x v ( k ( x ) ) x u .
So, by v i r t u e o f t h e e q u a t i o n ( 2 . 3 ) , we have
and
Upon performing t r a n s f o r m a t i o n s
the relation ( 2 . 3 ) for
k = 1 t a k e s t h e form u(x) = r ( k ( x ) ) ,
dI‘
- = v(k),
where
(2.9)
dR
Each s o l u t i o n ( 2 . 3 ) o f t h e system (1.1) can b e p r e s e n t e d i n t h e e q u i v a l e n t form (2.9).
The form ( 2 . 9 ) a p p e a r s t o be s i m p l e r from t h e p o i n t o f view o f c a l c u l a t i o n s .
There a r e two ways o f u s i n g Theorem 1. Method 1.
One can f i n d a v e c t o r f i e l d
ranklla?’yy311 < n .
Then one f i n d s
y(u)
u = f(R)
E
T,H
which i s c h a r a c t e r i s t i c , i . e
as a n i n t e g r a l curve o f t h e f i e l d
y
by means o f i n t e g r a t i n g t h e system o f O . D . E . ’ s df’ = dR Also one p u t s
y(R):= y(f(R))
t h e vector
y(R).
Method 11.
One d e t e r m i n e s
y’(f(R)). ~
and f i n d s a n a r b i t r a r y c o v e c t o r
u = f(R)
related t o
as an i n t e g r a l curve o f t h e system o f e q u a t i o n s
dl
ranklla?’((f(R)) df’ 7
X(R)
< n.
A.M. Grundland
130
I t i s i n g e n e r a l an o v e r d e t e r m i n e d s y s t e m , f o r example, f o r
m = n
i t may b e
reduced t o one p a r t i a l d i f f e r e n t i a l e q u a t i o n d e t (a?’ I
for
R
unknown f u n c t i o n s .
y ( R ) := d f and f i n d s
determines
Next one
h(R)
as
dR
was mentioned above (Method I ) . Let
M
g:= ( . 1 .) : MxM
(1,m).
=
m + 1 dimensional
+
R1
be m n 0
By
(x
(n 2 2 )
symmetric s c a l a r p r o d u c t w i t h s i g n a t u r e
be a degenerate
,. . . , xm )
Minkowski s p a c e , and l e t
we d e n o t e
Cartesian coordinates i n
M
so t h a t t h e
s c a l a r p r o d u c t i s d e f i n e d by (xly)
=
xvy
u
=
xOyO -
m Z x1. y 1. i=l
W e use a l s o a standard notation
Now we prove t h e f o l l o w i n g s t a t e m e n t . LEMMA 1.
If the function
o f Minkowski s p a c e
R(x)
d e f i n e d on some neighbourhood o f t h e p o i n t
satisfies
M
mZ h=O
then f o r
5
E
R’
x0
x
R R
x
(2.10)
=0,
and
m z. = 1
R.
R. . n . - Sni, j = 1 11 J C
n . :=
where
2 ,
(2.11)
Ro
t h e following i d e n t i t y
m
Z
(R. . - z . n . - z . n . 1 i , j = i 11 1 3 J
= 45
2
m
-45
Z R..n.n. i , j = l 11 1 3
+
m + 2
Z
i , j ,k,R=l
m - 2 holds.
R . R. n n . n n . + ik J Q k 1 R 1
m
R . .R. n . n + C R. .R.. i , j , k = l 11 I k 3 k i , j = i 11 11 Z
(2.12)
131
Riemann Invariants
From ( 2 . 1 0 ) and ( 2 . 1 1 ) we o b t a i n t h e f o l l o w i n g formulae 2 Z Ri 2 i = l n, = ~- 1, 1
Proof:
m z.n. =
C
k=l
R. n n . ik k 1
-
(2.13)
Hence
m
z
(R..
i,j=l
m
i , j ,k,L
m -
45
m
z
- 4
i , j ,k=l
R.R.nn.+ i j ik k 1
m m 2 2 2 R. R. n n . n n . + 4 < n . n . + 4<{ C R . . n . n . - 2 X R. n n . ) = ik J L k 1 L i 1 7 i,j=1 '1 3 k,i=l ik k 1
z
= 4cL
=
m
m I: R.R.nn Ik i,k,L=l
Z R..R..+2 i,j=l
+ 2
z . n . - z.n.1' 1 1 1 1
m
m
=
-
11
m
m
z
R. . n . n . i , j = l 11 I
+
2
'
c
R. R . n n . n n i , j , k , a l k I' I
-
z R R.nn + i , k , e = l i k iL k L
m +
C
i,j=l
Q.E.D.
R. .R. ij i j *
T h e r e f o r e t h e f o l l o w i n g theorem ( G . C i e c i u r a and A . Grundland 1982) h o l d s . Let M b e a Minkowski s p a c e . Then f o r a f u n c t i o n R : 0 c M + R' (of 3 t h e c l a s s C ) , d e f i n e d i n some neighbourhood of t h e p o i n t xo E 0 t h e f o l l o w i n g
THEOREM 3 .
conditions are equivalent : (i)
i n a c e r t a i n neighbourhood
a s o l u t i o n o f t h e system
where (ii)
03 x
0
ct
Ox c 0
of t h e point
xo
the function
R(x)
( V R ~ V R= ) 0,
(2.14)
OR= 0,
(2.15)
d e n o t e s t h e d'Alembert o p e r a t o r .
R(x)
may be d e f i n e d i n i m p l i c i t form by t h e e q u a t i o n R(x) = v ( ( X ( R ) l x ) ) ,
where
is
0
q(-)
i s a r e a l f u n c t i o n o f one v a r i a b l e and
(2.16)
R
)--t
one p a r a m e t e r f a m i l y o f non-zero i s o t r o p i c v e c t o r s , i . e .
X(R)
E
M
is a
i s such t h a t
132
A.M. Grundland
Proof:
k = 1 and t h u s ( 2 . 1 4 ) h o l d s
D i f f e r e n t i a t i n g ( 2 . 1 6 ) we o b t a i n ( 2 . 4 ) f o r
Then by s u c c e s s i v e d i f f e r e n t i a t i o n we o b t a i n
where
-+
c : = d ( X x) .
dX Now ( 2 . 1 7 ) i m p l i e s (-]A) dK
Therefore t h e function
= 0.
R
is
a s o l u t i o n o f t h e system (2.15) and (2.16) i m p l i e s ( 2 . 1 4 ) and ( 2 . 1 5 ) , s o i t remains t o be proved t h a t ( 2 . 1 4 ) and (2.15) imply ( 2 . 1 6 ) .
Differentiating t h e equation
(2.14) we have
m
c
m
X
z
= 0,
X=O
( R R~~~
+
R
X
X=O
= 0.
R
u xv
(2.18)
Thus t h e e q u a t i o n (2.15) may be w r i t t e n i n t h e form
If
R = const,
t h e n (2.16) h o l d s w i t h
cp = c o n s t .
T h e r e f o r e we may assume t h a t
Then from (2.18) f o l l o w t h e formulae l
m
R . = Z R..R 01 Ro j = l 1 1 j ’
(2.20)
So (2.19) by v i r t u e o f (2.11) may b e e x p r e s s e d as
--
m E
- 2
RijR,(ninjnknQ
i , j ,k,R=l
m C R..R. n.n i , j , k = l ’J l k J
+
m
C R..R. . . i , j = 1 11 11
(2.21)
Taking i n t o account Lemma 1 one can s e e t h a t e q u a t i o n (2.21) d e s c r i b e s t h e f a c t t h a t a l l l e v e l s of t h e f u n c t i o n (2.12) i n t o (2.21)
R
are h m e r p l a n e s i n E. I n f a c t , s u b s t i t u t i n g
133
Riemann Invariants
m
c
(R..
i,j=l
-
11
z.n. 1 1
-
m m C R . R n n n n c R.i j n i nj + z . n . 1 2 = 45* - 45 ik jQ k j L i 1 1 i,j=l i , j ,k,R=l = (25
m
z
-
,
R. . n . n . ) 2
i , j = i ’1
=
1 1
whichmeans t h a t R . . = z.n. + z.n
for
z.
j
i
’
R. . n . n . 1 1
.
I J
11
given by (2.11) w i t h
c
.
m
< = -
,
l , ] = l 11
Thus u s i n g (2.20) w e o b t a i n Z.
R
=
5 R
v u
llv
llv
Now we n o t i c e t h a t i f function
R,
(i . e .
5
where
+ 6 R
X
.=
0’
~
r Cn R. . n . n . 2 R 0 i , j = i 11 1 I l
i s a vector f i e l d i n
i s such t h a t
m
C R Xu = O),
u=o
VR
(2.22)
Bi:= RO’
tangent t o l e v e l s o f t h e
t h e n (2.20) g i v e s
ll
m
m
Hence g r a d i e n t
E
and
has a c o n s t a n t d i r e c t i o n on e a c h l e v e l of t h e f u n c t i o n
R (at
l e a s t i n t h e l o c a l s e n s e ) . Thus VR = $ - X ( R ) f o r a c e r t a i n f u n c t i o n JI f 0 and f o r some one-parameter f a m i l y o f i s o t r o p i c v e c t o r s X ( R ) E F.4 i . e . ( X ( R ) I X ( R ) = 0 . Therefore
and t h u s t h e f u n c t i o n
f o r a c e r t a i n function
R
may be e x p r e s s e d as a f u n c t i o n on
( X ( R ) 1x1, i . e .
Q.E.D.
cp(.).
Thus w e o b t a i n THEOREM 4.
Let
M
be a n n-dimensional Minkowski s p a c e .
Let
R
-+
X(R)
E
M be
a one p a r a m e t e r family o f non-zero i s o t r o p i c v e c t o r s , i . e . b e s u c h t h a t ( A ( R ) I h ( R ) ) I f t h e r e e x i s t s a r e a l s i m p l e element
y@X
*
E
TUH@E
s o l u t i o n s of t h e f o l l o w i n g overdetermined systems :
o f t h e system ( 1 . 4 ) t h e n
=
0.
134
A.M. Grundland
d xf(R)
t h e system o f 0 . D . E ' s :
(i) (ii)
= y(R) c 'Tf(R)H,
t h e system f o r t h e f u n c t i o n space
R:
0R
i n t h e hodograph s p a c e H .
(VRIVR) = 0 ,
0,
=
i n t h e Minkowski
M,
g i v e a s o l u t i o n o f homogeneous q u a s i l i n e a r h y p e r b o l i c system (1.1) i n t h e form u(x) = f(R(x)). The Minkowski s p a c e
M
i s a Cartesian product
where
E1xEm,
R
4
A(R)
X(R) =
M
E
E
1
R1
9
and
Em
The i s o t r o p i c v e c t o r
a r e E u c l i d e a n s p a c e s of dimension 1 and m r e s p e c t i v e l y .
can be ( a f t e r s u i t a b l e n o r m a l i z a t i o n ) w r i t t e n i n t h e form
( w , - F (R))
where
w
E
and
El
G(R)
E
Em
i s a v e c t o r , s o (2.16) t a k e s on
t h e form R = rp(w(xo-h(R)x),
I
/F(R) 2 = w 2
where
x0
The form o f s o l u t i o n (2.23) s u g g e s t s t h a t t h e v e c t o r
E
x
El,
X
E
(2.23)
Em.
s h o u l d b e t r e a t e d as an
analogue o f t h e wave v e c t o r ( w , - i ) , which d e t e r m i n e s v e l o c i t y and d i r e c t i o n o f t h e p r o p a g a t i o n o f t h e wave.
Here t h e wave v e c t o r
solution as well, therefore the profile
rp
depends on t h e v a l u e o f t h e
o f t h e wave i s changing d u r i n g i t s
propagation. Let us now c o n s i d e r t h e more g e n e r a l case, when space M has an a r b i t r a r y s i q r i a t u r e (p,n-p), p > 0
of t h e metric I.1.).
If a function
THEOREM 5 .
point
xo
where
q(.,..,.)
E
0
R: 0 c M
i s a real f u n c t i o n o f
b a s i s o f an i s o t r o p i c s u b s p a c e each
i,j
E
can b e d e f i n e d i n a neighbourhood of t h e
R1
-+
i n t h e i m p l i c i t form by t h e e q u a t i o n ( 2 . 3 )
{l,..,k}
T(R)
c
k M
(Ai(R)IAj(R)) = 0 ) ,
2:
v a r i a b l e s and o f dimension
k
(Al(R),..,Xk(R)) 5
then t h e function
min(p,n-p) R(x)
is a solution
o f t h e system
v 2R where
V2
(P ,n-p).
= 0,
is a
(i.e. for
(VR~VR = )o
d e n o t e s t h e Laplace-Beltrami o p e r a t o r w i t h r e s p e c t t o s i g n a t u r e
135
Riemann invariants
T h i s theorem may be e a s i l y proved by d i f f e r e n t i a t i o n o f t h e e q u a t i o n s ( 2 . 3 ) 2 . P h y s i c a l l y , t h e nonplanar s i m p l e Riemann wave [ 2 . 3 ) which a d m i t s a F o u r i e r t r a n s form ( e . g . i f
i s L1 i n each o f t h e v a r i a b l ’ e s ) can be c o n s i d e r e d as a s u p e r -
q
p o s i t i o n o f an i n f i n i t e number of p l a n e waves r e l a t e d t o t h e same c h a r a c t e r i s t i c curve given by O . D . E .
(2.1).
So we have
k [P(nl,.
.
f
’Tik)
=
___ $(yl,...,yk) ( 2 T )l l / k i
exp(-i
C
Y,q,ldYl...dYkj
c1=1
and
where
qa:=
A
c1U
xu.
NONLINEAR SUPERPOSITION OF RIEMANN WAVES I N A HOMOGENEOUS SYSTEM.
The a l g e b r a i z a t i o n o f t h e system o f homogeneous e q u a t i o n s (1.1) allows u s t o c o n s t r u c t more g e n e r a l c l a s s e s o f s o l u t i o n s , r e p r e s e n t i n g n o n l i n e a r s u p e r p o s i t i o n o f simple Riemann waves i . e . double waves and i n g e n e r a l k-waves. Let 1 ,. . . ,y @Ap be s i m p l e i n t e g r a l elements w i t h l i n e a r l y independent A 1, . ,A k P and yl, . . . , y k r e s p e c t i v e l y . We propose now a form o f t h e s o l u t i o n u f o r which
. .
ylaX
t h e t a n g e n t mapping
where
St f 0
du(x)
i s t h e sum o f simple i n t e g r a l e l e m e n t s .
i s c o n s i d e r e d as a f u n c t i o n of
x.
The s i m p l e i n t e g r a l elements
I t was proved (M.
Burnat 1969, 2. ys@Xs r e p r e s e n t a simple Riemann wave. P e r a d z y i n s k i , 1970) t h a t t h e s o l u t i o n s o f t h e p o s t u l a t e d form (3.1) e x i s t and can be w r i t t e n i n Riemann i n v a r i a n t s .
I n t h e l a t t e r case i t means t h a t w e make t h e
a d d i t i o n a l assumption t h a t t h e commutator o f t h e v e c t o r f i e l d s
ys,yt
is a linear
combination o f t h e s e f i e l d s , i . e .
If t h e s e c o n d i t i o n s a r e s a t i s f i e d we may change t h e l e n g t h of t h e v e c t o r s
y,
so
that
TY,.Y~I and t h e v e c t o r s
yl,...,yk
= 0,
c o n s t i t u t e a holonomic system and t h e r e e x i s t s a
(3.2)
136
A.M. Grundland
parametrization of t h e surface tangent t o t h e f i e l d s u(x)
=
f(R(x))
where
R(x):=
.
yl,.
,yk
1 k k (R ( x ) , . . . , R (x)) E R
,
(3.3)
such t h a t
The system (3.4) i s an over-determined solutions).
Consequently,
du(x) =
C
system of P.D.E.s
(which does not alwayshave
af
aRS
7 dRS, where dRS = 1 dx’, 11 ’=I ax
s=l aR
which t o g e t h e r w i t h t h e assumption t h a t
yl, ...,yk
a r e l i n e a r l y independent,
l e a d s t o a system o f P f a f f i a n forms dRs(x)
=
CsAs(R(x))
The f i e l d s of c o v e c t o r s AS =
As(R)dx’ II !J=l
E
E*).
As
where
cs(x) f 0,
s
E
{l,
. . . ,k l .
become f u n c t i o n s o f t h e p a r a m e t e r
(3.5)
R (i.e.
I n v o l u t i v i t y c o n d i t i o n s f o r t h e system (3.5) have been
a l r e a d y i n v e s t i g a t e d ( Z . Peradzynski 1970).
Namely, t h e system ( 3 . 5 ) h a s s o l u t i o n s
( i s i n t e g r a b l e ) i f t h e following conditions a r e s a t i s f i e d .
A
sftEI1, where
as
t
and
From e q s . (3.6)
f3:
axs = atA s s s t + BtA , t
(3.6)
..., k ) a R
a r e given f u n c t i o n s o f
by formulae,
R
t h e c o m p a t i b i l i t y c o n d i t i o n s ( t h a t i s , t h e symmetry o f t h e second
d e r i v a t i v e s ) are o f t h e form
If l i n e a r l y independent v e c t o r s
one v e c t o r
y
S’
As,.
t h e n t h e above problem f o r
( Z . Peradzynski 1971).
. . ,A?
1 (s)
k
’
s = 1,
...,k
are r e l a t e d t o
waves admits a g e n e r a l i z a t i o n
The system of P f a f f i a n forms (3.5) may b e r e p l a c e d by the
137
Riemann Invariants
following
where
E:[,)
$ 0,
R = (R
,... , R k ) ,
1
s
E
{l, ...,k 1
.
( 3 . 6 ) , we have t h e f o l l o w i n g c o m p a t i b i l i t y c o n d i t i o n s
In t h i s c a s e , i n s t e a d o f
= 0,
R‘
where j ( r ) = 1 ,..., i [ r ) , I t h a s been proved (H.
k ( s ) = 1 ,...,i(s).
Btjttger 1978, Z . Peradzynski 1978) t h a t i n t h e c a s e
d e s c r i b e d by Riemann i n v a r i a n t s t h e s o l u t i o n
u
decays i n an
e x a c t way i n t o
s i m p l e Riemann waves b e i n g o f t h e same type’)
a s a t t h e i n i t i a l moment.
Thismeans
t h a t t h e r e s u l t o f n o n l i n e a r s u p e r p o s i t i o n o f t h e simple Riemann waves c o n s i s t s o f
k
waves o f t h e same k i n d b u t w i t h d i f f e r e n t amplitudes and, i n g e n e r a l , with
d i f f e r e n t d i r e c t i o n s of propagation.
So w e have
and t h e t y p e of t h e s i m p l e Riemann waves. s u p e r p o s i t i o n o f t h e s i m p l e Riemann waves.
a
c o n s e r v a t i o n lawfor t h e number
I n t h i s case we can speak about e l a s t i c The Cauchy problem f o r t h e system ( 3 . 5 )
h a s been f o r m u l a t e d and s o l v e d u s i n g t h e Riemann f u n c t i o n s .
(A.
Grundland and
R . Zelazny 1983, Z . Peradzynski 1979).
We w i l l p r o v e t h a t f o r an a r b i t r a r y
k 2 1
a l l the solutions of t h e
system (3.5) can b e o b t a i n e d [ t a k i n g i n t o account ( 3 . 6 ) and ( 3 . 7 ) ) by r e s o l u t i o n 1 Rk o f t h e system with respect t o t h e variable R
,...,
AS(R)*xu = Ys(R),
u
where
Vs
R = (R
are t h e a r b i t r a r y f u n c t i o n s o f c l a s s
1
,...,Rk ) , C1
of
k
(3.8)
variables.
Indeed,
d i f f e r e n t i a t i o n o f t h e e q u a t i o n s (3.8) g i v e s
If t h e system (3.5) i s s a t i s f i e d , t h e n t h e l e f t hand s i d e i s p r o p o r t i o n a l t o
and t h e r i g h t hand s i d e i s a l i n e a r combination o f
f$
0,
then
Xr(R)
for
r # s.
As(R).
Thus i f
138
A.M. Grundland
(3.10) From (3.9) and from t h e c o m p a t i b i l i t y c o n d i t i o n s (3.6) w e have s s
u A (R)x’ ri.l
s r
+ B h (R)x’
=
r u
ws
(R) ___ aRr
,
t h e r e f o r e , by v i r t u e o f t h e e q u a t i o n (3.8) (3.11) Then from t h e e q u a t i o n (3.9) aAS(R)xU
(--a$’
-
a RS
’a RS
)dRS = A s ,
axs (R) xu where
(-
-
a RS
aRS
i s t h e resolvability condition.
I f 0
Thus w e have
equation ( 3 . 5 ) . Let us i n v e s t i g a t e now t h e c o m p a t i b i l i t y c o n d i t i o n s f o r t h e system ( 3 . 1 1 ) . I t i s a system of
for
s
# r #
t.
k(k-1)
e q u a t i o n s f o r k unknown f u n c t i o n s .
We have
Symmetry o f t h e second d e r i v a t i v e s g i v e s
Hence t h e s e c o n d i t i o n s a r e i d e n t i c a l l y s a t i s f i e d by v i r t u e o f ( 3 . 7 ) .
Let t h e f u n c t i o n s
be s o l u t i o n s o f t h e e q u a t i o n s
p s = ps(R)
aps= -u s a Rr
then t h e functions
for
r
+ s,
ns:= exp(pS) s a t i s f y
allS + a Rr Introducing the notation
$”:=
qsJISr
= 0.
from (3.11) we obtain
(3.12)
139
Riemann invariants
ajlS ~
s-r
for
B,$
=
# s;
r
(3.13)
aRr and t h e e q u a t i o n s ( 3 . 7 ) t a k e t h e form (3.14)
Let us n o t i c e , t h a t i n formulae (3.13) and (3.14) t h e v a r i a b l e
x
does n o t a p p e a r
explicitly.
5
I f f o r example
1
i n t h e equation (3.5), then
E 0
R
1
= const.
Y
From t h e
-
c o m p a t i b i l i t y c o n d i t i o n s (3.6) and ( 3 . 7 ) f o r t h e c o v e c t o r s Xs(R) := As(Rb,K), 2 k s E 12, k } where R = (R ,..., R ) we o b t a i n t h e f o l l o w i n g e q u a t i o n s ,
...,
where
-s
-S
- -- -Bst -Btr "r
- -s-r
a?
Functions
$
2
Brat
,..,qk
+
-s
aa
aa
a?
a 2
L E - - --
-5-5
Brat,
for
0
t
# s # r
E
12,
s a t i s f y t h e c o n d i t i o n s ( 3 . 8 ) and (3.11) f o r r 1 2 k R = R1 and by t h e f u n c t i o n R = (R ,..., R )
s o l u t i o n i s given by
...,k } . 2
0
2.
Thus t h e
o b t a i n e d by
r e s o l u t i o n o f t h e system
In p a r t i c u l a r , f o r
k
=
1 we o b t a i n t h e formula (3.8) f o r a s i m p l e Riemann
wave ( 2 . 9 ) .
REMARK 2 .
As we a l r e a d y know, s o l u t i o n o f t h e e q u a t i o n ( 3 . 5 ) can be o b t a i n e d by
i n t e g r a t i o n o f t h e system ( 3 . 1 1 ) . degenerate
Conversely, we w i l l show now, t h a t e v e r y non-
s o l u t i o n o f t h e system (3.5) can be o b t a i n e d u s i n g t h i s method.
Indeed, t h e e q u a t i o n (3.5) i m p l i e s t h a t f o r
k dqS = Xs(R) + If
gr f 0 ,
q S ( x ) : = AE(R(x))x'
aAs(R)x'
'-
C
CPXr(R),
R:= (R
1
w e have
,. . , R k )
then k dqs =
C
r= 1
pSdRr,
for
s
=
l,..,k,
(3.15)
140 where
A.M. Grundland
vs
a r e f u n c t i o n s d e f i n e d on
1
E.
Additionally functions R ,..,Rk are 1 k R k + l, . . ,Rn). f u n c t i o n a l l y i n d e p e n d e n t , t h u s we can add them t o t h e map (R , . . , R ,
Then (3.15) means, t h a t we have t h e f o l l o w i n g c o n d i t i o n s i n t h i s map.
So t h e f u n c t i o n s REMARK 3.
ns
can be e x p r e s s e d as f u n c t i o n s of R ,...
0
I f (on a c e r t a i n domain At(R)x’
= 0,
R:=
E)
c
(R
1
axs
.
t h e equation
,..,R k ) ,
can be r e s o l v e d with r e s p e c t t o t h e v a r i a b l e a s o l u t i o n of equations (3.5).
RK
s
E
{i ,..,k l ,
R( i . e .
(3.16)
R = R(x)),
t h e n we o b t a i n
Indeed, i f t h e e q u a t i o n s (3.16) are s a t i s f i e d , t h e n
k
(R) xu
dRS +
Xs(R) +
a RS
ahz(R)x’ dR’
C
=
0,
r+s=l aR’
b u t from t h e c o m p a t i b i l i t y c o n d i t i o n s (3.6) and t h e e q u a t i o n s (3.16) we have
axsu (R)xU
= 0.
a Rr Hence, i f t h e c o n d i t i o n o f r e s o l v a b i l i t y
axsu ( ~ ) x ’ l a RS i s s a t i s f i e d , then covector RS = Rs(x)
i s proportional t o
Xs(R)
s a t i s f y t h e equations ( 3 . 5 ) .
a f a m i l y (parametrized by p o i n t s
# 0,
a
E
E)
i . e . , functions
E we obtain
o f s o l u t i o n s o f t h e system ( 3 . 5 ) ,
These s o l u t i o n s have a s i n g u l a r i t y a t t h e p o i n t EXAMPLE.
dRS,
Through a t r a n s l a t i o n i n
x = a.
Let us c o n s i d e r now a system of P f a f f i a n forms dR1(x) = S1(x)X1(R’(x),R2(x)), (3.17) dR2(x) = ~ 2 ( x ) X 2 ( R 1 ( x ) , R 2 ( x ) ) ,
where :
141
Riemann Invariants
(a,B = a r e a r b i t r a r y f u n c t i o n s
of t h e i r arguments), f o r which t h e c o m p a t i b i l i t y
c o n d i t i o n s (3.6) a r e a u t o m a t i c a l l y s a t i s f i e d
ah2
-
1
1
1
B - a (cos R , s i n R , 0) + 0 1( - s i n ~ l , c o sR
aR2
1
,
0) =
(%-a) _ -- a
x1 +
- aA . 2
6-a
%-a
The g e n e r a l s o l u t i o n o f t h e system (3.17) has t h e form
where
1
1
2
2
$ (R ) , $ (R )
a r e a r b i t r a r y d i f f e r e n t i a b l e f u n c t i o n s o f t h e i r arguments.
Thus from e q u a t i o n (3.18) we o b t a i n 1
-atJ2 _
- 0,
aR
Hence t h e c o n d i t i o n s (3.11) are s a t i s f i e d .
To simply t h e formulae we use t h e
following n o t a t i o n
The s o l u t i o n s can be w r i t t e n i n t h e e q u i v a l e n t form $'(R1)
= a(R 1)Xu(R 1 1 1) x 1 ,
A. M. Grundland
142 that is
1
-1 1
( R ) = x cos R1 + x'sin
I$
2
(R
2
- 91(R1I
1 + x3i(R 1) ,
2 1 3 R~ + x cos R + ( B ( R ~ ) - ~ ( R ' ). ) ~
= -xlsin
1 cr(R ) = aR1
For t h e s a k e of s i m p l i c i t y we t a k e equations (3.17).
R
2 8(R ) = bR2
and
i n the
In t h i s c a s e e q u a t i o n ( 3 . 1 6 ) has t h e form 1 x cos R'
+
x 2 s i n R'
+ ax3 = 0,
1 . 2 2 1 - x s i n R1 + x cos R1 + (bR -aR )
=
0.
Hence we have
From t h e above e q u a t i o n s ,
and ax2 + (bR2 - aR1)xl - r x l + x 2 t g R1 =
+ (bR2 - aR2)x2 - x1 + r x
ax'
where
a r : = bR2 - aR
1
.
2’
So w e have
r =
1
((x112
+
(x
2 2
2
- a (x
ax
3 2 %
I I .
F i n a l l y w e o b t a i n a s y s t e m of e q u a t i o n s 1 + 2 1 R (x) = a r c t g " 1 2 ' x + rx R 2 (x) =
1
((x1)2+(x2)2-a2(x3)2)' +
bx
The l e v e l s of f u n c t i o n s
(3.19)
R1
and
o r i g i n of t h e c o o r d i n a t e system.
R2
1 2 arc tg rx + x 1 2 x + rx
.
are the straight lines intersecting a t the
143
Riemann Invariants
SIMPLE STATES I n t h e p a p e r (A. Grundland 1974) i t h a s been proved t h a t t h e r e e x i s t s o l u t i o n s o f t h e nonhomogeneous system (1.1) f o r which d e r i v a t i v e s du(x) mapping
u
of t h e
would be nonhomogeneous s i m p l e elements i f
du(x) = yo@ho where
a?’y;A: 7
=
bS,
yo
E
TuH, h
0
=
o
A dx
p
u
E
E
*
.
(4.1)
The c o n d i t i o n s o f c o m p a t i b i l i t y ( t h a t i s t h e symmetry o f t h e second d e r i v a t i v e s Schwarz lemma) a r e o f t h e form d(du(x))
dy0
=
A
A o + yo@dho = 0
modulo e q u a t i o n s ( 4 . 1 ) ,
where
:=E
Weuse t h e n o t a t i o n X ,
ax yi . -
Namely t h e system ( 4 . 1 ) h a s a s o l u t i o n ( i s
i = i aul
integrable) i f
hi)
A
ha = ( ) . T h i s means t h a t t h e d i r e c t i o n o f
i s constant
‘h
YO along t h e f i e l d where
a = a(u)
yo,
# 0,
and t h e n , by t h e n o r m a l i z a t i o n , one
gets
Ao
ho
-f
aho
c o n s t a n t along the f i e l d
Thus t h e image o f t h e s o l u t i o n i s a curve
u = f(R)
1 and y,, + a y,,, yo, i . e .
tangent t o t h e f i e l d
y
0;
moreover, one can choose i t s p a r a m e t r i z a t i o n s o t h a t
AS
0
A (f(R))
=
*fl A = c o n s t , and
i s a s o l u t i o n o f a nonhomogeneous e q u a t i o n (1.1) , t h e n
Hence, a s a Riemann i n v a r i a n t w e can choose l i n e a r f u n c t i o n
R(x)
=
-0 v A>,x
.
Equatior
A.M. Grundland
144 (4.4) determines a s o l u t i o n
u
=
f(R)
of the field
u = u(x)
c o r r e s p o n d i n g t o t h e i n t e g r a l curve
Such s o l u t i o n s w i l l be c a l l e d s i m p l e s t a t e s .
yo,
The n o t i o n o f t h e “ s i m p l e wave” we r e s e r v e e x c l u s i v e l y f o r s o l u t i o n s o f One o f t h e arguments f o r s u c h an approach i s t h a t t h e
homogeneous systems ( 1 . 1 ) .
a c c e p t e d d e f i - n i t i o n o f a s i m p l e s t a t e can be used f o r e l l i p t i c e q u a t i o n s a s w e l l . Such e q u a t i o n s have no s o l u t i o n s which can be i n t e r p r e t e d p h y s i c a l l y a s waves.
On
t h e o t h e r hand i t s h o u l d be n o t i c e d , t h a t some phenomena which a r e d e f i n e d a s waves i n t h e t r a d i t i o n a l ( p h y s i c a l ) terminology can b e d e s c r i b e d i n terms o f t h e notion of a simple s t a t e .
For example, a l o c a l i z e d d i s t u r b a n c e p r o p a g a t i n g i n a
nonhomogeneous system and c a l l e d a s o l i t a r y wave ( o r a s o l i t o n i n some c a s e s ) corresponds t o a s i m p l e s t a t e .
Physically, a simple s t a t e describes a disturbance
which p o s s e s s e s , u n l i k e a s i m p l e wave, a w e l l d e f i n e d p r o f i l e as w e l l as c o n s t a n t v e l o c i t y and d i r e c t i o n o f p r o p a g a t i o n .
Moreover a s i m p l e s t a t e h a s no auJ of the solution are
gradient catastrophy ( i . e . a l l p a r t i a l derivations
I----)
axu bounded i n t h e e n t i r e domain).
NONLINEAR SUPERPOSITION OF R I E M A ” WAVES I N A NONHOMOGENEOUS SYSTEM.
I t i s p o s s i b l e t o g e n e r a l i z e t h e proposed method o f c o n s t r u c t i n g s o l u t i o n s t o t h e c a s e o f many waves i n nonhomogeneous systems. I t r e q u i r e s t h e s o l u t i o n o f t h e system o f P f a f f i a n forms to be a sum o f homogeneous and nonhomogeneous s i m p l e i n t e g r a l elements i n t h e form:
gr f 0
where q u a n t i t i e s t h a t vectors simple element elements
yoah
x.
We assume
F i e l d s o f nonhomogeneous
( r e p r e s e n t i n g a s i m p l e s t a t d and o f homogeneous simple
The e x i s t e n c e c o n d i t i o n s f o r s o l u t i o n s o f t h e P f a f f i a n forms (5 . l )
with a r b i t r a r y v a r i a b l e s R. Zelazny 1982
1.
a r e l i n e a r l y independent.
( r e p r e s e n t i n g Riemann waves) s a t i s f y t h e e q u a t i o n s :
y,@A’
respectively.
are c o n s i d e r e d a s v a r i a b l e s dependent on
yo,yl, ...,yk
.
5‘
were c h t a i n e d b y A . Grundland 1974, A . Grundland and
I t i s n e c e s s a r y t o c o n s i d e r t w o cases:
The c a s e o f two independent v a r i a b l e s :
145
Riemann Invariants
where c o v e c t o r 2.
E
T*H
The c a s e o f many
has t h e p ro p ert y
(u > 2)
2)
0
=
<w,y,>
independent v a r i a b l e s .
ol,B
= 0,
E
{O,l,..,kl
r
E
(5.4)
tl,..,kI,
0
(5.5)
6a '
where t h e r e i s no summation w i t h r e s p e c t t o
a;.
These r e l a t i o n s a r e t h e n e c e s s a r y
and s u f f i c i e n t c o n d i t i o n s f o r e x i s t e n c e o f t h e s o l u t i o n s o f t h e system ( 5 . 1 ) .
They
g u a r a n t e e t h a t t h e s e t o f s o l u t i o n s o f t h e system (5.1) depends on k a r b i t r a r y f u n c t i o n s o f one v a r i a b l e a c c o r d i n g t o t h e C a r t a n t h e o r y of systems i n i n v o l u t i o n . A p h y s i c a l i n t e r p r e t a t i o n o f s o l u t i o n s ( 5 . 2 ) and (5.3) c o v e r s a l s o g e n e r a t i o n of
waves a s a r e s u l t o f t h e i r s u p e r p o s i t i o n on a s i m p l e s t a t e .
Generation can t a k e
p l a c e , f o r example, when t h e commutators (5.2) o f t h e v e c t o r f i e l d s
y,
and
y
B
are l i n e a r combinations (with c o e f f i c i e n t s n o t n e c e s s a r i l y c o n s t a n t ) o f t h e s e fields
y,
and
yB
and a l s o o f o t h e r f i e l d s
connected with t h e s e f i e l d s
y E , ...,yo.
I t means t h a t waves
yp,..,yu t a k e p a r t i n t h e s u p e r p o s i t i o n .
Thus, as
an e f f e c t o f t h e s u p e r p o s i t i o n o f two s i m p l e waves we o b t a i n new waves ( o f a t y p e o t h e r t h a n t h o s e a t t h e i n i t i a l moment).
I f t h e s e new waves do n o t v a n i s h asymptot-
n
i c a l l y f o r l a r g e time ( t : = x " f m ) ,
then t h e e f f e c t
o f g e n e r a t i o n i s permanent ( i n
accordance with t h e a c c e p t e d terminology i t i s "a n o n e l a s t i c s u p e r p o s i t i o n " ) .
In
t h e c a s e d e s c r i b e d by Riemann i n v a r i a n t s ( 5 . 4 ) and (5.5) t h e c o n s e r v a t i o n law f o r t h e k i n d and q u a n t i t y o f waves h o l d s ( s o c a l l e d e l a s t i c s u p e r p o s i t i o n ) ( A . Grundland and R . Zelazny 1982).
I t h a s been a l s o proved (A.
Grundland a R . Zelazny 1983)
t h a t such s o l u t i o n s , r e s u l t i n g from t h e s u p e r p o s i t i o n o f many simple waves p r o p a g a t i n g on t h e s i m p l e s t a t e , decay i n an e x a c t way i n t o simple waves on t h e s t a t e , of t h e same k i n d as t h o s e e n t e r i n g t h e i n t e r a c t i o n . Consider now, as a n i l l u s t r a t i o n , a p r o c e s s o f p r o p a g a t i o n o f t h e Riemann wave in the
nonhomogeneous system ( 1 . 1 ) by means o f t h e p r o c e d u r e s d e s c r i b e d above.
W e propose a form o f t h e s o l u t i o n
u
f o r which t h e t a n g e n t mapping
sum o f a homogeneous and a nonhomogeneous simple elements
du(x)
is a
A.M. Grundland
146
where yo
51 $ 0
and
y1
i s c o n s i d e r e d as a f u n c t i o n o f
c o n s t i t u t e a holonomic system, i . e . t h a t t h e p a r a m e t r i z a t i o n
=
0 1 f(R , R )
of t h e surface, tangent t o t h e f i e l d s
yo, y l ,
u
Consequently,
du(x) =
af a dRo + af dR1 1 aR
y o , y1
(5.7)
is such t h a t
which t o g e t h e r w i t h t h e assumption t h a t
aR
a r e l i n e a r l y i n d e p e n d e n t , l e a d s t o a system o f P f a f f i a n forms dR dR
0
0
0
As
=
1
= A (R , R ) ,
1
=
h:(f(R))dx’
where
A
0
A
A
1
# 0,
c1(x) f 0
(5.9)
5 1A 1(R 0, R 1) .
The f i e l d s o f t h e c o v e c t o r s (i.e.
We s h a l l assume t h a t t h e v e c t o r s
x.
E
AS
E*).
0 1 R : = (R , R )
become f u n c t i o n s o f t h e p a r a m e t e r
The c o m p a t i b i l i t y c o n d i t i o n s (5.5) f o r t h e system
(5.9) t a k e t h e form (5.10)
where
ao, a l , B0,
B1
are g i v e n f u n c t i o n s o f
consequence o f e q u a t i o n s (5.10)1 of t h e e q u a t i o n
% aR
= ao, t h e n
R.
and ( 5 . 1 0 ) 2 . ho(R)
If
a1 # 0 ,
then (5.10)3
Let the f u n c t i o n
= A(R1)exp q ( R ) .
v
is a
be a solution
Thus we c o n s i d e r t h e
system (5.9) i n t h e form dR 0 = A(Rl)exp(cp(R)),
where
X(R 1)
A
dA(R1) +
o,
dR1
(5.11)
f o r which t h e c o m p a t i b i l i t y c o n d i t i o n s (5.10) a r e a u t o m a t i c a l l y s a t i s f i e d . THEOREM 6 .
A l l s o l u t i o n s o f t h e system (5.11) can b e o b t a i n e d by r e s o l u t i o n , w i t h 1
respect t o the variables
Ro
and
R
, o f t h e system
Riemann Invariants
Xu(R1)xu = JI(R1)
+
1;
147
1 exp(-cp(r,R ) ) d r , (5.12)
dh (R1)xu
RU
dR1
0
u
where Proof:
1
1 exp(-cp(r,R ) ) d r ,
aR1
$(.) is a n a r b i t r a r y d i f f e r e n t i a b l e f u n c t i o n o f
R1
I n d e e d , i f (5.11) i s s a t i s f i e d , t h e n dh (R1)xu
11-1 dCX (R ) x 3 = dXu(R1)xu dR1 + u dR1
-
h(R1) = exp(-cp)dR 0
+
(
) dR1
)J
dR1
acp 0 7exp(-cp)dR
d2Xu(R1)xu +
aR
.
'
+ -)dR
(
d(R1)2 dXu (R1)xu
Therefore both functions
A (R1)xu
and
can b e e x p r e s s e d as f u n c t i o n s
u of
0 R (x)
and
dR1
1
R (x). (5.13)
D i f f e r e n t i a t i n g t h e e q u a t i o n s ( 5 . 1 2 ) we o b t a i n
ax+. O 1
X
aR1 (5.14)
ax
0
dX -
2
dR1
where t h e f o l l o w i n g n o t a t i o n
v
h a s been u s e d .
0
:=
1
X [R ) x u , u
1
dXu(R 1) Xu
S u b s t i t u t i n g (5.14)
dR1
d2Xu (R1) xu
2
, v
v :=
:=
d(R1I2
'
i n t o t h e system o f e q u a t i o n s (5.11) we o b t a i n
A.M. Grundland
148
(5.15) Note, t h a t ( 5 . 1 5 ) 3 (5.15)2.
i s t h e c o m p a t i b i l i t y c o n d i t i o n f o r t h e system ( 5 . 1 5 ) 1
The g e n e r a l s o l u t i o n o f e q u a t i o n s ( 5 . 1 5 ) 3 h a s t h e form
x 1(R)
= -
f 0
1
a'(r;R aR
1 d$ (R1) e x p ( - q ( r , R ) ) d r + -, dR1 0
i s a c o n s t a n t of i n t e g r a t i o n w i t h r e s p e c t t o R , depending on 1 Having t h e f u n c t i o n x (R) we can s o l v e t h e system (5.15)1 and (5.15)2.
where
and
Ji
Thus by v i r t u e of (5.131,
(5.16) R
1
.
(5.16) and ( 5 . 1 7 ) t h e g e n e r a l s o l u t i o n of t h e system
(5.11) h a s t h e form ( 5 . 1 2 ) .
Q.E.D.
I t h a s been proved ( A . Grundland and R. Zelazny 1983) t h a t t h e above s o l u t i o n s
can be g e n e r a l i z e d t o t h e c a s e o f many Riemann waves i n t h e nonhomogeneous system ( 1 . 1 ) and t h e Cauchy problem f o r t h e s y s t e m (5.1) h a s been f o r m u l a t e d and s o l v e d u s i n g t h e Riemann f u n c t i o n s .
Examples o f such s o l u t i o n s f o r t h e e q u a t i o n s of
gasdynamics and magneto-hydrodynamics can be found ( A . Grundland 1974 a , b , 1978, 1983, A . Grundland and R. Zelazny 1982, 1983). ACKNOWLEDGMENTS.
I am g r e a t l y i n d e b t e d t o P r o f e s s o r s B . Lawruk and P . W i n t e r n i t z
f o r very i n s p i r i n g d i s c u s s i o n s d u r i n g t h e study of this s u b j e c t .
149
Riemann Invariants
FOOTNOTES. 1)
A s i m p l e wave (2.9)
of matrix
2)
SU
a. X 7 v
i s of t h e i - t h t y p e i f
y =
i s the i - t h eigenvector
.
The p a r e n t h e s i s d e n o t e s t h e c o n t r a c t i o n o f t h e covector E
TtH
with t h e vector
y
a
t
T H. u
A.M. Grundland
150
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B u r n a t , M., 1971, The method o f Riemann i n v a r i a n t s and i t s a p p l i c a t i o n s t o t h e t h e o r y o f p l a s t i c i t y , P t . 1 , Arch. Mech. s t o s . , 23. C a r t a n , E . , 1953, Sur l a s t r u c t u r e des groupes i n f i n i s de t r a n s f o r m a t i o n s . C h a p i t r e I : Les systZmes d i f f g r e n t i e l l e s e n i n v o l u t i o n , G a u t h i e r - V i l l a r s , Paris. C i e c i u r a , G . Grundland, A.M.,1982 Geometrical a s p e c t s of t h e s y s t e m 1VvI2 = a ( v ) , 2 V v = B(v) and a p p l i c a t i o n s t o t h e n o n l i n e a r wave e q u a t i o n . P r e p r i n t I n s t i t u t e f o r T h e o r e t i c a l P h y s i c s , I . F . T . 9 , 1982 Warsaw U n i v e r s i t y , pp.1-14. Courant, R . , H i l b e r t , D . ,
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l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . C o n d i t i o n s of i n v o l u t i o n , B u l l . Acad. P o l . S c . S 6 r . t e c h . 4 , pp. 177-185.
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order p a r t i a l q u a s i - l i n e a r d i f f e r e n t i a l e q u a t i o n s - a l g e b r a i c a s p e c t s . Examples from gasdynamics, Arch. o f Mech. 26, pp. 271-296. Grundland, A . M . , 1975, A l g e b r a i c p r o p e r t i e s o f nonhomogeneous e q u a t i o n s o f magnetohydrodynamics i n t h e p r e s e n c e of g r a v i t a t i o n a l and C o r i o l i s f o r c e s . of s o l u t i o n s
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t h e "Cauchy problem f o r d i s s i p a t i v e media", Ed. W . K o s i n s k i . Wroclaw, P u b l i s h e d i n Ossolineum, i n P o l i s h , pp. 195-209. 1981. N o n l i n e a r s u p e r p o s i t i o n o f s i m p l e waves i n nonhomogeneous
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s y s t e m s , I n : P r o c . Conf. on N o n l i n e a r Phenomena i n P h y s i c s and B i o l o g y , ( B a n f f , Canada 1980). Ed. R . H . Enns. R e i d e l 1981. Also p r e p r i n t o f t h e I n s t . f o r N u c l e a r Research 1980, VII, P S , A , Warsaw 1981, p p . 1-46. Grundland, A . M . ,
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of magnetohydrodynamics.
Acta Geophys. P o l o n i c a 1983, no. 3 , p p . 4-29.
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t i o n s o f t h e I n s t i t u t e o f Geophysics A-14 ( 1 6 2 ) . Warsaw 1983, p p . 1-109. Grundland, A.M.
Zelazny, R . ,
1983. Simple waves i n q u a s i - l i n e a r h y p e r b o l i c s y s t e m s ,
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Examples o f a p p l i c a t i o n s .
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Simple waves i n q u a s i - l i n e a r h y p e r b o l i c s y s t e m s ,
P t . 11. Riemann i n v a r i a n t s for t h e problem o f s i m p l e waves i n t e r a c t i o n s . J .
of Math. Phys. 1983, Vol. 24, No. 9 , p p . 2315-2329. J e f f r e y , A . , 1976, Q u a s i l i n e a r h y p e r b o l i c systems and waves, Academic P r e s s , London. Jeffrey, A., Taniuti, A . ,
1964. N o n l i n e a r wave p r o p a g a t i o n , Academic P r e s s , New
York. Leibovich S . , S e e b a s s , A . R . , L i g h t h i l l , J. , 1968.
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N o n l i n e a r waves, Wiley - I n t e r s c i e n c e , London.
H y p e r b o l i c e q u a t i o n s and waves. W i l e y - I n t e r s c i e n c e , New York.
Lundford, G.S., 1955. G e n e r a l i z e d Riemann I n v a r i a n t s P a c i f . .J. Math. 5 p u . 441-450.
Mises, R . ,
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New York. Peradzynski, Z . , method.
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e,10,
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1971. Riemann i n v a r i a n t s f o r t h e n o n p l a n a r k-waves, B u l l . Acad.
Pol. Sc. S d r . tech. Peradzynski, Z . ,
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1974. N o n l i n e a r i n t e r a c t i o n s d e s c r i b e d by p a r t i a l d i f f e r e n t i a l
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Schwingungsweite, KGnigt. Gas. Wiss. (GGttingen), Rozhdestvenskiy, B . , Yanenko, N . ,
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a p p l i c a t i o n i n gasdynamics ( i n R u s s i a n ) , Publ. M i r . Moscow. Whitham, G . , 1974. L i n e a r and N o n l i n e a r h a v e s , John Wiley and Sons, New York. Zajaczkowski, W . ,
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153
VARIATIONAL FORMULATION OF THE SINGULAR SURFACE PROPAGATION I N NONSIMPLE ELASTIC MATERIALS Jacek T u r s k i Department o f Mechanical E n g i n e e r i n g * McGi 1 1 U n i v e r s i t y L l o n t r e a l , P . Q . H3A 2K6
The g e n e r a l i z e d framework o f t h e v a r i a t i o n a l formalism o f continuum mechanics i s a p p l i e d t o t h e f o r m u l a t i o n o f dynamical c o n d i t i o n s f o r t h e p r o p a g a t i o n o f a s i n g u l a r s u r f a c e i n nonsimple e l a s t i c m a t e r i a l s . C o n s i d e r i n g t h e f i r s t v a r i a t i o n o f t h e f u n c t i o n a l i n t h e doma i n where t h e s i n g u l a r s u r f a c e i s p r o p a g a t i n g , t h e W e i e r s t r a s s Erdmann c o n d i t i o n s a r e o b t a i n e d . These c o n d i t i o n s a r e e q u a t i o n s f o r t h e jump o f d e n s i t i e s o f momentum and o f energy across t h e s i n g u l a r s u r f a c e . F o r t h e case o f a m a t e r i a l o f grade 2 t h e s e r e l a t i o n s a r e o b t a i n e d h e r e f o r t h e f i r s t t i m e . Using t h e proposed v a r i a t i o n a l f o r m u l a t i o n an e x t e n s i o n o f r e s u l t s t o a m a t e r i a l o f any grade N can be o b t a i n e d . 1.
INTRQDUCTION
I n t h i s paper t h e t r e a t m e n t i s e x c l u s i v e l y i n E u c l i d e a n spaces u s i n g s t a n d a r d t u c l idean c o o r d i n a t e s . Consider a f u n c t i o n a l
where D if a f i n i t e domain i n R 3 , $!$(X,t) a dynamical v a r i a b l e $ E a $ / a t , $,,-_a$/ax . . and L t h e Lagrangian d e n s i t y . I n continuum mechanics L=T-W where
.
T=1/2p0(X)$’ i s t h e k i n e t i c energy p e r u n i t volume i n R 3 and W t h e deformation energy p e r u n i t volume i n R 3 . F o l l o w i n g N o l 1 , f o r s i m p l e e l a s t i c m a t e r i a l s
W=W(X,$,,) whereas f o r nonsimple e l a s t i c m a t e r i a l s W=W(X,$,,, Q’LK.. . , G * L ~ L ’ . . .iN). I f t i i s t h e o r d e r of t h e h i g h e s t g r a d i e n t a c t u a l l y p r e s e n t as an argument o f W, t h e c o r r e s p o n d i n g m a t e r i a l i s c a l l e d t h e m a t e r i a l o f grade N. The t h e o r y of e l a s t i c m a t e r i a l s o f grade 2 has been c o n s i d e r e d by Toupin [l]. L e t a r e g u l a r s u r f a c e C ( t ) moving i n t h e domain D a t t i m e t d i v i d e D i n t o t h e subdomains D + ( t ) and D - ( t ) . The s i d e o f t h e s u r f a c e which i s i n t o u c h w i t h t h e r e g i o n D + ( t ) i s l a b e l e d w i t h a s u p e r s c r i p t (+] w h i l e t h e o t h e r s i d e i s l a b e l e d w i t h ( - ) . F o r a g i v e n f u n c t i o n $(X,t) XED l e t @ and $- denote t h e f i n i t e values o f 4 a t any p o i n t o f t h e s u r f a c e C ( t ) which i s approached from D + ( t ) and D - ( t ) respect i v e l y . I t i s assumed i n g e n e r a l , t h a t t h e dynarnical v a r i a b l e @ of (1.1) i s cont i n u o u s i n D, d i f f e r e n t i a b l e i n D + ( t ) , D - ( t ) and $+#$-, ($,L)+#($,L)-. . . e t c , i . e . C ( t ) i s t h e s i n g u l a r s u r f a c e s r e l a t i v e t o t h e f i e l d $. I n o r d e r t o o b t a i n t h e v a r i a t i o n o f f u n c t i o n a l (1.1) f o r such a c l a s s o f f u n c t i o n $ one has t o express variational quantities
154
J. Turski
by surface operators: surface d i f f e r e n t i a l s , normal variation of s u r f a c e , . . . e t c . From these expressions one can get the variational conditions of compatibility. These conditions a r e complementary t o those known in t h e theory of s i n g u l a r surfaces, i .e. geometrical and kinematical conditions of compatibility. Note t h a t the concepts of t h e theory of s i n g u l a r surfaces and i t s applications t o continuum mechanics are a v a i l a b l e in numerous references, f o r example in [21,[31,[41.
The generalization presented here of t h e c l a s s i c a l Weierstrass-Erdrnann conditions 151 of the variational problem a r e t h e dynamical compatibility conditions i .e. conditions f o r jump of momentum a n d energy across s i n g u l a r surfaces C ( t ) . The general f e a t u r e of t h i s formulation of the dynamical conditions f o r t h e propagat i o n of s i n g u l a r surfaces (shock waves and acceleration waves) in nonsimple e l a s t i c materials a r e i l l u s t r a t e d s u f f i c i e n t l y well by materials o f grade 2 and t h e analysis of higher grade materials i s only t h a t much more complicated in d e t a i l . 2.
VARIATIONAL CONDITIONS O F COMPATIBILITY
Let RcR3 be a n open connected s e t and UcR2 an open s e t . The points o f R and U a r e denoted by ( X L ) , L=1,2,3 and ( u a ) a = 1 , 2 , respectively. Assume t h a t the mapping 3 x : U x T + R xL = xL( ua t )
(2.1
1
9
i s the one-parameter family of regular surfaces which a r e o r i e n t a b l e and f o r every t of the interval T divide R i n t o two non-empty subsets d ( t ) and Q - ( t )i . e . R=R*(t)UR-(t)UX(t). The p o s i t i v e unit normal NL t o E f t ) i s directed toward t h e region n*(t). For s i m p l i c i t y t h e s e t of p o i n t s of C ( t ) will be a l s o c a l l e d a surface. Let a mapping
€1
(-E,
3
s
&--
x(u,t,s)
E
R
x ( u . t . 0 ) :x ( u , t ) be a deformation of the surface ( 2 . 1 ) . The variation of the surface C i s defined as the d i f f e r e n t i a l of ( 2 . 2 ) a t s=O, i . e .
The normal variation of t h e surface (2.1) i s defined by L
Ix
=
L K N NKdx
For a given surface, tensor-valued functions ation defined by d @ = 9’s
i.e.
1
s=o
@(x L( ua , t , s ) , t , s ) = @ ( au , t , s )
have vari-
Singular Surface Propagation in Nonsimple Elastic Materials
155
The normal v a r i a t i o n o f t h e s u r f a c e t e n s o r @ i s d e f i n e d by
Note t h a t q u a n t i t i e s (2.4) and (2.5) a r e independent o f t h e c h o i c e o f t h e s u r f a c e parametrization. I n g e n e r a l , if t h e surface p a r a m e t e r i z a t i o n i s n o t t h e normal one, t h e n i t f o l l o w s t h a t
L L i a 6x ( 6 s ) = 6X (6s) + x adu 8@(6s) = 6 @ ( S s ) t @;a6ua
where @;a = @,L
x,,L
.
L t Consider a t e n s o r - v a l u e d f u n c t i o n @(X , t ) , XEQ which i s d i f f e r e n t i a b l e i n R ( t ) , R - ( t ) , a n d has t h e f i n i t e values $t and @- a t any p o i n t X k C ( t ) which i s approached from Q t ( t ) and 0 - ( t ) , r e s p e c t i v e l y . I n general @ has a jump across I ( t ) u s u a l l y denoted by
so [@I i s a f u n c t i o n o f p o s i t i o n on t h e s u r f a c e C ( t ) and t i m e t. When [I@n#O o r some d e r i v a t i v e s o f @ a r e d i s c o n t i n u o u s across C ( t ) , t h e n r e l a t i v e t o t h e f i e l d @ t h e s u r f a c e i s c a l l e d s i n g u l a r . Using t h e n o t a t i o n
K K K where D L @ = ( ~ L - N L N )@,L i s t h e surface g r a d i e n t o f @ and D@=N$ , K t h e normal grad i e n t of @, t h e n by Hadamard's Lemma: (DL@)*=DL@*, t h e f o l l o w i n g c o n d i t i o n s a r e obtained
K K where BL=-DLN i s t h e second fundamental form o f t h e surface C. From t h e d e f i n i t i o n o f t h e normal v a r i a t i o n o f t h e s u r f a c e t e n s o r s ( 2 . 5 ) t h e f o l l o w i n g c o n d i t i o n s a r e obtained
156
J. Turski
( 6 $ 1 , ~ ) ' = DL(b@')
-
DL{(D$)'8E1 + NLB(D@)'
JK
J K i
+ N ~ S D ~ ( ~ c ) D -~ $NL(@,JKN + N where 8C=NL6x
a
(2.9)
L
On t h e b a s i s o f e q u a t i o n s ( 2 . 8 ) and ( 2 . 9 ) t h e c o n d i t i o n s which a r e c a l l e d t h e v a r i a t i o n a l c o n d i t i o n s o f c o m p a t i b i l i t y a r e f o r m u l a t e d as f o l l o w s
(2.11) Note t h a t i f t h e parameter o f t h e d e f o r m a t i o n i s t i m e , t h e n r e l a t i o n s (2.10) and (2.11) become t h e well-known k i n e m a t i c a l c o n d i t i o n s o f c o m p a t i b i l i t y o f o r d e r one and two, r e s p e c t i v e l y . 3.
VARIATION
OF FUNCTIONALS I N A DOMAIN WITH THE PROPAGATING SINGULAR SURFACE
Assume t h a t t h e dynamical v a r i a b l e con t inuous f u n c t i o n R
3
3 Q 3X
L
i
-x
= $I
i
L ( X ,t)
9 o f the f u n c t i o n a l (1.1) i s a vector-valued,
E
E
3
where E 3 i s E u c l i d e a n space w i t h s t a n d a r d c o o r d i n a t e s . Consider a system (D,C,$) where C i s t h e s i n g u l a r s u r f a c e r e l a t i v e t o t h e v a r i a b l e $ i n t h e f i n i t e domain D c R . For t h e v a r i a t i o n o f t h e f u n c t i o n a l (1.1) t h a t i s g i v e n by 6L =
w ds j s=o
one can w r i t e
where dA i s a E u c l i d e a n area element o f C i n R 3 . By assuming f o r convenience i n t h e p r e s e n t a n a l y s i s , t h a t C ( t ) , t r T a r e c l o s e d s u r f a c e s such t h a t C ( t ) c D and by u s i n g r e l a t i o n s (2.8) and ( 2 . 9 ) as w e l l as t h e i n t e g r a l i d e n t i t y :
i DLfdA
c
=
- 1
z
K NLfBkdA
which h o l d s under above assumptions, t h e f o l l o w i n g r e s u l t s a r e o b t a i n e d :
Singular Surface Propagation in Nonsimple Elastic Materials
(A)
The e q u a t i o n ( 3 . 1 ) w i t h t h e Lagrangian o f t h e f o r m L(X,$,
becomes
dV 1 t2
J’
f
I)J,~)
157
D
It1
i 8 aL where E ( L ) = -(T)
a t oqi
aL - (-----) ’+i,L
,
,L
a
~
.
a r e t h e E u l e r e q u a t i o n s , U ( N ) = N L x x i s t h e normal v e l o c i t y o f C i n R 3 and t h e f o l l o w i n g n o t a t i o n s have been used 3.3)
3.4)
i s a q u a n t i t y independent o f t h e c h o i c e o f t h e s u r f a c e p a r a m e t r i z a (N) t i o n , and bri, and 81 a r e m u t u a l l y independent v a r i a t i o n s . A c c o r d i n g t o Du BoisReymond’s Lemma t h e v a r i a t i o n o f L g i v e n by ( 3 . 2 ) vanishes f o r a l l v a r i a t i o n s 6$ i such t h a t 6 i i I =0, =O if and o n l y i f
Note t h a t U
I
The e q u a t i o r s b ) and c ) of ( 3 . 5 ) a r e t h e g e n e r a l i z e d Weierstrass-Erdmann c o n d i t i o n s .
(B)
The e q u a t i o n (3.1) w i t h t h e Lagrangian o f t h e form L(X,$,$,L,$,LK)
6~ =
I 1 T D
E i ( L ) ~ $ i dVdt -
Ii T aD
(Tic‘@i
f
HiD6Gi)dAdt
becomes
J. Turski
158
where
i a aL E ( L ) =-(-) a t a+i
2L - [-aqi,L
1 1
aL
( p
aqi,LK ,K ,L
a r e E u l e r e q u a t i o n s and i T
iL
iLKN
T NL - DL(H
=
- Hi BK K
I n t h e above e x p r e s s i o n s t h e f o l l o w i n g n o t a t i o n s have been i n t r o d u c e d (3.9) (3.10)
(3.11)
According t o Du Bois-Reymond's.Lemrna t h e v a r i a t i o n o f L q i v e n by e q u a t i o n ( 3 . 6 ) vanishes f o r a l l v a r i a t i o n s 6JI' such t h a t 0 1 ) ~ = 0, D?J$~ = 0, 6$i) = 641 =
I,,
i f and o n l y i f
t2
tl
o
a)
E ~ ( L =)
b)
[Ti
c)
[(Ti+PiU(NjD$il
d)
[Hill
+ PiU(N)rI = 0
=
o
Hi+ = Hi-
+ ULI + fHi$i,LKNLNK]
if = 0
uoq if
= 0
(3.12)
o
=
iWill #
0
The c o n d i t i o n s b ) c ) and d) o f (3.12) a r e t h e g e n e r a l i z e d Weierstrass-Erdmann c o n d i t i o n s f o r t h e c o r r e s p o n d i n g v a r i a t i o n a l problem. 4.
DYNAMICAL CONDITIONS OF COMPATIBILITY I N MATERIALS OF GRADE 2
The m o t i o n o f QcR3i n t h e p h y s i c a l space E 3 ( E u c l i d e a n space w i t h s t a n d a r d c o o r d i n a t e s ) i s a one-parameter f a m i l y o f homeomorphisms
R
3X-
x = $,(X)
= @(X, t )
E
E
3
(4.1)
o
159
Singular Surface Propagation in Nonsirnple Elastic Materials
where R i s c a l l e d t h e space o f m a t e r i a l p o i n t s . Consider t h e system ( Q , X , $ ) where R i s t h e m a t e r i a l o f grade 2 and C i s t h e s i n g u l a r s u r f a c e r e l a t i v e t o t h e m o t i o n I+ o f s1. The dynarnical p r o p e r t i e s o f such syst i m e a r e p r e s c r i b e d by t h e f u n c t i o n a l
By t h e r e s u l t (B) o f t h e paragraph 3 t h e Weierstrass-Erdrnann c o n d i t i o n s i n (3.12)
f o r t h e f u n c t i o n a l ( 4 . 2 ) become
i i Here T and H a r e g i v e n by e x p r e s s i o n s ( 3 . 7 ) and ( 3 . 8 ) , r e s p e c t i v e l y , i n which t h e t e n s o r ( 3 . 9 ) g i v e n by
i s t h e g e n e r a l i z e d K i r c h h o f f s t r s s s t e n s o r and t h e t e n s o r (3.10) g i v e n by HiLK -
aw
(4.7)
a@i ,LK
i s the hyperstress tensor. By a p p l y i n g i n e q u a t i o n s ( 4 . 3 ) - ( 4 . 5 ) t h e formula f o r energy
and t h e f o l l o w i n g k i n e m a t i c a l c o n d i t i o n s o f c o m p a t i b i l i t y
one can c o n s i d e r two cases:
(i)
f o r t h e s i n g u l a r surface such t h a t [$,n#O, w a v e f r o n t one has
i . e . i n t h e case o f a shock
160
J. Turski
( i i ) f o r t h e s i n g u l a r s u r f a c e such t h a t IJ$iJ=O, 7.e. i n t h e case o f an a c c e l e r a t i o n wavefront one has [Tin
=
0
,
aHil
= 0
(4.11)
(4.12) The e q u a t i o n s ( 4 . 8 ) - ( 4 . 1 2 ) a r e t h e dynarnical c o n d i t i o n s o f c o m p a t a b i l i t y f o r t h e c o r r e s p o n d i n g cases. The e q u a t i o n s (4.8) - (4.12) can be w r i t t e n i n t h e s p a t i a l f o r m by u s i n g d e s c r i p t i o n i n t h e p h y s i c a l space E 3 . hus, l e t as i s usual (4.13) h j i k = J-l HiLK j
k
P y hK
(4.14)
be t h e g e n e r a l i z e d Cauchy s t r e s s t e n s o r and t h e h y p e r s t r e s s t e n s o r i n t h e space i Here J=det(q,,-) i s a J a c o b i a n . U s i n g r e l a t i o n s ( 4 . 1 3 ) and i (4.14), t h e s p a t i a l f o r m s of Ti and H t h a t a r e g i v e n by ( 3 . 7 ) and ( 3 . 8 ) , respect i v e l y , a r e o b t a i n e d by E3, respectively.
(4.15) hji = h j i k nk
(4.16)
where ti
= T
i dA
(4.17)
i s t h e t r a c t i o n p e r u n i t area i n E 3 and (4.18) i s t h e s u r f a c e q u a n t i t y whose mechanical meaning w i l l be discussed a t t h e end o f t h i s s e c t i o n . I n t h e above n . i s t h e u n i t normal t o t h e s i n g u l a r s u r f a c e u and da i s t h e E u c l i d e a n area e l e m e n t ’ o f u where u i s t h e image of t h e surface C g i v e n by m o t i o n $. The t e n s o r b i j which appears i n (4.15) i s t h e second.. fundamental f o r m of t h e surface ci and d.hJi denotes t h e s u r f a c e d i v e r g e n c e o f hJ’. Note t h e d i f J f e r e n c e between d e f i n i t i o n s (4.17) and (4.78) g i v e n above and t h e d e f i n i t i o n s (10.17) i n [l]. Using e q u a t i o n s (4.15)
-
(4.18) t h e dynarnical c o n d i t i o n s o f c o m p a t i b i l i t y f o r t h e
Singular Surface Propagation in Nonsirnple Elastic Materials
161
c o r r e s p o n d i n g cases ( i ) and ( i i ) have s p a t i a l forms as f o l l o w s (iv)
on a shock w a v e f r o n t
nup$!
=
- Itin
(4.19)
nuel = -
utiq
(4.20)
(hJin.)+ = J
ihJi
o
nj)- =
(4.21)
( i i v ) on an a c c e l e r a t i o n w a v e f r o n t ntin
=
unen =
o ,
-
[hJin.n J
.. hJ'l$.
I n t h e above p = J - ' p o
=
0
(4.22)
.I
(4.23)
1 ,J
i s t h e mass p e r u n i t volume i n t h e space E 3 and
i s t h e energy p e r u n i t valume i n E 3 and
u=
yn)-
.i
$ ni
i s t h e normal component o f t h e v e l o i s t h e l o c a l speed o f p r o p a g a t i o n where u (n) c i t y o f t h e s u r f a c e a. If i t i s assumed t h a t t h e d e f o r m a t i o n energy W(X,$,L, that
$,)
has t h e f o r m such
(4.24)
where parentheses e n c l o s i n g i n d i c e s i and j i n d i c a t e s y m m e. .t r i z a t i o n w i t h r e s 7 e c t t o t h e s e i n d i c e s , t h e n i t f o l l o w s t h a t i n (4.23) one has hJ1=-hiJ e q u a t i o n becomes
ilcre
W.
.=-w..
1J
J1
and t h i s
i s t h e v o r t i c i t y t e n s o r t h a t i s g i v e n by
1 ' w.. = -($ ij
2 i , j - 'j,i)
.. and. hJ' i s i n t e r p r e t e d as a c o u p l e - t r a c t i o n p e r u n i t a r e a o f t h e s u r f a c e a. (See e q u a t i o n ( 4 . 1 8 ) ) . The r i g h t hand s i d e o f e q u a t i o n ( 4 . 2 5 ) i s t h e n e g a t i v e o f t h e jump o f r a t e o f w o r k i n g o f t h e c o u p l e - t r a c t i o n . Note t h a t t h e general form o f t h e
162
J. Turski
d e f o r m a t i o n energy W(X,+,,,
@,,,I such
t h a t ( i ) c o n d i t i o n (4.24) and ( i i ) r o t a -
t i o n a l i n v a r i a n c e o f W h o l d , was d i s c u s s e d i n [l]. I n g e n e r a l , e q u a t i o n (4.23) can be w r i t t e n as
.. where d . . = l / Z ( $
.+$. . ) i s t h e d e f o r m a t i o n r a t e t e n s o r and mJi=hCJ1’,
i , J .J,1 symmetric p a r t o f hJ1, i s t h e c o u p l e - t r a c t i o n . 1J
*
i.e. theanti-
T h i s i n v e s t i g a t i o n was c a r r i e d o u t w h i l e a u t h o r was r e s i d e n t i n t h e Department o f Mathematics a t M c G i l l U n i v e r s i t y .
REFERENCES Toupin, R.A. T h e o r i e s o f E l a s t i c i t y w i t h Couple-stress. Mech. Anal. 17 (1964) 85-112.
Arch. R a t i o n a l
The C l a s s i c a l F i e l d T h e o r i e s . Handbuch d e r T r u e s d e l l , C. and Toupin, R.A. P h y s i k , V o l . III/l ( S p r i n g e r - V e r l a g , B e r l i n and New York 1960). Thomas, T.Y. Concepts f r o m Tensor A n a l y s i s and D i f f e r e n t i a l Geometry. (Academic Press, New York, London, 1961). Eringen, A.C. and Suhubi, E.S. New York, London 1974) Gelfand, 1963).
I.M. and Fomin, S.V.
Elastodynamics, Vol. 1 (Academic Press, C a l c u l u s of V a r i a t i o n s ( P r e n t i c e - H a l l , I n c .
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
163
FORCED INTEGRABLE SYSTEMS David J. Kaup Clarkson Colleqe of Technology Potsdam, New York 13676 U.S.A.
A discussion is given of boundary value problems for nonlinear integrable systems. The forced nonlinear Schroedinger equation and the forced Toda lattice are used as examples. It is demonstrated that one can quite well solve such systems by using reasonable approximations. As an example, the birthrate for solitons in the forced Toda lattice is solved approximately and it is shown that the result compares quite favorably with the actual birthrate. ITJTRODUCTION It has now been some tine (Gardner et al. (1967)), (Gardner et al. (1974)) since we have been able to solve one-dimensional evolution equations. And it also has been some time, albeit shorter, since we have been able to solve integrable multidimensional problems (Ablowitz an$ Haberman (1975)), (Kaup (1981)) , (Fokas and Ablowitz (1984)). So where do we go from here? One way is to study integrable systems when they are being forced by localized sources. And that is what I wish to report on here. In a sense, this i s not really new. There are already examples of systems similar to forced systems being solved, although the forced nature of the system was not necessarily obvious earlier on. What I shall attempt to do first is to discuss three of these examples and to explain why they are analogous to forced integrable systems. For example, the phenomenon known as "self-induced transparency" ( S I T ) (PlcCall and Hahn (1969)), (Lamb (1973)),(Kaup (1977)) can be considered. to be a forced system. To make this clearer, consider Figure 1 which illustrates the geometry of the SIT system. Along the horizontal axis lies the material which is composed of simple two-level atoms. From the left comes an electromagnetic wave whose frequency can resonate with the two-level atoms. The standard manner for solving this problem is to solve an eigenvalue problem in time for the scattering data, using the envelope of the incident electromagnetic field as a potential. Thus one maps the electromagnetic field envelope into a scattering space, which is analogous to taking a Fourier transform. Now one can propagate this scattering data forward in space in the standard manner, and how this scattering data evolve in space is not a simple harmonic motion as in the case of the KdV equation or the nonlinear Schroedinger equation. Rather its exact evolution depends on the initial state at t = 0 of these two-level atoms. Even in the case where all the atoms are considered to be initially in the ground state, the time-dependence of this scattering data is not simple. As it evolves in x, the continuous spectrum will
164
D.J. Kaup
t {Vacuum
}
Figure 1 The geometry of the SIT system. A coherent electromagnetic pulse is directed from the left onto the two-level atoms located in the shaded half-space x > 0. The final state of the atoms will be determined by the envelope, b(x,t-x/v ) , of the EM pulse and the initial state of 9 the atoms. v is the group velocity of the EM pulse. g
be selectively absorbed. Thus one may consider the initial state of the atoms to be "forcing" the evolution of the scattering data. In other words, the evolution of the scattering data is being determined by a localized (here, at t = 0) quantity. The reader is warned not to become confused by the interchange of space and time in the treatment of the SIT system. Normally, as in solving the nonlinear Schroedinger equation or the KdV equation, one would solve an eigenvalue problem in space, and evolve it forward in time. But in solving the S I T system, we solve an eigenvalue problem in time and evolve it forward in space. This method is used simply because we do not know how to solve the much more complicated spatial eigenvalue problem of SIT. But let me point out that when one considers general forced systems, one must be prepared to tackle such ’complicated’ eigenvalue problems. More shall be said on this later. For now, let me give a couple more examples. Another example is "simple harmonic motion" (Kaup (1978)). This system consists of a fundamental wave and its harmonic interacting in a nonlinear medium. When the group velocities of the fundamental and the harmonic are significantly different, then the problem becomes a two-dimensional problem where the natural coordinates are the characteristic coordinates. This case is illustrated in Figure 2 where the fundamental wave propagates along the horizontal axis (x) and the harmonic propagates along the tilted axis ( T ) . This system is solvable by using an inverse scattering transform (IST) in the X-direction, where one would solve for the scattering data from the X-eigenvalue
Forced Integrable Systems
165
Figure 2 The geometry of the simple harmonic generation system. The envelope fundamental, ql, propagates along the X-axis while the envelope of the harmonic, q 2 , propagates along the tilted -r-axis. The harmonic is generated by the nonlinear interaction of the fundamental with itself. In this figure, the wavy lines indicate the general direction of flow of the two types. In the above example, no harmonic (q2) is present initially (at T = 0 ) . problem, and then propagate it forward in T. In this case, the initial envelope of the harmonic, if any, is the initial data. The initial envelope of the fundamental (its value along x = 0 ) then acts as a forcing agent on the scattering data, forcing it to change as it evolves in T . Again we have the evolution of the scattering data being determined by another or second envelope. Thus this second envelope can be said to be forcing the system. This forcing term is localized along the -r-axis. And this is what I mean by forcing. The forcing term is to be localized. Better examples will follow. But I wished to bring out this example because it is similar to SIT and the next example which is stimulated Raman scattering (SRS) (Kaup (1983)), (Steudel (1983)) . The SRS system and the similar Two-Photon Propagation system have been solved more recently. This system has a very high degree of symmetry between the time and the space coordinates. In fact, one can exchange the two and the equations remain form-invariant. As in SIT, we have an incident electromagnetic wave and a material composed of two-level atoms. But now the interaction is second-order, and we also have a second electromagnetic wave called the Stoke s wave. As in the SIT system we can consider the incident electromagnetic wave to be incident from the left and then to be forcing the SRS system to respond. But most significantly, the method of solution of the SRS system requires one to solve two eigenvalue problems. Thus, there are two IST s involved instead of only one. Now this system is one step up from the previous two examples. In those examples, the evolution of the scattering data could be determined by a simple
166
D.J. Kaup
quadrature. But in the SRS system, one uses the first IST to determine the initial scattering data. Then a second IST is required for determining how these initial scattering data are to be evolved. Note that the time evolution of this system is not of the conventional type. The time-dependence of the initial scattering data is determined by a second independent set of scattering data. And in this respect, the SRS system resembles and really is a forced integrable system. However, it is also a very special forced system. In fact, it is so special that one could even say that it was a degenerate forced system, and it is exactly this degeneracy which allows this system to be solved so easily. This degeneracy arises because in order to specify the time-dependence of the scattering data, one needs to specify only quantity (the polarization state of the atoms) and no more. This comes from the fact that the SRS equations are only first-order partial differential equations. On the other hand, if they were of a higher order, then more than one quantity would have to be specified in order to determine the time-dependence of the scattering data. But as we shall see later, these profiles would then be independent and thus one would be immediately into a nonlinear consistency problem, and it is this consistency problem which has always stifled other previous attempts to treat forced integrable systems .
one
AN EXAMPLE
-
THE FORCED NONLINEAR SCHROEDINGER EQUATION
To illustrate the last comment, let us look at an example. take the nonlinear Schroedinger equation (NLS)
Let us
how since the nonlinearity is not essential for this example, we shall first omit it and review the linear solution. Let us take q to be driven by some source at x = 0 where for t > O , q(O,t) = Q(t)r
2
and let the initial value for x
(2)
0 be
q(x,Of = S(X).
(3)
As is known, the specification of these two functions uniquely determingthe solution of the linear equation. An outline of the linear solution will be of value for understanding what is required for the nonlinear solution. So we start with expressing q(x,t) for x 2 0 as a Fourier transform m
l l
q(x,t) = 2Tr
-m
A(k,t)eikxdk,
where we take A to be given by m
A(k,t) =
1
q(x,t)e 0
From iqt = -qxx, it follows that
- ikxdx
(4)
167
Forced Integrable Systems
2 iA - k A = P(t) t where Q(t) is given by Eq.(2) and
+
ikQ(t),
(7)
P(t) = qx(O,t).
Now note that, according to Eq.(6), it seems as though we may need to independently specify P as well as Q and S in order to determine how A evolves in time. But such is not the case. Instead, one can show (for the linear case at least) that P(t) is indeed dependent on Q(t) and S(x). To demonstrate this, the solution of Eq. (6) is
where by Eqs.(3) and ( 5 1 , m
A(k,O) =
1
S(x)e-ikxdx. 0
Now from Eqs.(4) and (7) P(t) =
&
m
ikA(k,t)dt. -m
which from Eq.(8) and due to the symmetry in the k-integral, reduces to m
+
&
m
lmk2dk ltdsQ ( s )exp [ik2 (s-t)1
,
LII
-m
,
’0
which gives P(t) wholly in terms of S and Q. So, although Eq.(6) may require that one knows what P(t) is, one can eventually evaluate P(t) in terms of S and Q. Now, what changes occur when we consider the nonlinear case? Well, similar to Eq.(9), we may map the initial data, S(x), into the scattering data. This is accomplished by solving for the t = 0 eigenfunctions of the Zakharov-Shabat (ZS) equations vlx
+
icvl = S(x)v2,
(lla)
between x = 0 and x = +a. From the values of these eigenfunctions at x = 0 and as x + m , one can then construct the scattering data (Ablowitz et al. (1974)). The scattering data are the nonlinear generalizations (Ablowitz et al. (1974)) of the Fourier amplitude, A(k), defined by Eq.(5). To determine the time-dependence of the scattering data, we must solve the nonlinear generalizations of Eq.(6), which are the "ABC" equations. These equations describe how the ZS eigenfunctions evolve in time, and for the NLS they are
168
D.J. Kaup
These e q u a t i o n s a l s o d e t e r m i n e t h e e v o l u t i o n o f t h e s c a t t e r i n g d a t a s i n c e t h e s c a t t e r i n g d a t a a r e d e f i n e d i n t e r m s of t h e s e eigenfunctions. For t h e t i m e e v o l u t i o n o f t h e s c a t t e r i n g d a t a , w e o n l y r e q u i r e E q . ( 1 2 ) e v a l u a t e d a t x = 0 and f o r x + +a. S i n c e w e demand the f i e l d s to vanish a s x +m, this part is trivial. On t h e o t h e r hand, a t x = 0 , Eq. ( 1 2 ) becomes -f
2
5
=
iv
= +(2i5Q* + P*)vl
2 t
(25
Q*Q)v,
+
ivlt
-
(2icQ
-
-
P)v2
(2c2 + Q*Q)v2
(13b)
w i t h P ( t ) and Q ( t d ) e f i n e d a s b e f o r e i n Eqs. ( 2 ) a n d ( 7 ) . T h i s i s c e r t a i n l y n o t s o t r i v i a l t o s o l v e a s was E q . ( l O ) . One s h o u l d n o t e t h e a n a l o g y between Eqs. ( 6 ) a n d ( 1 3 ) . Each d e p e n d s on P and Q. But as w e have a l r e a d y d e m o n s t r a t e d , P i s n o t i n d e p e n d e n t of Q and S i n t h e l i n e a r case. Presumably t h e s a m e i s t r u e i n t h e n o n l i n e a r case, n.nd o n e now h a s t h e problem o f how c a n o n e d e t e r m i n e P i n t h e nonl i n e a r c a s e , g i v e n Q and S. Although a f i n a l answer i s s t i l l t o b e g i v e n , p r e l i m i n a r y c a l c u l a t i o n s have a l r e a d y v e r i f i e d t h a t d i r e c t l y from t h e a n a l y t i c p r o p e r t i e s o f t h e ZS e i g e n f u n c t i o n s one may d e t e r mine a l l t h e i n i t i a l d e r i v a t i v e s o f P ( t ) , namely P t ( 0 ) , P t t ( 0 ) , .
. ..
So i t i s e x p e c t e d t h a t it w i l l be o n l y a matter o f t i m e b e f o r e w e w i l l b e a b l e t o r e c o v e r P ( t ) i t s e l f d i r e c t l y from S and Q .
THE FORCED TODA LATTICE AND THE BIRTHRATE OF ITS SOLITONS Although w e e x p e c t e v e n t u a l l y t o b e a b l e t o o b t a i n P ( t ) d i r e c t l y f r o m Q ( t ) a n d S ( x ) , t h e r e q u i r e d p r o c e d u r e m i g h t be s u f f i c i e n t l y complex so a s t o be a l s o u s e l e s s from a n u m e r i c a l o r a p p l i e d p o i n t of view. I n t h i s c a s e , w e would want t o a s k w h e t h e r o r n o t t h e r e i s some o t h e r method whereby w e c o u l d b y p a s s h a v i n g t o d e t e r m i n e P ( t ) . I t t u r n s o u t t h a t one c a n d o t h i s t o a c e r t a i n d e g r e e and t h a t i n many cases t h i s d e g r e e might b e q u i t e a d e q u a t e . The b a s i c i d e a i s t o r e a l i z e t h a t t h e o n l y t h i n g p r e v e n t i n g u s from o b t a i n i n g t h e time-dependence of t h e s c a t t e r i n g d a t a i s knowing what P ( t ) i s . So i f w e c o u l d make a r e a s o n a b l e g u e s s a s t o what P ( t ) w a s , t h e n w e c o u l d a p p r o x i m a t e P ( t ) by u s i n g t h i s g u e s s , and t h e n d i r e c t l y s o l v e f o r t h e t i m e dependence of t h e s c a t t e r i n g d a t a . Of c o u r s e , t h i s would o n l y b e a n a p p r o x i m a t e s o l u t i o n , b u t it c o u l d w e l l be a d e q u a t e and c o u l d a l s o p e r h a p s g i v e some i n s i g h t i n t o t h e a c t u a l s o l u t i o n . A t t h e moment w e h a v e no r e s u l t s f o r t h e NLS;
however, w e do h a v e s u c h n u m e r i c a l r e s u l t s f o r t h e f o r c e d Toda l a t t i c e (Kaup ( 1 9 8 4 ) ) , (Kaup and Neuberger ( 1 9 8 4 ) ) . So I s h a l l now d i s c u s s t h o s e r e s u l t s and u s e them t o i l l u s t r a t e s u c h a p p r o x i m a t e methods.
By t h e f o r c e d Toda l a t t i c e , I mean t h e s e m i - i n f i n i t e ( n = 0 , 1 , 2 , . . . ) Toda l a t t i c e (Toda ( 1 9 7 0 ) ) where t h e z e r o t h p a r t i c l e i s c o n s t r a i n e d These e q u a t i o n s a r e t o move i n some p r e s c r i b e d manner.
Qn = exp(C?nn-l-Qn) where n = 1,2,3,
...,
-
exp(Qn-Qn+l)
Q o ( t ) i s t o b e g i v e n and a s n
-+
+m,
Qn
-+
0.
169
Forced Integrable Systems
B (t)is the forcing term. T h i s l a t t i c e h a s b e e n s t u d i e d a s a model 0 f o r " m o l e c u l a r d y n a m i c s " ( H o l i a n and S t r a u b ( 1 9 7 8 ) ) . I n t h i s c a s e , Then a t t = 0 , one g i v e s t h i s o n e l e t s Qo ( t ) b e z e r o f o r t < 0 . z e r o t h p a r t i c l e a l a r g e f o r w a r d and c o n s t a n t v e l o c i t y . This causes t h e z e r o t h p a r t i c l e t o ram i n t o t h e o t h e r p a r t i c l e s , t h e r e b y c r e a t i n g a s h o c k wave. The s i z e o f t h i s s h o c k wave d e p e n d s on t h e v i o l e n c e w i t h which t h e z e r o t h p a r t i c l e i s rammed i n t o t h e l a t t i c e . T h i s s h o c k wave h a s a s i z e which a l s o i n c r e a s e s l i n e a r l y i n t i m e . The s h o c k i s s u p e r s o n i c and c o n s i s t s m o s t l y o f s o l i t o n s ( H o l i a n e t a l . ( 1 9 8 1 ) ) . And a s i t e v o l v e s , more a n d more s o l i t o n s a r e formed i n t h e s h o c k . - A t y p i c a l example o f a wide s h o c k i s shown i n F i g u r e 3 where b (= -?jQn-l) i s shown v s n . The q u a n t i t y b l ( = -+ao) i s j u s t t h e n n e g a t i v e o f o n e - h a l f o f t h e v e l o c i t y w i t h which t h e z e r o t h p a r t i c l e i s b e i n g rammed i n t o t h e l a t t i c e . The s h o c k i s t h e r e g i o n from What one would l i k e t o d o i s t o be a b l e t o n = 75 up t o n = 1 4 0 . p r e d i c t t h i s s t r u c t u r e . A s a s t a r t , w e s h a l l a d d r e s s simply t r y i n g t o p r e d i c t t h e b i r t h r a t e of t h e s o l i t o n s i n t h i s s h o c k wave. (I s h o u l d p o i n t o u t t h a t from l o o k i n g a t F i g u r e 3 i t i s not a t a l l o b v i o u s t h a t t h e r e a r e s o l i t o n s p r e s e n t i n t h e shock. That i s t r u e b e c a u s e t h e y h a v e b e e n rammed t o g e t h e r i n t o a n a p p a r e n t i n c o h e r e n t lump. However, t h e y a r e t h e r e ( H o l i a n a n d S t r a u b ( 1 9 7 8 ) ) , and i n t h i s one shock, t h e r e a r e a b o u t 60 s o l i t o n s p r e s e n t . ) ~
~
_.
0 1
t = 64.0
b,= -1.95
t '
I
Figure 3 P l o t of bn v s t i n t h e f o r c e d Toda l a t t i c e f o r bl = -1.95 where -2bn+l
i s t h e v e l o c i t y of t h e nth p a r t i c l e .
envelope s t r u c t u r e t o t h e r i g h t .
a t t
=
64.0
Note t h e r e g u l a r
D.J. Kaup
170
The r e q u i r e d m a t h e m a t i c s f o r t h i s a n a l y s i s is t o be p u b l i s h e d i n J . Math. Phys. (Kaup (1984)) , (Kaup and Neuberger (1984)), s o h e r e I s h a l l o n l y d e s c r i b e t h e s e c a l c u l a t i o n s and g i v e t h e m a j o r r e s u l t s . I n o r d e r t o d e t e r m i n e t h e time-dependence o f t h e Toda l a t t i c e scatt e r i n g d a t a , one must s o l v e t h e e q u a t i o n
where
x
='i(z
+
1
-).
(16)
I n E q . ( 1 6 ) , z i s t h e e i g e n v a l u e f o r t h e Toda l a t t i c e s c a t t e r i n g d a t a . 1. The s o l i t o n s p e c t r u m o c c u r s a l o n g t h e r e a l z - a x i s where I z I (The c o n t i n u o u s s p e c t r u m o c c u r s on t h e u n i t c i r c l e i n t h e complex z - p l a n e where I z I = 1 . ) Equation required t i o n can (15) w e
(15) i s e x a c t l y a n a l o g o u s t o E q . ( 6 ) and a s i n E q . ( 6 ) , w e are t o know o n e a d d i t i o n a l p i e c e o f i n f o r m a t i o n b e f o r e a s o l u be o b t a i n e d . I n Eq.(6) w e h a d t o know P(t). Here i n Eq. a r e r e q u i r e d t o know (Qo- Q1) ( t ) i n s t e a d . W e d o know w h a t
Q o ( ti)s b e c a u s e t h e z e r o t h p a r t i c l e i s f o r c i n g t h e s y s t e m . B u t w e do n o t know b e f o r e h a n d what Q , ( t )i s . However, w e may a p p r o x i m a t e it. And t o see how b e s t t o a p p r o x i m a t e i t , l e t u s l o o k a t F i g u r e s 4 and 5 2 where w e p l o t 4 a 2 = e x p ( Q - Q ) a s a f u n c t i o n of t f o r d i f f e r e n t
0
1
v a l u e s of bl.
-
2.a-
1
-
bd
=
-0.5
1.5;
-
4a;
1 1.0-
-
-
a.5-
-
8.0
~
E
i n i t i a l ringing.
l
l
I0
l
20
~
l
30
l
l
40
l
[
50
l
l
60
l
~
~
171
Forced Integrable Systems
12.5-
18.0.
ti
.s
4a;
S O
2.5
0.0
A p l o t of
2
4a2 vs t when bl
A s i s c l e a r f o r bl
= -0.5
=
Figure 5 -2.0 showing t h e a s y m p t o t i c o s c i l l a t i o n s
i n F i g u r e 4 , t h e q u a n t i t y e x p ( Q o - Q,)
r a p i d l y a p p r o a c h e s a c o n s t a n t v a l u e of 2 . 2 5 ,
2 2 in
( = 4a )
w h e r e a s f o r bl = -2.0
F i g u r e 5 w e see t h a t i t d o e s n o t a p p r o a c h a c o n s t a n t v a l u e b u t r a t h e r So t h i s s u g g e s t s o s c i l l a t e s a b o u t an a v e r a g e v a l u e of a r o u n d 9 . 0 . t h a t w e c o u l d a p p r o x i m a t e e x p ( Q - Q,) by c o n s t a n t v a l u e s u s i n g 2 . 2 5 0 When w e do t h i s , w e may t h e n f o r b l = - 0 . 5 and 9 . 0 f o r bl = - 2 . 0 . s o l v e E q . ( 1 5 ) , from w h i c h o n e may p r e d i c t t h e s o l i t o n s p e c t r u m a s a f u n c t i o n o f t i m e (Kaup and N e u b e r g e r ( 1 9 8 4 ) 1 c o m p a r i n g i t t o t h e a c t u a l s p e c t r u m . When w e d o t h i s f o r t h e s e t w o v a l u e s of bl, w e o b t a i n t h e r e s u l t s shown i n F i g u r e s 6 and 7 . I n F i g u r e 6, w e see t h a t f o r bl = - 0 . 5 , t h e a g r e e m e n t between t h e a p p r o x i m a t e s p e c t r u m ( s o l i d l i n e ) and t h e a c t u a l s p e c t r u m ( d a s h e d l i n e ) i s i n d e e d q u i t e good. T h i s is n o t t o o s u r p r i s i n g s i n c e o u r a p p r o x i m a t i o n f o r e x p ( Q o - Q1) w a s i n d e e d q u i t e e x c e l l e n t ( s e e F i g u r e 4 ) t o s t a r t w i t h . F o r bl = - 2 . 0 ,
F i g u r e 7 shows t h a t t h e a g r e e m e n t is now n o t q u i t e s o
g o d , a l t h o u g h t h e p r e d i c t e d spectrum d o e s c l o s e l y f o l l o w t h e a c t u a l spectrum. Presumably had w e i n c l u d e d t h e o s c i l l a t i o n s t h a t a r e p r e s e n t i n exp(Qo- Ql) (see F i g u r e 51, t h e n t h e a g r e e m e n t would h a v e b e e n much b e t t e r .
S t i l l , f r o m t a k i n g e x p ( Q o - Q,)
t o be a c o n s t a n t
172
D.J. Kaup
one d o e s o b t a i n a r e a s o n a b l e p r e d i c t i o n o f t h e a c t u a l
f o r bl = - 2 . 0 ,
s o l i t o n spectrum.
---Actual -Predicted
(dashed) (solid)
Actual (dashed)
//
_____--
---.
_____---
w -10
Z
I
-08
-06 -04 - 0 2
Figure 6 The s o l i t o n b i r t h r a t e when bl = - 0 . 5 a s p r e d i c t e d ( s o l i d
Figure 7 The s o l i t o n b i r t h r a t e when bl = - 2 . 0 a s p r e d i c t e d ( s o l i d
l i n e ) and a s it actually is (dashed l i n e ) .
l i n e ) and a s it a c t u a l l g i s (dashed l i n e ) .
SUMMARY I n c o n c l u s i o n , I hope t h a t I h a v e d e m o n s t r a t e d t o you t h a t t h e r e a r e s t i l l some i n t e r e s t i n g a s p e c t s o f i n t e g r a b l e s y s t e m s t o b e s t u d i e d . Here I have d i s c u s s e d f o r c e d i n t e g r a b l e s y s t e m s which a r e a form o f boundary v a l u e problems f o r n o n l i n e a r i n t e g r a b l e s y s t e m s . A s a c l o s i n g t h o u g h t , I should l i k e t o pose t h e q u e s t i o n of whether or n o t s u c h s y s t e m s a r e i n t e g r a b l e o r c o u l d t h e y d e m o n s t r a t e any of t h e p r o p e r t i e s of n o n i n t e g r a b l e s y s t e m s ? A t t h e moment I d o n ' t t h i n k t h a t For i n s t a n c e , a s demonstrated f o r t h e r e i s a c l e a r answer t o t h i s . t h e sine-Gordon s y s t e m , i f one f o r c e s i t w i t h i n a f i n i t e r e g i o n (Eilbeck e t a l . ( 1 9 8 1 ) ) chaos develops. But, a s I have i n d i c a t e d h e r e , g i v e n l o c a l i z e d f o r c i n g , t h e s y s t e m j u s t m i g h t w e l l be f u l l y integrable. C l e a r l y , more work n e e d s t o b e d o n e , a n d u n d o u b t e d l y i t w i l l be. ACKNOWLEDGMENTS The a u t h o r w i s h e s t o t h a n k D r . on t h e f o r c e d NLS e q u a t i o n .
P a u l Hansen f o r v a l u a b l e d i s c u s s i o n s
T h i s r e s e a r c h w a s s p o n s o r e d by t h e A i r F o r c e O f f i c e of S c i e n t i f i c R e s e a r c h , A i r F o r c e Systems Command, USAF, u n d e r G r a n t o r C o o p e r a t i v e Agreement Number AFOSR-82-0154. The U n i t e d S t a t e s Government i s
Forced Integrable Systems
173
authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints for governmental purposes. This material is based upon work supported by the National Science Foundation under Grant No. MCS-8202117. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation. REFERENCES Ablowitz, M.J. and Haberman, R., Resonantly coupled nonlinear evolution equations, J. Math. Phys. 16 (1975) 2301-2305. Ablowitz, M.J., Kaup, D.J., Newell, A.C. and Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249-315. Eilbeck, J.C., Lomdahl, P.S., and Newell, A.C., Chaos in the inhomogeneously driven sine-Gordon equation, Phys. Lett. 87A (1981) 1-4. Fokas, A.S. and Ablowitz, M.J., On the inverse scattering of the time-dependent Schroedinger Equation and the associated Kadomtsev-Petviashvili (I) Equation, INS Preprint # 2 2 , Clarkson College (to appear in Stud. Appl. Math., 1984). Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M., Plethods for solving the Korteweg-deVries equation, Phys. Rev. Lett. 19 (1967) 1095-1097. Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M., Kortewep-deVries equations and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974) 97-133. Holian, B.L. and Straub, G.K., Molecular dynamics of shock waves in one-dimensional chains, Phys. Rev. B18 (1978) 15931608. Holian, B.L., Flaschka, H. and McLaughlin, D.W., Shock waves in the Toda lattice: Analysis, Phys. Rev. A24 (1981) 25952623. Kaup, D.J., Coherent pulse propagation: A comparison of the complete solution with the McCall-Hahn theory and others, Phys. Rev. A16 (1977) 704-719. Kaup, D.J., Simple harmonic generation: An exact method of solution, Stud. Appl. Math. 59 (1978) 24-35. Kaup, D.J., The solution of the general initial value problem for the full three-dimensional three-wave resonant interaction, Physica 3D (1981) 374-395. Kaup, D.J., The method of solution for stimulated Raman Scattering and two-photon propagation, Physica 6D (1983) 143-154. Kaup, D.J., The forced Toda lattice: An example of an almost integrable system, INS Preprint #32, Clarkson College (to appear in J. Math. Phys., Jan. 1984).
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[14] Kaup, D.J. and Neuberger, D.H., The soliton birthrate in the forced Toda lattice, INS Preprint #33, Clarkson College (to appear in J. Math. Phys., Jan. 1984). [15] Lamb, G.L., Jr., Phase variation in coherent-optical-pulse propagation, Phys. Rev. Lett. 31 (1973) 196-199. [16] McCall, S.L. and Hahn, E.L., Self-induced transparency, Phys. Rev. 183 (1969) 457-485. [17] Steudel, H., Solutions in stimulated Raman Scattering and resonant two-photon propagation, Physica 6D (1983) 155-178. 1181
Toda, M., Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. No. 45 (1970) 174-200.
Wave Phenomena: Modern Theory and Applications
C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Hollaml), 1984
175
INHOMOGENEOUS PLANE WAVES I N INCOMPRESSIBLE ELASTIC MATERIALS Michael Hayes Department of Mathematical Physics University College Dub1 i n
Gibbs b i v e c t o r s [l] a r e used t o g i v e a d e s c r i p t i o n of inhomogeneous plane waves i n a n i s o t r o p i c homogeneous I t i s shown
incompressible l i n e a r e l a s t i c m a t e r i a l s .
t h a t i f t h e slowness b i v e c t o r i s n o t i s o t r o p i c then t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y polari s e d wave propagates, and conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e corresponding wave i s c i r c u l a r l y polarised. INTRODUCTION
1.
This i s a sequel t o a previous paper
121 where t h e corresponding problem
of inhomogeneous plane waves i n compressible a n i s o t r o p i c e l a s t i c and compressi b l e i s o t r o p i c v i s c o e l a s t i c m a t e r i a l s was considered. t h a t instead of specifying a d i r e c t i o n
a s i n d e a l i n g with homogeneous
p l a n e waves, f o r inhomogeneous plane waves a p a i r
n
where
i s a u n i t v e c t o r and
m
The e s s e n t i a l i d e a i s
@, E)
is perpendicular t o it.
is specified,
Associated with
t h i s p a i r i s an e l l i p s e , t h e d i r e c t i o n a l e l l i p s e o f t h e b i v e c t o r
1. + IIJ. Thus i n s t e a d o f s p e c i f y i n g a d i r e c t i o n 3, a d i r e c t i o n a l e l l i p s e i s s p e c i f i e d . The corresponding slowness b i v e c t o r and
4
I t i s seen t h a t
are real.
is written Te"
S = Te [m
+
11.)where T
i s determined from t h e s e c u l a r
equation and t h e corresponding amplitude b i v e c t o r i s determined as an eigenb i v e c t o r of t h e a c o u s t i c a l t e n s o r . I t i s seen, i n t h e u s u a l way t h a t t h e r e a r e i n general j u s t two waves which may propagate f o r a given choice of t h e d i r e c t i o n a l e l l i p s e of t h e slowness bivector.
The corresponding amplitude b i v e c t o r s
arthogonal:
A.B =
0.
&,
(say) a r e mutually
Hence f o r a given d i r e c t i o n a l e l l i p s e of t h e slowness
b i v e c t o r t h e p a r t i c l e displacements corresponding t o t h e two waves l i e on p l a n e s which may not be orthogonal.
Also t h e e l l i p s e s corresponding t o
e i t h e r displacement when p r o j e c t e d onto t h e p l a n e of t h e d i r e c t i o n a l e l l i p s e
M. Hayes
of t h e slowness b i v e c t o r a r e s i m i l a r and s i m i l a r l y s i t u a t e d with major a x i s perpendicular t o t h e major a x i s of t h e e l l i p s e of t h e slowness b i v e c t o r . On t h e assumption t h a t t h e slowness b i v e c t o r i s not i s o t r o p i c , it i s shown
t h a t t h e a c o u s t i c a l t e n s o r has double r o o t s i f a c i r c u l a r l y p o l a r i s e d wave propagates.
Conversely, i f t h e a c o u s t i c a l t e n s o r has double r o o t s then t h e
corresponding wave i s c i r c u l a r l y p o l a r i s e d .
Thus a s t h e r o o t s coalesce t h e
amplitude b i v e c t o r s merge and t h e o r t h o g o n a l i t y condition A.A- = -
0, so t h a t t h e wave i s c i r c u l a r l y p o l a r i s e d .
A.B
= 0
becomes
The condition t h a t t h e
r o o t s be double leads t o a q u a r t i c with complex c o e f f i c i e n t s .
If t h i s q u a r t i c
has a r e a l r o o t then a c i r c u l a r l y p o l a r i s e d inhomogeneous plane wave may propagate with a p p r o p r i a t e slowness.
The superposition of t h e displacements corresponding t o t h e p a i r of waves with given d i r e c t i o n a l e l l i p s e i s a l s o considered.
For motion p a r a l l e l t o t h e
plane o f t h e amplitude b i v e c t o r of e i t h e r wave it i s seen t h a t f o r e q u i d i s t a n t p o i n t s on c e r t a i n l i n e s t h e t o t a l displacement e l l i p s e s a r e similar and s i m i l a r l y s i t u a t e d with r e s p e c t t o each o t h e r and with r e s p e c t t o t h e p r o j e c t i o n of t h e slowness b i v e c t o r onto t h e plane of t h e amplitude b i v e c t o r .
52. -
EQUATIONS OF MOTION The c o n s t i t u t i v e equations a r e taken t o b e t.. = 11 =
u . .
' d i j k e Uk,,t
-p 6 . ij
(2.11
'
0 ,
1,1
where dijk9. =
djik,t
Here t h e s t r e s s e s a r e denoted by displacements by
dk,tij
=
=
dij,tk
*
t i j , t h e e l a s t i c c o n s t a n t s by
and t h e comma denotes d i f f e r e n t i a t i o n : u
u.
The summation convention i s used throughout.
Also
p
dijka.
,
the
~ E, auk/axk ~
.
is a scalar, the
h y d r o s t a t i c p r e s s u r e which i s t o be determined from t h e equations of motion and t h e boundary conditions.
Equation ( 2 . 2 ) expresses t h e f a c t t h a t a l l deformations
i n t h e body a r e i s o c h o r i c .
The equations of motion a r e given by 2 a u
at.. *
=
I
P
y
i
at
i n t h e absence o f body f o r c e s , where
(2.41
>
p
is t h e material density.
Inserting
177
lnhomogeneous Plane Waves
(2.1) i n t o ( 2 . 4 ) g i v e s dijki
13. -
k , a j - P , ~=
a 2u i / a t
P
2
.
ISHOMOGEiiEOUS I'LAiiE WAVES
Now i t i s assumed t h a t t h e displacements
a r i s e due t o t h e propagation ui of an i n f i n i t e t r a i n of inhomogeneous p l a n e waves i n t h e body. Thus
Here
A i s a bivector:
&
&+ +
=
The planes of constant phase a r e amplitude a r e = constant.
z-.~
I&-
2,
and so i s
s+.x=
t h e slowness b i v e c t o r .
c o n s t a n t , and t h e planes of constant
The period of t h e wave is
Equation
27r/w.
(3.1) d e s c r i b e s an i n f i n i t e t r a i n of e l l i p t i c a l l y p o l a r i s e d inhomogeneous
plane waves.
For f i x e d
x(= x*
say) t h e displacement v e c t o r
’1
e l l i p s e which i s s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of
A+ exp(- w
t h e e l l i p s e whose conjugate semi-diameters a r e
A-expc-
uZ-.f).
l i e s on an
&,
z-.~*) and
namely
A t any given time t h e displacement v e c t o r i s along one semi-
diameter of t h e e l l i p s e and t h e p a r t i c l e v e l o c i t y i s p a r a l l e l t o i t s conjugate semi-diameter.
t
As
t h e e l l i p s e i s from
i n c r e a s e s t h e sense i n which t h e p a r t i c l e moves along
&+ t o
A-.
The p l a n e of p o l a r i s a t i o n i s determined by
t h e plane of t h e e l l i p s e of t h e amplitude b i v e c t o r
Circularly polarised i f Linearly p o l a r i s e d i f
Te"
S = where
5
The slowness b i v e c t o r
T
vector.
g
angles t o problem.
and
$
(5+,):I
are real.
and t a k i n g
m.:
.
The wave i s
-A.A _
=
0 ;
(3.2)
&A&
=
0.
(3.33
When equation ( 3 . 2 ) is s a t i s f i e d t h e b i v e c t o r "nullt'.
A -
A -
i s s a i d t o be " i s o t r o p i c "
or
may be w r i t t e n
__
= 0, n.n = 1
,
(3.4)
By f i x i n g on a p a r t i c u l a r choice of t h e u n i t
5 t o be o f a r b i t r a r y [chosen) magnitude and a t r i g h t
1 , i t will be seen t h a t T and 4 a r e determined by an eigenvalue 5 and may be regarded a s t h e p r i n c i p a l axes of an e l l i p s e .
I 78
M. Hayes
Then
S/T
(z+,g-) i s
and
13i s
el$@ +
=
a p a i r of conjugate semi-diameters of t h i s e l l i p s e
t h u s a p a i r of conjugate semi-diameters of a similar and s i m i l a r l y
s i t u a t e d e l l i p s e whose major and minor axes a r e
(m .+):1
axes of t h e e l l i p s e of
times t h e major and minor
Also
S+
= T(cos$m
-
sin$n-),
S-
= T(sin@
+
cos@),
The angle
T
2 3 ; = T(cos2$m2 + s i n $)
12-1 =
,
z f .
T(sin2$m2 + cos $)
(3.5)
between t h e planes o f constant phase and t h e planes of
8
constant amplitude i s given by tan 8
=
[Z]
m
(3.6)
(m 2 - l ) c o s $ s i n $
The condition (2.2) gives
A.
2
=
0.
(3.7)
This means i n general t h a t t h e e l l i p s e of t h e amplitude b i v e c t o r
2
e l l i p s e o f t h e slowness b i v e c t o r l a r t o each o t h e r .
S
A
and t h e
may not l i e on p l a n e s which a r e perpendicu-
Also, t h e p r o j e c t i o n of e i t h e r
(5 say)
upon t h e plane o f
i s an e l l i p s e whose aspect r a t i o ( r a t i o of major t o minor a x i s ) i s equal t o
t h e aspect r a t i o of t h e e l l i p s e of t h e minor a x i s o f t h e e l l i p s e of
5
and whose major a x i s i s perpendicular t o
2.
A A. A =
Exceptionally it may happen that
i s a s c a l a r and a l s o t h a t polarised.
84. -
I t i s seen i n
2. 55
=
0
i s p a r a l l e l t o 2: A = as, where so t h a t t h e wave i s c i r c u l a r l y
a
that t h i s i s a p o s s i b i l i t y f o r i s o t r o p i c bodies.
THE PROPAGATION CONDITION
Here it i s assumed t h a t t h e d i r e c t i o n a l e l l i p s e of eigenvalue equation f o r t h e determination of that t h e r e a r e j u s t two non-zero eigenvalues.
Te"
5
i s given.
i s obtained.
The
I t i s seen
I f t h e s e a r e n o t equal it i s
shown t h a t t h e corresponding eigenbivectors a r e orthogonal. has t h e form (3.1), t h e n from the equations o f motion (2.5)
Now i f follows t h a t
p
must have a s i m i l a r form.
Thus write
it
179
lnhomogeneous Plane Waves
p
=
P expiw
(5. x
.
-t)
(4.1)
I n s e r t i n g t h e expressions (3.1) and (4.1) i n t o t h e equations of motion and P
using c3.5) t o e l i m i n a t e
2. 2 SO,
and assuming
leads t o
Using (3.4) t h i s may be w r i t t e n
(4.3) where
Q.
lk
*
m,
1 ~ k R ks j
=
'
%i
=m+ip,m .n = O , n .n = l .
S Thus, i f
* *
= d..
(4.41
a r e given, equation (4.3) i s an eigenvalue problem f o r t h e
determination of
and t h e corresponding eigenbivector
T, $
A.
The propagation condition (4.3) l e a d s t o t h e s e c u l a r equation d e t (pT-' f o r the determination o f det(Q)=0
e-21'
-
6ik
[Te'$)
= 0,
f o r given
*
*
s .s - * *
since det
6.i k)
<.
) = 0,
Thus equation (4.5) i s a q u a d r a t i c i n Let t h e r o o t s
eigenbivectors be
T-2e-21' and
T 2 e210
be denoted by Alp-', respectively.
I t i s c l e a r from (4.4)
and hence one r o o t of (4.5)
rns m zero.
(4.5) that
is
.
h2p
-1
.
Let the corresponding
Then
(4.6) (4.7) I t i s now shown t h a t if h l
11.B
=
A2
0.
,
then
(4.8)
180
M. Hayes
Now s i n c e
A. S* = B. BiQikAk
4
since
*
-
i s symmetric.
= 0, i t follows from equation (4.4) t h a t
A.Q. 1 ik Bk
X1 f
A. g
A,
and
The e l l i p s e s of
B
e l l i p s e of
(4.9)
(4.10)
= 0.
A
S
a r e on planes no two of which may be
S
and o f
when p r o j e c t e d on t h e plane of t h e
a r e s i m i l a r and s i m i l a r l y s i t u a t e d .
r o t a t e d through a quadrant. t i o n s of t h e e l l i p s e s o f t i o n s of t h e e l l i p s e s o f
B &
Examples o f such t r i a d s [3],
The p r o j e c t i o n s a r e s i m i l a r
i n t h e p l a n e of
and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of
For example
,
X2,
Hence t h e e l l i p s e s of orthogonal.
= 0
Hence from (4.6) and (4.7)
(A1, - X2) Thus, f o r
- AiQikBk
BiQikAk
3
when they a r e
A similar statement may be made about t h e projec-
and
5
and
upon t h e plane o f
A
upon t h e plane of
2.
and about t h e projec-
s a t i s f y i n g (4.11) a r e e a s i l y constructed.
A, B,
take
c4.12) a, 6, 6
where
are scalars, satisfying
ci
2
-
B2
=
Returning t o equation (4.3) i t i s seen t h a t i f eigenbivector f o r given of
Te'+
i s known t o be an
and i s not i s o t r o p i c , then t h e corresponding value
i s given by p ~ - 2 e-21'
since
*
1.
A.5=
0.
If
&
A1 . A .1 = Q i k ~ i ~ =k Qi kA iA k '
i s i s o t r o p i c , then it w i l l be shown i n
corresponding eigenvalues a r e double and hence t h e v a l u e of half t he t race of
(4.13)
Te"
86
that the
i s given by
Q/p:
2pT
-2 -214
e
=
A
Qii*
(4.14)
181
Inhomogeneous Plane Waves
ISOTROPIC MATERIALS
55. -
In t h i s s e c t i o n t h e propagation of inhomogeneous plane waves i n i s o t r o p i c homogeneous incompressible e l a s t i c bodies i s considered. o f waves.
There a r e two c l a s s e s
In t h e f i r s t t h e slowness b i v e c t o r i s i s o t r o p i c , t h e waves a r e
c i r c u l a r l y p o l a r i s e d and t h e s o l u t i o n i s u n i v e r s a l i n t h e sense t h a t t h e slowness does not depend upon t h e shear modulus which d e s c r i b e s t h e e l a s t i c response of
In t h e second c l a s s , t h e p r e s s u r e term i s zero, t h e waves a r e
the material.
t r a n s v e r s e , and t h e amplitude b i v e c t o r may be chosen so t h a t t h e wave is c i r c u l a r l y polarised. The c o n s t i t u t i v e equation (2.1) now reads t.
=
Ij
where
p
i s a constant.
-
p6ij
+
p(ui,j
,
+ uj,i)
(5.1)
Using t h i s and equation (4.1) i n equation (2.4)
leads
to
Now
AiSi
=
0
and hence
either
ci)
(5.3)
where
a
is a scalar
Cii) =
P
In
c a s e Ci),
usj sj
0 ,
p
s.s-
=
aSi expiw@. 5 - t )
=
-
ipwaexpwC2.
=
0
,
a
This i s a c i r c u l a r l y p o l a r i s e d wave. p l a n e s of constant amplitude:
5'. 5-
r e s t r i c t i o n on t h e magnitude of case ( i i ) ,
.
(5.4)
we have t h e s o l u t i o n ui
In
= p
the solution is
2.
,
-t) , arbitrary.
Planes of constant phase a r e orthogonal t o = 0.
Also
Is+( =
15-1
.
There i s no
I t i s independent of t h e shear modulus P
.
M. Hayes
182
A
where
where
From (5.6)2
0.
m2 < 1 ,
Thus, i f
and i f
A. 5 =
i s any b i v e c t o r s a t i s f y i n g
m2 > 1
y, y'
,
a r e a r b i t r a r y and 2.2 = 1,
1.11=
r.fl= 0 .
I n both c a s e s t h e
planes o f constant phase a r e orthogonal t o t h e planes of constant amplitude.
5
c Cp/p) homogeneous t r a n s v e r s e wave i s recovered. As
m
-+
a,
then from ( 5 . 9 ) ,
For given " +
15,and m
-+
a, m, A
s a t i s f y i n g (S.?),
the waves are c i r c u l a r l y p o l a r i s e d .
-f
y
12 +
or
and t h e usual
y'f,
y'
may be chosen s o t h a t
Thus, from equation (5.8),
take
y
given
bY (5.10)
Then (5.11) corresponds t o a c i r c u l a r l y p o l a r i s e d wave. c i r c l e i n t h e p l a n e spanned by
m
2
> 1,
take
yt
given by
m
and ;{
The displacement v e c t o r lies on a
+- (-2-1) 1 4E} m
.
Similarly, f o r
(5.12)
Then (5.13)
183
Inhomogeneous Plane Waves
corresponds t o a c i r c u l a r l y p o l a r i s e d wave, t h e c i r c l e of p o l a r i s a t i o n l y i n g i n a plane spanned by
86.
m m-m
1 and
1 4 IS}
{ -i (1 - -2)
m
STRUCTURE OF THE ACOUSTICAL TENSOR.
.
CIRCULARLY POLARISED WAVES.
Here t h e s t r u c t u r e o f t h e a c o u s t i c a l t e n s o r i s considered more f u l l y .
In
i s not i s o t r o p i c , it i s shown t h a t i f t h e s e c u l a r
particular,assuming t h a t
equation has a double r o o t then a c i r c u l a r l y p o l a r i s e d wave may propagate i n t h e material.
Also, it i s shown, assuming
S
.S 9
0, t h a t i f a c i r c u l a r l y p o l a r i s e d
wave propagates then t h e s e c u l a r equation has a double r o o t .
i s n o t i s o t r o p i c i s made, f o r , i n g e n e r a l , i n
The assumption t h a t
d e r i v i n g t h e form of t h e a c o u s t i c a l t e n s o r expressions o f t h e form enter.
I t was seen i n
o b t a i n a s o l u t i o n with
[s@SJ/(s.s)
f o r an i s o t r o p i c m a t e r i a l t h a t i t i s p o s s i b l e t o
35
5.2 =
0.
However, i n g e n e r a l , i f t h e d e t a i l a d s t r u c t u r e
of t h e e l a s t i c c o e f f i c i e n t s i s n o t given then it is e s s e n t i a l t o assume t h a t
S
i s not i s o t r o p i c . Now s i n c e any b i v e c t o r
where
a
may be w r i t t e n [l]
5
=
+
ib), a.b =
0,
i s a s c a l a r , i n general complex, we may, without l o s s of g e n e r a l i t y ,
write
where
T,
0,
respectively.
m
a r e r e a l and Then, assuming,
defined by equation (4.4),
i, 2.S
a r e u n i t v e c t o r s along t h e x and y axes 2 0 so that m $: 1, it i s seen t h a t Q,
may b e w r i t t e n * *
and has components given by
184
M. Hayes
Thus t h e matrix
4
has t h e form
where
One eigenvalue of
X1, AZ,
(0)
191
i s zero s i n c e
and t h e o t h e r two, denoted by
a r e given by
A,
t h e corresponding e i g e n b i v e c t o r s being
where
= 0
given by
tl, t2 a r e given by
Now tl t2 =
and hence
m2
-
1
,
(6.9)
185
Inhomogeneous Plane Waves
=
A . B-
(6.10)
0.
A1so
A.2 -
B.5
0,
=
=
(6.11)
0.
In t h e s p e c i a l case when (6.12) it follows from equation (6.4) t h a t
6
Cii)
=
(:i ; Q3i)
and t h e non-zero eigenvalues a r e bivectors
CO,
0, y)
and
Q,,
(1, m, 0)
and
(6.13)
,
a + imB
where
w i t h corresponding eigen-
is arbitrary.
y
The f i r s t of
t h e s e corresponds t o a l i n e a r l y p o l a r i s e d wave, t h e second t o an e l l i p t i c a l l y p o l a r i s e d wave.
(Recall
m
2
9 1).
CIRCULARLY POLARISED WAVES
F i n a l l y , the p o s s i b i l i t y of equa
eigenva ies i s considered.
The eigenvalues a r e equal provided L
a
(6.14)
so that (6.15) Then A1
=
From equation (6.4),
A2
=
(a + imB + Q3,)/2
(Q) i s now given by
.
(6.16)
M. Hayes
186
where the upper and lower s i g n s correspond t o t h e upper and lower s i g n s i n (6.15).
C(say) i s given by
The corresponding eigenbivector
(6.18) It is clear that
5.C =
0,
and thus the wave corresponding t o t h e double r o o t
is c i r c u l a r l y polarised. I t may be noted t h a t t h e c i r c l e s of p o l a r i s a t i o n corresponding t o t h e d i f f e r e n t s i g n s i n equation (6.15) and t h e r e f o r e a l s o i n (6.18), a r e g r e a t c i r c l e s i n t h e u n i t sphere.
They a r e described i n opposite senses.
most e a s i l y seen i n t h e s p e c i a l c a s e
_C = _i r i-k
are unit circles i n the
-
xz
if it i s p o s s i b l e
-
when m = 0.
This i s Then
plane, described i n opposite senses.
Also, r e t u r n i n g t o equation (.6.14), which i s t h e condition f o r double r o o t s ,
this condition may be w r i t t e n , using equation (6.5),
CQ,,
+
2 1 m Q12
-
m
2
Q2,
-
-
(1
2
as 2
m 1 Q33}
(6.19) This is a q u a r t i c i n real root for
m
m
with complex c o e f f i c i e n t s .
I f t h i s equation has a
then t h e r e a r e two corresponding c i r c u l a r l y p o l a r i s e d waves.
In general it w i l l not possess r e a l r o o t s f o r
m.
Assuming t h a t 5 i s not i s o t r o p i c it has been shown t h a t i f t h e a c o u s t i c a l tensor Q) has double r o o t s then t h e corresponding eigenbivector i s i s o t r o p i c and accordingly a c i r c u l a r l y p o l a r i s e d inhomogeneous wave may propagate. shown, assuming t h a t possesses an
Now, it i s
S
i s not i s o t r o p i c , t h a t i f t h e a c o u s t i c a l t e n s o r (Q) i s o t r o p i c eigenbivector then 9 has a double eigenvalue. n
Now from (4.4) t h e a c o u s t i c a l t e n s o r
9
has t h e form (6.20)
cij
=
6ij
-
SiSj
‘mSm
(6.21)
187
Inhomogeneous Plane Waves
and (6.22)
0
Without l o s s of g e n e r a l i t y , l e t
be given by
(6.23)
i s t o be an i s o t r o p i c eigenbivector of
Now
@)
so t h a t it s a t i s f i e s (6.24)
C6.25)
(6.26)
without l o s s of g e n e r a l i t y .
_A.S_
5
= 0,
I t i s assumed that
2.5
0
and since
may be assumed t o have t h e form (6.27)
where
i s some s c a l a r .
6
2.2
Now
i’ 6 -1
6
2
=
-1
-6
and
=
(
and
&:)
-1
62+1
0
( 6 * + l ) b -la -16c
- t b +(6’
+tb)
,
(6.28)
-16
2 (6 -1)b -if -6g
(6 -1)a - t b -6c -:(a
62,
=
2
(6 - 1 ) ~- t g -6h
+ l ) f -16g
-6(b + t f )
-6(c
+tg)
(6.29)
Then from (6.24) it follows t h a t
,
2tb
=
f - a
61
=
6(a + tb) - cc + 18)
,
(6.301
188
M. Hayes
and (6
-1)a - t b -6c
(S2 + l ) b - i a -16c
62G =
[-&:a On expanding r o o t Csince
17. -
+
-1
a -6g
(6
2 6 (a+2tb)+Ca+ib)-t6g
= 0,
2
-1)c-Ig-dh
-tc+(6
-
-t6Ca + ib)
tb)
IQ - e r l 191
( ~ 5+ l~) b
2
6(c + 1g)
i t i s seen t h a t t h e s e c u l a r equation has a zero
X
= 0 ) , and a double r o o t
given by ( 6 . 3 0 ) 2 .
SUPERPOSITION OF WAVE TRAINS WITH COMMON DIRECTIONAL ELLIPSE
In t h i s s e c t i o n t h e s u p e r p o s i t i o n of two wave t r a i n s with common d i r e c t i o n a1 e l l i p s e i s considered.
S
For given f i x e d
p a r a l l e l t o t h e plane of t h e e l l i p s e of
*
t h e motion i n any plane
is examined.
The displacement o f
a p a r t i c l e a r i s e s a s a l i n e a r combination of t h e two b a s i c displacements and (The sum of two b i v e c t o r s i s
hence, i n general, w i l l a l s o be e l l i p t i c a l . also a bivector.)
The e l l i p s e s f o r d i f f e r e n t p a r t i c l e s w i l l g e n e r a l l y d i f f e r .
However, it i s seen t h a t t h e e l l i p s e s a t c e r t a i n p o i n t s on a c e r t a i n l i n e are i d e n t i c a l both i n o r i e n t a t i o n a n d i n t e r m s of t h e lengths of t h e i r p r i n c i p a l axes. I t i s assumed t h a t
where
a,,
B,
a, b
5, A,
a r e given by
I t i s e a s i l y checked that
a r e assumed t o be r e a l .
A.B=
A.S-1 = B.S-2 = 0. Also Tle 161 , T2e1" of t h e s e c u l a r equation ( 4 . 5 ) f o r given 2 = corresponding eigenbivectors a r e The t o t a l displacement
E* =
A
and
a r e assumed t o be t h e s o l u t i o n s (ol/B)
u_"Csay) in t h e p l a n e of
A expiw C Tle'"(crx
+ iBy]
+ {B - (B.n) - - n) expiu{T2e"2(ax
+
ii,
and t h e
respectively.
-
&
may be w r i t t e n
t 1 + ’By)
-
t)
,
(7.21
189
lnhomogeneous Plane Waves
A
2 is t h e u n i t normal t o t h e p l a n e of
where
given by
- s i n d i + acossk
z =
2 2 2 l ( s i n S+a cos 6)
(7.3)
’
Now
Bcos6
-
bfisind[ii
=
IacosSj + sin8k) ]
,
(7.41
(1+B c o s S ) and
-s* -
=
(s”.fl)” 5 6
l[lL
-
’c0s6 ~ a c o s +~ si i n & & )I . 2 2 (1+B cos 6)
Notice t h a t t h e e l l i p s e s of t h e p r o j e c t i o n s of
and
5
(7.5)
upon t h e plane of
are s i m i l a r and s i m i l a r l y s i t u a t e d . Now
u*
may be w r i t t e n
+ a-’b@sin6{iL
-
(acossi
BCOSG
+
sin6k) 1 exp
IWP]
(1+6 cos 6)
,
(7.61
where (7.71 I t i s c l e a r t h a t f o r given
x, y,
t h e displacement
%* l i e s on an e l l i p s e .
The axes of t h e e l l i p s e w i l l v a r y from p o i n t t o p o i n t f o r p o i n t t o point. the ellipse a t
However, if exp(iwb) M
has p r i n c i p a l axes along
t h e ellipses at the points axes of t h e e l l i p s e of
5
M
u v a r i e s from
i s purely r e a l a t a point
i
M(say)
and (ctcosdi + sin@).
then
Thus
have p r i n c i p a l axes p a r a l l e l t o t h e p r i n c i p a l
and a l s o of course t o t h e p r i n c i p a l axes of t h e
e l l i p s e of t h e p r o j e c t i o n of
2.
Of course t h e s e e l l i p s e s a t t h e p o i n t s
M
will not be i d e n t i c a l but they do have t h a t one f e a t u r e i n common, t h a t t h e i r p r i n c i p a l axes a r e p a r a l l e l .
The p o i n t s
M
a r e any p o i n t s o n t h e e q u i d i s t a n t
190
M. Hayes
d
parallel lines
2
:
where
k,
:
wk[a[cos$)x $
-
p(sin$)y]
=
qs,
q
0,
...
+1, f 2 ,
(7.8)
a r e given by T e 2
‘42
- Tle
Consider t h e p o i n t s o f i n t e r s e c t i o h
32 with t h e l i n e s
:
.
(7.9) M*(say)
(asin+)x + ( ~ c o s $ ] y = The term
exp(iwu)
p o i n t s and has t h e same value a t each.
of a l i n e
7
constant
(7.10)
i s p u r e l y r e a l a t each of t h e s e
Thus f o r t h e s e p o i n t s
M
*
the
displacement e l l i p s e s a r e a l l i d e n t i c a l and are s i m i l a r and s i m i l a r l y s i t u a t e d t o t h e e l l i p s e of t h e p r o j e c t i o n of
ACKNOWLEDGMENT.
5
upon t h e p l a n e of
A.
This work was supported by t h e National Board f o r Science
6 Technology under Grant 19/79.
191
Inhomogeneous Plane Waves
REFERENCES [l]
Gibbs, J.W. Elements of Yector Analysis, 1881, 1884 ( p r i v a t e l y p r i n t e d ) E pp 17 - 90, Vol. 2 , p a r t 2 , S c i e n t i f i c Papers, Dover Publications, New York, 1961.
[Z]
Hayes, M. Inhomogeneous Plane Waves. appear)
[3]
Synge, J . L . The Petrov C l a s s i f i c a t i o n of G r a v i t a t i o n a l F i e l d s . Dublin I n s t . f o r Adv. S t u d i e s , No. 15, Dublin, 1964.
Arch. R a t ' l Mech. Anal. 1984 ( t o
A,
Corn.
This Page Intentionally Left Blank
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Mwdie (eds.)
0 Elsevier Science Publishers B.V. (North-Holland). 1984
193
CAN ACOUSTIC FOCUSING GENERATE TURBULENT SPOTS?
Gregory A. Kriegsmann and Edward L . R e i s s Department of E n g i n e e r i n g S c i e n c e s and Applied Mathematics Northwestern U n i v e r s i t y Evanston, I l l i n o i s 60201 USA
Experimental r e s u l t s , see e . g . [1-3] and r e f e r e n c e s g i v e n t h e r e , s t r o n g l y s u g g e s t t h a t c o h e r e n t f l u i d s t r u c t u r e s , which o r i g i n a t e i n t h e v i s c o u s s u b l a y e r s of w a l l t u r b u l e n t boundary l a y e r s , b u r s t and v i o l e n t l y e j e c t f l u i d i n t o t h e t u r b u l e n t regime of t h e boundary l a y e r . F u r t h e r m o r e , i t is b e l i e v e d t h a t t h i s b u r s t i n g produces most of t h e t u r b u l e n c e i n t h e boundary l a y e r . These e x p e r i m e n t s f u r t h e r s u g g e s t t h a t t h e c o h e r e n t s t r u c t u r e s , which a r e sometimes c a l l e d t u r b u l e n t s p o t s , are l o c a l i z e d r e g i o n s of i n t e n s e v o r t i c i t y t h a t p r o p a g a t e streamwise i n t h e sublayer. Because of t h e i r v o r t i c i t y , t h e s p o t s g r a d u a l l y l i f t as t h e y propa g a t e downstream where t h e y suddenly b u r s t and e j e c t t h e i r f l u i d upward. The p r e c i s e mechanism f o r t h e f o r m a t i o n of t h e s e s p o t s h a s n o t been c l e a r l y a s c e r t a i n e d , a l t h o u g h s e v e r a l c o n c e p t s have been proposed [l-31. I n t h i s p a p e r , w e d i s c u s s a p o s s i b l e new mechanism f o r t h e f o r m a t i o n of t h e s e s p o t s which w a s o r i g i n a l l y p r e s e n t e d i n Ref. [ 4 ] . S i n c e t h e b u r s t i n g p r o c e s s i s v i o l e n t , i t g e n e r a t e s a p r e s s u r e f i e l d w i t h a r e l a t i v e l y broad frequency spectrum, as e x p e r i m e n t a l r e s u l t s confirm. In [ 4 ] , we considered t h e b u r s t s a s localized s o u r c e s of energy which, among o t h e r i m p o r t a n t e f f e c t s , p r o p a g a t e a c o u s t i c waves i n t h e s u b l a y e r . The waves produced by t h e h i g h f r e q u e n c y components of t h e b u r s t s w e r e a n a l y z e d i n [ 4 ] u s i n g t h e methods of g e o m e t r i c a l a c o u s t i c s , see e . g . [ 5 ] . For s i m p l i c i t y , w e c o n s i d e r e d a two d i m e n s i o n a l a c o u s t i c p r o p a g a t i o n probl e m i n a s h e a r l a y e r which i s p a r a l l e l to t h e w a l l z = 0. The l a y e r a t t a i n e d a uniform f l o w p a r a l l e l t o t h e w a l l f o r z > D , where D i s t h e "depth" of t h e s u b l a y e r . The b u r s t w a s modeled a s a l i n e s o u r c e p e r p e n d i c u l a r t o t h e f l o w d i r e c t i o n , o r more g e n e r a l l y as a s o u r c e t h a t is l o c a l i z e d a l o n g t h i s l i n e , b u t i s spanwise uniform. Then w e l i n e a r i z e d t h e two d i m e n s i o n a l f l u i d dynamics e q u a t i o n s f o r i n v i s c i d , i s e n t r o p i c , c o m p r e s s i b l e f l o w a b o u t t h i s s h e a r flow t o o b t a i n t h e a c o u s t i c p r o p a g a t i o n e q u a t k o n s . The l i n e s o u r c e was assumed t o l i e i n s i d e t h e s h e a r l a y e r a t x = 0, z = z < D , as shown i n F i g u r e 1.
In o u r a n a l y s i s w e s t u d i e d t h e downstream a c o u s t i c f o c u s i n g o f t h e high-frequency components of t h e s o u r c e , by t h e flow g r a d i e n t of t h e s h e a r l a y e r . Thus, t h e s o l u t i o n o f t h e a c o u s t i c p r o p a g a t i o n problem w a s c o n s t r u c t e d i n t e r m s of t h e r a y s and w a v e f r o n t s of g e o m e t r i c a l a c o u s t i c s . The r a y s emanating from t h e s o u r c e , which are c a l l e d t h e primary r a y s , are shown i n F i g u r e 1 f o r a t y p i c a l set of f l u i d p a r a m e t e r s and s o u r c e l o c a t i o n . S i n c e t h e r a y s c a r r y energy ( n o i s e ) from t h e s o u r c e t o t h e f i e l d , i t i s s e e n t h a t some o f t h e energy i s e i t h e r r a d i a t e d d i r e c t l y t o t h e f a r f i e l d away from t h e w a l l , o r i t i s f i r s t p r o p a g a t e d upstream and t h e n " t u r n e d " by t h e f l o w t o t h e downstream f a r f i e l d . However, some of t h e energy i s t r a n s m i t t e d by t h e primary r a y s d i r e c t l y t o ' t h e w a l l and t h e n r e f l e c t e d a s shown i n F i g u r e 2 . Some of t h e r e f l e c t e d energy i s t h e n r a d i a t e d t o t h e f a r f i e l d . However, t h e r e are two one-parameter f a m i l i e s of r e f l e c t e d r a y s t h a t a r e t r a p p e d i n a c h a n n e l a d j a c e n t t o t h e w a l l which do n o t e s c a p e t o t h e f a r f i e l d a s i s i n d i c a t e d i n F i g u r e 3. Each r a y i n t h i s channel i s a l t e r n a t e l y r e f r a c t e d toward t h e w a l l by t h e s h e a r f l o w g r a d i e n t , and t h e n
194
G.A.
Kriegsmann and E. L. Reiss
reflected from the wall. These two families of rays then form an infinite sequence of caustics downstream of the source, as shown in Figure 3. The amplitude of the acoustic field is calculated by the method of geometrical acoustics along the rays and is given in 1 4 1 . However, as is well known, the geometrical acoustics approximation for the amplitude is unbounded on the caustics, s o that this approximation is invalid on and near the caustics. A boundary layer method, similar to the one developed in [ 6 ] for the Helmholtz equation, is then employed to determine the acoustic amplitude on and near the caustics. It is shown in [ 4 ] that the amplitude of the field on and near the caustics is O(k2l3) larger than the geometrical acoustics field away from the caustics for large k. Here k is the dimensionless wave number that is defined by k Z wD/c, where c is the ambient sound velocity in the shear layer, and w is the circular frequency of the time periodic component of the source. More significantly, the vorticity of the acoustic field on and near the caustics is O ( k 7 / 6 ) larger than the vorticity of the geometrical acoustics field away from the caustics. Thus, our analysis shows that a localized disturbance (the line source, which models a bursting spot) of unit amplitude can generate an array of narrow regions downstream of the source and surrounding each caustic in which the vorticity intensity is extremely large in the acoustic approximation. These local regions of intense vorticity can serve as nuclei for the "birth" of future turbulent spots. That is, the process of turbulent spot formation and hence of turbulence production in the boundary layer may be a self-induced phenomenon - old bursts give birth to new spots. To substantiate this conjecture it is necessary to consider the three-dimensional propagation from a point source in the shear flow, possibly including the low frequency spanwise vorticity in the sublayer flow. Since the amplitudes on the caustics are unbounded as k + m , this suggests that a nonlinear analysis is required to describe the further evolution of the local vortex nuclei of the acoustic theory. We are presently studying the possibility of eliminating or diminishing the caustic fields by appropriate wall motions which may be induced by driven or compliant walls. Furthermore the formation and bursting of the turbulent spots is closely related to the noise field emitted by turbulent boundary layer fluctuations of a moving vehicle submerged in a fluid. Thus, any mechanism for reducing the bursting of turbulent spots may substantially reduce the viscous drag of the vehicle and its concomitant radiated noise. Finally, if the sound source is interpreted as noise produced by components, such as propulsors or local vibrations of the vehicle structure, then the results presented in [ 4 ] give estimates of the near and farfield noise resulting from the vehicle& motion. The large self-noise corresponding to the caustic field, which is close to the wall, may mask weak signals received from distant sources. Then knowing the caustic field it can be filtered from the total received signal to determine the weak incident signal. ACKNOWLEDGEMENTS This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 80-0016A, by the National Science Foundation under Grant No. MCS8300578 and the Office of Naval Research under Contract No. N00014-83-C-0518,
195
Can Acoustic Focusing Generate Turbulent Spots?
REFERENCES
1.
Offen, G . R . and K l i n e , S . J., A proposed model of t h e b u r s t i n g p r o c e s s i n t u r b u l e n t boundary l a y e r s , J. F l u i d Mech. 70 (1975) 209-228.
2.
B u s h n e l l , D . M . , Hefner, J . N . and Ash, R. L . , E f f e c t of compliant w a l l motion on t u r b u l e n t boundary l a y e r s , Phys. of F l u i d s 20 (1977) S 3 1 - S 4 8 .
3.
W i l l m a r t h , W. W. and Bogar, T . J . , Survey and new measurements of t u r b u l e n t s t r u c t u r e s n e a r t h e w a l l , Phys. of F l u i d s 20 (1977) S9-S21.
4.
Kriegsmann, G . A. and Reiss, E . L . , A c o u s t i c wave p r o p a g a t i o n i n a s h e a r f l o w , J. Acoust. SOC. of A m e r . , i n p r e s s .
5.
K e l l e r , J . B., L e w i s , R. M. and S e c k l o r , B. D., Asymptotic s o l u t i o n s of some d i f f r a c t i o n problems, Comm. on Pure and Appl. Mech. 9 (1956) 207-265.
6.
Buchal, R . N . and K e l l e r , J . B . , Boundary l a y e r problems i n d i f f r a c t i o n t h e o r y , Comm. on Pure and Appl. Math. 1 3 (1960) 85-114.
FIGURE CAPTIONS
F i g u r e 1:
The primary r a y s emanating from t h e s o u r c e p o i n t x = 0, z = z , f o r a t y p i c a l s u b s o n i c Mach number o f the s h e a r flow.
Figure 2:
Reflected rays.
Figure 3 :
R e f l e c t e d r a y s i n w a l l c h a n n e l t o show c a u s t i c f o r m a t i o n .
196
G.A. Kriegsmann and E. L. Reiss
Can Acoustic Focusing Generate Turbulent Spots?
197
WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
Wave Phenomena: Modern T h e o r y and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
199
NONLINEAR WAVES I N THE PELLET FUSION PROCESS
V.
J . E r v i n , W.
F.
and
Ames
E.
Adams
I n s t i t u t f c r Angewandte Mathematik Universitat Karlsruhe 75 Karlsruhe, F e d e r a l R e p u b l i c of Germany
School o f Mathematics G e o r g i a I n s t i t u t e o f Technology A t l a n t a , GA 3 0 3 3 2 U.S.A.
A g a s dynamic model o f t h e p e l l e t f u s i o n p r o c e s s
having a t i m e - i n v a r i a n t s o u r c e t e r m is s t u d i e d Some e x a c t s o l u t i o n s by means of g r o u p a n a l y s i s . of t h i s n o n l i n e a r s y s t e m a r e c o n s t r u c t e d f o r The s p e c i f i c ( p h y s i c a l ) v a l u e s of p a r a m e t e r s . d e v e l o p m e n t o f m u l t i p l e s h o c k waves i s demonstrated i n several cases analytically. Additional numerical r e s u l t s i l l u s t r a t e t h e e v o l u t i o n of s i n g u l a r i t i e s .
1.
INTRODUCTION
In t h e p e l l e t fusion process, a spherical p e l l e t is f i r e d i n t o a c o n t a i n m e n t chamber and t h e n bombarded w i t h p u l s e s of l a s e r energy.
When h i t by a l a s e r p u l s e , a s h o c k wave p r o p a g a t e s t h r o u g h
t h e p e l l e t a s t h e temperature rises.
[A t y p i c a l p e l l e t configura-
t i o n i s s k e t c h e d below.] 2 2 5 . 7 q @,
c7-0 67.2 q
\\
L PA
c 4 -,
’B,
;%, s
\
void
\\
II Q3*““8%
,I
\ \
/
\
%
I
I
I
/
\
I \
/ \ I
F i g . 1.
A Typical P e l l e t Configuration.
The o u t e r m o s t l a y e r of t h e p e l l e t , c o n s i s t i n g of l e a d , p r e v e n t s t h e o u t w a r d e x p a n s i o n of t h e i n n e r l a y e r s : h e n c e , a s t h e
Lithium-Lead
l a y e r h e a t s u p , it e x p a n d s inward f o r c i n g t h e DT f u e l t o w a r d t h e center. The s h o c k wave, a f t e r p a s s i n g t h r o u g h t o t h e i n n e r l a y e r , i s reflected.
O n s t r i k i n g t h e o u t e r s u r f a c e of t h e p e l l e t ,
another
p u l s e a r r i v e s and c o n s t r u c t i v e l y a d d s t o t h e s h o c k wave a l r e a d y i n
200
V.J. Ervin et a/.
the pellet. The p r o c e s s c o n t i n u e s u n t i l t h e c o n c e n t r a t i o n of f u e l and t h e t e m p e r a t u r e , a t t h e c e n t e r , i s h i g h enough f o r t h e f u s i o n r e a c t i o n
t o t a k e place. 2.
THE MODEL
I n t h i s a n a l y s i s , i n s t e a d of m o d e l i n g p u l s e s o f e n e r g y , w e assume t h a t t h e p e l l e t is s u b j e c t t o a t i m e - i n v a r i a n t e n e r g y s o u r c e n g i v e n by B0r
.
Using t h e p r i n c i p l e s o f f l u i d ( g a s ) d y n a m i c s , and u t i l i z i n g t h e s p h e r i c a l symmetry of t h e p r o b l e m , t h e e q u a t i o n s d e s c r i b i n g t h e process are:
aU +
au +
at
:E
-
ar
*+
RTp-'
ar
-1
aT
+ u ar + z ( y - 1 ) T u r
+
R aT =
ar
(y-l)T
o
2U at =
B 0 rn
where p ( r , t ) i s t h e d e n s i t y i n s i d e t h e p e l l e t , u ( r , t ) i s t h e v e l o c i t y of t h e s h o c k wave, T ( r , t ) is t h e temperature i n s i d e t h e p e l l e t ,
r
is the radial variable,
t
is time,
R
i s t h e g a s c o n s t a n t for l e a d , and
y, B0
a r e t h e r a t i o o f s p e c i f i c h e a t s and s o u r c e s t r e n g t h , respectively.
The b o u n d a r y / i n i t i a l p(r,O)
c o n d i t i o n s are: u(r,O) = 0 ,
= p0(r),
T(r,O)
= To(')
and u ( 0 , t ) = 0 3.
WAVE VELOCITIES
The wave v e l o c i t i e s f o r e q u a t i o n s ( 1 ) - ( 3 ) a r e
c1 = u,
C2
= u-(yRT) 1 / 2
,
c 3 = u+(yRT)1 / 2
.
Thus f o r y , R , T # 0 t h e s y s t e m i s TOTALLY HYPERBOLIC.
4.
APPLICATION O F THE DILATATION GROUP ( A m e s [l], Bluman and C o l e [21, O v s i a n n i k o v I311 Assume p =
asp,
a-
s-
E-
x-
u = a u, T = a T, r = a r, t = a t ,
201
Nonlinear Waves in the Pellet Fusion Process
with parameter a .
The invariants of the group are
E Z O ( 1 ) - ( 3 ) imply that
The invariance of the P.D.E.’s
E = -n+1 3
- -_2- (n+l) A
-
2-n 3
Following from the transformation described by ( 4 ) and subject to the conditions (5)- ( 7 ) , equations (1 - (3) become f’
h’
+
(2+@+E)fg E
E
f’ h g’ - A n(gg’ + R E f
+
Rh’
n(gh’ + (y-1)g’h)
+
( -6+
-
2(y-1) + E a (y-1))gh = O o ,
(10)
where f = f(q), g = g ( n ) , h = h(n). NOW, let us investigate the boundary/initial conditions under the dilatation transformation (i) p(r,O) = po(r) From ( 4 ) p (r,t) = r"IEf ( n ) , so that p(r,O) = ralEf ( 0 ) . a Hence assuming that p 0 (r) = Ar gives a/& = a and f ( 0 ) = A. (ii) u(r,O) = 0 Since u(r,t) = ralEg(q) it follows that u(r,O) = r’"g(O), whereupon g(0) = 0. (iii) T(r,O) = To(r) then T(r,O) = r2/3 (n+l)h (0) Hence Since T(r,t) = r"’h(q)
.
with To(r) = Br 2/3(n+1) it follows h(0) = B. (iv) u(0,t) = 0 Recall u(r,t) = rBiEg(q) = r R / E 9(=)t = r (n+l)/3g (tr(n-2)/3) r whereupon lim r(n+1)/39(t,r(n-2)/3) = r+O > 2. provided n -
202
V.J. Ervin et al.
CONSTRUCTION O F EXACT SOLUTIONS
5.
Now
C o n s i d e r t h e c a s e when n = 2 .
B = 1 n = 2 implies -
-
6 -
'
E
= 2
;
-
E q u a t i o n s ( 8 ) (10) t h e n become
+
f'
+
g'
(3
+
g2
+
h'
+
E)fg = 0
(2
+
E)Rh = 0
(3y-l)gh
Bo
=
w i t h i n i t i a l c o n d i t i o n s f(0) = A , g ( 0 ) = 0 and h ( 0 ) = B . From (11) it f o l l o w s t h a t r)
f ( r ) ) = A e x p [ ( 3 + ):
g(SIdS1
’J 0
From ( 1 2 ) h = - 4 ' (2
2
+
+
g E)R
whereupon
-
=
h’
g" (2
Equation
+
+
2gg' F)R
(12) gives
gl ( 0 ) = - ( 2 + ~ ) R B .
(17)
S u b s t i t u t i n g ( 1 5 ) and (16) i n t o ( 1 3 ) y i e l d s g"
+
( 3 Y + l ) g ' g + ( 3 y - l ) g 3 = -8,(2+:)R,
with i n i t i a l conditions
g(0) = 0
,
g ' ( 0 ) = -(2
(18)
.
+
(19)
The change o f v a r i a b l e s'
= ugs,
(20)
gives = lls(g'
s"
2 + IJg ) ,
(21)
and = IJs(g"
s'"
I n p a r t i c u l a r , f o r y = 5/3
v
= 2,
2 3
+ 3 u g ' g + LJ g 1 .
(22)
( c o r r e s p o n d i n g t o a monatomic gas) and
e q u a t i o n ( 1 8 ) becomes s"'
+ 280 ( 2 +
:)Rs
= 0,
w i t h i n i t i a l c o n d i t i o n s s (0) = 1, s ' ( 0 ) = 0 , s" ( 0 ) = - 2 ( 2 The s o l u t i o n of
( 2 3 ) and ( 2 4 ) i s
s(r)) =
where
cle -"
+
z r
e ( F ' 2 ) r ) ( c2c o s Fr) 2
+
+
(23 1 9)RB. (24)
c3 s i n Fr)), ( 2 5 ) 2
203
Nonlinear Waves in the Pellet Fusion Process (r
F = ( 2 f i O ( 2 + ,)R)
1/3
and c 1 , c 2 , c 3 a r e g i v e n by
From e q u a t i o n ( 2 0 ) ri
s(n) = e x p [ 2
.f
g(S)dSI
0
and
where c4
For y =
2 -, 3
=d-
w e choose 1~
and
Cp
= Tan -1 c2
1 i n equations
=
c3 (20)-(22).
(30) Equation
(18) t h e n becomes s"'
(n) +
a0(2
with i n i t i a l conditions s ( 0 )
=
+
(31)
Z)Rs(n) = 0,
1, s ' ( 0 )
=
0 , s" ( 0 ) = - ( 2 t
E ) R B .( 3 2 )
With G:?-,G#O,
(33)
t h e s o l u t i o n f o r s ( q ) becomes
where t h e c o n s t a n t s c 1 , c 2 , c 3 a r e g i v e n by
where
Jd-
4 = Tan -1 c2 and 2 c4 = C, I n t h e case y = 1 / 3 , e q u a t i o n ( 1 8 ) becomes
.
'
g " ( n ) + 2g' ( n ) g ( q ) = - B 0 ( 2 + F ) R . Integrating, we obtain t h a t f i r s t order d i f f e r e n t i a l equation g'(n) + (g(q))' = -B0(2
+ Fa )
R ~+ c .
(37) (38)
204
V.J. Ervin et a/.
Applying t h e i n i t i a l c o n d i t i o n s g ( 0 )
+
we o b t a i n t h a t c = - ( 2
:)RB.
0 and g ’ ( 0 ) = - ( 2
=
+
a)RB, E
t h e s e i n t o equation (38)
Substituting
there results g ’ (II)
+
(g(II)l2= -D
0
(2
+
E)R(n
+
B r)
(39)
0
Upon a p p l y i n g e q u a t i o n s ( 2 0 ) and (21) w i t h il = 1, e q u a t i o n ( 3 9 )
becomes a
+ B0(2 + r ) R ( r i
S”(II)
B
+ ~ ) s ( I I ) = 0,
(40)
0
w i t h i n i t i a l c o n d i t i o n s s ( 0 ) = 1, s ’ ( 0 ) = 0. Making t h e change of v a r i a b l e
5
+
= q
becomes
B
,
B , equation (40) 60
B s ’ (--) a0
= 0 (42) 60 E q u a t i o n ( 4 1 ) i s a B e s s e l e q u a t i o n of o r d e r 1/3, whose s o l u t i o n i s
s(-)
=
1
s ( C ) = C1’2[clJl/3(fi(C))
+ C~J-~,,~(O(E))I,
(43)
where
For t h e a p p l i c a t i o n of i n i t i a l c o n d i t i o n s , it i s c o n v e n i e n t t o s e t
no
=
=
(6 ( 2 + Ia) R ) 1/2 2 ( B ) 3 / 2
(46)
0 60 = 1 implies C ~ J ~ , , ~ + (CQ~~J )- ~ , ~= ( (-B-) Q 6 0 ~1 /)2
60
Then s ( - B) and
(47)
BO B
s ’ (--)
= 0 gives
(48)
BO [C1J1/3 (Qo)+C2J-1/3
( n o )I
+
~ ~ ~ O O [ C ~(RO)+C2J-1/3 J ; / ~
(no)I
= 0-
U s i n g e q u a t i o n ( 4 6 ) and ( 4 7 ) e q u a t i o n ( 4 8 ) r e d u c e s t o C1Ji/3(Qo)
+
C2J-1/3(Q
Hence c1 and c 2 a r e g i v e n by
0) =
-
1 60 1 / 2 3 R 0 (--)B
.
(49)
205
Since t h e functions J
(n) and
( Q ) a r e l i n e a r l y independent 1/3 s o l u t i o n s of ( 4 1 ) t h e n t h e Wronskian ( J1 / 3 ( Q ) r ~ - 1 / 3(a)) R=fi 0. 0 Hence c1 and c2 a r e u n i q u e l y d e t e r m i n e d by ( 5 0 ) . Using t h e r e l a t i o n s h i p
113
(5)
=
1
1
- Jr+,(z)
Jr-l(Z)
S’
J-
5 - 1 / 2 {C1[JlI3
(\))
+
(51)
= 2Jk(Z)
+ 3 RJ-
z
2/3
3 (0) - 7 RJ
4/3 (n) (52)
3 + c2[J(Q) + 1/3 2 and t h e r e f o r e g(u) =
1
q(n +
-e/3
-
(fi)
- 2 3
+ 3 RJ-2,3(R)
-1(c1[J1/3(h2) -)a 0
QJ
c1JlI3
(a) +
3 RJ (a) I } 2 2/3 f2J4/,(Q)l
c2J-1,3 ( Q ) (53)
where
6.
SINGULARITIES OF g ( q ) I n e a c h of t h e cases p r e s e n t e d a b o v e ,
f u n c t i o n g ( n ) w a s g i v e n by s (q)/ps(q). whenever s(n) = 0 .
(y = 5/3,
2/3,
1/31,
the
Thus g ( n ) i s s i n g u l a r
The s i n g u l a r i t i e s o f g ( q ) may b e i n t e r p r e t e d a s
t h e “ t i m e “ when t h e f u s i o n p r o c e s s o c c u r s .
The p r o c e d u r e would be
v i r t u a l l y u s e l e s s i f , a f t e r two atoms f u s e d t o g e t h e r , t h e p r o c e s s What w e would l i k e t o see happen i s a c o n t i n u a t i o n of
stopped.
f u s i o n a f t e r an i n i t i a l f u s i o n h a s t a k e n p l a c e . T h i s l e a d s u s t o i n v e s t i g a t e t h e z e r o s of s(n).
For y
=
5 3
F = (2a0(2
y = 2
+
s(n) = cle-Fn CY
+
;)R)
s(q) = cle -GQ G =
( a0 ( 2
a
+
+ -)R)
c 4 e ( F / 2 ) Q s i n (J5 T
~n + 4);
1/3 c4e(G/2)Qsin(q
+
4);
1/3 t
and f o r
Y
=
1
3
s(n) = ( n +
&)1’2[~lJl,,3(Q(n))
+
C2J-1/3(f2(Q))I;
206
V.J. Ervin et a/. =
Q(n)
(B0(2
+ *),'I2
$(q
E
+
a) B 3/2 . 0
I n t h e f i r s t two c a s e s i t i s o b v i o u s t h a t s ( n ) w i l l have i n f i n i t e l y A c t u a l l y a s q becomes l a r g e t h e s e z e r o s w i l l
many z e r o s f o r q > 0 .
o c c u r a l m o s t p e r i o d i c a l l y w i t h p e r i o d s 4 n / n F and 4 n / f i G tively.
respec-
F o r y = 1 / 3 l e t u s examine e q u a t i o n ( 4 0 )
U n l i k e t h e p r e v i o u s t w o cases, i n which t h e z e r o s o c c u r almost periodically, Theorem'' [ c f . 4 1 Theorem: U"
+
the zeros of
S(Q)
( f o r y = 1/3)
o c c u r "more
To see t h i s w e a p p l y t h e " I n t e r l a c i n g of Zeros
frequently."
.
I f f ( x ) and g ( x ) a r e l i n e a r l y i n d e p e n d e n t s o l u t i o n s o f
p(x)u'
+
g ( x ) u = 0,
t h e n f ( x ) must v a n i s h a t o n e p o i n t between
any two s u c c e s s i v e z e r o s o f g ( x ) .
I n o t h e r words, t h e z e r o s of f ( x )
and g ( x ) o c c u r a l t e r n a t i v e l y . Observe t h a t
i s a s o l u t i o n t o equation ( 4 1 ) . a r e s e p a r a t e d by a p p r o x i m a t e l y n.
i n c r e a s e s by T , d e c r e a s e s .
5 t h e z e r o s o f J 1 / 3 ( ') A s q i n c r e a s e s t h e amount by
For l a r g e
Hence t h e z e r o s of w ( q ) o c c u r "more
S i n c e t h e z e r o s of s ( n ) and w ( q ) a r e i t f o l l o w s t h a t t h e z e r o s of s ( q ) must a l s o o c c u r "more f r e q u e n t l y " a s rl i n c r e a s e s .
f r e q u e n t l y " as rl i n c r e a s e s . interlaced,
7.
NUMERICAL RESULTS
I n a d d i t i o n t o t h e a n a l y t i c r e s u l t s , p r e v i o u s l y d i s c u s s e d , some numerical s o l u t i o n s f o r t h e f i r s t o r d e r system ( e q u a t i o n s ( 8 ) - ( 1 0 ) ) have b e e n c a l c u l a t e d on t h e G e o r g i a Tech CYBER u s i n g t h e DGEAR package.
The g r a p h s w e r e made by a TERAK micro-computer.
F i g u r e 2 p r e s e n t s t h e s o l u t i o n s for f , g , and h w i t h i n i t i a l d a t a f ( 0 ) = 5.10, and p,
g ( 0 ) = 0 and h ( 0 ) = 0 . 0 1 ,
for n
=
2,
CX/E
= 2
= 3.47.
F i g u r e 3 p r e s e n t s t h e s o l u t i o n s f o r f , g , and h w i t h i n i t i a l d a t a f ( 0 ) = 8.16,
g ( 0 ) = 0 , and h ( 0 ) = 0 . 0 1 ,
for n = 2,
CX/E
= 5
and O o = 3 . 1 0 . The l a s t F i g u r e , 4 ,
r e s u l t s from t h e same i n i t i a l d a t a a s t h a t
f o r F i g u r e 2 b u t w i t h n = 7, a / ~= 2 a n d (3, = 5.95. The m u l t i p l e p e a k s o f t h i s f i g u r e a r e n o t o f p h y s i c a l o r i g i n . They w e r e i n t r o -
207
Nonlinear Waves in the Pellet Fusion Process
d u c e d by t h e TERAK c o m p u t e r a s i t a t t e m p t e d t o " s t e p " o v e r t h e singularity. REMARKS
The s u b s t i t u t i o n d e s c r i b e d i n e q u a t i o n ( 2 0 ) e n a b l e d u s t o
r e w r i t e t h e n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s ( 1 8 ) and ( 3 9 ) a s l i n e a r e q u a t i o n s o f one h i g h e r o r d e r . U n i q u e n e s s of t h e s o l u t i o n s f o r p r o b l e m s ( 2 3 ) - ( 2 4 )
,
( 3 1 )- ( 3 2 ) ,
a n d ( 4 1 ) - ( 4 2 ) i m p l i e s u n i q u e s o l u t i o n s o f e q u a t i o n s ( 1 8 )- ( 1 9 ) f o r t h e cases y = 5 / 3 , I n t h e case n = 2 , p ( r , t ) =r"/'f(t),
2/3,
and 1/3.
t which i m p l i e s t h e s e p a r a b l e s o l u t i o n 2 u ( r , t ) = r g ( t ) , and T ( r , t ) = r h ( t ) . T-
=
REFERENCES
1.
A m e s , W. F . , N o n l i n e a r P a r t i a l D i f f e r e n t i a l E q u a t i o n s i n E n g i n e e r i n g , Academic P r e s s , N e w York, V o l . I ( 1 9 6 5 ) , Vol. I1 (1972).
2.
Bluman, G. W . a n d C o l e , J . D . , S i m i l a r i t y Methods f o r D i f f e r e n t i a l E q u a t i o n s , S p r i n g e r , 1974.
3.
O v s i a n n i k o v , L . V . , Group A n a l y s i s of D i f f e r e n t i a l E q u a t i o n s , ( W . F. Ames, E d . ) Academic P r e s s , N e w York, 1982.
4.
B i r k h o f f , G. a n d R o t a , G . - C . , Ordinary D i f f e r e n t i a l Equations ( 3 r d E d . ) , W i l e y , N e w York, 1 9 7 8 ( p . 3 8 ) .
V.J. Ervin et al.
208
150
Figure 2.
Nonlinear Waves in the Pellet Fusion Process
Figure 3 .
209
210
V.J. Ervin et al.
F i g u r e 4.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
21 1
ENTROPY PRINCIPLE, SYMMETRIC HYPERBOLIC SYSTEMS AND SHOCK WAVES
Tommaso Ruggeri D i p a r t i m e n t o d i Matematica - U n i v e r s i t a d i Bologna V i a V a l l e s c u r a 2 , 40136 Bologna, I t a l y
INTRODUCTION The o b j e c t o f t h i s p a p e r i s t o g i v e r e l a t i o n s between two arguments a p p e a r i n g i n t h e l i t e r a t u r e w i t h d i f f e r e n t c o n c l u s i o n s and a p p a r e n t l y w i t h o u t common features. The former argument i s r e l a t e d t o a g e n e r a l q u e s t i o n of s y s t e m s o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f h y p e r b o l i c t y p e ; t h e l a t t e r c o n c e r n s t h e r o l e of t h e " e n t r o p y p r i n c i p l e " i n continuum t h e o r i e s . W e s t a r t with a b r i e f survey of these s u b j e c t s . W e c o n s i d e r a g e n e r i c q u a s i - l i n e a r f i r s t o r d e r s y s t e m of t h e t y p e :
AO(U)
au/at +
A'(%)
ayax,
=
gcg
+
-t
f o r t h e unknown N-column v e c t o r u ( t , x ) : u : ( u ~ , u ~ , . . . , u; ~x ) ~b e l o n g i n g t o are r e a l N X N m a t r i c e s which are f u n c t i o n s of u , t h e source ; Ao , A' term f ( 5 ) i s a l s o a v e c t o r of RN , T i n d i c a t e s t h e t r a n s p o s e and t h e E i n s t e i n c o n v e n t i o n on t h e i n d e x i s u s e d . By i d e n t i f y i n g t h e t i m e t w i t h xo , i t i s p o s s i b l e t o w r i t e t h e s y s t e m i n t h e a b b r e v i a t e d form: Rn
D e finition of hyperb o li cit y : The s y s t e m (1) i s h y p e r b o l i c i n t h e t - d i r e c t i o n , i)
d e t A"
#
0
-f
ii) Vv
E
Rn
:
if:
,
11 +v 11
= 1
,
t h e f o l l o w i n g e i g e n v a l u e problem
has only r e a l proper values h ( c h a r a c t e r i s t i c v e l o c i t i e s ) and N l i n e a r l y i n d e p e n d e n t e i g e n v e c t o r s r (Av = AiVi)
.
D e f i n i t i o n o f c o n s e r v a t i v e system: The s y s t e m (1) i s s a i d t o b e " c o n s e r v a t i v e " ( o r b e t t e r : a s y s t e m o f b a l a n c e l a w s ) , i f t h e r e e x i s t f o u r v e c t o r s Fa of RN , such t h a t t h e m a t r i c e s A" are gradients of w i t h r e s p e c t t o a s u i t a b l e f i e l d u : Aa = , i . e . e x h i b i t t h e form:
.ca
ac"/ag
T. Ruggeri
212
W e o b s e r v e t h a t a s y s t e m of t y p e ( 2 ) is v e r y common i n Mathematical P h y s i c s , b e c a u s e i t r e p r e s e n t s t h e l o c a l form (when d i f f e r e n t i a b i l i t y c o n d i t i o n s h o l d ) o f an i n t e g r a l b a l a n c e . O n t h e o t h e r hand i t i s w e l l known t h a t f o r t h i s k i n d o f system i t i s p o s s i b l e t o c o n s i d e r a more g e n e r a l c l a s s of s o l u t i o n s : " t h e weak Shock waves, t h a t a r e v e r y s o l u t i o n s " , f o r which c o n t i n u i t y i s n o t n e c e s s a r y . i m p o r t a n t i n some p h y s i c a l p r o b l e m s , b e l o n g t o t h i s c l a s s .
D e f i n i t i o n of symmetric h y p e r b o l i c s y s t e m s : A system o f t y p e (1) i s s a i d t o b e symmetric i f
is positive definite; i) A" = (Aa)T; ii) A" (from l i n e a r a l g e b r a any symmetric s y s t e m i s a u t o m a t i c a l l y h y p e r b o l i c ) The symmetric s y s t e m p l a y s an i m p o r t a n t r o l e c o n c e r n i n g t h e w e l l - p o s i n g o f t h e Cauchy problem; i n f a c t t h e r e e x i s t some theorems e n s u r i n g e x i s t e n c e , u n i q u e n e s s For i n s t a n c e w e quote t h e and c o n t i n u o u s dependence f o r t h e l o c a l Cauchy problem. r e s u l t due t o F i s c h e r and Marsden 111 i n 1 9 7 2 : "Any symmetric s y s t e m , t h e i n i t i a l d a t a of which a r e chosen i n a Sobolev s p a c e HS w i t h s 2 4 , h a s a u n i q u e s o l u t i o n 2 E Hs , i n t h e neighbourhood o f t h e i n i t i a l manifold". The r e a d e r who i s i n t e r e s t e d may a l s o see, i n a more g e n e r a l c o n t e x t , t h ? i m p o r t a n t r e s u l t by V o l p e r t and Hudiaev 12
1.
Supplementary c o n s e r v a t i o n law ( e n t r o p y p r i n c i p l e ) : I n 1 9 7 1 F r i e d r i c h s and Lax 131 f o r m u l a t e d t h e f o l l o w i n g q u e s t i o n : o f system ( 2 ) i s compatible w i t h t h e following assumptions?
which t y p e
i ) a l l t h e s o l u t i o n s o f ( 2 ) a l so s a t i s f y a n o t h e r s u p p l e m e n t a r y c o n s e r v a t i o n law: aaha(:) ii) choosing
N D F R
.
u -
=
Fo , ho -
= gcg
(3)
t
i s a s t r i c t l y convex f u n c t i o n of
u i n a convex domair) -
Under t h e s e h y p o t h e s e s t h e y p r o v e d t h a t t h e r e e x i s t s a p o s i t i v e d e f i n i t e NxN m a t r i x H ( u ) , s u c h t h a t t h e new s y s t e m H ( g ) . { a aFa(%) o b t a i n e d from ( 2 ) t i m e s H
,
-
f(g)}=
0
(4)
i s a symmetric h y p e r b o l i c one.
The i m p o r t a n t r e s u l t o f t h e s e a u t h o r s h a s , however, t h e f o l l o w i n g d i s a d v a n t a g e s :
1) The c o n s e r v a t i v e form of t h e new symmetric s y s t e m ( 4 ) i s l o s t ; 2 ) W e have n o t an e x p l i c i t e x p r e s s i o n f o r t h e f u n c t i o n a l dependence o f respect the f i e l d g
Fa
with
.
The former r e s t r i c t i o n d o e s n ' t a l l o w u s e o f t h e p e c u l i a r p r o p e r t i e s o f symmetric s y s t e m s c o n c e r n i n g weak s o l u t i o n s and,in p a r t i c u l a r , s h o c k s . I n f a c t , many s y s t e m s o f e v o l u t i o n i n t h e macroscopic t h e o r i e s of c o n t i n u a are w r i t t e n i n t h e form ( 2 ) , b u t t h e f u n c t i o n a l dependence o f w i t h r e s p e c t t o t h e f i e l d u i s completely a s s i g n e d o n l y when t h e c o n s t i t u t i v e r e l a t i o n s a r e known. One o f t h e m o s t i m p o r t a n t problems i s t o e s t a b l i s h t h e g e n e r a l c r i t e r i a f o r c o n s t i t u t i v e relations.
ra
I n t h e modern approach t o thermodynamics t h e e n t r o p y i n e q u a l i t y i s r e g a r d e d as a c o n s t r a i n t o n t h e c o n s t i t u t i v e r e l a t i o n s and n o t as an i d e n t i f i c a t i o n of a privileged t i m e orientation. F i r s t s u p p o r t e r s o f t h i s p o i n t o f view were
213
Entropy Principle and Shock Waves
l a t e r , a more g e n e r a l p r i n c i p l e w a s proposed by M i l l e r i n
Coleman and No11 141;
151. The s i t u a t i o n i s s u b s t a n t i a l l y t h e f o l l o w i n g : b o t h t h e e n t r o p y d e n s i t y and t h e e n t r o p y f l u x are t h o u g h t o f a s t h e c o n s t i t u t i v e r e l a t i o n s of t h e s e t o f f i e l d v a r i a b l e s and for a l l thermodynamical p r o c e s s e s i . e . , f o r a l l t h e s o l u t i o n s of t h e p a r t i a l d i f f e r e n t i a l s y s t e m , t h e f o l l o w i n g i n e q u a l i t y must b e i d e n t i c a l l y satisfied ~~
2
t
pS
+ a.
Qi
= s t 0
is t h e m a s s density).
(p
I n t h e l i m i t o f t h e c o n s t i t u t i v e f u n c t i o n s t h a t l o c a l l y depend on t h e f i e l d , t h i s problem i s c o m p l e t e l y t h e same a s t h e p r e v i o u s one, t h r o u g h t h e i d e n t i f i c a t i o n s ho = -pS , and h' = -0'
.
I n t h i s case i t i s no l o n g e r i m p o r t a n t t h a t t h e s y s t e m b e symmetric b u t on t h e c o n t r a r y i t i s i m p o r t a n t t o g i v e an e x p l i c i t c h a r a c t e r i z a t i o n o f f u n c t i o n a l and ha w i t h r e s p e c t t o t h e f i e l d 5 and t h e r e f o r e a complete dependence of c h a r a c t e r i z a t i o n of t h e compatible c o n s t i t u t i v e e q u a t i o n s .
c"
Now we q u o t e a theorem t h a t g i v e s a d i f f e r e n t p r o o f t o t h e , o n e b y F r i e d r i c h s and I t i s f r e e from t h e p r e v i o u s o b j e c t i o n s and a t t h e same t i m e , p e r m i t s a Lax. connection w i t h t h e entropy p r i n c i p l e . M A I N FIELD AND GENERATORS OF A SYMMETRIC SYSTEM
Theorem:
" A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a c o n s e r v a t i v e s y s t e m
t o be compatible w i t h a supplementary c o n s e r v a t i o n i n e q u a l i t y
i s t h a t t h e c o n s t i t u t i v e r e l a t i o n s are a l l o n e s f o r which f o u r s c a l a r s and a p r i v i l e g e d s e t o f f i e l d v a r i a b l e s x' , e x i s t s u c h t h a t :
F"
= ahi"/a
The components o f t h e "main f i e l d "
11’
h'"
are r e l a t e d t o
Moreover i f i t i s p o s s i b l e t o choose
u’
o
.
(5)
are t h e "Lagrange m u l t i p l i e r s " , s u c h t h a t :
u ' . d F-a and t h e ( p o t e n t i a l s
=
;
h'"
2 dh"
ha
,
by:
as f i e l d = aho/au
5 , we
h a v e from ( 6 )
,
(8)
ro
and if ho i s a convex f u n c t i o n o f 5 = ( o r l i k e w i s e , i f h'O i s a convex u’) , t h e n t h e o r i g i n a l s y s t e m i s a v e r y s p e c i a l , symmetric one f u n c t i o n of i n t h e f i e l d 5':
T h e r e f o r e t h e l o c a l Cauchy problem i s w e l l p o s e d " .
u’
which S u b s t a n t i a l l y , t h i s theorem g u a r a n t e e s t h a t there e x i s t s a f i e l d symmetrizes t h e o r i g i n a l system w i t h o u t l o s i n g t h e c o n s e r v a t i v e s t r u c t u r e and w i t h respect t o t h i s f i e l d , t h e s y s t e m assumes t h e s p e c i a l form ( 9 ) . W e o b s e r v e t h a t a l l matrices i n ( 9 ) are H e s s i a n o n e s and h q 0 i s t h e Legendre c o n j u g a t e
214
T. Ruggeri
function of
ho ( s e e ( 7 ) ,
(8) f o r
c1 =
0).
The systems of t y p e ( 9 ) have been considered f i r s t by Godunov 161 i n 1961, who observed t h a t t h e system f o r a p e r f e c t f l u i d and the Euler-Lagrange equations a r e s u s c e p t i b l e t o t h i s form. Subsequently t h e same author noted t h a t t h e systems of this form (9) always admit an equation ( 3 ) a s a consequence 17
I.
The i n t r o d u c t i o n of t h e f i e l d u’ through t h e eq. system i s due to B o i l l a t 181 in-1974.
( 8 ) f o r a g e n e r i c hyperbolic
5’ a r e introduced a s t h e Lagrange m u l t i p l i e r s and new p r o p e r t i e s appear i s due t o Ruggeri and Strumia 19 i n 1981 and a technique which holds a l s o f o r non-hyperbolic systems was presented q u i t e r e c e n t l y in )10).
A more general approach, i n which
1
D e f i n i t i o n of "Generators" : The systems of type (2) with t h e supplementary conservation l a w ( 3 ) a r e completely determined i f w e know t h e 2N+4 o b j e c t s : [E' , h " , f-} In f a c t from (5.1) we determine and from ( 7 ) and (5.2) h" and g Therefore w e c a l l t h e s e q u a n t i t i e s "generators".
c"
.
.
Now it i s necessary t o d i s t i n g u i s h two d i f f e r e n t kind of problems. Problem I I f one knows t h e supplementary conservation law and t h e c o n s t i t u t i v e r e l a t i o n s , i t i s very easy t o e v a l u a t e t h e generators from (8) and ( 7 ) and t h e r e f o r e t h e symmetric form of the system. I n t h i s s i t u a t i o n it is p o s s i b l e to use t h e s p e c i a l s t r u c t u r e of t h e system ( 9 ) and prove some important theorems on shock waves. Problem I1 I f we regard the supplementary law a s an "entropy p r i n c i p l e " t h e theorem becomes only q u a l i t a t i v e , because "a p r i o r i " 5’ and h ' " a r e not known. I n o t h e r words we know t h a t t h e system i s i n t h e form ( 9 ) , o r e q u i v a l e n t l y
b u t what i s not evident a r e t h e r e l a t i o n s between t h e q u a n t i t i e s t h e physical q u a n t i t i e s appearing i n t h e problem.
g’
and
hIa
and
I n t h e next s e c t i o n , w e p r e s e n t f i r s t some r e c e n t r e s u l t s on shock waves r e l a t e d t o problem I and then w e p r e s e n t some p o s s i b l e approaches t o t h e second problem. Most of t h i s l a s t p a r t i s completely new.
CONSEQUENCES FOR SHOCK WAVES Under t h e hypotheses o f t h e previous theorem, i t i s p o s s i b l e t o prove some i n t e r e s t i n g p r o p e r t i e s on shock waves:
i) t h e shock v e l o c i t i e s
s
a r e proved t o s a t i s f y t h e i n e q u a l i t i e s :
i n f min u6D k where
X
c s < sup Max
(k)
ucD k -
a r e the c h a r a c t e r i s t i c v e l o c i t i e s .
This theorem ( B o i l l a t and Ruqqeri 1111)
, ensures
t h a t t h e Rankine-Hugoniot
21 5
Entropy Principle and Shock Waves
equations :
([:
1
d e n o t e s t h e jump; +(xu)
=
0
i s t h e e q u a t i o n of t h e shock s u r f a c e
r
;
and n . = $i/lV$l , s = - $ t / / V $ / a r e r e s p e c t i v e l y t h e u n i t normal and t h e v e l o c i t y o f l t h e shock s u r f a c e ) , admit o n l y t h e t r i v i a l s o l u t i o n [%I = 0 T h e r e f o r e n o n - v a n i s h i n g s h o c k s take p l a c e o n l y i f t h e i r i f (11) i s n o t s a t i s f i e d . s p e e d i s g r e a t e r t h a n t h e s m a l l e s t c h a r a c t e r i s t i c s p e e d and smaller t h a n t h e g r e a t e s t one.
0,
= a$/axa
A c o v a r i a n t v e r s i o n o f t h i s theorem i s s t a t e d i n 191 and as a consequence i n a r e l a t i v i s t i c t h e o r y Is1 2 c , where c i s t h e l i g h t v e l o c i t y i n vacuum.
ii) F o r m a l l y t h e Rankine-Hugoniot e q s . are o b t a i n e d from t h e f i e l d e q s . ( 2 ) t h r o u g h t h e c o r r e s p o n d e n c e r u l e aa -f [ However t h i s r u l e d o e s n o t work when it i s a p p l i e d t o t h e s u p p l e m e n t a r y e q . ( 3 ) ; i n f a c t
.
d o e s n o t , i n g e n e r a l , v a n i s h . I t i s known t h a t t h e p o s i t i v e s i g n a t u r e on q f o r a f l u i d implies t h e growth o f thermodynamic e n t r o p y a c r o s s t h e s h o c k . T h i s i s t h e r e a s o n why t h e c o n d i t i o n I- Z 0 i s o f t e n c a l l e d i n t h e l i t e r a t u r e t h e " e n t r o p y growth c o n d i t i o n " . Here, t h i s c r i t e r i o n p i c k s up t h e p h y s i c a l s h o c k s among t h e s o l u t i o n s o f t h e Rankine-Hugoniot e q s . i n s t e a d of s e l e c t i n g t h e c o n s t i t u t i v e e q s . , a s i n t h e d i f f e r e n t i a b l e c a s e . U s u a l l y t h i s growth i s p r o v e d t h r o u g h a r t i f i c i a l v i s c o s i t y c o n s i d e r a t i o n s I12 ; a p r o o f which u s e s o n l y c o n v e x i t y arguments f o r k-shocks i s made i n 1131 , 191.
I
iii) The knowledge o f t h e jump o f t h e f i e l d Example:
o n l y t h e s c a l a r f u n c t i o n q on r i s enough t o d e t e r m i n e b e h a v e s a s a s o r t of " p o t e n t i a l " f o r t h e shock 1131. : q
u’
Perfect Fluid
A s a t e s t of t h e p r e v i o u s c o n s i d e r a t i o n s , w e p r e s e n t t h e s i m p l e and w e l l known
example o f t h e e q s . o f a p e r f e c t f l u i d . The e q s . i n c o n s e r v a t i v e form are:
+
atp
[at - + +
( p +v )
atE
+
+ div +
o
d i v (pv) =
( m a s s balance)
T = 0
(momentum eq. ) -f
d i v { ( ~ + p ) v=} 0
(energy balance)
2
-f
where T = pv ti v + p I ; E = p ( e + v /2) and p , v , e mass d e n s i t y , v e l o c i t y , i n t e r n a l energy and p r e s s u r e . I n t h i s c a s e t h e supplementary l a w i s t h e at(ps)
The s y s t e m (13) and t h e e q . following identifications:
+
,p
(13)
,
are r e s p e c t i v e l y t h e
entropy balance:
+
d i v (psv) =
o
(14) are e q u i v a l e n t t o ( 2 ) and ( 3 ) , t h r o u g h t h e
I n t h i s case, s i m p l e c a l c u l a t i o n s g i v e t h e f o l l o w i n g g e n e r a t o r s :
216
T. Ruggeri
where 0 and G a r e r e s p e c t i v e l y t h e absolute temperature and t h e f r e e enthalpy ( t h e chemical p o t e n t i a l : G = e - 8s + p/p) . The convexity of ho = -pS respect t o (p,B) , i . e . : c > 0 P
,
(aV/apg)
< 0
can be proved i f
(V
= l/p
,
c
P
G(p,0)
i s a concave function with
is the s p e c i f i c heat a t constantpressure).
These c o n d i t i o n s are t h e u s u a l thermodynamic s t a b i l i t y conditions a t equilibrium. and a l l previous Therefore t h e system (13) i s symmetric i n t h e main f i e l d u ' considerations hold (on t h i s s u b j e c t s e e 161, 1121, 1141, -191, 1151). Several o t h e r examples i n t h i s a r e a are considered i n t h e r e c e n t l i t e r a t u r e v i z : h y p e r e l a s t i c s o l i d s I16 ; r e l a t i v i s t i c f l u i d s and magnetofluids 19 , I17 ; nonl i n e a r electrodynamics 118 , 1 9 , I20 , s u p e r f l u i d s 121 , mixture of f l u i d s I 2 2
I
I I I
I
I
I
I
I.
The reader who i s i n t e r e s t e d i n more a n a l y t i c a l arguments on t h e s e s u b j e c t s about the entropy growth and t h e weak s o l u t i o n s , may read t h e important n o t e s by Dafermos 1231 . CONSTRUCTION OF SYSTEMS OF BALANCE LAWS WHIM ADMIT A SUPPLEMENTARY LAW: REPRESENTATION THEOREMS The p h y s i c a l l y s i g n i f i c a n t examples examined s o f a r were p e r t i n e n t t o t h e f i r s t problem, where t h e supplementary law was "a p r i o r i " known t o g e t h e r with t h e associated c o n s t i t u t i v e equations. For i n s t a n c e , i n t h e previous c a s e of a p e r f e c t f l u i d we have followed t h e c l a s s i c a l t h e o r y ; i n o t h e r words t h e Gibbs r e l a t i o n w a s supposed t o hold and t h e concept of absolute temperature, entropy and i n t e r n a l energy was "a p r i o r i " accepted. The a i m was t o o b t a i n a new symmetric hyperbolic conservative system i n o r d e r t o have q u a l i t a t i v e information on t h e s o l u t i o n (well-positing of t h e Cauchy problem, some p r o p e r t i e s o f shock waves and so on ) . A l t e r n a t i v e l y i f we look a t t h e problem from t h e p o i n t o f view of " r a t i o n a l thermodynamics", w e d o n ' t accept t h e s e concepts a s p r i m i t i v e ones and by considering t h e entropy i n e q u a l i t y a s a c o n s t r a i n t on t h e c o n s t i t u t i v e equations, t h e Gibbs r e l a t i o n emerges a u t o m a t i c a l l y a s an i n t e g r a b i l i t y c o n d i t i o n , and t h e absolute temperature a s an i n t e g r a t i n g f a c t o r . Clearly t h i s approach provides a q u a l i t a t i v e jump from t h e view p o i n t o f t h e axiomatic thermodynamics, even though it does not change t h e p r a c t i c a l r e s u l t s . Moreover, s o m e t i m e s t h e t r u e equations a r e not known "a p r i o r i " because they r e p r e s e n t new models of some p h y s i c a l phenomena. For i n s t a n c e , f o r a viscous, h e a t conducting f l u i d , t h e problem i s t o -f determine t h e c o n s t i t u t i v e r e l a t i o n s f o r t h e h e a t f l u x q and f o r t h e viscous tensor C 5 ( C , ,) .
...
13
I n t h e c l a s s i c a l approach t h e s e axe t h e Fourier and Navier-Stokes equations which, a s it i s w e l l known, d e s t r o y t h e h y p e r b o l i c i t y of t h e f i e l d system with t h e consequence t h a t an i n f i n i t e wave propagation speed i s obtained. In o r d e r t o e l i m i n a t e t h e paradox and hyperbolize t h e system, t h e non-equilibrium thermodynamics or, following M u l l e r ' s d e f i n i t i o n , t h e "extznded i r r e v e r s i b l e thermodynamics" 1241, aims a t determining e v o l u t i o n s f o r q and C , preserving t h e h y p e r b o l i c i t y of t h e e n t i r e system. The b a s i c i d e a of t h i s approach is: one c o n s i d e r s t h e non-equilibrium entropy d e n s i t y ( d i f f e r e n t from t h e equilibrium one) n o t only t o depend on t h e two thermodynamic v a r i a b l e s b u t a l s o on $ and C ,
217
Entropy Principle and Shock Waves
c o n s e q u e n t l y m o d i f y i n g t h e Gibbs r e l a t i o n . As t h i s i s not the r i g h t place t o p u r s u e s u c h a s u b j e c t , we r e f e r t h e i n t e r e s t e d r e a d e r t o t h e work 1101 where t h e a u t h o r u s e s a t e c h n i q u e connected w i t h t h e g e n e r a t o r s t o set up a n e v o l u t o r y symmetric h y p e r b o l i c s y s t e m . However, w e want t o show how t h e s t r u c t u r e theorem, mentioned above, can g i v e some i n f o r m a t i o n a b o u t new models (which are c o m p a t i b l e w i t h wave p r o p a g a t i o n ) d e s c r i b e d b y more g e n e r a l b a l a n c e e q u a t i o n s which admit a supplementary c o n s e r v a t i o n l a w ( n o t n e c e s s a r i l y t h e e n t r o p y o n e ) . The t e c h n i q u e i s b a s e d on some simple h" w i t h r e s p e c t t o t h e components r e p r e s e n t a t i o n theorems f o r t h e f o u r - v e c t o r of t h e main f i e l d 5’
.
L e t u s b r i e f l y i l l u s t r a t e t h e methodology w i t h an example, i n o r d e r t o make t h e method c l e a r e r .
F o r example, w e s e e k t o d e t e r m i n e t h e most g e n e r a l s t r u c t u r e of a d i f f e r e n t i a l s y s t e m o f b a l a n c e e q u a t i o n s where t h e components o f t h e f i e l d a r e a s c a l a r f a c t o r and a R3-vector; moreover the s y s t e m must b e c o m p a t i b l e w i t h a s u p p l e mentary l a w t o b e d e t e r m i n e d . A t f i r s t s i g h t t h e problem i s e x t r e m e l y complex. I n f a c t one must w r i t e a s y s t e m o f t h e t y p e ( 2 ) , formed b y a s c a l a r e q u a t i o n and a v e c t o r i a l o n e , and t h e n must c h a r a c t e r i z e t h e s u b c l a s s which i s c o m p a t i b l e w i t h a s u p p l e m e n t a r y l a w ( 3 ) . One c o u l d u s e theorems on r e p r e s e n t a t i o n , b u t t h e s e would b e e x t r e m e l y complex even i n t h i s s i m p l e c a s e , and what i s more, it would b e e x t r e m e l y d i f f i c u l t t o c h a r a c t e r i z e t h e i n t e g r a b i l i t y c o n d i t i o n s so t h a t e v e r y s o l u t i o n o f ( 2 ) s a t i s f i e s ( 3 ) . Now w e show how w e can make u s e o f the q u a l i t a t i v e theorem a c c o r d i n g t o which t h e m o s t g e n e r a l s y s t e m ( 2 1 , when i t i s compatible w i t h a supplementary l a w , h a s t h e s t r u c t u r e such t h a t is t h e g r a d i e n t o f h q a w i t h r e s p e c t t o t h e f i e l d '; F o r t h e c a s e u n d e r examination, w e know c e r t a i n l y t h a t u’ must have f o u r components formed by a s c a l a r f a c t o r and a R3-vector: '; :
ca
.
(F,l)T .
3
h t 0 i s a s c a l a r f a c t o r and h I i i s t h e g e n e r i c component o f a R - v e c t o r , the unique p o s s i b l e r e p r e s e n t a t i o n f o r them i s :
As
h'O = p ( S , z )
;
i 2 hQi= v(6,z)h ; z = A /2
.
I n t h i s way, t h e problem i s c o n s i d e r a b l y s i m p l i f i e d , as we have had t o r e p r e s e n t only the p o t e n t i a l s h'" and n o t t h e complete s e t o f t h e v e c t o r s Fa Now from (5.1), i t f o l l o w s t h a t t h e most g e n e r a l s y s t e m i s o f t h e form:
.
and e v e r y s o l u t i o n of (15) w i l l s a t i s f y , by ( 7 ) , ( 3 ) , ( 5 . 2 ) , t h e s u p p l e m e n t a r y law:
(where t h e s u b s c r i p t s d e n o t e t h e d e r i v a t i v e s w i t h r e s p e c t t o t h e a r g u m e n t s ) . I n t h i s way, even though t h e q u a n t i t i e s which are i n v o l v e d h a v e , a s o f y e t , n o p h y s i c a l meaning, w e h a v e e s t a b l i s h e d the s p e c i a l i z e d s t r u c t u r e ( i t i s i m p o r t a n t to n o t i c e t h a t t h e c o e f f i c i e n t o f t h e main p a r t s are r e l a t e d t o t h e d e r i v a t i v e s of o n l y two s c a l a r f u n c t i o n s p , u) f o r t h e s y s t e m (15) which a u t o m a t i c a l l y and "a p r i o r i " s a t i s f i e s t h e s u p p l e m e n t a r y l a w (16). F u r t h e r m o r e , t h i s makes u s u n d e r s t a n d t h a t , i n d e p e n d e n t of t h e p h y s i c s o f t h e d i f f e r e n t q u a n t i t i e s i n v o l v e d , s y s t e m s w i t h t h e same number o f c o n s e r v a t i v e e q s . and c o m p a t i b l e w i t h a
21 8
T. Ruggeri
supplementary law, must have t h e same s t r u c t u r e . Now, i f w e wish t o s p e c i a l i z e t h e s y s t e m (15) t o t h e s y s t e m f o r t h e i s e n t r o p i c p e r f e c t f l u i d , t h e n e c e s s a r y i d e n t i f i c a t i o n s are: P = p5
Set
E
=
,
pv
i
5lJ + 2zFi 5 E =
= v
5’A
-
l~
= pzA
i
, pv iv j
= vsij
+
+
~
z
niAj , a = B = O .
, and w e o b t a i n , a f t e r s t r a i g h t f o r w a r d c a l c u l a t i o n s ,
2 P ( e + v / 2 ) , w i t h e such t h a t :
d e = p dp/p
2
: p = w = p
;
+
2 l = v ; ~ = e + p / p - v / 2 .
I n o t h e r w o r d s , we f i n d t h a t t h e supplementary e q u a t i o n i s t h e e n e r g y conservat i o n one ( i n t h e m e c h a n i c a l case, t h e e n e r g y e q u a t i o n p l a y s t h e same r o l e as t h e e n t r o p y one i n thermodynamical case) and t h e c o n s t i t u t i v e c o n s t r a i n t a r i s e s a u t o m a t i c a l l y from t h e i d e n t i f i c a t i o n . Heat e q u a t i o n w i t h f i n i t e v e l o c i t i e s
Now, w e w i s h t o c o n s i d e r a d i f f e r e n t problem, d e s c r i b e d by t h e same number o f b a l a n c e e q u a t i o n s : t h i s i s , the case o f a r i g i d c o n d u c t o r w i t h h e a t p r o p a g a t i o n . I f w e a i m t o o b t a i n h y p e r b o l i c e q u a t i o n s , i n t h i s case, t h e staze o f t h e s y s t e m i s c h a r a c t e r i z e d by t h e t o t a l energy 6 and by t h e h e a t f l u x q
.
The f i r s t e q u a t i o n of (15) must b e i d e n t i f i e d w i t h t h e e n e r g y c o n s e r v a t i o n equation
+
a&/at +
div q
o ,
=
w h i l e t h e remaining o n e s of ( 1 5 ) are t h e e q u a t i o n s f o r I t i s s e e n t h a t ( 1 7 ) can a l s o b e w r i t t e n a s
+
aE/at
-+
div
(EW)
=
o
( n o t known "a p r i o r i " ) .
q
+ ;
(17)
+
+
w = q/E
.
(18)
T h i s is s i m i l a r t o t h e mass b a l a n c e e q u a t i o n , p r o v i d e d w e c o n s i d e r t h e f o l l o w i n g duality:
+
p + E ;
+
V + W .
From w h a t h a s been s a i d , ( 1 5 . 2 ) w i l l be s i m i l a r t o t h e e q u a t i o n o f momentum f o r t h e f l u i d , by making u s e o f t h e d u a l i t y mentioned above. In f a c t , equation ( 1 5 . 2 ) becomes:
a
gj
+ aiiqiqjlE
+
o(E)
t
2'1 =
gq j
.
(19)
The e q u a t i o n ( 1 6 ) w i l l assume t h e mzaning of t h e e n t r o p y i n e q u a l i t y . Then, i f one r e q u e s t s t h a t a t e q u i l i b r i u m ( q = 0) t h e e n t r o p y s a t i s f i e s t h e Gibbs r e l a t i o n , and s e t t i n g , as u s u a l , E = p c 0 , one f i n d s , a f t e r s t r a i g h t f o r w a r d o v is t h e following: c a l c u l a t i o n s , t h a t t h e unique c o n s t i t u t i v e e q u a t i o n f o r O ( E ) 0 = (p
C /5)
o v
109 8
and moreover
8 (po
T
= -p
c
/ X T ~
o v
i s t h e c o n s t a n t m a s s d e n s i t y , cv i s t h e s p e c i f i c h e a t a t c o n s t a n t volume, i s a c o n s t a n t parameter, and x i s t h e thermal c o n d u c t i v i t y ) .
I t f o l l o w s t h a t t h e e q u a t i o n ( 1 9 ) becomes:
Entropy Principle and Shock Waves
21 9
I t i s r e m a r k a b l e t o o b s e r v e t h a t when T -t 0 e q . ( 2 0 ) g o e s o v e r + i n t o t h e F o u r i e r e q u a t i o n , and i f w e n e g l e c t i n e q . ( 2 0 ) t h e q u a d r a t i c t e r m i n q it becomes t h e Maxwell-Cattaneo e q u a t i o n ( w i t h a n o n - c o n s t a n t r e l a x a t i o n t i m e ) :
+
T(e)a q
t
+ xve
-+
= -9 ; T = x T e / p o c v
.
The s u p p l e m e n t a r y law becomes:
.
The e n t r o p y i n e q u a l i t y i s a u t o m a t i c a l l y f u l f i l l e d u n d e r t h e (Seq = cv l o g 8) u s u a l c l a s s i c a l c o n d i t i o n : t h e t h e r m a l c o n d u c t i v i t y must b e n o w n e g a t i v e . It i s e a s y t o p r o v e t h a t t h e system (171, ( 2 0 ) i s a symmetric and h y p e r b o l i c o n e , when T > 0
.
T h e c o n s t r u c t i v e arguments which w e have u s e d h e r e are t h e s u b j e c t o f a work i n p r o g r e s s w h e r e i n t h e i n t e r e s t e d r e a d e r can f i n d d e t a i l e d c a l c u l a t i o n s which h e r e h a v e been o m i t t e d .
The a i m o f t h i s work w a s o n l y t o g i v e t h e b a s i s of t h e method and t o show how a p p r o a c h e s a r i s i n g i n wave t h e o r y can b e u s e d t o s e t up m a t h e m a t i c a l w e l l - p o s e d models. Acknowledgement T h i s work was s u p p o r t e d by t h e C . N . R . and by INFN - S e z . di Bologna.
(Gruppo N a z i o n a l e p e r l a F i s i c a Matematica)
T. Ruggeri
220 REFERENCES :
151
28, 1 (1972). Volpert, A.I. and Hudiaev, S.I., Math. USSR Sbornik 2, 571 (1972). Lax, P.D. and Friedrichs, K.O., Proc. Nat. Acad. Sc. USA 68, 1686 (1971). Coleman, B.D. and N o l l , W., Arch. Rat. Mech. Anal. 2, 1 (1963). Mcller, I . , Arch. Rat. Mech. Anal. 40, 319 (1971).
161
Godunov, S . K . ,
Sov. Math. 2n 947 (1961).
171
Godunov, S . K . ,
Russ. Math. Surv.
181
Boillat, G., C. R. Acad. Sc. Paris 278-A , 909 (1974).
191
Ruggeri, T. and Strumia, A., Ann. Inst. H. Poincar6
111 121 131
141
Fischer, A. and Marsden, D.P., Comm. Math. Phys.
1101 Ruggeri, T . , Acta Mech.
47,
17,145
(1962).
2, 65
(1981).
163 (1983).
111) Boillat, G. and Ruggeri, T., C.R. Acad. S c . Paris
z ,257 (1979).
1121 Lax, P.D., in: Contributions to non linear functional analysis; Zarantonello (ed.) (Academic Press, New York 1971). 1131 Boillat, G., C.R. Acad. Sc. Paris
E ,409
1141 Fusco, D . , Rend. Sem. Mat. Modena
2, 223
(1976).
(1979).
1151 Ruggeri. T . , Appunti di propagazione ondosa, ba Scuola Estiva di Fisica Matematica del C.N.R. - Dip. di Mat. - Univ. Bologna (Sett. 1981).
2, 271 (1980). Phys. 2, 1824 (1981).
1161 Boillat, G. and Ruggeri, T., Acta Mech. 1171 Ruggeri, T. and Strumia, A., J. Math. 1181 Boillat, G., C.R. Acad. Sc. Paris 1191 Strumia, A., Lett. Nuovo Cimento
E ,259
36, ( 1 7 ) ,
(1980).
569 (1983).
1201 Boillat, G. and Venturi, G., I1 Nuovo Cimento
3, (3),358
(1983).
1211 Boillat, G. and Muracchini, A., On the symmetric conservative form of Landau’s superfluid equations, (to appear in ZAM?, 1984). 1221 Virgopia, N. and Ferraioli, F., On the shock wave generating function in a simple mixture of gases (submitted in I1 Nuovo Cimento).
14,
1231 Dafermos, C . , Arch. Rat. Mech. Anal. 0, 167 (1979); J . Diff. Eqs. 202 (1973); Hyperbolic systems of conservation laws, Div. of Appl. Math. Brown Univ. Providence USA (March 1983). 1241 Liu, I-Shih and Miiller, I., Arch. Rat. Mech. Anal. 83, ( 4 ) , 285 (1983).
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland).1984
22 1
DIFFERENTIAL AND DISCRETE SPECTRAL PROBLEMS AND T H E I R INVERSES P.J. Caudrey Department o f Mathematics U.M.I.S.T. P.O. Box 88, Manchester M60 1QD U.K.
1Je c o n s i d e r t h e d i f f e r e n t i a l and d i s c r e t e s p e c t r a l problems
A
and g a r e m a t r i c e s which where u i s a column v e c t o r and d e p e n d o n t h e complex s p e c t r a l parameter 5. M a t r i x a l s o depends on e i t h e r t h e r e a l v a r i a b l e x o r t h e i n t e g e r v a r i a b l e n as shown. I n each case a s e t o f s p e c t r a l d a t a i s d e f i n e d and a method o f s o l v i n g t h e i n v e r s e problem, i . e . t h a t o f r e c o n s t r u c t i n g B from t h i s data, i s d e s c r i b e d . 1.
INTRODUCTION
The success o f t h e s p e c t r a l t r a n s f o r m o r " i n v e r s e s c a t t e r i n g method" i n s o l v i n g v a r i o u s n o n l i n e a r e v o l u t i o n e q u a t i o n s i s w e l l known E l - 3 1 . Many s p e c t r a l problems have been used f o r t h i s purpose 14-141. N e a r l y a l l o f them can be w r i t t e n i n one o f two forms depending on whether i t i s a d i f f e r e n t i a l o r a d i f f e r e n c e problem. These two forms a r e examined i n t h i s paper. I n 5 2 t h e d i r e c t s p e c t r a l problems a r e c o n s i d e r e d and t h e s p e c t r a l d a t a i s d e f i n e d i n 53. A method o f s o l v i n g t h e i n v e r s e s p e c t r a l problem i s e x p l a i n e d i n 54. The p r a c t i c a l use o f t h e s p e c t r a l t r a n s f o r m i s demonstrated i n 555 and 6. The Boussinesq e q u a t i o n i s s o l v e d i n 5 5 u s i n g a d i f f e r e n t i a l s p e c t r a l t r a n s f o r m and t h e Toda l a t t i c e e q u a t i o n s a r e s o l v e d i n 56 u s i n g a d i s c r e t e s p e c t r a l t r a n s f o r m . 2.
THE DIRECT SPECTRAL PROBLEMS
We s h a l l c o n s i d e r t h e s p e c t r a l problem 1 4 1
a
u(x) = ax -
Id(<)+
(2.1)
g(x,c)l.~(x)
and i t s " d i s c r e t i s e d " v e r s i o n
u(n+l)
=
IA(t)
t
,
g(n,c)}-!(n)
(2.2)
a r e square m a t r i c e s depending on t h e complex where u i s a column v e c t o r and A and s p e c t r a l parameter 5. B a l s o depends on t h e r e a l v a r i a b l e x ( i n t h e d i f f e r e n t i a l case) o r on t h e integer=n ( i n t h e d i s c r e t e case) and
B-tO -
E x a c t l y how
+
0will
as
x
or
n - + -
.
(2.3)
be s p e c i f i e d l a t e r .
We s h a l l assume f o r s i m p l i c i t y t h a t the e i g e n v a l u e s x i ( < ) o f & ( i )a r e d i s t i n c t
222
P. J. Caudrey
r e g u l a r functi-ons o f 5 t h r o u g h o u t t h e complex <-plane and t h a t t h e l e f t and r i g h t eigenvectors l i ( 5 ) and l i ( r ) are also regular.
<)V(<) -v . (<).A(<) xixi ((<)$<) -
-1
(2.4)
(i) =
=
(2.5)
*
N.B. These c o n d i t i o n s can be r e l a x e d . A n a l y t i c i t y i s s u f f i c i e n t i n most cases and r e p e a t e d e i g e n v a l u e s can be accommodated p r o v i d e d t h e r e e x i s t s a f u l l complement o f e i g e n v e c t o r s . It i s c o n v e n i e n t t o i n t r o d u c e t h e p r o j e c t i o n m a t r i c e s
and (2.7) i n the d i f f e r e n where t h e summation i s over a l l j such t h a t Re A,(<)> Re A.(C) t i a l case and lh,(<)I > lh;(<)l i n t h e d i s c r e t e J c a s e . We Ean a l s o d e f i n e
where this summation i s o v e r a l l j such t h a t ihi(c)i. This gives Ibj(i)l <
or
Re A , ( < ) < Re h i ( < ) J
(2.9) It should be n o t i c e d t h a t w h i l e P+(<)
PJU)(r) i s not.
=;1
i s a n a l y t i c t h r o u g h o u t t h e complex <-plane,
The <-plane i s d i v i d e d i n t o r e g i o n s .
o f the q u a n t i t i e s
Within a region the order
i s f i x e d and hence t h e P . ( < ) s which
Re x i ( < ) ( o r I h i ( < ) l ) 3 appear i n (2.7) do n o t change. Hence p . ( ' ) ( ~ i)s a n a l y t i c . However, on c r o s s i n g a boundary f r o m one r e g i o n t o a n o t h e r d e o r d e r changes and one o r more p.(<)can -J e n t e r o r leave t h e sum i n ( 2 . 7 ) . Thus P+(")(<) can have a jump s i n g u l a r i t y on t h e boundary between two r e g i o n s .
I f x o r n i s l a r g e so t h a t B c a n be neglected,any s o l u t i o n o f ( 2 . 1 ) o r ( 2 . 2 ) i s a l i n e a r combination o f or
We d e f i n e a s e t o f J o s t s o l u t i o n s k i ( x , < ) conditions (i) and
(ii)
2i(x,<)
o f ( 2 . 1 ) which s a t i s f y t h e boundary
= exp { - h i ( c ) x 1 k i
2 i ( x $ r ) remains bounded as
S i m i l a r l y i n t h e d i s c r e t e case ki(n,C)
(x,r)
+
y i ( c ) as x
x
+
m.
-+
+
-
-
(2.10) (2.11)
s a t i s f i e s (2.2) and (2.12)
and
(ii)
>i ( n , i )
remains bounded as
n
-+
+
=.
(2.13)
Spectral Problems and their Inverses
223
The s p e c t r a l problem (2.1) p l u s boundary c o n d i t i o n s i s e q u i v a l e n t t o t h e m a t r i x / Fredholm e q u a t i o n @.(XiS) = l i ( i )
c
+
-1
where t h e k e r n e l i s
!$(x,Y,S)*2i(Y~S)dy
(2.14)
(2.15)
A s u f f i c e n t ( b u t n o t n e c e s s a r y ) c o n d i t i o n f o r t h e Fredholm d e t e r m i n a n t and f i r s t Fredholm m i n o r t o e x i s t f o r (2.14) i s t h a t ll~(X*Y,S)ll where
k(Y9r;)
(mk(y,c)dy = M ( c )
<
(2.16) (2.17)
m
L and IlKll i s t h e modulus o f t h e l a r g e s t element o f &. I t i s t h i s c o n d i t i o n ( o r any o t h e r which ensures t h a t t h e Fredholm d e t e r m i n a n t and f i r s t m i n o r e x i s t ) which governs t h e way i n which B + 0 as x t
-.
-f
E q u a t i o n (2.14) has an e x a c t p a r a l l e l i n t h e d i s c r e t e case m
2i(n,c)
= yi(c)
1
+
&(n,m,c)-2i(m,c)
(2.18)
m=-m
where
= K. = ( n l,m,
L)
= (hi *
{A(
( i ) lm-n i) }n-m-l
te(n
The analogue o f ( 2 . 1 6 ) and (2.17)
-m)L -
~ ( ~ ) ( i .g(rn,c) ) } .
(2.19)
is (2.20) (2.21)
When we come t o i n v e r t t h e s p e c t r a l t r a n s f o r m we s h a l l r e q u i r e t h e p r o p e r t y
1i m
5-xo
or
1i m
e-
[a$X,S)
-
C?i(n,c)
- xi
(2.22)
3(5)1 = 0
(ill
=
0
.
(2.23)
T h i s can c o n v e n i e n t l y be ensured i n b o t h t h e d i f f e r e n t i a l and t h e d i s c r e t e case i f
M(c)
+
0
as
-f
m
.
While convenient, t h i s c o n d i t i o n again i s n o t necessary. 3.
'THE SPECTRAL DATA
The s p e c t r a l d a t a c o n s i s t s of c e r t a i n i n f o r m a t i o n a b o u t t h e s i n g u l a r i t i e s of t h e There a r e two types of s i n g u l a r i t i e s . On
Jost f u n c t i o n s i n t h e complex <-plane.
224
P.J. Caudrey
t h e boundaries between t h e r e g i o n s mentioned e a r l i e r t h e k e r n e l s K.(x,y,c) or K.(n,m,i) have jump s i n g u l a r i t i e s and hence s o do t h e J o s t f u n c t i o s . W i t h i n =1 these r e g i o n s t h e Fredholm d e t e r m i n a n t s and m i n o r s a r e r e g u l a r . Hence so a r e t h e J o s t f u n c t i o n s e x c e p t where a Fredholm d e t e r m i n a n t vanishes. A t such a p o i n t t h e corresponding J o s t f u n c t i o n w i l l have a p o l e . For s i m p l i c i t y we s h a l l assume t h a t these poles a r e simple, do n o t occur on a boundary and no two J o s t f u n c t i o n s have poles i n t h e same p l a c e . ( A l l these c o n d i t i o n s a r e a g a i n c o n v e n i e n t b u t n o t necessary). F i r s t we examine t h e p o l e s . W
To do t h i s we i n t r o d u c e t h e wronskian
= d e t I$ ,$ 1 -2
.
-..I
I n t h e d i f f e r e n t i a l case
a
W
= Tr{A(c)
+ g(x,c)lW
which can be i n t e g r a t e d t o g i v e X
W(x,c)
=
exp I T r t A ( c ) } x + X det
I,
TrIg(y,c)}dyl
[ ~ l ( c ) , ~ J c ) , - *1 * .
(3.3)
S i m i l a r l y i n t h e d i s c r e t e case W(n+l)
=
det
LA(<) t
B(n,c)lW(n)
which l e a d s t o
X det
Cxl(c),yJc), -.-I
.
(3.4)
The i m p o r t a n t t h i n g i s t h a t ( p r o v i d e d &(c) and &(c) + k(m,c) a r e n o n - s i n g u l a r i n t h e d i s c r e t e case) W i s f i n i t e and non-zero even when c approaches a p o l e o f one o f the Jost functions. L e t ki(x,c)
have p o l e s a t c = c 1J k ) k = 1,2,
~ i ( ~W =) d)e t ~ i ( we~ g)e t
- - -(i ,-
Then ( c -
and l e t t i n g
-f
0
=
,
det
Res
$, -.-I
... ,mi.
~i(~))$~,-*-l
.
(3.5)
or where t h e c o e f f i c i e n t s
(3.7) 1 j
a r e independent o f x o r n .
The d i s c r e t e p a r t of t h e s p e c t r a l d a t a c o n s i s t s o f t h e l o c a t i o n s o f a l l t h e p o l e s i n t h e c-plane t o g e t h e r w i t h t h e c o e f f i c i e n t s y .(.k 1 . 1J SD = { ( c i ( k ) , ( y i j( k ) , a l l j # i ) , k = 1,2, ,mi), a l l il , (3.8)
---
Now we examine t h e jump s i n g u l a r i t y i n $ . ( x , c ) o r $ . ( n , c ) when c crosses a b o r d e r between two r e g i o n s . C o n s i d e r i n g t h e d T l f e r e n t i a l - ’ c a s e we w i 11 a t f i r s t assume t h a t t h e r e i s o n l y one j # i such t h a t Re A i ( i ) > Re X . ( c ) on one s i d e (labelled +) J
225
Spectral Problems and their Inverses
on t h e o t h e r (labelled - ) . U s i n g t h e s u p e r f i x i t o and Re h i ( < ! < Re A . ( < ) denote t h e l i m i t o f d q u a n t i t y as t h e boundary i s approached f r o m t h e i s i d e we have
K++ (x,Y,<) -
q- (x,y,c)
= exp
{ ( ~ ~ ( c- )~ ~ ( i ) ) ( x - y ) )
x -
exp { ( k j ( c )
=
Lj(e)*E(Y,i) ~ ~ ( c ) ) x ~ x ~ ( i ) ; ~ ( ~ , c )( 3 . 9 )
where
5+ (x,Y,c)
and
= exp { ( AJ. ( c ) - h 1. ( i ) ) ( x - ~ ) } ; K _ j ( x , y , i )(3.11) .
Using these r e s u l t s we f i n d m
t
!$ (x,y).exp { ( A ~ ( c )- A j ( C ) ) Y J {ti ( ~ 3 -~ 2; 1 (y,<)ldy
I-!&+
-m
=
exp t ( ~ ~ ( -c )k j ( c ) ) x }
-m
(X,Y)-{?;
+
{2i
= exp I ( A ~ ( c -) A j ( s ) ) x J
(x.c)
(y.5)
exp { ( h i ( < )
-
x
(<))XI
{F; ( x , ~ ) -
'zi- (Y,C) Thus
exp { ( ~ -~ h j (( c i) x) J
(y,c))dy (y,c)dy}
!{
lm-J V.(L);.(Y,<)*@; -J
(x,i)l +
(Y,<) dy
4 1J . . ( i ) -Jv . ( i )
(3.12) (3.13)
*
-
+
tFi (x,c) - ?i ( x , i ) l = Q .1J. ( e l-J@ . ( x , i ) $i
i.e.
dY
2;
+
=
-
+
(x,<)
-
&i-(~,<)=Qij(~)kJ:
(x,i)
(3.14)
.
(3.15)
I n t h e more c o m p l i c a t e d case where t h e r e i s more t h a n one j # i such t h a t Re A . ( c ) - Re A . ( c ) changes s i g n as 5 crosses t h e boundary, t h e r e s u l t i s j u s t as s i m p l e a l t h o u g h J t h e p r o o f i s even more t e d i o u s . Thus, $.t -1
(x,i)
- k c (x.5)
= IQij(<)&J!
(x,i)
(3.16)
j
where t h e c h o i c e o f $ t ( x , ~ ) o r +.- ( x , ~ ) on t h e r i g h t hand s i d e i s t h e l i m i t as t h e boundary i s a p p r d c h e d from -%he s i d e on which Re A ~ ( < <) Re A , ( < ) (The J l a b e l s i are f i x e d a r b i t r a r i l y ) . For t h e d i s c r e t e s p e c t r a l problem t h e r e s u l t i s j u s t t h e same, namely +.+ -1
(n,z;)
- ii (n,c) -=
I Q ~ ~ ( c ) ~( n~ ,!i )
(3.17)
j
where i n t h i s case i t i s I x i ( i ) l - I A . ( < ) / which changes s i g n on t h e boundary and t h e terms on t h e r i g h t hand s i d e a r e t h e J l i m i t s from t h e s i d e on which l A i ( c ) l < l A j ( i ) l . The continuum s p e c t r a l d a t a i s Si
j # i, and 5 runs i , j = 1,2, over a l l the boundaries)
= {Qij(i),
.
226
4. THE
P.J. Caudrey
INVERSE SPECTRAL PROBLEM
We now c o n s i d e r t h e problem of r e c o n s t r u c t i n g t h e m a t r i x B from t h e s p e c t r a l data. This w i l l be s t r a i g h t f o r w a r d i f we can f i n d t h e v e c t o r s i , so l e t us r e v i e w t h e p r o p e r t i e s o f these i n terms o f t h e s p e c t r a l d a t a (i)
T~ has
s i m p l e p o l e s a t 5 = 5 J k ) , k = 1,2, 1 w i t h r e s i d u e s g i v e n by e i t h e r
- - - mi,
or
(ii)
On a boundary between two r e g i o n s t h e r e i s a jump s ingu 1a r it y g i ven by
or
(iii)
Elsewhere O . i s r e g u l a r and as 5 -1
O.
-1
( x o r n,c)
---
-
v.(2;)
-1
-f
+
(4.4)
-
0.
from these p r o p e r t i e s i s a Riemann-Hilbert problem. F i n d i n g o . , i = 1,2, Complex d r i a b l e t h e o r y enables us t o w r i t e
or
where t h e i n t e g r a t i o n i s a l o n g a l l boundaries i n t h e d i r e c t i o n which p u t s t h e t s i d e on t h e l e f t . By p u t t i n g and < ' % a s e t o f s i n u l a r m a t r i x / equal t o (k)) Fredholm e q u a t i o n s can be o b t a i n e d i n t h e unknowns Q . ( x , ~ ~ and @:(x,c') or -J -J
~i(~)
s
227
Spectral Problems and their Inverses
o (n,si(k))
and 0: ( n , c ' ) . The q u e s t i o n o f e x i s t e n c e and uniqueness o f t h e s o l u t'llon o f these ha'i;lnot been answered i n g e n e r a l . However, i n p a r t i c u l a r cases t h e r e appears t o b e no problem.
-
5.
THE BOUSSINESQ EQUATION
To i l l u s t r a t e how t h e s p e c t r a l t r a n s f o r m can be used i n p r a c t i c e we s h a l l c o n s i d e r two examples. F i r s t we examine t h e d i f f e r e n t i a l s p e c t r a l problem 4uxXx
+
( 1 + 6 w X ) uX + ( 3 w X x - n i w t ) u = A U
where w ( x , t ) (and hence u ) depends on t as w e l l as x t o t according t o u
t
If u evolves w i t h respect
Di ( u x x + wxu)
=
(5.1)
(5.2)
then t h e c o m p a t i b i l i t y c o n d i t i o n i s Wtt
so t h a t f = w
X
= (wxxx + 3w;
(5.3)
+Wxlx
satisfies ftt = (fxx + 3f2
+ f)xx
(5.4)
which i s t h e Boussinesq e q u a t i o n . The s p e c t r a l problem c a n ' b e c a s t i n t o t h e form (2.1) by p u t t i n g
giving
A= and
g=
1
l,i
0
(5.6)
-h
i
i
0’
0
0 3
- ?yJx
(5.7)
0,
It i s convenient t o p u t
so t h a t t h e e i g e n v a l u e s o f & a r e hi(<) where w = 1, 1
vectors are
w = - $ +$ 2
1
w.i
= __
zfi
ni and
. w = 3
'
'1
-- ,
i = 1,2,3
(5.9)
Oi'
-i - 3 fli
a r e t h e cube r o o t s o f u n i t y .
The e i g e n -
228
P.J. Caudrey
and t h e complex <-plane i s d i v i d e d i n t o t w e l v e r e g i o n s as shown
The i n v e r s e t r a n s f o r m can be s i m p l i f i e d by u s i n g e x t r a p r o p e r t i e s which a r e s p e c i f i c t o t h i s example. I n view o f (5.5) we need o n l y e v a l u a t e t h e f i r s t element o f
(5.11)
(5.12) we need o n l y c o n s i d e r $ ( x , ~ ) . 1
The symmetry (5.13) t e l l s us t h a t t h e p o l e s o f $ ( x , s ) o c c u r i n p a i r s s ( k ) 5 (k) <(k') 1
1
=
-1
1
and A1 so
and 5 ( k ' ) ,
where
1
(5.74) (5.15)
.[- I
Q 1J(
1 5
= Qlj(5).
Equation ( 4 . 5 ) can now be reduced t o
where
1
(5.16)
229
Spectral Problems and their Inverses
For d e f i n i t e n e s s we s h a l l take t h i s
and k i s summed over one pole from each p a i r . t o be the one f o r which I r ( k ) l > 1. 1
If w ( x ) i s real the locations o f these poles a r e very r e s t r i c t e d .
Either (5.19 (5.20
ck
where in both cases e i t h e r Ck'
and e i t h e r n / 3
=
sk
(5.21
= B[ 71
o r - 12’. < arg 5 The time evolution of the spectral data can be deduced from the f a c t t h a t <
exp s a t i s f i e s (5.2)
arg
ok,
and
:C
and B~ are r e a l or they occur i n p a i r s with
<
{ai ( ~
k
2
(
-
7
j
.
(5.22)
~ ( r ) ) ~( xt, tl, i+) ~
(5.23)
and
Qij(t,c)
exp { f l i C ( A j ( r ) ) 2 - ( A i ( r ) ) 2 1 t } Q i j ( 0 , r )
=
The N-soliton s o l u t i o n i s found by putting Q . . ( t , r ) manipulation the r e s u l t i s 1J
=
0.
.
(5.25)
After some tedious
a2 I n (det M(x,t)) ax2 where the kp,th element of the N x N matrix M(x,t) i s f(x,t) = 2
6.
~
THE TODA LATTICE
The Toda l a t t i c e 1151 i s probably the b e s t known example of a problem which can be solved using a d i s c r e t e spectral transform. The Toda l a t t i c e equations d2Yn - dt2
-
exp (Yn-l
- Yn) -
exp ( Y n
- Y,+1)
(6.1)
a r e the compati bi 1i t y conditions of the s p e c t r a l equation U
5
dt
and the time evolution equation
(6.2)
230
P.J. Caudrey
By p u t t i n g u(n) =
i
un (6.4)
Ynl!
(6.1) t a k e s t h e form o f (2.2) w i t h
In t h i s case t h e e i g e n v a l u e s and e i g e n v e c t o r s o f A ( 5 ) a r e
and
- (c)
-v1
=[1
-;]
-v
,
-2
=
(1
-5)
.
The complex <-plane i s d i v i d e d i n t o t w o r e g i o n s by t h e u n i t c i r c l e 151 = 1. Again t h e i n v e r s e t r a n s f o r m can be s i m p l i f i e d . where
We need o n l y f i n d one q u a n t i t y
$,,(el
(6.10)
(6.11) (6.12) (6.13) (6.14)
on 151 = 1. Thus t h e s p e c t r a l d a t a reduces t o S = { c ( ~ ) ,Y(~), k = 1,2,
If, as i s u s u a l l y t h e case, y,
Q(c*)
=
N;
Q(r), )el
= 1)
(6.15)
i s r e a l t h e n s ( k ) and y(k) a r e r e a l and
CQ(c)l*
Equation (4.6) now reduces t o
**-,
.
(6.16)
231
Spectral Problems and their Inverses
(6.17) where
q 5 )
=
n
5 q c )
(6.18)
*
S i n c e o n ( < ) i s r e g u l a r f o r 151 > 7 i t may be expanded as a L a u r e n t s e r i e s (6.19) and (6.17) becomes n (6.20)
S u b s t i t u t i n g (6.18) and (6.19) i n ( 6 . 2 ) g i v e s (6.22) Hence t h e i n v e r s e s p e c t r a l t r a n s f o r m can be e v a l u a t e d b y u s i n g (6.21), f i n a l l y (6.22).
(6.20) and
The o n l y gap r e m a i n i n g i n t h e argument i s t i m e e v o l u t i o n o f t h e s p e c t r a l data. T h i s e a s i l y f o l l o w s from t h e f a c t t h a t (6.3) i s s a t i s f i e d by exp
and
I - $(c - l / 5 ) t l $ n (t,c) .
(6.23) (6.24)
<(k)(t)
=
p ( 0 )
Y(k)(t)
=
exp
E
Q(t,c)
=
exp
{ i
5(k)
-
- l/c(k))tl
y(k)(o)
(6.25)
.
(6.26)
l / < ) t l Q(0,c)
F i n a l l y t h e N-sol it o n s o l u t i o n i s found by p u t t i n g Q ( L ) :0. y,
=
I n d e t [Mn
-
I t takes the form
I n d e t IMn+ll
(6.27)
REFERENCES:
C11 Gardner, C.S., Greene, J.M., 19, 1095 (1967). C21
Zakharov, V.E.
C31 A b l o w i t z , M.J., 449 (1974).
K r u s k a l , M.D.
and Shabat, A.E., Kaup, D.J.,
and Miura,
R.M., Phys. Rev. L e t t .
Sov. Phys. JETP 34, 62 (1972).
Newell, A.C.
and Segur, H . ,
Stud. Appl. Math. 53,
232
P.J. Caudrey Physica D6, -
C41
Caudrey, P.J.,
51 (1982).
C57
Wadati, M., S o l i t o n s , Topics i n Current Physics 17, Bullough, R . K . and Caudrey, P.J., eds. (Springer, B e r l i n , Heidelberg, New York, 1980) pp. 287-299.
C61
Calogero, F. and Degasperis, A.,
C71
Kaup, O.J.,
Stud. Appl. Math.
i b i d . pp. 301-323.
62,
189 (1980).
L81 Caudrey, P.J., Phys. L e t t . 79A, 264 (1980),
and Proc. Roy. I r i s h Acad. 83A, 23 (1983). I
C97
Mikhailov, A.V.,
Physica 03, - 73 (1981).
1107 Case, K. and Kac, M., 1113 Case, K.,
J. Math. Phys.,
14,594
(1973).
J. Math. Phys. 14, 916 (1973).
1121 Flaschka, H., C131 Manakov, S.,
Prog. Theor. Phys. 51, 703 (1974). Sov. Phys. JETP 40, 269 (1975).
C147 Ablowitz, M.J. and Ladik, J.F., 3. Math. Phys. 16, 598 (1975), 55, 213 (1976). and Stud. Appl. Math. C157 Toda, M., S o l i t o n s , Topics i n Current Physics 17, Bullough, R.K., and Caudrey, P.J., eds. (Springer, B e r l i n , Heidelberg, New York, 1980) pp. 143-155.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) @ Elsevier Science Publishers B.V. (North-Holland), 1984
233
WAVEFRONT FIELDS IN THE SCATTERING O F TRANSIENT ELASTIC WAVES BY A SURFACE-BREAKING CRACK Julius Miklowitz and Constantine Chazapis Division of Engineering and Applied Science California Institute of Technology Pasadena, California 91 125
Presented here is a n application using a method f o r deriving the wavefront fields in the elastodynamic quarter-plane due t o Miklowitz [l]. The application concerns the scattering of transient elastic waves and wavefront fields by a surface-breaking crack. Such wavefront fields, generated by the interaction of incident surface o r body waves, can carry important information about the geometries of the surface-breaking crack. Information of thls type is invaluable, in particular, to quantitative nondestructive evaluation (QNDE). The application splits into two parts, the work of the first part which is a very recently published paper on this problem by Miklowitz [2], and will be very useful to the present work (the second part of the work). The scattered wavefront fields in the present surface-breaking crack problem a r e generated by an incident line load on the free surface of a half-plane a distance, say a , from the crack ( a Lamb type problem). The present method ultimately gives all of the scattered wavefront fields, yielding the magnitudes and geometric positions of these wavefronts for the surface-breaking crack.
INTRODUCTION The present crack problem falls into a class that is nonseparable classically and as such i s difficult t o solve, the nonseparability stemming from media corners and crack edges. Mathematically, however, the elastodynamic quarter-plane boundary-value or boundary-initial value problems involved can be handled through integral transforms, integral equations, asymptotlcs and numerical analysis. Recently, t h e present crack problem has been addressed in the literature for the case of timeharmonic loads and waves by Achenbach, Keer and Mendelsohn [3], Mendelsohn, Achenbach and Keer [4] and Kundu and Ma1 [ 5 ] , and very useful information for QNDE was obtained, In [3] and [4] the surface-breaking crack is assumed to be the two-dimensional normal edgecrack of depth d in an elastic half plane, cf. Fig. 1. It i s assumed also in these works that (1) the two faces of the crack involved do not touch one another, hence this crack never closes completely, and (2) the loads and waves are time-harmonic in nature. The total field in the half plane for each of the probIerns in [3,4] is composed of the superposition of a specific incident field (say free surface or body waves) in the uncracked halfplane and t h e scattered field in the cracked half-plane generated by suitable surface tractions on the crack faces. Through equilibrium arguments in the plane of the crack these surface tractions are equal and opposite t o the tractions generated by the incLdent wave in the uncracked half plane. Then by decomposing the scattered field into symmetric and
234
J. Miklowitz and C. Chazapis
antisymmetric fields with respect t o the plane of the crack, one obtains a pair of boundary value problems for the quarter plane. Integral transform methods then reduce the t w o boundary value problems t o two uncoupled singular integral equations which are solved numerically. The present work and the earlier work by Miklowitz [Z],involving transient loads, waves and wavefronts, are in effect the counterpart of the foregoing time-harmonic cases in the literature for the high frequencies (wavefronts). Numerical results are presented graphically in [3],[4],and [ 5 ] for the surface-breaking crack, [2] gives a n abstract of the numerical results for each of these papers, and a reference to an important recent genera1 work on the present subject, the ASME monograph on "Elastic waves and non-destructive testing of materials," Y.H. Pao Ed., in 1978 [6].
Fig. 1. Waves incident on a surface-breaking crack of depth d (after Achenbach, Mendelsohn and Keer). WAVEFRONT ANALYSIS IN THE NONSEPARABLE ELASTODYNAMIC QUARTER-PUNEPROBLEMS. We now draw on the work [I] by Miklowitz, a two-Part paper on wavefront analysis in the nonseparable elastodynamic quarter-plane problems. This work involves loadings and waves (wavefronts) that are transient in nature, and theretore differs from the time-harmonic wave nature of the works [3,4,] and 151. However, the basic quarter-plane features of 13.41,and [ 11 are similar, i.e., the boundary value problems of [3,4] are. in [ 11, boundary-initial value problems, the boundary (edge) conditions in [3,4], are like those in [l] except for the latter's time dependence. Integral transforms and related singular integral equations are a property of the solution techniques in both works, in which the solutions for these integral equations a r e found by numerical analysis [3,4]and by direct simple integrations yielding algebraic equations with the following numeric evaluations [ l]. I t follows that the techniques in [ l] can be applied t o the quarter-plane problems in [3.4],hence giving the transient wavefront fields a t their fronts and just behind them for regular wavefronts, as well as two-sided wavefronts involving a precursor, for these problems, i.e., the counterparts (high frequency pulses) of the solutions in [3,4].The scattered transient wavefront flelds obtained here then will be very valuable to the QNDE methods based on scattering of ultrasonic elastic waves by cracks, with interest towards solving the inverse problems of obtaining the crack geometries from these
Wavefront Fields in Elastic Wave Scattering
235
scattered transient wavefront fields Method of Solution. I t has been pointed out earlier that t h e techniques in [l] c a n be applied
hence giving t h e transient wavefront fields for these to t h e quarter-plane problems in [3,4], problems, i.e.. the counterparts of t h e solutions i n [ 3 , 4 ] . In order t o reduce the t e z t of [ I , P a r t s I a n d Z ] f o r the reader, the following sections contain main point conszlltations providi n g the necessary background f o r treating the p r e s e n t problem.
General Quasi-Formal Solutions. First the governing equations for t h e general elastodynamic quarter-plane problem for plane strain are discussed, i.e., t h e coupled displacement partial differential equations of motion in u(x,y,t) and v(x,y,t) corresponding stress-strain relations, initial conditions and radiation conditions. This is followed by applying one-sided Laplace transforms i n t. parameter p, and x, parameter s, t o t h e displacement p.d. equations of motion in t h e now double-transformed displacements %(s.y.p) and%(s,y,p),cf. (5),: where t h e bar and tilda over quantities indicate L.T.'s on t and x, respectively. Solutions t o these equations give t h e general quasi-formal solutions i n terms of a double inversion integral involving b o t h Bromwich paths, Brp and BT,, t h e p- a n d s-planes, respectively, cf. ( 6 ~ ) Equations ~ . ( 6 b ) 1 - ( 6 d ) , give %(s,y.p), %(s , y , p ) with its complementary functions Aka, A,@, a n d particular integrals p e a ,,u*~. The p's a r e defined by integrals which contain t h e time-transformed edge (x=O) unknowns (displacements and displacement gradients) through f(s,y,p) a n d g(s,y,p) t h e r . h . sides of (511. These unknowns do become k n o w n quantities for particular problems later in t h e method, which t h e n makes ( 6 ~a )formal ~ solution.
Associated with ( B a ) l - ( 6 d ) , a r e t h e radiation conditions on y, according t o (4)1,which c a n be ~ . interest in a n s-plane m e t with suitable values involving t h e a and B functions in ( 6 ~ )With inversion, we fix p and require that it be real a n d >O. Then as shown in [ l , P a r t 1. Fig. 21, taking c u t s to t h e left from branch points s = ik,, * k d , we c a n represent analytic branches of a(.) and P ( s ) through t h e equations ('?'),-( lo), f o r t h e BT,L a n d B T , ~ contours, respectively t h e lower a n d upper halves of B r S . I t follows t h e n from (9)1, t h a t t h e multivalued functions
a(.) and p ( s ) satisfy Re t o ( 6 b ) , ,will satisfy
(4)1 for
1 2 0 , f o r Re I $ { >O, on
BT,L ( a n d B r P ) , a n d hence, according
y--, provlded &(a)
=
-Ka~s)(s~P)
and
from (1 (12)1 for
which leads to t h e double-transformed displacements 3 s ,yp ) a n d %(s ,y, p ) in BSL.
Similarly, from Re 1 vided
'
> 0 $ , on
t h e multivalued functions a(.)
and
satisfy Re f ; [ : ] G O
Br-y (and B r P ) . and hence, according t o ( 6 b ) , , will satisfy (4),for
j , for
v 4 m v pro-
Here and hereafter subscript 1 o n equation numbers indicates equations f r o m [ l , P a r t 11, e.g., (5)1.
J. Miklowitz and C. Chazapis
236
and
from (13)1, which leads t o the double -transformed displacements 3 s ,y , p ) and %(s ,y , p ) in (14)]for B . 0 .
I t is important t o point out now t h a t the branch functions a(.)
and @ ( s ) . defined in (7)1 and
(8)1, respectively, are analytic all along Brs, having Re[$;]]>O Re[$E]]
on Br,t from (9)1 and
Equations (12), f o r % a n d %are bounded as y+m, the critical
terms being those involving exp [ a ( s ) y ] and exp[g(s)y] where Re[a[i]]>O
B
and the
coefficients of these exponential functions vanish as y+=. Similarly, (14), for % and % a r e bounded as y +-, the critical terms being those involving exp[-a(s)y] and exp[-p(s)y] where
Re[i[Z]].cO and the coefficients of these exponential functions vanish as y-r-. I t is of further
-
that according t o the principle of reflection interest t o note from the results of (9)1 and on BrsO, so t h a t (14)1 forms a natural conjugate of ( l Z ) l . This leads to a real formal solution f o r the dispIacements over BrsL,as well as other conjugations in the s-plane. a ( s ) = - a ( s ) and @(s)=-@m
T h e Surjuce-Breaking Crack P r o b l e m . We now apply the basic techniques of [I], involving transient wavefront approximations t o elastic waves scattered by cracks, t o the problem of [3,4],i.e., the surface-breaking crack depicted in Fig. 1. Then, as pointed out earlier, the surface-breaking crack problem has a scattered Aeld t h a t decomposes into symmetric and antisymmetric fields with respect t o the plane of the crack, generating a pair of boundaryvalue problems f o r the quarter plane. For the surface-breaking crack we have the following symmetric and antisymmetric boundary-initial value problems for the quarter plane, (cf. Fig.
2). U,(O.Y
f)
= 0.
uz(O,y,t)= -uST(y,O,t)
and
osy <=
(1)
OCry
( 2)
dsy
<-
I
respectively, with the boundary conditions on y =O.
237
Wavefront Fields in Elastic Wave Scattering
applying t o both of t h e above quarter plane problems
The Surface -Breaking Crack Problem
If+ - e\
/
Fig. 2. The quarter-plane f r e e surface line loading a n d (1) the incident and reflected dilatational waves PI and PR it generates on the y-axis below t h e crack, and (2)the system of incident surface pulses, Dilatation D, Equivoluminal E and Rayleigh R it generates. For t h e incident field at y = 0, shown above, we have: UUJ(Z,O,t) =
-ug6(x - a ) H ( t ) ,
z2=0,t>O
.
~ ~ Z J ( ~ = P 0O J ~ ~ )
In equations (1)-(7) u and v a r e t h e displacement components, a n d u= , ow, etc. a r e t h e stress components. In (2) a n d (5) t h e r . h . sides involve t h e total incident stress wave disturbances uz' a n d u27, subscript I indicating t h e incident compressional and vertically polarized shear wave n a t u r e of these stresses, respectively, t h e superscript H for H ( t ) t h e Heaviside s t e p function, t h e time source in ( 8 ) . a n d T for t h e total incident stress waves comprised of t h e incident and reflected (along t h e y-axis) components. Also in (8),uo is t h e positive stress magnitude constant of t h e source of dimensions force per unit length, a n d S(z - a ) is t h e delta function representing the spatial singular normal line loading a t z = a , Analogous t o t h e double transformed boundary conditions @)I on y = 0 , a n d t h e timetransformed boundary conditions (lo), on x = 0 fur t h e pressure shock problem, cf. [I,P a r t 1, Fig. 11, here we have from ( 7 ) and (8). ? , ( s , O , p ) / ( h + 2 p ) = (k’
- 2 ) k - ’ [ ~ , , O , p ) ~ O , O , pf)$]( ~ , O , p ) = 0
1
( 9)
I
where k 2 = cf/ c," with c d = ( h + 2 / ~ / p and ) ~ cr)= ( p / p ) ? t h e body wave speeds, a n d h and p a r e Lame's constants and p is t h e material density, and from (l), ( 2 ) a n d (3) of t h e symmetric
238
J. Miklowitz and C. Chazapis
boundary-initial value problem [the antisymmetric boundary-initial value problem will be similar involving (4)-(6)], we have
u,(O,y.p)/p
= V ; ( O , y g ) + aM(o.y.p)= 0, osy<= ,
= -&T(O,yg)/(X and
(11)
+ 2/4, Ol;y
c ( 0 , y . p )= 0 , d.cy<-
The stress 3$T(0,y,p)in (12) c a n be derived from t h e formal solution of Lamb’s (half plane) problem with a line load surface source, represented here by t h e r.h.s. of ( 8 ) . This solution i n Cagniard-de Hoop form c a n be found in a variety of places including this writer’s book [ 7 ] , which is essentially de Hoop’s treatment. To reduce t h e text here we call on [2, p.421, eq. (14) through p.424, eq. (21) end Para] t o obtain -uzyTfO,y,p). To this point, we have dealt with t h e front running incident dilatational displacements a n d displacement gradient wavefronts, a n d t h e array of wavefronts they generate. Shortly after in time, similarly the incident equivoluminal displacement and displacement gradient wavefronts arrive, generating another system of wavefronts. Still later, the incident Rayleigh wave displacement and displacement gradient wavefronts arrive, generating still another system of wave f r o nt s . The later arriving incident equivoluminal a n d Rayleigh wavefronts c a n be handled like the foregoing dilatational wavefronts, cf. eq. (14)-(21) and related t e x t . The reader is further referred t o [7, pp, 307 - top 309,eqs. (6.28), (6.32), and pp. bottom 310 - t o p 313, eqs. (6.39) (6.42), (6.47)]. With these three different types of inputs f o r example, it will b e of interest t o learn, for the case of the free surface pulse incident t o t h e quarter-plane corner, what t h e reflected wavef r o n t i s like for each of t h e incident surface pulses, e.g., surface pulses D, E. and R in Fig. 2. These surface pulses will be studied in later work. Now, as i n [ l , P a r t 13, f r o m (12), we d r i v e $ ( s , y , p ) a n d % ( s , y , p ) , and substituting these a n d (12), into (9) and ( l o ) , we f l n d A ( s , p ) and B ( s p ) t o be (17)1. Further, a s in [l, P a r t 11,where we used t h e x = 0 edge conditions (lS), i n J’ ( s ,y , p ) and g ( s ,y, p ) which reduced t h e latter t o ( l B ) , , here t h e analog is t h e edge conditions (1) a n d (2) which give
f o r O s y < d . Now for t h e rest of t h e y-axis, d ly<-, we have t h e x = 0 edge conditions (11) and (13) which lead t o
Wavefront Fields in Elastic Wave Scattering
2 39
It follows that substituting t h e present A(s,p) a n d B(s,p) a n d (14)-(17) into (12)1 deflnes t h e doubly transformed displacement solutions @ s ,yp ) and %(s ,y, p ) , and hence the quasiformal solution ( 6 a ) l for t h e surface-breaking crack problem. This becomes the formal solution when t h e two time-transformed edge unknowns E ( 0 , y, p ) a n d D ( 0 , y , p ) become known, a process t o be discussed later. I n d i c a t e d C o n t o u r I n t e g r a t i o n f o r S o l u t i o n . As discussed in [ 1, P a r t 11, one needs a somewhat
special contour in t h e s-plane t o solve t h e present class of quarter-plane probIems, one in which the so-called nonphysical roots and associated poles a n d residues of t h e generalized Rayleigh function a r e generated a n d , hence, can be exploited in t h e form of conditions on t h e solution ruling out unbounded and inadmissible events. A general contour t h a t c a n do this is depicted in Fig. 3 of [ l , Part 13. The detail on t h e c u t s and contours of this general contour will be left t o t h e reader. Most important, however, a r e the contours involving t h e nonphysical zeros of t h e generalized Rayleigh function. Figures 4 and 5 of [ I , P a r t 11. a n d related discussion, point out how the Fig. 3 of [ 1. P a r t 11 general contour is reduced through arguments of solution boundedness and wave speed restrictions (c > c d ) t o p a t h integrals. Consistently, we now require t h a t t h e contributions (residues) from t h e nonphysical poles, generated by t h e zeros of t h e generalized Rayleigh function, stjfp) = * k R j . j = 2.3 and s l ( p )= k R 1 m u s t be ruled o u t of t h e solution ( k R , = p / c ~ , where , c ~ ,is t h e wave speed). Of these, the residues a t the poles sj(p) = k R j , j = 1,2,3 a r e ruled out by t h e boundedness condition on t h e solution. The residues a t s - j ( p ) = - k R j , j = 2 , 3 are ruled out because t h e corresponding speeds of these Rayleigh wave events a r e greater t h a n cd. Hence, a s t h e next section points out t h e residues for these flve poles a r e set equal t o zero. This eliminates t h e circular paths in Fig. 5 of [l, P a r t 11 for these contributions, generating four coupled integral equations that guarantee boundedness for t h e solution and t h e elimination of inadmissible events. Solution of these integral equations determines t h e edge unknowns in o u r problem, a t least for wavefront (high frequency) events. C o n d i t i o n s f o r S o l u t i o n B o u n d e d n e s s a n d I n a d m i s s i b l e E v e n t s . In [ 1, Part 11 this section is concerned with t h e derivation of the four integral equations for t h e edge (z= 0) unknowns which begins with first establishing t h a t s * j = i k R j , j = 2 , 3 a n d sl(p) = k R 1 a r e zeros of t h e generalized Rayleigh function. An example, s2 = kR2 is considered, and shown t o be a zero of this Rayleigh function, cf. (21)l. The o t h e r nonphysical zeros, a n d t h e physical zero S ( p ) = -kR1 follow Suit.
The residues a t s = k R 1 , kkR, = 2,3 can be shown t o be non-vanishing through t h e usual calcu-
240
J. Miklowitz and C. Chazapis
1ations.f These residues can then be set equal to zero by requiring that
where A ~ ( s , p ) , B ~ ( s are , p ) given in ( 1 7 ) L . Now with s * j ( p ) = *kR,, j = 2,3 and sl(p)= k R 1 representing the five Rayleigh poles here, the two equations (22), are expanded using (17)1 and ( 6 d ) l . After some algebra, and a simple integration o n y for the coefficient of the input term u,,/pp, the general integral equation is found t o be
where s, = s i b ) , j Consider al and
-
= 1. i 2 , +3. The s j are basic to the definitions
PI first.
From (7)1 and
(B)1
we have, when
s1
-
and
= i(p3p4)'=
= kR1 (cf. Fig.
3 of [l. Part l]),
1
1
a l = i(pIp2): = i ( k 3 ,
of the ar and p i ,
i ( k R I - kz)'.
Likewise, when s *2 = kkR2
1
s + =~i k R 3 , along paths Z3 and L , , ( 7 ) ~and 1 -
-ah2
-1
show t h a t
a12
= ( p I p 2 ) ' = ( k g - ’)&k
,
1
and
'
= -(p3p4) = = - ( k t - k& ) with a *3 and 8 13 having the same forms as a +2 and @ k2 except that k,, replaces k R z . Equation (23), represents five coupled integral equations for the two time-transformed edge displacement unknowns a(0,y , p ) ,O ( 0 , y p ) and their corner values (o,O,p). (0,Op ) . @ .tz
Now substituting the j = 1 terms, s l ,a,,and fil above, into (23)1.one finds it splits into two equations, the real and imaginary parts created by the imaginary nature of al and PI. These two real integral equations provide the definition of the time-transformed corner displacements S ( O . 0 , p ) which is given by ( 2 2 ) ~ . I t follows that if these integral expressions for G(O.O,p) and G ( 0 , O p ) are substituted into (23),. then there are just four integral equations involving only the time-transformed edge unknowns tZ(0.y , p ) ,V ( 0 . y , p ) , and related derivatives %(O,y , p ) and Gv(O,y 9).These four remaining equations are parametric ins t2 and s f 3 , respectively, with related a k 2 ,@ * z . a * 3and pfs. This solution provides the information reqyired t o reduce the quasi-formal solution to the formal solution which can then be inverted by known approximate solution techniques. The analog of the above f o r the surface-breaking crack problem involves the same A ( s . p ) and B ( s , p ) mentioned earlier, with AN and BN in (22),, (17)1, and ( 6 d ) i . The m a j o r difference occurs, however, in the analog of (23), which stems from the spatially m i z e d b o u n d a r y ~~
These calculations are like those in the latter section dealing with the physical Rayleigh waves on the free edge
y = 0, cf. [I, Part 21.
241
Wavefront Fields in Elastic Wave Scattering
conditions along the y-axis for the present problem, i.e., O S y < d and d S y < m . The replace-
ment, guided by the boundary conditions (11). (12), and (13). and the f(s.y.p) and g(s.y,p) terms, (14)-(17), along the y-axis, is
+ (k2 - 2)k,2a(O,O,p) +
kZ(k.2-2Sf) 2s,aj
-
(O,O,p) =0
where sj = s j ( p ) ,j = 1, i 2 , +3. The sj are basic t o the definitions of the aj and of (24)1 for the present problem is
pi.
The analog
242
J. Miklowitz and C.Chazapis
where
Now if 1 1 ( 0 , 0 p ) and C ( O , O , p )from (19) a r e substituted into (le), t h e n t h e r e a r e just four integral equations, parametric in S * Z , s + 3 , respectively, with related ak2,@ * z , a t 3 , a n d pk3, involvlng only t h e time- transformed edge unknoums U ( 0 , y, p ) ,U ( 0 . y , p ) a n d related derivatives % ( O , y , p ) , V v ( O , y , p in ) t h e present problem As in [l, P a r t 11,t h e solution of these four integral equations provldes t h e time-transformed knowns needed t o reduce t h e quasi-formal solution t o t h e formal solution which can then be inverted by known approximate solution techniques Determination of T i m e - P a n s f o r m e d Edge (x=O) a n d Corner (x=y =0) Unknowns f O T Wavef r o n t E v e n t s a n d their Inverses. A s in [l, P a r t 11,t o derive t h e time-transformed edge ( x = 0)
unknowns representing wavefront (high frequency events) we will use as a guide similar general features of related events from t h e elastic half-plane a n d waveguide problems, Indeed, we can construct forms for t h e time-transformed edge unknowns which will reduce t h e four integral equations to algebraic equations for the time-transformed p-dependent coefficients of these unknowns. In t h e present problems the time-transformed edge (x=O) unknowns lead t o diffracted waves propagating along t h e x=O edge (y-axis). From t h e wavefront analysis work [ E l on t h e plane strain semi- infinite plate, involving mixed pressure edge conditions, we assume a t t h e dilatational wavefront, t h e time-transformed displacements ~ ( 0 , py) , G ( O , y , p )a r e , respectively,
f r o m which we also have
both in O < y < d , i.e., along the crack faces. It may be noted t h e y-1/2 in these expressions is in agreement with t h e classical form for t h e propagation of two-dimensional surfaces of discontinuity in elastodynamics. The exp ( - k d Y ) through its shift operator n a t u r e guarantees the expected one-dimensional wave nature of these quantities. e.g..
1
a d ( 0 , y 9 S>ud (0 , y st
-y
cd),
Consider next t h e Rayleigh wave disturbance on t h e edge x = 0. Here as i n the half-plane problem one would expect a nondecaying in space ( y ) one-dimensional disturbance. Hence, it is assumed t h a t =R(OIY,P) = C ( p ) e x p ( - k R l ' ! / ) DRiR(osYIp)= D t P ) exP(-kRiY), I
from which one also has
(22)
Wavefront Fields in Elastic Wave Scattering
243
both in O s y < d , i.e., along the crack faces. The edge of the crack a t y = d is a source of diffracted dilatational and Rayleigh surface wavefronts that propagate from y=d t o y=O (cf Fig 2). and indeed keep diffracting from y=O and y=d ad inflnitum as time progresses Furthermore, the crack edge a t y=d generates the diffracted dilatational wavefronts that propagate in the domain d S y < m from y =d to -, ad infinitum being generated by the diffracted dilatational wavefronts on the crack faces when they reach the crack edge y = d as time progresses Since the boundary conditions in d c y <m are mixed [o,(O,y.t)=u(O,y t ) = 01, the Rayleigh surface waves are not present In d ~ y < = cf ,
"?, Fig
B 51
For these additional diffracted wavefronts, assuming their time behavlor is always the same as the basic wavefronts derived from the time transformed edge (x=O)unknowns (20)-(23), we can derive these additional wavefronts from forms similar to (20)-(23) First, we have for the domain d Iy <=,
V,(O,y,p)N B(p)(y-d)-1/2e-*d('-d) , and
The time-transformed edge (x = 0) unknowns for the first and htgher order diffracting wavefronts in t h e domain of the crack are for the dilational wavefronts. ?Zdi(o,Y,p)-A(p)y-'/Z6kd' , O C y c d ,
where subscript 13.3 ...n indicate the first, second, third, etc. transits of the associated wavefronts. Clearly, (27)is a series of terms for .ah,(o.y p ) with n being finite, since the inverse series will be finite f o r any fixed t. We note replacing A(p) in (27) with B(p) will result in a similar series of terms for adn(O,y,p).
I t follows that the Rayleigh surface wavefronts have essentially the same form of timetransformed edge unknowns as in ( 2 7 ) , but based on (22). Concerning the time-transformed edge ( z = O ) unknowns for the Arst and higher order diffracting dilational wavefronts in the domain below the crack, we have
244
J. Miklowitz and C. Chazapis
where subscripts 1,3,5, ...n , indicate the first, second, third, etc. transits of the associated wavefronts. Again here we have a series of terms for fZ‘,(O,y,p) and Udn(0,y,p)hence fJu,(O,y,p)Prom (26) [with B(p) for A(p) in (28)]. It follows then, neglecting the equivoluminal disturbance on the edge (x = 0) for the moment, that the r.h. sides of the expressions (20)-(28) may be substituted into the four integral equations obtained from ( l a ) and (19),as discussed after (19). The simple integrations these terms provide reduce the four integral equations to four algebraic ones that determine the time-transformed coefficients A(p), B(p), C ( p ) , and D(p) through Cramer’s rule. The four algebraic equations are of the form Jj(Sjjp)A@) + gj(SjnP)B(P) + ?j(SjP)C(P) + aj(sj/ P)D(P) = T j ( S j P )
9
129)
where ij,Bj,?,# and 3j are the diffractive coefficients of A ( p ) . B ( p ) , C @ ) ,and D ( p ) , respectively, and Tj represent the loading terms, with j = i2,-t3. The diffractive coefficients and the Tj are given in the Appendix of this paper. I t is of principal interest now to derive the timetransformed coefficients corresponding t o the first and second transits of the wavefronts in 01y s d , and then inverting to get the time nature of these wavefronts. The first transit case is the simplest. We find for the dilatational displacement wavefronts on the edge x = 0 ,
where subscripts ‘il and 21 indicate the Coefficients of u d and ’ud respectively, and subscripts O , l r . , . ,Nd indicate the terms associated with the different gdi’s that are given in t h e appendix (qdo = 0). Setting Nd = 40 we have 40 non-zero q d i ’ s involved in (30). DT and sample coefficient (nu),,can be found in the appendix. Similarly the Rayleigh displacement wavefronts,
where qm‘s are given in the appendix.
245
Wavefront Fields in Elastic Wave Scattering
The derivation of the time-transformed coefficients corresponding to the second and higher order transits of the wavefronts over the crack, O S y < d , and over the domain below the crack, d i y 5 - , is algebraically more complicated due to the multiple exponential decay terms occurring in the coefficients of A ( p ) , B b ) . C @ ) , and D@). associated with each higher order wavefront transit. e g ,
where subscripts on the 1's indicate dilational wavefronts, and super cr and super bc indicate "along the crack" and "below the crack," respectively. Note t h a t the first term in (32),associated with the first wavefront transit does not have exponential decay. It and other similar terms will be associated with first order terms in the solution which have y-l/z decay, e.g.,
(30). Concerning the two-sided equivoluminal wavefronts on the edge, z = O , one would expect the present method to yield them as in [l, Part 21 in terms of the now time-transformed coefficients A @ ) , B ( p ) , C(p) and Dk).In effect this says the diIatationa1 and Rayleigh wavefronts (on x = 0) generate these equivoluminal wavefronts on the crack faces and below, as well as the other wavefronts in the problem. Also, as pointed out in [l. Part 17. interesting is the fact t h a t the two-sided equivoluminal wavefronts o n the crack faces have the same time behavior as the dilatational wavefronts there, which is also true in Lamb's plane strain problem, cf. [ 7 , Eqs. (6.52). (6.53)].One might expect then, in the present problem, to find the same behavior over the first transit of the crack. Further, we have the input of the two-sided equivoluminal wavefronts on the crack faces, stemming from the source function a t the point x = a ,y = O shown in Fig. 2 . In this case, T in this figure touches a point y on the crack face associated with the incident two-sided equivoluminal wavefronts SVj (E shown in Fig. 2 at an earlier position on the surface t o the left of the source) and the reflected SVR and PR wavefronts. Recalling from (11) t h a t the shear stress vanishes everywhere on the edge x = 0, we have -
u,(o,YP)/P=qfY(04/9)
+%(O,y,p) = o , osy<-
I
(34)
and so over the crack faces O l y < d . Hence, we can prove (34) and in the process set down the nature of the incident and total incident system of the two-sided equivoluminal wavefronts represented byu$l(z,y,t), ugY(z,y, t ) ,v & ( z , y , t ) , and vZ;(x,y$).
246
J. Miklo witz and C. Chazapis
To determine the corner displacements u ( O , O , t ) and v ( O , O , t ) .the r.h. sides of ( 2 0 - 2 8 ) are substituted into (19), resulting in real integrals of the types,
where a and rn are >O. The r.h, sides of (35) therefore reduce (19) to algebraic equations tor n(O,O,p)and U ( O , O , p ) involving A(p), B(p), C(p). and D(p) which, using the r.h. sides of thesr four time-transformed coefficients will yield a(O,O,p)and G ( 0 , O . p )where constants like Q,v^ in (A.2) of the Appendix of [ l , Part 1 3 would be involved. I t follows t h a t the corner displacements are of the form u(O,O,t)
= U(t)
, v(O,O,t) = V ( t )
(36)
with corresponding velocities and accelerations. The Formal Solution f o r t h e Time-Transformed %placements +Z(z#yp ) and U(z ,y, p ) . With the time-transformed edge (x=O) unknowns in (20)-(28) now determined by the solution of the four algebraic equations (29) for A ( p ) . B k ) , Cfp), and D ( p ) , and similarly the timetransformed corner displacements iZ(O,O,p) and U ( O , O , p ) .the quasi-formal solutions for the displacements u (z ,y d ) and v (z ,y, t ) , like (612, b , c , d ) l become the formal solutions f o r wavefront (high frequency) events for these displacements. It follows that these formal solutions in the present problem can be written in the form of those in ( 3 5 ~ and ) ~ (35b),. Further, with the aid of the principle of reflection in complex variables, the present analog of ( 3 5 ~ ) ~ and (35b), can be reduced to real equations for Q(z.y,p)and Q ( z , y , p )through complex coni.e,, jugation. The result in [ I , Part l]is given by (43)1.
) ~ Q(z,y,p) where & and A,, are the integrands of t h e first integral on the r.h.s of ( 3 5 ~ for and Q(z.y, p ) , respectively. The present problem would have a similar result. Inversion of the R m e - W a n s f o r m e d Displacements a ( z ,y, p ) !'u(z ,p) b y the Cagniard-de Hoop Technique. The formal solutions for a ( z , y p ) and U(z,yp) of the surface-breaking
crack problem can be inverted by the now well-known Cagniard-de Hoop technique which was used for the problem in [l, Part 11 and discussed there. Most important for the reader is the material associated with (44),-(47)1 and Fig. 6 there, setting down the integrals in (45), for Q(z,y,p) and Q(z,y,p) along with discussions of inversions of these integrals through transformations generating new paths of integration and associated singularities. Wavefront Events in the Surface-breaking Crack A o b L e m . In [ 1, Part 21 the wavefront events in the edge uniform pressure problem, cf. Fig. 1 of [l, Part 11, were derived through further
Wavefront Fields in Elastic Wave Scattering
247
steps i n t h e Cagniard-de Hoop inversion of (45), a n d t h e related asymptotic method in [a]. In t h e interior of t h e q u a r t e r plane the events included for t h e displacements, velocities and accelerations, t h e dilatational wavefronts, t h e two-sided equivoluminal and head wavefronts in t h e two critical shear regions associated with t h e edges x = 0 and y = 0 , a n d of course t h e plane dilatational wavefront from the edge x = 0 input. The events on t h e edges of t h e quart e r plane included, for y = 0 , t h e wavefronts just discussed for t h e interior and, in addition, t h e Rayleigh wavefronts. Concerning t h e loaded edge x = 0, (30), and (31), have already defined t h e dilatational a n d Rayleigh wavefronts. However, a s pointed out i n [I,P a r t I ] , t h e present asymptotic method also yields the two-sided equivolumina1 and head wavefronts on the loaded edge, x = 0 . Further, we have t h e input of t h e two-sided equivoluminal wavefronts on t h e crack faces. stemming from t h e source function at t h e point x = a ,y = 0 shown i n Fig. 2 . This work may be found in the material associated with (43)-(57) in ref. [2]. Finally, we will be seeking similar wavefront events f o r the present problem a n d t h e subsurface crack, which will be the subject of forthcoming papers. REFERENCES 1. Miklowitz. J., 1962, Wavefront analysis in t h e nonseparable elastodynamic quarter-plane problems, I. P a r t 1: The general method; P a r t 2: Wavefront events in t h e edge uniform pressure problem; J. Appl. Mech.. Trans. ASME, 49:797 and 49:BOB. 2. Miklowitz, J.. Wavefront Fields in t h e Scattering of Elastic Waves by Surface-Breaking and Sub-surface Cracks, Review of Progress in Quantitative Nondestructive Evaluation Vol. 2A, 413-440, Eds., D.O. Thompson and D.E. Chimenti, 1983 Plenum Press, New York. 3. Achenbach, J.D., Keer, L.M., and Mendelsohn, D.A., 1980, Elastodynamic analysis of a n edge crack, J. Appl. Mech., Trans. ASME. 47:551.
4. Mendelsohn. D.A., Achenbach, J.D., and Keer, L.M., 1980, Scattering of elastic waves by a surface-breaking crack, Wave Motion, 2:277. 5. Kundu, T., and Mal, A.K., 1981, "Diffraction of elastic waves by a surface crack o n a plate, J.
Appl. Mech., Trans. ASME, 48:570. 6. Pao, Y.H., editor, 1978, Elastic waves and non-destructive testing of materials, ASME Mono-
graph, AMD-Vol. 29. 7. Miklowitz, J., 1978, "The Theory of Elastic Waves a n d Waveguides," North-Holland Publishing Co., Amsterdam, New York, and Oxford. 8.Rosenfeld, R.L., a n d Miklowitz, J., 1962, Wavefronts in elastic rods and plates, Proc.4 t h U.S. National Congress of Appl. Mech. 1, ASME, N.Y., pp. 293-303.
248
J. Miklowitz and C. Chazapis
Wavefront Fields in Elastic Wave Scattering
249
250
J. Miklowitz and C. Chazapis
I t is important t o note that DT and the coefficients of the wavefronts (e.g. (xu).) all contain trigonometric terms of the form s i n b a l d + q ) and sin(pb,d + &) , k = l,,..,Y . To achieve a proper inversion that leads t o (30) and (31)oi the text, these trigonometric terms must be evaluated in such a manner that we can establish a maximum value for the coemcients A ( p ) ,B ( p ) , C ( p ) and D ( p ) before inversion.
Wavefront Fields in Elastic Wave Scattering
25 1
- C,’)E 1
gj
= (bj
c,
=
1
TR,
+ ( - - 1 ) ~ - 1 c , p J,~k = 1,2
= c a 2 - ZCFf I
q = ( b f c j+ l)h
=
~ ~ ( k r *tyics )
I
= 1,2,3
Where k 2 with c d and cs are defined after (10) of the text. Further the y r (1 = 1,2,3)depend on Poisson’s ratio only. For Poisson’s ratio v = 1/4 , y 1 4 . 9 2, y2= 1.78and y3 = 2.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.8. Moodie (eds.) @ Elsevier Science Publishers B.V. (North-Holland), 1984
253
WIENER-HOPF ANALYSIS: AN APPRAISAL AND A NON-STANDARD APPLICATION llarold Levine Department of Mathematics Stanford University Stanford CA 94305 USA
A boundary value problem relating to electromagnetic wave
excitation inside and outside an open-ended, semi-infinite, perfectly conducting circular guide is analysed by the Wiener-Hopf method. The focus on excitations which are not axisymmetric, and thus involve both axial and azimuthal electric currents at the guide wall, presents a simultaneous pair of complex transform relations that necessitate a sequential resolution procedurc. Explicit representations are obtained for the distant radiation field associated with a mode incident on the open end of the guide from its interior as well as for a reflection coefficient of the same modal type.
51.
Introduction
The Wiener-Hopf method or technique, which culminated an effort to resolve a particular integral equation arising in astrophysical radiation theory, has acquired maturity after a 50-year span of existence and earned a reputation for remarkable and diverse accomplishments. Noteworthy application and development of the intricate though systematic technique dates from the 1940s, under the stimulus of wartime studies concerning neutron diffusion and microwave propagation problems; a measure of inherent capability can be gleaned from the fact that it has provided exact and explicit solutions to various boundary value problems which arc otherwise unattainable! It is a reasonable proposition that Wiener-Hopf and related function-theoretic methods represent a mathematical advance comparable in importance with the operational methods pioneered by Heaviside. Utilization of the method commences after the establishment of one o r more relations involving unknown complex functions that are analytic in different, say the U(pper) and Lfower), half-planes whose intersection is a common strip of regularity. For example, in one of the simplest diffraction problems which envisages a plane time-periodic sound wave incident on a rigid plane screen with an infinite straight edge, the pair of unknown functions are identifiable as half-range complex Fourier transforms (relative to a coordinate variable normal to the edge) of the pressure discontinuity at the screen and the normal velocity in the complementary planar domain, respectively. The next stage requires the split or factorization of known functions of the transform variable into analytic and non-zero factors within t h e respective half-planes, o r else’their resolution into a sum of U and L functions; then, a rearrangement which secures manifest analyticity of individual terms in either U or L permits the identification of an entire function throughout the complex plane and separate determinations of the transforms follow. The whole analysis, involving a factorization that is generally obtainable by means of contour
H. Levine
254
integrals and a specification of the entire function through its asymptotic behavior necessitates due consideration of all aspects in each problem, and especially those bearing on asymptotic transform estimates which ensure uniqueness of the desired solution. If two or more transform relations are involved the Wiener-Hopf factorization poses a more intricate problem, unless these relations can be uncoupled or analysed successively; that is, a given matrix whose elements define analytic functions within a strip of the complex plane requires display as a product of two non-singular matrices with analytic elements in U and L , respectively. The unresolved technicalities of general matrix factorization thus far preclude the attainment of solutions to numerous interesting boundary value problems in wave theory. Wiener-Hopf analysis has furnished, in particular, precise results for the reflection within and radiation from semi-infinite open-ended circular guides o r tubes. The early success (Levine and Schwinger, 1948; Vainstein, 1948) in securing an explicit frequency and correction and other details for plane acoustic wave incidence on the open end of a rigid circular pipe has been impressively generalized (Munt, 1977; Cargill, 1982) to accomodate different (steady) axial flows inside and outside the pipe and thereby elucidate the interaction between sound and flows. Analogous problems that pertain to electromagnetic excitations originating within a perhectly conducting circular guide are likewise amenable to resolution and the analysis is simplest when the condition of axial symmetry obtains; for the lowest (Eo or Ho) symmetric modes, there is but one component of current at the guide wall, namely in the axial o r azimuthal direction, respectively, and thus a standard Wiener-Hopf procedure suffices. A significantly more complicated situation arises, however, f o r the asymmetric H 1 mode which features both axial and azimuthal currents at the guide wall; and the following (old, albeit unpublished) calculation exemplifies the style of a non-standard o r twostage Wiener-llopf procedure.
§2. The Boundary Value Problem and its Transform Equivalent Consider a semi-infinite circular guide within which an individual E o r II mode propagates toward the open end (at z = 0) from the remote interior (z = - m ) ; two aspects of the resultant time-periodic excitation may be singled out for study, viz. the reflected modal z amplitude(s) at z = -a and the distant
( ’ ’
(m
-t -) outside radiation field the guide. The guide wall, assumed to have infinitesimal thickness, constitutes a curvilinear current sheet S(p = a , -- < z < 0 , 0 c 4 < 2 ~ r ) and an
appropriate source representation of the electric field Re [ E ( ~ ) e - ~ ~ ~ anywhere l is given by
in terms of the free-space dyadic Green’s function,
255
Wiener-Hopf Analysis
r(c)
and t h e c u r r e n t d e n s i t y (which i n c o r p o r a t e s v a r i o u s c o n s t a n t f a c t o r s ) . Determination of t h e c u r r e n t d i s t r i b u t i o n h i n g e s on t h e r e q u i r e m e n t t h a t t h e t a n g e n t i a l components o f t h e e l e c t r i c f i e l d E , E ) v a n i s h on t h e g u i d e s u r f a c e
$ z
0 , a s b e f i t s the hypothesis of p e r f e c t conductivity. Since h a s no component i n t h e r a d i a l d i r e c t i o n , o n l y t h e d y a d i c elements p = a
r
$z
,
,z < and
r
$$
p r e s s ion
it f o l l o w s t h a t
and
_I ( _r )
rzz’rz$ =
,
with
p = p' = a
,
a r e involved.
Taking a c c o u n t of t h e ex-
H. Levine
256
-00
T h e fields and currents with a particular angular dependence e ( 6 ) , through the relations coupled, via (1)
im@
-
from which the argument of the cylinder functions, m
a
,
are
is omitted; here
denote complex Fourier transforms of the respective electric field components on the geometrical continuation of the guide surface (p = a , z > 0) and
represent the transforms of t h e currents at the guide surface
(P = a
,z<
An uncoupling of the simultaneous transform relations (7) is directly achieved for axisymmetric modes (m = 0) and the pertinent individual expressions are
0)
.
257
Wiener-Hopf Analysis
D e f i n e , i n c o n n e c t i o n w i t h non-symmetric modes
(m # 0)
,
and t h e n t h e t r a n s f o r m e q u a t i o s ( 7 ) a r e r e p l a c e d by
a l t e r n a t i v e l y , i n t r o d u c i n g t h e l i n e a r combination o f e l e c t r i c f i e l d t r a n s f o r m s
i t f o l l o w s from (7) t h a t
and
(14)
The E modes o f a c y l i n d r i c a l g u i d e w i t h i n d e f i n i t e e x t e n s i o n a l o n g t h e p o s i t i v e and n e g a t i v e z - d i r e c t i o n s f e a t u r e a v a n i s h i n g a x i a l component of t h e magnetic f i e l d , and t h u s I (5) = 0 ; t a k i n g i n t o a c c o u n t t h e n u l l v a l u e s o f $ b o t h t a n g e n t i a l e l e c t r i c f i e l d components E Z , E and t h e i r t r a n s f o r m s a t t h e $ g u i d e w a l l , t h e second o f t h e r e l a t i o n s ( 1 2 ) i s a u t o m a t i c a l l y s a t i s f i e d , w h i l e = 0 ( i f I(<) # 0 ) and i m p l i e s t h e e x i s t e n c e of the f i r s t requires that
Jm
a spectrum of modal wave numbers
where
Jm(y;,=)
=
0
.
H
modes, l a c k i n g a x i a l components o f e l e c t r i c f i e l d and
wall c u r r e n t , a r e c h a r a c t e r i z e d by t h e p r o p e r t y
H. Levine
258
on t h e b a s i s o f t h e f i r s t r e l a t i o n i n ( 1 2 ) ; and t h e second l e a d s t o t h e i d e n t i f i c a t i o n o f modal wave numbers
93.
A n a l y s i s o f Simultaneous Transform Equations A r e s o l u t i o n o f t h e t r a n s f o r m e q u a t i o n s ( 1 2 ) , assuming t h a t o n l y t h e
H1
mode can p r o p a g a t e w i t h i n t h e s e m i - i n f i n i t e c i r c u l a r g u i d e , commences by g i v i n g a t t e n t i o n t o t h e f i r s t e q u a t i o n of t h e p a i r , v i z .
and n o t i n g t h a t , s i n c e
v a n i s h e s f o r any
f o l l o w s , where of angular index
H
mode, t h e e s t i m a t e
> 0
i s t h e wave number a p p r o p r i a t e t o t h e lowest
. Accordingly, I(() = 1 e-;xt
m = 1
E
mode
t h e transform
I(tl b t
'00
d e f i n e s an a n a l y t i c f u n c t i o n i n t h e h a l f - p l a n e estimate
Im 5 >
- X I1,1
.
The a s y m p t o t i c
O ;p t
E
p hl
uniierlies a deduction t h a t moreover, b o t h
H:l)(ma)
e , -
3300
E (5) e x i s t s i n t h e h a l f - p l a n e and
(21)
2"
J l [ m a )
Im 5 <
E
(= I m kj ;
a r e analytic within the s t r i p
I I m 51 < E , which t h u s c o n s t i t u t e s t h e domain o f v a l i d i t y o f t h e t r a n s f o r m r e l a t i o n (18). On i n v o k i n g t h e decomposition
259
Wiener-Hopf Analysis
w i t h (non-vanishing) a n a l y t i c f u n c t i o n s planes
Im
<
> -k , < k
,
K+(r)
,
K-(<)
i n the respective half-
i t f o l l o w s from (18) t h a t
A c o n s t a n t v a l u e o f t h e i n t e g r a l f u n c t i o n , which may be s e t e q u a l t o u n i t y , i s i n d i c a t e d by t h e i n d i v i d u a l a s y m p t o t i c b e h a v i o r o f t h e v a r i o u s f a c t o r s i n (231, v i z .
and
whence t h e r e s u l t a n t s p e c i f i c a t i o n s
The second of t h e coupled t r a n s f o r m e q u a t i o n ( 1 2 ) now t a k e s t h e form
and t h e a p p e r t a i n i n g decomposition
where
~ y d e, n o t e~s
L+(<) , L (z;)
t h e ( r e a l ) wave) number o f t h e p r o p a g a t i n g
p o s s e s s t h e same c h a r a c t e r i s t i c s a s
conversion of (25) i n t o a r e l a t i o n
K+(C)
H1
, K-(<) ,
mode and enables t h e
260
H. Levine
which d e f i n e s a n o t h e r i n t e g r a l f u n c t i o n . s a y A , i n keeping w i t h t h e e s t i m a t e s
and a v a l u e f o r
A
The l a t t e r p r o v e s t o be a c o n s t a n t ,
i s s e c u r e d from t h e p r e v a i l i n g e x p r e s s i o n of ( l l ) ,
on a p p l y i n g t h e c o n d i t i o n of r e g u l a r i t y f o r
I (5)
at
5
=
k ; t h i s yields
and, f u r t h e r m o r e ,
I t appears t h a t t h e function H
mode p r o p a g a t i o n c o n s t a n t s ,
L+(-k)
-
I
@
(c)
h a s s i m p l e p o l e s a s s o c i a t e d with t h e
+Y;',~ ,
and i s r e g u l a r a t
5
=
-k
since
(<+k)-' ; t a k i n g n o t e o f t h e a s y m p t o t i c b e h a v i o r
a representation of the (voltage
-.Hz
-
I ) r e f l e c t i o n coefficient follows,
@
Wiener-Hopf Analysis
where
The f a c t o r s
K+(k)
,
L+(k)
,
L + ( L K ~ , ~ a) r e e x p r e s s i b l e , w i t h r e l i a n c e on i n -
f i n i t e p r o d u c t s and t h e Cauchy i n t e g r a l formula, by means o f s e r i e s and s i n g l e d e f i n i t e i n t e g r a l s . r e s u l t i n g in t h e d e t e r m i n a t i o n s
26 1
262
H. Levine
which employ a conventional notation for the cylinder functions I1 , K1 , involve the Euler-Mascheroni constant y = 1.781 , and the complex number 5 O such that
...
Explicit values of the (complex) voltage reflection coefficient, in correspondence with appropriate specification of the parameter ka , are obtainable from the preceding formulas through numerical analysis.
94. Radiation Patterns
A characterization of the electromagnetic field and thence of energy flux outwards at large distances from the open end of the guide is secured after utilizing the asymptotic form of the dyadic Green’s function ( 2 )
where n = r/r , r = Irl , designates a unit vector that is directed towards Thus, (1) and (35) provide an estimate the remote Field point-fl
.
which, coupled with the fact that the proportionate electric and magnetic fields E , H in the far field are transverse to each other and the vector 2 , underlies the expression
for the time-average radial energy flux. arbitrary vector)
The identity (wherein V denotes an
263
Wiener-Hopf Analysis
p r o v e s c o n v e n i e n t as r e g a r d s t h e e v a l u a t i o n o f ( 3 6 ) , y i e l d i n g f o r a s t a t e of c i r c u l a r p o l a r i z a t i o n c h a r a c t e r i z e d by c u r r e n t components w i t h t h e common azimuthal angle f a c t o r
ei@ -
i n terms of t h e c u r r e n t transforms
Ib(k cos Q)
t h a t a r e f u n c t i o n s of t h e p o l a r a n g l e ,
Q
and ( r e c a l l ( 1 1 ) )
, reckoned
i(Q)
from t h e g u i d e a x i s .
The power g a i n f u n c t i o n , , which measures t h e r a d i a t i o n i n t e n s i t y a l o n g a p a r t i c u l a r d i r e c t i o n a i n s t t h e t o t a l o u t p u t , i s found (on u t i l i z i n g the p r e v i o u s s p e c i f i c a t i o n o f c u r r e n t t r a n s f o r m s t o g e t h e r w i t h t h a t o f t h e t o t a l r a d i a t e d power through t h e d i f f e r e n c e of i n c i d e n t and r e f l e c t e d amounts) t o have t h e r e p r e s e n t a t i o n
where
264
H. Levine
and
I
proof, in the analytical sense, that energy conservation, v i z .
A
9
(€I)
(39)
conforms with the requirement of
Tt
s go, 5&9 43 = 2, 0
has been secured on the basis of contour integration. If the wave guide field is rectangularly polarized, as manifest by the azimuthal angle variations
the corresponding power gain function, and polar angles, assumes the form
9
(@,@) , dependent on both the azimuthal
Wiener-Hop f Analysis
265
The l a t t e r f u n c t i o n , whose d o u b l e i n t e g r a l o v e r t h e f u l l r a n g e s o f 8 , c$ must e q u a l TI , remains f i n i t e f o r a l l a n g l e s 0 5 Cl < TI i n b o t h t h e e l e c t r i c ( @ = ~ / 2 ) and m a g n e t i c (I$= 0) p l a n e s , p u r s u a n t t o t h e assumption o f a s i n g l e p r o p a g a t i n g mode w i t h i n t h e c i r c u l a r g u i d e .
References C a r g i l l , A . M.
1982
Levine, H. and Schwinger, J . Munt, R. M.
1948
Physical
1977 J o u r n a l of F l u i d Mechanics
V a i n s t e i n , L. A.
83, 339-354 Review 2, 384-400
J o u r n a l o f Sound and V i b r a t i o n
1948
83,
609-640
J o u r n a l o f T e c h n i c a l P h y s i c s (USSR)
Is, 1543-1564
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
267
STRESS WAVE PROPAGATION I N D I S C R E T E RANDOM SOLIDS M a r t i n Ostoja-Starzewski Micromechanics Research L a b o r a t o r y Department o f Mechanical E n g i n e e r i n g McGi 11 U n i v e r s i t y M o n t r e a l , Quebec Canada
A s t o c h a s t i c t h e o r y i s p r e s e n t e d f o r t h e wave p r o p a g a t i o n i n a one-dimensional model o f a s o l i d w i t h a d i s c r e t e random m i c r o s t r u c t u r e . The w a v e f r o n t e v o l u t i o n a t a m i c r o s c a l e i s developed t h r o u g h t h e wave m o d u l a t i o n and d i s p e r s i o n t i m e processes l e a d i n g t o a f o r m u l a t i o n i n terms o f two Markov o p e r a t o r s p a r a m e t r i z e d by t h e i n t e r n a l and macroscopic times, r e s p e c t i v e l y . An a p p r o x i m a t i o n i s g i v e n i n terms o f a s e t o f two Fokker-Planck e q u a t i o n s and a p p l i e d t o model an a c c e l e r a t i o n w a v e f r o n t as a domain o f f i n i t e t h i c k n e s s undergoing random e v o l u t i o n . F i n a l l y , t h e random geometry e f f e c t s a r e discussed. 1.
INTRODUCTION
I n t h i s work we f o c u s o u r a t t e n t i o n on t h e t r a n s i e n t s t r e s s wave p r o p a g a t i o n i n
sol i d s w i t h m i c r o s t r u c t u r e s such as t h o s e encountered i n p o l y c r y s t a l l i n e and l a y e r e d media. F o l l o w i n g p r o b a b i l i s t i c micromechanics r1.21 we aim a t t h e i n c l u s i o n o f t h e d i s c r e t e n e s s and randomness o f m i c r o q u a n t i t i e s i n t h e governi n g r e l a t i o n s and t h e i r s o l u t i o n s . We r e c o g n i z e t h a t due t o t h e random p h y s i c a l p r o p e r t i e s ( c o n s t i t u t i v e m o d u l i , mass d e n s i t y ) as w e l l as t h e random geometry o f t h e microelements o f t h e s t r u c t u r e o f t h e s o l i d (e.g. c r y s t a l s , l a y e r s ) , a c o m p l i c a t e d s c a t t e r o f even t h e s i m p l e s t i n i t i a l p u l s e s w i l l t a k e p l a c e t h r o u g h o u t t h e whole m i c r o s t r u c t u r e . While i t i s obvious t h a t t h e random geometry o f t h e g r a i n boundaries w i l l g i v e r i s e t o r a t h e r c o m u l i c a t e d phenomena i n wave proDaqation, i t i s n e v e r t h e l e s s c o n s i d e r e d t h a t t h i s r e p r e s e n t s o n l y a secondary e f f e c t w i t h r e s p e c t t o t h e s i g n i f i c a n c e o f t h e randomness o f t h e p h y s i c a l p r o p e r t i e s o f t h e medium. Thus, an i n i t i a l f o r m u l a t i o n o f wave p r o p a g a t i o n i n s t r u c t u r e d s o l i d s can be g i v e n by assuming a l l microelements t o have i d e n t i c a l c u b i c shapes d x d x d and t o be a r r a n g e d i n a l a t t i c e - l i k e s t r u c t u r e i n t h e bodydomain. I n t h i s s e t t i n g we can t h e n c o n s i d e r p r o p a g a t i o n o f p l a n e waves whose p r o p a g a t i o n v e c t o r i s p a r a l l e l t o one o f t h e axes o f t h e l a t t i c e . A l t h o u g h i t i s c l e a r t h a t a p e r f e c t p l a n e wave generated a t a boundary o f t h e macroscopic body domain w i l l soon b e g i n t o l o s e i t s p e r f e c t f o r m due t o t h e i n t e r a c t i o n s between t h e microelements, i t w i l l c o n t i n u e t o propagate as a macrodisturbance i n t h e p r i n c i p a l d i r e c t i o n b e f o r e d i f f u s i n g c o m p l e t e l y t h r o u g h t h e e n t i r e body. I t i s t h e e v o l u t i o n o f such a d i s t u r b a n c e which i s o f main i n t e r e s t t o us h e r e .
A more g e n e r a l t h e o r y o f s t r e s s wave p r o p a g a t i o n i n such a 3-0 model o f a s t r u c I n t h i s paper we t u r e d s o l i d has r e c e n t l y been proposed i n [31 (see a l s o [41). p r e s e n t and f u r t h e r develop t h e 1-0 model o f t h e wave p r o p a g a t i o n . We f i r s t f o r m u l a t e t h e e v o l u t i o n o f t h e w a v e f r o n t a t t h e microelement s c a l e ( e x c l u d i n g shocks) i n terms o f two Markov o p e r a t o r s p a r a m e t r i z e d by t h e i n t e r n a l ( r e a l ) and macroscopic (average) t i m e s , r e s p e c t i v e l y . Then we pass t o an approximate, b u t a n a l y t i c a l l y more t r a c t a b l e , f o r m u l a t i o n i n terms o f two d i f f u s i o n e q u a t i o n s o f t h e Fokker-Planck t y p e . Subsequently, by a p p l y i n g t h i s f o r m u l a t i o n t o t h e a c c e l e r a t i o n waves we a r e a b l e t o model t h e w a v e f r o n t as a domain o f f i n i t e t h i c k n e s s undergoing a random e v o l u t i o n . F i n a l l y , we d i s c u s s t h e e f f e c t o f t h e i n t r i n s i c random geometry o f t h e m i c r o s t r u c t u r e i n t h e wave p r o p a g a t i o n i n s o l i d s .
268
2.
M. Ostoja-Starzewski
STOCHASTIC MODEL OF THE WAVEFRONT PROPAGATION
We c o n s i d e r t h e wave p r o p a g a t i o n i n a s e m i - i n f i n i t e b a r o f a c o n s t a n t a r b i t r a r y cross-section. The b a r i s t a k e n as an ensemble M o f K n o n - i n t e r a c t i n g sequences sk (k=1,2 ,...,K ) , v i d e F i g . 1 . The sequences c o n s i s t o f microelements a=1 ,2 ,..., each o f which f o r t h e s i m p l i f i c a t i o n o f t h e a n a l y s i s i s assumed t o be a l i n e a r e l a s t i c continuum w i t h random p h y s i c a l p r o p e r t i e s 'p. "E, CIG. These l a t t e r q u a n t i t i e s a r e assumed t o be c o n s t a n t t h r o u g h o u t t h e domain o f t h e g i v e n m i c r o element and t o be d i s t r i b u t e d homogeneously i n t h e macroscopic body o f t h e b a r M, i . e . t h e p r o b a b i l i t y d i s t r i b u t i o n s P("p), P("E), P("G) do n o t depend on t h e p o s i t i o n i n t h e c o o r d i n a t e space ( X i , X2, X3). The microelements a r e a l s o charact e r i z e d by t h e i r random l e n g t h "d i n t h e X1 d i r e c t i o n (P("d) - space homogeneous), which however i s assumed t o be c o n s t a n t i n t h e i n i t i a l f o r m u l a t i o n o f t h e model. T h i s c o n d i t i o n w i l l be r e l a x e d i n t h e l a s t s e c t i o n o f t h i s paper.
x2
T
/M
P
Fig.1 I-D model of the discrete random solid We s t u d y t h e response o f t h e b a r i n i t i a l l y a t r e s t w i t h t h e f o l l o w i n g f o r c i n g condition:
I t i s a l s o assumed t h a t df/dt(t,O
= 0
, which
excludes t h e shock-wave-type
f o r c i n g . Furthermore, we c o n s i d e r waves o f t h e c h a r a c t e r i s t i c l e n g t h l a r g e r t h a n t h e microelement s i z e d. C o n s i d e r i n g t h a t t h e c r y s t a l s i z e may, i n g e n e r a l , v a r y i n p o l y c r y s t a l l i n e s o l i d s between 0.1 and 5.0 rnrn, and t h e wave p r o p a g a t i o n v e l o c i t y between 2.103-7.103 m/s, e l e m e n t a r y c a l c u l a t i o n s show t h a t t h i s w i l l correspond t o t h e f r e q u e n c i e s below 106-107 Hz . T h i s shows t h a t t h e proposed t h e o r y can s t i l l cover most o f t h e f r e q u e n c i e s n o r m a l l y encountered i n t h e engineering applications. We look a t t h e s e t o f p o i n t s i n t h e domain M t o which a p u l s e can t r a v e l i n t i m e l e s s t h a n o r equal t o t>O , due t o t h e d i s t u r b a n c e s p e c i f i e d by ( 1 ) . C l e a r l y , due t o t h e random p r o p a g a t i o n speed i n t h e microelements o f t h e s o l i d , t h e
269
Stress Wave Propagation in Discrete Random Solids
boundary o f t h i s s e t w i l l n o t be a p l a n e surface i n (X,,X,,X3) s i n g l e w a v e f r o n t s i n a l l t h e p a r t i c u l a r sequences sk
E
M
.
b u t a c o l l e c t i o n of Since t h e p h y s i c a l
domain i s d i s c r e t i z e d h e r e by t h e s i z e d we d i s t i n g u i s h between t h e microelements r a t h e r t h a n p o i n t s i n t h e continuum mechanics sense, and hence d e f i n e t h e following sets: (i)
Y(t)
=
{microelements
0:
disturbed i n time
st3
.
( i i ) O ( t ) = a Y ( t ) - boundary o f t h e s e t y ( t ) , a l s o c a l l e d t h e wavefront. The l a t t e r i s t h e s e t o f t h o s e microelements t o which t h e p u l s e can t r a v e l i n t i m e t b u t n o t f a s t e r , v i d e t h e space-time graph i n F i g . 2 .
/
average path, internal time t
+x
0 I
0
----aXI average time i = (c> I
Fig. 2 Space-time graph of the wavefront propagation
It i s e v i d e n t f r o m t h i s f i g u r e t h a t due t o t h e random p r o p a g a t i o n speed t h e O ( t ) domain has a tendency t o spread, and hence t o account f o r t h i s phenomenon we introduce the dispersion time variable: .d,
= t -
t
(3)
I n t h e above, t i s t h e i n t e r n a l ( r e a l ) t i m e and 5 i s t h e average (macroscopic) t i m e . The e x p r e s s i o n s " i n t e r n a l t i m e " and "macroscopic t i m e " f o l l o w t h e
270
M. Ostoja-Starzewski
t e r m i no1 ogy o f P r i gog ne [51.
Both t h e s e t i m e s a r e d e f i n e d here as f o l l o w s :
N N t = C %
t
1
(4)
T2
a=1
where a~ ="d/ % i s t h e passage t i m e t h r o u g h a s i n g l e microelement o f t h e s o l i d , The phenomenon o f and N i s t h e number o f microelements d i s t u r b e d i n t i m e 5 t t h e random a r r i v a l t i m e s i n t h e space-time graph i s t h e n d e s c r i b e d by t h e d i s p e r -
.
s i o n t i m e process function o f the dTt
Thus,i f o r d. ~
dTt
each w1
process, " b e i n g
E
R1, Cd.~(wl),
a p r o b a b i l i t y space.
t
E
TI}
i s a sample
F u r t h e r , f o r each
t ,
i s an Fd- -measurable f u n c t i o n , FdE b e i n g t h e 0 - a l g e b r a o f t h e d i s p e r s i o n t Df ; { F d t , &Tl)is an i n c r e a s i n g f a m i l y o f t h e a - a l g e b r a s such t h a t
t i m e s space f o r each
t , dTt
i s F d t -measurable.
C l e a r l y , d ~ =f 0 f o r t h e average path, whereby we r e c o v e r t h e macroscopic d e t e r m i n i s t i c d e s c r i p t i o n , and f u r t h e r
which shows t h a t t h e d i s p e r s i o n t i m e process i s e r g o d i c . However, e r g o d i c i t y i s n o t c h a r a c t e r i s t i c o f t h e p r i n c i p a l e f f e c t i n t h e wave p r o p a g a t i o n , namely t h e m o d u l a t i o n o f t h e s t a t e v a r i a b l e o f t h e elements o f t h e @ ( t ) domain due t o t h e g r a i n boundary e f f e c t s between a and a + l c r y s t a l s . We d e s c r i b e t h e mechanical s t a t e by t h e d e f o r m a t i o n r a t e o f t h e microelement a av
a"u at
(7)
which i s t a k e n as an average o v e r t h e p h y s i c a l domain o f a ; av i s a r e a l - v a l u e d random v a r i a b l e ( 1 - 0 model) and r e p r e s e n t s t h e f o r w a r d g o i n g wave o n l y . Thus t h e wave m o d u l a t i o n phenomenon i s d e s c r i b e d i n terms o f t h e vt process and f o r each
, t E T21 i s a sample f u n c t i o n o f t h i s process. F o r each t, v t F, - measurable f u n c t i o n , where Fv i s t h e u - a l g e b r a o f t h e w a v e f r o n t v e l o c i t y space V ( @ ( t ) ) , a l s o t o be denoted by Vt . Furthermore, {Fvt , t E T z }
w2
E
Q2 , {vt(w2)
i s an
i s an i n c r e a s i n g f a i n i l y o f u - a l g e b r a s such t h a t f o r each t, vt i s Fvt -measurable. C l e a r l y then, t h e f u l l d e s c r i p t i o n o f t h e w a v e f r o n t p r o p a g a t i o n i n t h e b a r t h e microelement s c a l e i s g i v e n by a random v e c t o r
, . L
M at
J
E
M , i.e. f o r
f
E
T1
( ~ 2 ), t
6
TJ
and i n any sequence sk d T t ( ~ l ), = [vt
It f o l l o w s t h a t { d T t
v a r i a b l e s , indexed by
,f
E
f and
T1) and { v t
w1
,
,t
w2 f i x e d , by
E
T2} a r e two f a m i l i e s o f random
t r e s p e c t i v e l y , and d e f i n e d on a p r o b a b i l i s t i c measure
27 1
Stress Wave Propagation in Discrete Random Solids
P}, where R = R1 x R2 and F = F1 x F2 f {Fdf , & T i } x {Fvt , tET21. Now, i n t h e case o f small v a r i a t i o n s i n p h y s i c a l p r o p e r t i e s , such as i n p o l y c r y s space {R, F,
t a l l i n e s o l i d s f o r example, b o t h t h e u - a l g e b r a s Fdf and Fvt f o r any f i x e d g e n e r a l i z e d t i m e ( t , t ) a r e assumed t o be independent, and hence a unique p r o b a b i l i t y measure on F f , t
= F d i x Fvt
can be i n t r o d u c e d as f o l l o w s :
P(Di x Et) = Pd(Dt)*Pv(Et) , where Dt
E
Fdf
and
Et
Fvt
E
.
(9)
The t e r m on t h e l e f t expresses t h e p r o b a b i l i t y
o f f i n d i n g v i n a s t a t e E a t t i m e t and p o s i t i o n X i = ( c ) domain, i . e .
f ,in
the physical
P { V ( f , t ) } E PtV(X1 ,t)EF} = PdjdTEDt} PvCvEEtl , F E F E , ~ (9' ) It i s t o be n o t e d t h a t t h i s assumption o f t h e q u a l l t a t i v e independence has been supported by t h e Monte C a r l o experiments [3,61.
On h e u r i s t i c grounds b o t h random processes i n v o l v e d a r e r e c o g n i z e d as Markov processes w i t h t h e one-step t r a n s i t i o n f u n c t i o n s df E D I d ~ ( f )= 2 ) P ( f , Z , F+("T), D) = Pdid,(t+(%)) =
p{"T
:
dT(f+(aT))
'T-('T)=
- dT(?)}
(10)
df p ( t , x, t+"T,E)
=
Pv { v ( t + % )
=
P tc : v ( t
E f
E l v ( t ) = x}
'T)
= C.v(t)}
Furthermore, s i n c e t h e p r o b a b i l i t y d i s t r i b u t i o n s o f t h e p h y s i c a l p r o p e r t i e s were assumed t o be space homogeneous i n (X1,X2,X3),the above t r a n s i t i o n f u n c t i o n s become time-homogeneous, i . e .
P ( t , z,t+("-r), P ( t , x. t+"T,
D) E)
=
P(("T),
=
P("T,
z , D)
(10')
x, E )
(11')
I n r e l a t i o n ( 1 1 ) we have made use o f t h e t r a n s m i s s i o n c o e f f i c i e n t C f o r t h e wave passage between two m i c r o e l e m e n t s . For a p a r t i c u l a r s o l i d under c o n s i d e r a t i o n t h i s c o e f f i c i e n t would have t o be determined f r o m t h e experiments, such as on b i c r y s t a l s . C l e a r l y t h e c o e f f i c i e n t C accounts f o r t h e p a r t i a l wave r e f l e c t i o n as w e l l as energy d i s s i p a t i o n e f f e c t a t i n t e r n a l g r a i n boundaries. However, a s i m p l e f o r m o f C can be o b t a i n e d by c o n s i d e r i n g g r a i n boundary t o be a r i g i d p l a n e s u r f a c e s e p a r a t i n g two e l a s t i c c o n t i n i l a a and a f l Then under t h e c o n d i t i o n s o f s t r e s s c o n t i n u i t y and v e l o c i t y c o n t i n u i t y a t t h i s boundary t h e t r a n s m i s s i o n c o e f f i c i e n t becomes ","+lC = 2 1 + ","+lX
.
where a."+lx
=
"+l p a+l c "P
"c
i s t h e r e l a t i v e impedance.
Also, we n o t e t h a t t h e { v t , t
E
T 2 ) process i n t h e
p r e s e n t c o n t e x t i s a t i m e c o n t i n u o u s jump process. However, i f t h e random - f o r w a r d and backward v a r i a b l e v were t o r e p r e s e n t t h e cummulative s t a t e o f g o i n g waves i n c l u d e d - we would have a process c o n t i n u o u s i n t h e s t a t e space t o o (a finer description).
M. Ostoja-Starzewski
272
A general f o r m u l a t i o n can be o b t a i n e d by i n t r o d u c i n g t h e Flarkov t r a n s i t i o n o p e r a t o r s , o r more p r e c i s e l y t h e c o n d i t i o n a l expected v a l u e o p e r a t o r s , on t h e L1(D) and L1(V) spaces as f o l l o w s df 'T(f)[f(z)l = f ( w ) P ( f , Z , dw) , f E L1(D) (14) df 'T(t)[g(x)l = g ( y ) P ( t , X, dY) E L1(V) (14' )
5
1' V
These o p e r a t o r s possess t h e semi-group p r o p e r t y , i . e .
With t h i s i n t r o d u c t i o n we can now propose ( s e e a l s o [3,2]) : Theorem
I f {lT(?),
f
E
T13 and { 2 T ( t ) , t
E
1 2 ) a r e one-parameter semi-groups o f
c o n t r a c t i o n o p e r a t o r s on L1(D x V , Pd x Pv) d e f i n e d by ( 1 4 ) and ( 1 4 ' ) , t h e n t h e r e e x i s t s a two-parameter semi-group { T ( f , t ) ,
(t,t)
E
T1 x T2} such t h a t :
T(i,t)
i s t h e d i r e c t p r o d u c t o f t h e o p e r a t o r s 1T and 2T so t h a t
T(t,t)
= lT(E) 2T(t)
(ii)
T(0,O)
= I
(iii)
llT(f,t)l/5 1
(i)
,
(16)
,
.
P r o o f o f t h e above f o l l o w s by t h e T o n e l l i ' s Theorem.
3.
DIFFUSION APPROXIMATION
We observe t h a t w h i l e t h e above theorem g i v e s a n e a t f o r m u l a t i o n o f t h e wave p r o p a g a t i o n process i n a d i s c r e t e random s o l i d , many advantages o f t h e a n a l y t i c p r o p e r t i e s o f t h e Markovian semi -groups a r e missed h e r e s i n c e t h e I T ( t ) o p e r a t o r i s p a r a m e t r i z e d by t h e d i s c r e t e t i m e . Hence, i n o r d e r t o o b t a i n a more e a s i l y t r a c t a b l e f o r m u l a t i o n we s h a l l now approximate t h e d i s p e r s i o n t i m e process Idrt, f ~ T 1 1by a t i m e c o n t i n u o u s Markov process (TI = IR' h e n c e f o r t h ) r e p r e s e n t a b l e by t h e d i f f u s i o n e q u a t i o n . Thus under s t a n d a r d c o n d i t i o n s on t h e t r a n s i t i o n p r o b a b i l i t y P ( f , z, D) ( d i f f e r e n t i a b i l i t y i n t and z ) we have t h e Fokker-Planck equation:
a at
-p(t,
-
z, w l =
a - %
[A(w)p(f, z, w ) l
t1
2
a2 [ B ( w ) p ( t , z , w ) l 9
where
p ( E , z, w, =
P ( f , z, dw) dw
and s u b j e c t t o t h e i n i t i a l c o n d i t i o n lim
f+O
p ( t , z, w) = p o ( z )
The c o e f f i c i e n t s Afw) and B(w) a r e t o be o b t a i n e d as f o l l o w s [ 7 ] : A(w) = - (1d ~ ( t
P.)
since P(%)
+(%))
-
d-r(f)) = 0
and hence P ( f , z , 0) a r e symmetric, and
(17)
Stress Wave Propagation in Discrete Random Solids
273
Furthermore, assuming t h a t
t h e equati.on ( 1 7 ) reduces now t o a s i m p l e d i f f u s i o n e q u a t i o n
a at
- p(f,
Z,
l a2 W) = B ~2
p(f,
Z,
W)
, B
= 28
(18)
Since i n t h e w a v e f r o n t e v o l u t i o n process t h e i n i t i a l c o n d i t i o n i s g i v e n by PO(Z) = 6 ( z ) t h e s o l u t i o n t o (18) w i l l be
(18‘
1
The above r e s u l t c l e a r l y shows t h e square r o o t - t y p e dependence o f t h e s t a n d a r d d e v i a t i o n o f t h e d i s p e r s i o n t i m e d i s t r i b u t i o n s on t h e d i s t a n c e t r a v e l l e d ( r e c a l l ( 5 ) ) ; t h i s has a l s o been c o n f i r m e d by t h e Monte C a r l o s i m u l a t i o n [81.
As was mentioned e a r l i e r , t h e vt process i s a c o n t i n u o u s t i m e Markov jump process. Here we s h a l l approximate i t as a sample-continuous process and assume t h a t t h e necessary c o n d i t i o n s f o r t h e Fokker-Planck e q u a t i o n h o l d , so t h a t
and s u b j e c t t o t h e i n i t i a l c o n d i t i o n
The d r i f t and d i f f u s i o n c o e f f i c i e n t s i n ( 2 0 ) a r e now c a l c u l a t e d as f o l l o w s [71 :
where
Furthermore, s i n c e t h e f l u c t u a t i o n s i n t h e p h y s i c a l p r o p e r t i e s a r e assumed t o be small, i . e .
1
([v(t+(ar))-
v ( t ) ~ n )=
o
for
n > 2
we j u s t i f y t h e a p p l i c a t i o n o f (20) t o t h e vt process.
Thus, e q u a t i o n ( 2 0 ) t a k e s on
274
M. Ostoja-Starzewski
the f o l l o w i n g form
where a = b ' - a ' , b = 2 b ' - a ' , c = t b ' Equation ( 2 2 ) i s a l i n e a r c a l c u l a t e d as f o l l o w s [ 9 1
Fokker-Planck e q u a t i o n .
A i s a B o r e l subset o f R ,
where
!J
The s o l u t i o n t o i t may be
a B o r e l measure, and $ A
, I J J ~a r e
both solutions
o f the Sturm-Liouville equation !'y2f"
+
b'
(2-d)yf'
+
Af =
0
I n view o f Theorem o f 4 2 t h e above r e s u l t s can be f o r m u l a t e d i n terms o f t h e new p r o b a b i l i t y d e n s i t y (two-parameter g e n e r a l i z a t i o n o f t h e Gauss-Weierstrass semigroup [ l o ] ) P ( t , t, w, y ) = P ( f , w) P ( t . Y) =
1 1 P(f,
vv
z w) P ( t , x. Y ) P,(z)
P,(x)
dx dz
(24)
where t h e e x p l i c i t forms o f t h e d e n s i t i e s on t h e r i g h t may be e s t a b l i s h e d f o r t h e s p e c i f i c m a t e r i a l s under i n v e s t i g a t i o n . We n o t e here t h a t even though t h e e x p l i c i t s o l u t i o n o f t h e l i n e a r Fokker-Planck e q u a t i o n ( 2 2 ) i s n o t easy t o o b t a i n , t h e c h a r a c t e r o f t h i s s o l u t i o n can be assessed a p p r o x i m a t e l y . C l e a r l y , t h e mean o f v t decreases w i t h t i m e t a c c o r d i n g d(;it))
to
(v(t+"T)
-
v(t))
=
all
v(t)
"T
so t h a t (v(0)) w i t h a " < 0 (v(t)) = Furthermore, t h e d i s t r i b u t i o n o f v t f o r any f i x e d t i m e t may be assessed f r o m t h e C e n t r a l L i m i t Theorem t o be a p p r o x i m a t e l y Gaussian ( f l u c t u a t i o n s a r e smal 1 : IC-(C)I<
ACCELERATION WAVEFRONT PROPAGATION
The f o r m u l a t i o n developed i n t h e p r e v i o u s two s e c t i o n s w i l l a l s o a p p l y t o t h e e v , o l u t i o n o f any p u l s e i n t h e 1 - 0 model o f a s t r u c t u r e d s o l i d where t h e m i c r o element s i z e d i s s m a l l e r t h a n t h e c h a r a c t e r i s t i c l e n g t h o f t h e p u l s e . I n view o f e q u a t i o n ( 2 ) t h e i n i t i a l c o n d i t i o n may be expressed as V(X1 = 0, t ) = ~ ( = f 0, t ) = R * f ( t )
,
(25)
where
It s h o u l d be n o t e d t h a t t h e s p e c i f i c e f f e c t o f - t h e random a r r i v a l t i m e s , a s m o d e l l e d by t h e d i s p e r s i o n t i m e process t E T ~ }, i s a l s o r e s p o n s i b l e f o r t h e o v e r l a p , o r m i x i n g i n t h e space-time graph, o f m i c r o w a v e f r o n t s which were
Stress Wave Propagation in Discrete Random Solids
generated a t d i f f e r e n t t i m e i n s t a n t s a t t h e f r o n t f a c e . v e l o c i t y Mv a t a g i v e n p o s i t i o n X i = L , ( f L = (c).L)
where
t'E
(1t-fLI-E.
It-fLIfE).
( v i d e ( 1 9 ) ) , t h e e f f e c t i v e range
275
Hence, t h e macroscopic
and t i m e t w i l l be
C o n s i d e r i n g t h a t d ~ ( f )i s a p p r o x i m a t e l y Gaussian E
may be t a k e n = 3 . 5 * G d ( t L ) .
macroscopic p u l s e a t f L and t i s seen as an average o v e r
I n any case, t h e
K sequences
sk o f t h e
b a r M, i . e .
c
Rk(fL+dTk)'f(t-tL-dTk) k o r e q u i v a l e n t l y , employing (24) M V ( f ~ ,t ) =
5
M ~ ( f t~) , = R S y.f(t-fL-w) D V
p(Rk) p(dTk)
p ( f , W ) p ( t , y ) dy dw
(27)
(27'
1
W h i l e t h e d i s p e r s i o n e f f e c t s may n o t be v e r y s t r o n g , t h e y w i l l a f f e c t any sharp p o i n t s o f t h e i n i t i a l p u l s e and cause them t o d i f f u s e . C l a s s i c a l l y i n continuum mechanics (see f o r example [ l l l ) , an a c c e l e r a t i o n w a v e f r o n t i s d e f i n e d as a s u r f a c e o f d i s c o n t i n u i t y such t h a t av [ a l l = a2 - a1 , a 5 -
at
s u b s c r i p t s 1 and 2 r e f e r r i n g t o t h e q u a n t i t i e s b e h i n d and ahead o f t h e w a v e f r o n t , r e s p e c t i v e l y . I n r e a l i t y however, l a g i s r a t h e r a c o n t i n u o u s change i n t h e p a r t i c l e a c c e l e r a t i o n i n a domain o f f i n i t e t h i c k n e s s 6 @ ( t ) , o r e q u i v a l e n t l y T h i s t h i c k n e s s i s c o n s i d e r e d here t o c o v e r , a l r e a d y a t t h e 6 @ ( t ) (see F i g . 3 ) . a p p l i c a t i o n o f t h e p u l s e , s e v e r a l ( n ) microelements o r c r y s t a l s , i . e .
&(O)
= n
(5)
The f u n c t i o n f ( t ) i n c o n d i t i o n ( 2 5 ) may be t a k e n as a c e r t a i n p o l y n o m i a l , most s i m p l y as 0, t < O f(t) =
i
t*, t
C1t
E
+
and
(0, 60(0))
c2. t 2 6@(6)
Then,by u s i n g t h e above i n ( 2 7 ' ) we can o b t a i n t h e Mv d i s t r i b u t i o n i n @ ( t ) , o r d i f f e r e n t i a t i n g i n t h e mean square sense, t h e macroscopic a c c e l e r a t i o n Ma i n t h e w a v e f r o n t domain, whereby a numerical i n t e g r a t i o n has t o be employed. Thus, i n a p p l y i n g t h e a n a l y s i s o f t h e f o r e g o i n g s e c t i o n we a r e a b l e t o s t u d y b o t h , t h e d i s p e r s i o n o f t h e w a v e f r o n t and i t s a t t e n u a t i o n . The f o r m e r e f f e c t , as m o d e l l e d b y t h e d i s p e r s i o n t i m e process, g i v e s f o r t h e t h i c k n e s s o f t h e w a v e f r o n t a f t e r t h e passage o f a d i s t a n c e L = (c) f~ &@(EL) ^1 6@(6) + 2.3.5 o d ( t L )
which may be used as a b a s i s f o r a c r i t e r i o n o f t h e passage o f an a c c e l e r a t i o n w a v e f r o n t i n t o a weaker one. We n o t e t h a t s i n c e t h e d i f f u s i o n c o n s t a n t B ' i n ( 1 8 ) i s r a t h e r s m a l l - o f t h e order o f f o r p o l y c r y s t a l l i n e s o l i d s - t h e d i s p e r s i o n e f f e c t does n o t become i m p o r t a n t u n t i l a l a r g e d i s t a n c e is t r a v e l l e d by t h e p u l s e . I n t h a t case, and e s p e c i a l l y f o r s t r o n g e r randomness i n t h e p h y s i c a l p r o p e r t i e s o f t h e m i c r a s t r u c t u r e we may e x p e c t t h i s d i s p e r s i o n t o be o f t h e o r d e r o f 6 0 ( 0 ) l e a d i n g t h e n t o t h e random s e l f - s i m i l a r ( f r a c t a l ) s t r u c t u r e o f t h e macropulse.
M. 0 stoja-Starzewski
276
t '
c C
2
c Q,
> 0
3
Fig.3 Random evolution of the acceleration wavefront
Stress Wave Propagation in Discrete Random Solids
277
5. REMARKS ON RANDOM GEOMETRY I t has been mentioned e a r l i e r t h a t the i n t r i n s i c random geometry of microstruct u r e s in t h e case of p o l y c r y s t a l l i n e s o l i d s i s considered t o be of a secondary importance as compared t o the randomness in t h e physical properties. The only way t o bring in t h e random geometrical e f f e c t s i n the 1 - D model i s t o consider the microelements t o have the random length (see Fig.1). Although we recognize t h a t in t h i s case the d ~process t formulation becomes only approximate and t h a t t h e hypotheses taken f o r the application of the equation (18) should be borne out exactly in experiment, i t i s useful t o employ t h i s simple diffusion model t o assess the random a r r i v a l time e f f e c t . With a d random, t h e probability d i s t r i b u t i o n of the passage time BT will be wider spread than f o r "d = const. In t h a t case the d i f f u s i o n constant B ' and hence t h e standard deviation q will become l a r g e r , leading through ( 1 9 ' ) t o a wider dispersion of t h e a r r i v a l times and hence t o the stronger mixing e f f e c t s of the microwavefronts as discussed in the foregoing section. A Monte Carlo simulation has been atvised f o r t h e a n a l y s i s of such e f f e c t s , whereby P(E), P ( p ) and P ( d ) f o r aluminum were taken a s Gaussian d i s t r i b u t i o n s w i t h t h e following c h a r a c t e r i s t i c s : UE =
7.06*1010 N/m2
The simulation r e s u l t s have shown t h a t t h e diffusion constant B ' in the dispersion time process i s roughly 1 . 5 t h a t of B ' f o r the case of t h e constant microelement length a d = )Jd ( s e e a l s o [81). We note t h a t f u r t h e r research i s needed t o more r e a l i s t i c a l l y model the random geometry e f f e c t s i n s o l i d s . 6. CONCLUSION I n t h i s paper we have proposed a s t o c h a s t i c theory of s t r e s s wave propagation i n s o l i d s with d i s c r e t e random microstructures. The formulation takes i n t o account the e f f e c t s of random passage time through t h e basic element of t h e microstructure ( e . 9 . s i n g l e c r y s t a l ) as well as p a r t i a l r e f l e c t i o n and d i s s i p a t i o n a t the i n t e r nal boundaries. These e f f e c t s lead t o a formulation of t h e wave propagation p r o cess in terms of two Markov processes parametrized by t h e average and internal times, respectively. For small f l u c t u a t i o n s in t h e physical properties of t h e s o l i d the evolution of the dynamical system can be expressed in terms of a twoparameter Markovian t r a n s i t i o n operator taken as a product of two operators parametrized by the two times independently. An approximation of t h i s model i s then given in terms of a s e t of two Fokker-Planck equations. The dispersion time process displays t h e square root-type behavior of the standard deviation with t h e distance t r a v e l l e d by the wavefront, w h i l s t the wave modulation process evolves approximately as a Gaussian packet whose mean decreases exponentially. This a n a l y s i s , when applied t o t h e acceleration waves permits modelling of t h e wavef r o n t as a domain of f i n i t e thickness r a t h e r than a s a surface of d i s c o n t i n u i t y as i n the c l a s s i c a l formulation. F i n a l l y , by introducing the random length of t h e microelements t h e aspect of random geometry i s modelled, showing a stronger dispersion e f f e c t than in case of the d e t e r m i n i s t i c geometry. ACKNOWLEDGEMENT
The author i s grateful t o Prof. D.R. Axelrad f o r the various suggestions, c r i t i c a l reading o f t h e manuscript and continuing support provided t o him d u r i n g the preparation of t h i s work. He would a l s o l i k e t o thank Mr. 3. Turski f o r the stimulating discussions and remarks.
278
M. Ostoja-Starzewski
REFERENCES
1.
D.R. Axelrad,
2.
D.R. Axelrad, "Foundations o f t h e P r o b a b i l i s t i c Mechanics o f D i s c r e t e Media", Pergamon Press, Oxford, 1984
"Micromechanics o f S o l i d s " , PWN-Elsevier, Amsterdam, 1978
3.
M. O s t o j a - S t a r z e w s k i , "Microdynamics o f S t r u c t u r e d S o l i d s " Ph.D. t h e s i s , Department o f Mechanical E n g i n e e r i n g , M c G i l l U n i v e r s i t y , 1483
4.
D.R. A x e l r a d and M.Ostoja-Starzewski, " P r o b a b i l i s t i c Approach t o t h e Wave Propagation i n S t r u c t u r e d S o l i d s " , Workshop on 'Media w i t h M i c r o s t r u c t u r e and Wave P r o p a g a t i o n ' , Houghton, Mich., 1983, t o appear i n I n t . J. o f Eng. S c i .
5.
I. P r i g o g i n e , "From Being t o Becoming", W.H. Freeman, San F r a n c i s c o , 1980; a l s o I . P r i g o g i n e , "Entropy, Time and K i n e t i c D e s c r i p t i o n " , i n 'Order and F l u c t u a t i o n s i n E q u i l i b r i u m and N o n e q u i l i b r i u m S t a t i s t i c a l Mechanics', 1 7 t h I n t . Solvay Conf. on P h y s i c s , eds. G. N i c h o l i s , G. Dewell and J.W. Turner, 3. Wiley, New York, 1981
6.
D.R. A x e l r a d and M. O s t o j a - S t a r z e w s k i , "Plane Wave P r o p a g a t i o n i n D i s c r e t e S o l i d s by t h e Monte C a r l o S i m u l a t i o n " , ' 4 t h I n t e r n a t i o n a l Conference on Mathematical Model 1 i n g I, Z u r i c h , 1983
7.
N.G. Van Kampen, " S t o c h a s t i c Processes i n - P h y s i c s and Chemistry", N o r t h H o l l and, Amsterdam, 1983
8.
9.
B.K. Biswas, "Monte C a r l o S i m u l a t i o n o f Wave P r o p a g a t i o n i n P o l y c r y s t a l l i n e S o l i d s " , M.Eng. T h e s i s , Department o f Mechanical E n g i n e e r i n g , M c G i l l U n i v e r s i t y , 1983 E.Wong,
" S t o c h a s t i c Processes i n I n f o r m a t i o n and Dynamical Systems", McGraw-
H i l l , New York, 1971
10. P.H. B u t z e r and H.Berens, "Semi-groups o f Operators and A p p r o x i m a t i o n " , S p r i nger V e r l a g , New York, 1967 11. P.J. Chen, " S e l e c t e d Topics i n Wave P r o p a g a t i o n " , Noordhof, Leyden, 1976
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Ekevier Science Publishers B.V. (North-Holland), 1984
279
DO LIOUVILLE'S SOLUTIONS POSSESS TOPOLOGICAL CHARGE?
George L e i b b r a n d t Department of M a t h e m a t i c s & S t a t i s t i c s U n i v e r s i t y o f Guelph Guelph, O n t a r i o Canada
W e employ t h e method of B:cklund transformations t o generate, i n t h r e e s p a t i a l dimensions, exact a l b e i t c o p l a n a r s o l u t i G n s of L i o u v i l l e ' s e q u a t i o n V2a = e x p a. I t i s f o u n d t h a t t h e t h r e e - w a v e i n t e r a c t i o n f u n c t i o n a 3 and t h e five-wave i n t e r a c t i o n f u n c t i o n a5 a r e c h a r a c t e r i z e d by r i n g e n t i t i e s which a p p e a r t o be c o n s e r v e d . The e v i d e n c e i n f a v o u r o f some t y p e of c o n s e r v e d q u a n t i t y i s s u p p o r t e d by t h e r e s u l t s from d e t a i l e d n u m e r i c a l work. Several three-dimensional graphs a r e shown.
1.
INTRODUCTION
The h i s t o r y of L i o u v i l l e ' s e q u a t i o n i n two s p a c e d i m e n s i o n s and no t i m e d i m e n s i o n ( 2 + 0 D) ,
g o e s back w e l l o v e r a c e n t u r y and i s a s r i c h and f a s c i n a t i n g a s t h a t of any r e s p e c t a b l e n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n . J. Liouv i l l e f i n a l l y s o l v e d i t i n 1853 u s i n g complex v a r i a b l e t h e o r y ( L i o u v i l l e ( 1 8 5 3 ) ) . H e had worked w i t h e q . ( l . l ) i n t h e c o n t e x t o f hydrodynamics t o a n a l y z e t h e v o r t e x m o t i o n of an i n c o m p r e s s i b l e f l u i d , e x p ( a $1 b e i n g o f t e n r e f e r r e d t o a s t h e " v o r t i c i t y " . The s c a l a r s t r e a m f u n c t i o n @ ( x , y ) must s a t i s f y (1.1) i n a c e r t a i n domain D of t h e XY-plane; f o r e x a m p l e , ji -+ - m a s ( x , y ) + a D . Once $ ( x , y ) is known, o t h e r q u a n t i t i e s of p h y s i c a l i n t e r e s t may b e d e d u c e d , s u c h a s the velocity
Many renowned mathematicians and physicists have studied ea. fl.1) over the years. F o r i n s t a n c e , i n 1 9 1 5 t h e B r i t i s h c o s m o l o g i s t Walker ( 1 9 1 5 ) employed i t t o i n v e s t i g a t e t h e d i s t r i b u t i o n of i n t e r s t e l l a r m a t e r i a l i n t h e c o n t e x t o f what was t h e n c a l l e d t h e " n e b u l a r t h e o r y " . S h o r t l y t h e r e a f t e r , von Laue ( 1 9 1 8 ) i n Germany and R i c h a r d s o n (1921) i n England made s i g n i f i c a n t c o n t r i b u t i o n s t o t h e t h e o r y o f e l e c t r o s t a t i c s when t h e y a p p l i e d e q . ( l . l ) i n t h e i r work on t h e d i s t r i b u t i o n of e l e c t r i c c h a r g e a r o u n d a g l o w i n g w i r e . L i o u v i l l e ' s equation has a l s o p l a y e d a k e y r o l e i n s t u d i e s o f i s o t h e r m a l g a s s p h e r e s (Davis
G. L eibbrandt
280
(1962)). Among the eminent mathematicians investigating (1.1) were Picard (1893), Poincar6 (1898), Lichtenstein (1915) and Bieberbach (1916). The advent of soliton physics in the 1970’s contributed also to a revival of Liouville’s equation, especially among those physicists interested in dual resonance models (Neveu (1982)) and monopole The two theories (Olive (1980), Polyakov (19Sl), Mansfield (1982)) principal objectives were (1) to find exact solutions of eq.(l.l) in more than two spatial dimensions and (2) to formulate eq.(l.l) and its solutions in the framework of quantum theory.
.
Our aim here is to examine the equa ion of Liouville in three spatial dimensions, 2 (ax +
:a +
2 aZ)+
=
k exp (a q
and to discuss a number of multiple solutions which can be generated from a new Bscklund-like transforma ion. In particular, we would like to know whether any of these multiple solutions carry topological charge.
2.
LIOUVILLE’S EQUATION IN 3
+
0 DIMENSIONS
The purpose of this section is to review briefly our method of treating Liouville’s equation in three spatial dimensions (Leibbrandt et _ al. (1982)), _
$2x
exp
=
x,
domain D,
(2.1)
where we have chosen, for convenience, k = a = 1 in eq.(1.3) and The appropriate Bzcklund transformation for eq. $2 E :a + :a + 2;. (2.1) reads K(if3
-
a) = f i exp
1 (a
+ if311 exp (i
0 S)
,
(2.2)
with differential operators K
=
S =
~ +i(aa + u a ) , i z J q , l y 3 2 u1 exp (-i X u2), I = ( 1o 0 1): I
X
a l l a2, a3 are Pauli matrices, while 0 , X are Bscklund parameters satisfying 0 5 8 2 2n, 0 < X < 2 ~ . The scalar function a ( x , y , z ) obeys Liouville’s equation, +2
V a = exp
a,
domain D,
(2.3)
281
Do Liouville's Solutions Possess Topological Charge?
w h e r e a s f3 ( x , y , z ) must s o l v e L a p l a c e ' s e q u a t i o n '2 V B = 0 , domain B . Moreover, t h e f u n c t i o n s a and B h a v e t o s a t i s f y t h e f o l l o w i n g i n t e g r a b i l i t y c o n d i t i o n s ( L e i b b r a n d t ( 1 9 8 0 ) ):
a
=
a
XY
a
XZ
=
a
=
a
c1
YZ
YX ZX
ZY
- Byx
Rxy
x x
exp a ,
= 2-l
s i n 28 c o s
= 2-1
s i n 28 s i n
= 2-l
s i n 2 8 s i n 2~ e x p a ,
= 0,
Bxz - B z x
= 0 ,
exp a,
-
Byz
=
0.
ZY
I t i s c l e a r t h a t t h e s o l u t i o n o f ( 2 . 3 ) h i n g e s d e c i s i v e l y on o u r a b i l i t y t o f i n d f i r s t e x a c t s o l u t i o n s of L a p l a c e ' s e q u a t i o n ( 2 . 4 ) . W e a l s o e m p h a s i z e t h a t ( 2 . 2 ) i s n o t a Bscklund t r a n s f o r m a t i o n ( B k k l u n d ( 1 8 7 6 , 1 8 8 0 , 1 8 8 2 ) ) of t h e t r a d i t i o n a l t y p e , s i n c e i t i s To g e t solutions n o t s u f f i c i e n t l x p o w e r f u l o n i t s own t o s o l v e ( 2 . 1 ) . o f ( 2 . 1 ) , t h e B a c k l u n d t r a n s f o r m a t i o n ( 2 . 2 ) must be augmented, i n g e n e r a l , by a s e t of c o n s t r a i n t e q u a t i o n s a s d e m o n s t r a t e d b e l o w . I t i s c o n v e n i e n t t o r e p l a c e t h e complex m a t r i x e q u a t i o n ( 2 . 2 ) by t h e f o l l o w i n g s y s t e m o f two r e a l e q u a t i o n s : I
ax
a
+
P B = JT IS s i n
e
s i n (8/2)
I cos
e
c o s ( @ / 2 )1 e x p ( a / 2 ) , (2.6a)
I
ax
6
-
P a = 43- c s s i n
e
c o s ( ~ / 2 )+ I c o s
e
s i n ( B / ~ ) Ie x p ( a / 2 ) , (2.6b)
-
CT 3 + a 3 a z . The s i m p l e s t n o n t r i v i a l s o l u t i o n o f ( 2 . 3 ) i s l Y o b t a i n e d by c h o o s i n g @ E @ = 0 i n e q . ( 2 . 6 ) and s o l v i n g f o r a. Note
where P E
0
t h a t Bo = 0
a s o l u t i o n of L a p l a c e ' s e q u a t i o n ( 2 . 4 ) . = Rn ( 2 / T i 2 ) ,
a 1( i )
T~ = x cos 0
i
+
sin
Then
.
i = 1,. .,5,
ei
(y cos Xi
+
z s i n 1.) + b,
with
+2
v
B o = 0.
H e r e b is a n a r b i t r a r y , f i n i t e c o n s t a n t . N o t i c e t h a t t h e r e i s no Since we intend t o c o n s t r a i n t e q u a t i o n l i n k i n g t h e 8 ' s and A's. c o n s t r u c t a five-wave i n t e r a c t i o n f u n c t i o n a s , w e r e q u i r e i n t h e 1
s p i r i t of Backlund-Bianchi
(Bianchi (1922) )
which h a v e b e e n l a b e l l e d a1 ( i )
, i
= 1,2,.
,
f i v e a1 s o l u t i o n s
.. , S ,
accordingly.
282
G. Leibbrandt
I t i s e v i d e n t f r o m F i g u r e 1, where t a n h
i n t h e domain D1 =
c ( x , v ) 1-45
5
x
5
( a (2)/4) 1
1 5 , -55
y
5
h a s been p l o t t e d 2
51,
t h a t a1 i s
c h a r a c t e r i z e d by a l i n e s i n g u l a r i t y . I n F i g u r e 1, 4 i s t h e v i e w i n g a n g l e , w h i l e y i s t h e a z i m u t h o r a n g l e o f r o t a t i o n i n t h e XY-plane. Next w e employ t h e p r i n c i p l e o f n o n l i n e a r s u p e r p o s i t i o n t o g e n e r a t e 8, from two uI s o l u t i o n s . The r e s u l t i s
where
and '2 82
3.
(')
= 0,
THE THREE-WAVE
domain B 2 .
INTERACTION FUNCTION a 3 ( x , y , z )
A d e c i s i v e a d v a n t a g e of t h e B k k l u n d method i s t h a t i t p e r m i t s t h e
c o n s t r u c t i o n o f o t h e r , more c o m p l i c a t e d s o l u t i o n s o f t h e u n d e r l y i n g equation.
E y combining,
o b t a i n t h e three-wave (1) -a
t a n h ( "3
4
a1(2) and a1 ( 3 ) , w e
f o r example, al(’),
f u n c t i o n a3 i n t h e form (2) = R~~
tan (
4
+2
9 a 3 = e x p u 3 , domain D 3 '
where
with
one c o n s t r a i n t 1
+
2 L12
equation : 2
L23 L 3 1 = L 1 2
2 +
L23
+
2 L13 '
(3.3)
-
e cos 8 . + s i n e s i n e cos (Xi Xj), i , j = 1,2,3. i 3 i j i s c l e a r from e q . ( 3 . 1 ) t h a t t h e L i o u v i l l e s o l u t i o n a 3 deDends
L~~ = c o s
i n t i m a t e l y on B 2 ( l )
It
and % 2 ( 2 ) from L a p l a c e ' s e q u a t i o n .
The c o n s t r a i n t e q u a t i o n ( 3 . 3 ) p l a y s a d u a l r o l e . F i r s t , it e n s u r e s t h a t t h e f u n c t i o n a 3 i n (3.1) i s a s o l u t i o n of ( 3 . 2 ) . T h i s i s a crucial point: unless t h e parameters 0,A are r e s t r i c t e d according t o ( 3 . 3 ) , t h e function a3 w i l l n o t s a t i s f y L i o u v i l l e ' s equation. Second, t h e c o n s t r a i n t (3.3) f o r c e s t h e o r i g i n a l l y t h r e e - d i m e n s i o n a l Accordingly, a l l m u l t i p l e s o l u t i o n s f u n c t i o n a 3 t o become c o p l a n a r . d e r i v e d i n t h i s manner a r e c o p l a n a r s o l u t i o n s .
Do Liouville's Solutions Possess Topological Charge?
in D1.
Fig.1.
The function tanh(w,(*)/4)
Fig.2.
A p l o t of the function tanh w3(l) in the domain D1, for @ = 8 0 ° and y = 1 3 5 O .
283
284
G. L eibbrandt
The g e n e r a l p r o c e d u r e f o r f i n d i n g m u l t i p l e ci s o l u t i o n s i s i l l u s t r a t e d i n T a b l e I . Note t h a t t h e d e r i v a t i o n o f t h e f i r s t L i o u v i l l e s o l u t i o n al depends on o u r a b i l i t y t o g u e s s a s i m p l e s o l u t i o n of L a p l a c e ‘ s e q u a t i o n , namely
Bo
= 0.
T a b l e I . SCHEMATIC DIAGRAM
+2
Laplace: V B = 0
+2
V
Liouville:
ci =
exp N
Bo = 0
N
2
1
ci3
= Rn(2/T )
=
al(2) + 4 t a n h - ’ ( ,
a5 = a 3 ( 2 )
The r e s u l t f o r
ci3,
+
4 tanh-’(.
.. )
.
.)
shown i n F i g u r e 2 , i s q u i t e s u r p r i s i n g ; t h i s
s o l u t i o n i s o n l y c h a r a c t e r i z e d by a r i n g s i n g u l a r i t y - t h e r e i s no e v i d e n c e of a n y l i n e s i n g u l a r i t i e s . Something u n e x p e c t e d h a s and cil ( 3 ) happened : t h e t h r e e l i n e s i n g u l a r i t i e s from c1 (1), h a v e c o m p l e t e l y d i s a p p e a r e d and a s i m p l e r i n g s t r u c t u r e h a s emerged. The l o g i c a l q u e s t i o n i s : what happens t o t h e r i n g s t r u c t u r e of t h e three-wave f u n c t i o n a 3 u n d e r a f u r t h e r n o n l i n e a r s u p e r p o s i t i o n ? To answer t h i s q u e s t i o n , w e s h a l l combine t h e t h r e e a 3 s o l u t i o n s a (1) 3 ’ ( 2 ) , a 3 ( 3 ) n o n l i n e a r l y t o form t h e five-wave i n t e r a c t i o n f u n c t i o n
a3 a
5‘
4.
The r e s u l t s a r e d i s c u s s e d i n t h e n e x t s e c t i o n .
THE FIVE-WAVE INTERACTION FUNCTION a 5 ( X , y , Z )
The c o n s t r u c t i o n of t h e f i v e - w a v e f u n c t i o n a5 i s d e s c r i b e d i n d e t a i l i n Leibbrandt e t a l .
(1982).
The answer i s
285
Do Liouville's Solutions Possess Topological Charge?
Fig.3.
The f u n c t i o n B 4 ( ' )
y
Fig.4.
=
135’
in D
i n t h e form sin B 4
(I) f o r
@ = 80'
I’
The f u n c t i o n t a n h a 5 i n t h e domain D 5 for $I = 80'
y = 900.
and
and
286
G. Leibbrandt
where
and (4.3)
(4.4) The g r a p h o f R 4 ( l )
i s shown i n F i g u r e 3 .
constraint equations f o r Lij,
+
S i n c e t h e r e a r e now
ten
namely 2
2
L12 L23 L31 - L 1 2
iL23
2 +
L31’
2 2 2 1 + 2 L12 L 2 4 Lql = L12 + L 2 4 + L 4 1 '
(4.5)
................................... +
2
L 1 4 L45 L51 - L 1 4
t h e t a s k of s o l v i n g f o r O i l
Xi
2 -F
'45
+
2 L51'
becomes much m o r e o n e r o u s a n d w e
r e f e r f o r d e t a i l s t o Leibbrandt e t a l .
(1982). ( W e f o u n d it c o n v e n i e n t
The b e h a v i o u r o f a5 i s shown i n F i g u r e s 4 - 6 .
t o p l o t t a n h u5 r a t h e r t h a n a5.) s e e m p a r t i c u l a r l y worth noting:
T h e r e a r e t w o f e a t u r e s o f a5 w h i c h
( i ) a l t h o u g h a 5 d e r i v e s f r o m f i v e ul’s t h e r e a r e n o l i n e singularities: ( i i ) t h e f u n c t i o n t a n h a5 i s d i s t i n c t l y c h a r a c t e r i z e d by t h r e e r i n g s i n g u l a r i t i e s , which c o r r e s p o n d r e s p e c t i v e l y t o t h e r i n g s i n tanh a 3 ( ' ) , t a n h a 3 ( 2 ) and t a n h a3 ( 3 )
.
I t ' w o u l d a p p e a r , t h e r e f o r e , t h a t t h e i n d i v i d u a l r i n g s have been " p r e s e r v e d " i n t h e s u p e r p o s i t i o n p r o c e s s and t h e q u e s t i o n i s : w h a t d o t h e s e r i n g s r e p r e s e n t ? Is t h e r e s o m e t h i n g e q u i v a l e n t t o t o p o l o g i c a l c h a r g e t h a t c o u l d be a t t r i b u t e d t o each r i n g ? To g a i n some i n s i g h t i n t o t h i s p r o b l e m w e r e c a l l t h a t a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e of t o p o l o g i c a l charge i s a set of d e g e n e r a t e vacuum s t a t e s C P a t a n i e t a l . ( 1 9 7 6 ) l . By way o f i l l u s t r a t i o n c o n s i d e r t h e sine-Gordon e q u a t i o n i n o n e s p a c e and one t i m e dimension,
287
Do Liouville's Solutions Possess Topological Charge?
w i t h b o u n d a r y c o n d i t i o n 1 - c o s x = 0 , s o t h a t x(-", t ) = 0 and x(+m, t ) = 271. The sine-Gordon L a g r a n g i a n l e a d s t o a n i n t r i n s i c 2 e n e r g y E = s i n ( x / 2 ) which i m p l i e s t h e p r e s e n c e o f i n f i n i t e l y manydeg e n e r a t e vacua :
x
= 0,
EVaC(x)
=
i 2 n ~ r ,n = 0 , 1 , 2
,...
x
= $'
L e t u s make a f i e l d t r a n s f o r m a t i o n c o s -f
t h a t $ = ($
1
, 4
cal current , ’ j j'
2
) with
52 =
1.
(4.7)
and s i n
x
= $L,
SO
Accordingly, t h e r e e x i s t s a topologi-
d e f i n e d by
= ( 2 ~ r ) - l E'"
E~~
4a
3" Q b ,
a , b = 1 , 2 ; p , v = 0,l
and a t o p o l o g i c a l c h a r g e Q :
I
00
Q =
jodx
-m m
-m
m
-00
.*.
Q =
+l.
-1 Hence t h e o n e - s o l i t o n s o l u t i o n x = 4 t a n ( e x p x), d e f i n e d f o r t = 0 , is s a i d t o possess t h e topological charge ( o r topological i n v a r i a n t )
-1
Similarly, the anti-soliton solution x = 4 tan ( e x p -x) h a s t o p o l o g i c a l c h a r g e Q = -1, w h i l e t h e t w o - s o l i t o n s o l u t i o n c a r r i e s Q = 2. Q = +l.
R e t u r n i n g t o e q . ( 2 . 3 ) , w e see t h a t t h e L i o u v i l l e s y s t e m f o r cil h a s no d e g e n e r a t e v a c u a , s o t h a t t h e r e i s no t o p o l o g i c a l c h a r g e c o n n e c t e d w i t h cil. T h i s c o n c l u s i o n seems p l a u s i b l e : w e h a v e a l r e a d y o b s e r v e d t h a t t h e r e i s " n o t h i n g permanent" a b o u t t h e l i n e s i n g u l a r i t i e s associated with the
The s i t u a t i o n i s l e s s clear f o r t h e
t h r e e - w a v e i n t e r a c t i o n f u n c t i o n a 3 whose r i n g s t r u c t u r e r e m a i n s s t a b l e under s u b s e q u e n t s u p e r p o s i t i o n w i t h o t h e r a 3 r i n g s .
Either
t h e r i n g s i n t a n h c i 3 and t a n h n 5 c o r r e s p o n d t o c e r t a i n ( n o n t o p o l o g i c a l ) i n v a r i a n t s of t h e u n d e r l y i n g t h e o r y o r t h e y formally, equivalent t o topological charges. In would f i r s t h a v e t o d e m o n s t r a t e t h e e x i s t e n c e o f W e are working on t h i s problem, b u t a t current. n o t c l e a r which of t h e s e c o n c e p t s i s c o r r e c t .
are, a t least t h e l a t t e r c a s e one a topological t h i s p o i n t it i s
G. Leibbrandt
I
Fig.5.
Fig.6.
The f u n c t i o n t a n h a5 in D5 f o r $I = 70°
A " c u t v i e w " of t a n h ci5 i n D -25
2 y
5
6
and y = 1 3 5 O .
= { ( x , y ) 1-30
3 5 1 , y = 135O, and @ = 60'.
'
5
x
5
30,
Do Liouville's Solutions Possess Topological Charge?
5.
289
ACCURACY AND ACCURACY C H E C K S
T h e s o l u t i o n s f o r a 1,
ci3
and
ci5
i n eqs. (2.71,
( 3 . 1 ) and (4.1),
r e s p e c t i v e l y , were o b t a i n e d w i t h t h e a i d o f a B:cklund-like Thus t h e a i , f o r m a t i o n , i . e . by p u r e l y t h e o r e t i c a l means.
i
trans= 1,3,5,
a r e e x a c t s o l u t i o n s o f +2 B ci = exp a . The r e a d e r mav, t h e r e f o r e , r i g h t f u l l v wonder a b o u t t h e p u r p o s e o f u s i n g a computer f o r t h i s p r o j e c t and a b o u t o u r i n s i s t e n c e on e x t r e m e l y h i g h a c c u r a c y . The answer is s i m p l e : t h e n e s t e d f u n c t i o n a l forms of a 3 and a 5 a r e s o complicated t h a t it i s d i f f i c u l t , without accurate p l o t t i n g routines, t o g e t an i d e a of t h e i r s t r u c t u r e a n d , e s p e c i a l l y , of t h e l o c a t i o n o f their singularities. But b e f o r e a 3 l a 5 c a n b e p l o t t e d w e have t o solve t h e appropriate c o n s t r a i n t equations numerically.
Moreover, i t
is essential t h a t the plotted functions from t h e s y s t e m
,
B:cklund
ci
3
transformation
plot
and a p l o t 5
derived
(2.2)
i
c o n s t r a i n t equations
(4.5)
a r e r e a s o n a b l y p r e c i s e r e p r e s e n t a t i o n s o f t h e e x a c t s o l u t i o n s a 3 and i.e. t h a t
ci5,
;2
@ot
plot ). exp ( a
I f t h e a c c u r a c y f o r a l ( i ) i s less t h a n f o r example, t h e s m a l l e s t of t h e t h r e e r i n g s of t a n h a5 b a r e l y shows up - i f i t shows up a t a l l ! are:
(i)
To summarize, t h e main s t e p s i n t h e n u m e r i c a l program
t o solve the c o n s t r a i n t eqs. (4.5) f o r t h e L . . t o high accuracy; 17
( i i ) t o u s e t h e L . . from ( i )t o compute O i l X i t o h i g h a c c u r a c y ; 17 ( i i i ) t o s u b s t i t u t e O i l X . i n t o c i l , a 3 , a5 and t h u s v e r i f y t h a t +2 V am = exp am,
m = 1,3,5,
and
'2
v
k = 2,4.
Bk = 0,
For instance, using quadruple precision throughout w e obtained t h e following 0 , X v a l u e s (only the f i r s t twenty s i g n i f i c a n t f i g u r e s are listed) :
X1
= 0.066
391
602
978
664
308
98
...,
.................................................. X 5 = 0.725
240
492
717
253
436
05
...,
290
G. Leibbrandt
while
el
= 65.740
776
764
2-12
056
193
DO,
...................................................... e5
= 104.683
428
073
006
160
110
62O
. ..
T a b l e I1 shows how t h e a c c u r a c y d e c r e a s e s w i t h t h e number of conI t is n o t d i f s t r a i n t s and a s t h e m u l t i p l i c i t y of c1 g e t s l a r g e r .
m
f i c u l t , though e x p e n s i v e , t o c o n v i n c e o n e s e l f t h a t t h e a c c u r a c y of ci c a n b e improved s y s t e m a t i c a l l y . 5-
TABLE 11.
DEPENDENCE OF ACCURACY ON MULTIPLICITY AND NUMBER OF
CONSTRAINTS Function
Number of
Number of constraints
e i t xi 2
cil
0
84
8
10
c15
o
10-20
1
6
a3
G2a - e x p c1 = is satisfied
0
4
a2
Accuracy t o which
4
10-10
10
10-6
SUMMARY
W e have s t u d i e d t h e L i o u v i l l e e q u a t i o n , ?2a = exp a , i n t h r e e s p a t i a l dimensions. Using t h e t r a n s f o r m a t i o n
cIax +
i (u 3 1 Y
+
u3aZ) I ( i 6 - a)
=
45 e x p ((a +
iB)/2)
exp ( i e S ) ,
w e w e r e a b l e t o g e n e r a t e s e v e r a l m u l t i p l e s o l u t i o n s which w e r e l a b e l l e d a3 and a 5 . The t h r e e - w a v e i n t e r a c t i o n f u n c t i o n a 3 t u r n s o u t t o p o s s e s s a s i m p l e r i n g s i n g u l a r i t y when p l o t t e d a s t a n h a 3 , w h i l e t h e five-wave f u n c t i o n
cis,
c o n s t r u c t e d from t h r e e a 3 s o l u t i o n s , was
shown t o c o n t a i n t h r e e d i s t i n c t r i n g s (when p l o t t e d i n t h e form tanh a 5 ) . Since L i o u v i l l e ' s equation i s n o t expected t o give rise t o s o l i t o n - l i k e s o l u t i o n s , it c a m e as a s u r p r i s e t o d i s c o v e r t h e e x i s t e n c e of s t a b l e e n t i t i e s o r r i n g s , a s we c a l l e d them. These r i n g s s e e m t o be conserved i n t h e n o n l i n e a r s u p e r p o s i t i o n p r o c e s s , j u s t a s t h e s h a p e s of two c o l l i d i n g sine-Gordon s o l i t o n s a r e p r e s e r v e d asymptotically (Leibbrandt e t a l . (1980)). I t i s n o t c l e a r a t t h i s
29 1
Do Liouville's Solutions Possess Topological Charge?
p o i n t j u s t what t h e n a t u r e of t h e s e r i n g s i s , whether they merely r e p r e s e n t k i n e m a t i c a l i n v a r i a n t s of t h e t h e o r y o r whether t h e y have It i s worth mentioning i n t h i s connection t h a t t h e a d e e p e r meaning. s i n e - G o r d o n e q u a t i o n and L i o u v i l l e e q u a t i o n a r e g o v e r n e d , i n 3 + 0 d i m e n s i o n s , by i d e n t i c a l c o n s t r a i n t e q u a t i o n s .
ACKNOWLEDGMENT
T h i s r e s e a r c h w a s s u p p o r t e d i n p a r t by t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada u n d e r G r a n t N o . A8063.
REFERENCES Bicklund, A.V., 297-320.
Uber Flschentransformationen, Math. Ann.
2
(1876)
B i c k l u n d , A.V. , Zur T h e o r i e d e r P a r t i e l l e n Dif f e r e n t i a l g l e i c h u n g e n e r s t e r Ordnung, Math. Ann. XVII ( 1 8 8 0 ) 285-328. B i c k l u n d , A . V . , Zur T h e o r i e d e r Fl:chentransformationen, XIX ( 1 8 8 2 ) 387-422. B i e b e r b a c h , L . , Au = e (1916) 1 7 3 .
U
Math. Ann.
u n d d i e a u t o m o r p h e n F u n k t i o n e n , Math. Ann.
B i a n c h i , L . , L e z i o n i d i Geometria D i f f e r e n t i a l e , V o l . S p o e r r i , P i s a , 1 9 2 2 ) 743-747.
71
I (Enrico
Davis, H . T . , I n t r o d u c t i o n t o Nonlinear D i f f e r e n t i a l and I n t e g r a l E q u a t i o n s ( D o v e r , N e w Y o r k , 1 9 6 2 ) 377-380. L a u e , M . v o n , J a h r b u c h d e r R a d i o a k t i v i t s t und E l e k t r o n i k 205.
(1918)
L e i b b r a n d t , G . , R . Morf a n d S . S . Wang, S o l u t i o n s o f t h e s i n e - G o r d o n e q u a t i o n i n h i g h e r d i m e n s i o n s , J . Math. P h y s . ( 1 9 8 0 ) 16131624.
21
Leibbrandt, G., Nonlinear s u p e r p o s i t i o n for L i o u v i l l e ' s equation i n ( 1 9 8 0 ) 317-321. t h r e e s p a t i a l d i m e n s i o n s , L e t t . Math. P h y s .
4
.
solutions L e i b b r a n d t , G . , S S . Wang a n d N . Zamani , B:cklund-generated o f L i o u v i l l e ' s e q u a t i o n and t h e i r g r a p h i c a l r e p r e s e n t a t i o n s i n t h r e e s p a t i a l d i m e n s i o n s , J . Math. P h y s . 25 ( 1 9 8 2 ) 1566-1572. L i c h t e n s t e i n , L . , I n t e g r a t i o n d e r D i f f e r e n t i a l g l e i c h u n g Au = e A c t a M a t h e m a t i k a 40 ( 1 9 1 5 ) 1. g e s c h l o s s e n e n Fl:chen,
U
L i o u v i l l e , J . , Sur l ' g q u a t i o n aux d i f f g r e n c e s p a r t i e l l e s n
d'log dudv
i
4 2a
= 0 , Journ.
d e Math.
5 (1)
( 1 8 5 3 ) 71-72.
M a n s f i e l d , P . , L i g h t - c o n e q u a n t i z a t i o n o f t h e L i o u v i l l e a n d Toda f i e l d t h e o r i e s , Cambridge U n i v e r s i t y p r e p r i n t DAMTP 8 2 / 2 3 (Cambridge, 1 9 8 2 ) .
auf
G. Leibbrandt
292 Neveu, A . ,
R e v i v a l o f t h e s t r i n g m o d e l , i n : P e t i a u , P . a n d M.
Porneuf
( e d s . 1 , 21St I n t e r n a t i o n a l C o n f e r e n c e on High E n e r g y P h y s i c s (Les E d i t i o n s d e Physique, P a r i s , 1 9 8 2 ) . C l a s s i c a l s o l u t i o n s i n g a u g e t h e o r i e s - s p h e r i c a l l y symLax P a i r s a n d Toda l a t t i c e s " , I m p e r i a l C o l l e g e p r e p r i n t ICTP/80/81 ( I m p e r i a l C o l l e g e , London, 1 9 8 1 ) .
Olive, D.,
metric monopoles
-
P a t a n i , A., M . S c h l i n d w e i n a n d Q . S h a f i , T o p o l o g i c a l c h a r g e s i n f i e l d t h e o r y , U n i v e r s i t y o f F r e i b u r q p r e p r i n t THEP76/1 ( F r e i b u r g , February 1976). U
P i c a r d , E . , D e l ' b q u a t i o n Au = e s u r une s u r f a c e d e Riemann f e r m b e , J o u r n a l d e M a t h b m a t i q u e 2 ( 4 ) ( 1 8 9 3 ) 273. U
P o i n c a r b , H . , L e s f o n c t i o n s f u c h s i e n n e s e t l ' b q u a t i o n hu = e J o u r n a l d e Mathbmatique 4 ( 5 ) ( 1 8 9 8 ) , 1 3 7 .
,
P o l y a k o v , A . M . , Quantum g e o m e t r y o f b o s o n i c s t r i n g s , P h y s . L e t t . ( 1 9 8 1 ) 207-210; Quantum g e o m e t r y of f e r m i o n i c s t r i n g s , P h y s . Lett. ( 1 9 8 1 ) 211-213. R i c h a r d s o n , O . W . , The E m i s s i o n o f E l e c t r i c i t y f r o m H o t B o d i e s man G r e e n , N e w York 1 9 2 1 ) 50-54.
(Long-
W a l k e r , G.W., Some p r o b l e m s i l l u s t r a t i n g t h e f o r m s o f n e b u l a e , P r o c . Roy. SOC. A91 - ( 1 9 1 5 ) 410-420.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
293
DISLOCATIONS IN THE DYNAMIC ANALYSIS OF CRACKS Louis 11. Brock Denartment of Enoineerinq Mechanics University of Kentucky Lexington, Kentucky U.S.A.
I n q u a s i - s t a t i c f r a c t u r e a n a l y s i s , cracks a r e often represented mathematically as dislocation d i s t r i b u t i o n s . This approach has t h e advantages t h a t d i s l o c a t i o n problems a r e generally e a s i e r t o solve. and enforcement of t h e zero t r a c t i o n crack surface conditions often leads t o standard s i n g u l a r integral equations of the Cauchy type. Despite t h e complications of the time variable and t h e wave nature of t h e s o l u t i o n , including boundary r e f l e c t i o n s , t h i s basic approach i s extended here t o an example of dynamic crack propaqation. INTRODUCTION As a physical e n t i t y , d i s l o c a t i o n s a r e c r y s t a l l a t t i c e defects (Read ( 1 9 5 3 ) ) , while in continuum theory, they a r e represented as d i s c o n t i n u i t i e s in t h e material displacement f i e l d (Kroner (1962)). I n continuum f r a c t u r e theory, cracks a r e s p e c i f i e d mathematically a s surfaces of zero t r a c t i o n , b u t t h i s requirement gives r i s e t o disnlacement d i s c o n t i n u i t i e s which a r e taken t o renresent crack opening. This s i m i l a r i t y has been exnloited in q u a s i - s t a t i c analyses of f r a c t u r e : The crack i s replaced mathematically by an a r b i t r a r y d i s l o c a t i o n d i s t r i b u t i o n and t h e zero t r a c t i o n conditions on t h e crack surface a r e relaxed. The resultincl dislocat i o n problem does not involve t h e mixed boundary conditions a r i s i n g on t h e crack plane and can, t h e r e f o r e , often be e a s i l y solved by standard q u a s i - s t a t i c solution methods, e.g. Dundurs (1971). The zero t r a c t i o n conditions a r e then enforced, r e s u l t i n g i n what a r e , in many i n s t a n c e s , standard s i n q u l a r integral equations of t h e Cauchy tyoe (Pluskhelishvili (1975)) f o r the d i s l o c a t i o n d i s t r i b u t i o n function. In dynamic crack pronaqation a n a l y s i s , however, two d i f f i c u l t i e s a r i s e in applying t h i s approach: First, the wave nature of t h e d i s l o c a t i o n solution f i e l d s can, e s p e c i a l l y i f wave r e f l e c t i o n s from boundaries a r e involved, be unwieldy combinat i o n s o f terms, some o f which cannot be written as e x p l i c i t l y a s o t h e r s . Moreover, i t would appear t h a t the a r b i t r a r y dislocation d i s t r i b u t i o n must be over time and s p a t i a l variables, so t h a t t h e r e s u l t i n g integral equations involve multinle integrals. In t h i s a r t i c l e , a crack nroblem examnle i s examined which shows t h a t both d i f f i c u l t i e s can often be overcome. CRACK PROBLEH
Consider an i n i t i a l l y undisturbed i s o t r o p i c , homogeneous, l i n e a r l y e l a s t i c halfplane defined by the Cartesian coordinate y>O. As seen in Figure 1 , a t s=O ( s i s t h e time mulitnlied by t h e half-nlane P-wave speed) a crack appears a t ( x , y ) = O and extends alonq the n o s i t i v e y-axis with a non-dimensionalized s u b - c r i t i c a l constant
294
L.M. Brock
speed c. (Here a l l speeds a r e non-dimensionalized w i t h r e s p e c t t o t h e P-wave The crack i s d r i v e n by equal b u t o p p o s i t e l y - d i r e c t e d p o i n t f o r c e s P speed t r a v e i n g along t h e crack surfaces a t t h e constant non-dimensionalized speed ecc.
Figure 1 Problem Geometry By symmetry, t h i s problem can be described by t h e f o l l o w i n g equations on t h e quarter-plane (x,y)>O f o r s>O:
v 2-u+(m2 - ~ ) V ( V .-U ) = a
Y
uX
= -P&(y-es),u
XY
= u
XY
2 m u
-
,
~
(1 1
~
= O(x>O,y=O)
= o(x=O,y
u XY
(2)
= ux = 0 (x=O ,y>cs
)
(3)
i s t h e xy-plane displacement w i t h components where y-0 f o r SLO and !(x,y,s) (ux,u ) i n t h e ( x , y ) - d i r e c t i o n s , 02 and c7 a r e t h e xy-plane Laplacian and g r a d i e n t opera!ors, t h e comma s u b s c r i p t denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e v a r i a b l e f o l l o w i n g , ( - ) denotes t h e i n n e r product and 6 i s t h e D i r x f u n c t i o n . The stresses a r e d e f i n e d by t h e c o n s t i t u t i v e r e l a t i o n s 1 ux = ~
m2 u x,x
+
(m2 - Z ) U ~ ,1~
=
rn2 u
+ (m2 - Z ) U ~ , ~
Y3Y
(4
Here p and l/m
Dislocations in the Dynamic Analysk of Cracks
1 1 U =tl(t-k) 2 2
= 0, u
u
XY
(x=O,y>O)
295
(6)
where H i s t h e H e a v i s i d e f u n c t i o n and t = s / y , k = l / c . The s o l u t i o n t o t h i s problem can r e a d i l y be o b t a i n e d a n a l y t i c a l l y , e . ? . Brock (1979, 1983), and, i n p a r t i c u l a r , t h e normal s t r e s s f o r x=O, y>O i s g i v e n by n
2
;m yax
A1
K2
= - B
A
1
B1
c1
(m+ t+k + B+k)H(t-1)
=
2a’
=
1
-A
2 2 4a Bt+M2, a 4a2Rt-r1
=
Am Bm ‘m t t k + -)H(t-m) a+k + (-t - k + -
(7)
2 2 2 / ( t -11, a = J(t -m +I)
The A1 and A,-terms r e p r e s e n t P and S-waves r a d i a t i n v f r o m t h e d i s l o c a t i o n edge, t h e B1 and Cm-terms r e p r e s e n t PP and SP-waves due t o r e f l e c t i o n s f r o m t h e s u r f a c e y=O w h i l e t h e Bm and C1-terms r e p r e s e n t SS and PS-waves due t o r e f l e c t i o n s f r o m y=o.
A comment on t h e f o r m o f ( 7 ) i s i n o r d e r : As a l r e a d y i n d i c a t e d , each t e r m r e p r e s e n t s one o f t h e s i x wave s i g n a l s p o s s i b l e a l o n f l x=O and each i s a n a l y t i c . T h i s i s i n c o n t r a s t t o s i m i l a r e x p r e s s i o n s i n o t h e r sources, e.g. Burgers and Freund (1981), w h i c h do n o t a l l o w s p e c i f i c s i g n a l i d e n t i f i c a t i o n and i n c l u d e terms which a r e w r i t t e n as c o m p l i c a t e d i n t e q r a l s . T h e r e f o r e , t h e p r e s e n t r e s u l t s show t h a t t h e u n w i e l d i n e s s of dynamic d i s l o c a t i o n s o l u t i o n s mentioned above can be p a r t i a l l y overcome s i m p l y by decomposing s o l u t i o n s i n t o s p e c i f i c wave terms. Equations ( 6 ) and ( 7 ) show t h a t t h e displacement f i e l d f o r t h i s d i s l o c a t i o n problem i s a l s o homogeneous o f degree 0 i n (x,y,s). Indeed, i f t h e s i n g l e d i s l o c a t i o n i s now r e p l a c e d by a c o n t i n u o u s d i s t r i b u t i o n o f d i s l o c a t i o n s +
U
=
lL F(v)dv k
t h i s degree i s preserved. T h i s o b s e r v a t i o n and comparison o f ( 1 1 ) w i t h ( 6 ) shows c l e a r l y t h a t any f i e l d v a r i a b l e f o r t h i s d i s l o c a t i o n d i s t r i b u t i o n problem can be o b t a i n e d f r o m t h e c o r r e s p o n d i n g v a r i a b l e f o r t h e f i r s t d i s l o c a t i o n p r o b l e m by p e r f o r m i n g t h e ( x ,y ,s ) - i ndependent o p e r a t i o n Jm
k
( )F(v)dv
(12)
The i n t e g r a t i o n l i m i t k shows t h a t t h e d i s l o c a t i o n s a r e d i s t r i b u t e d w i t h r e s p e c t t o a n o n - d i m e n s i o n a l i z e d i n v e r s e speed v i n terms o f an a r b i t r a r y f u n c t i o n F. S i n c e U i s analogous t o a crack-opening gap, t h e s o l u t i o n f o r t h e d i s l o c a t i o n d i s t r i b u t i o n w i l l be t h e c r a c k p r o b l e m s o l u t i o n i f F ( v ) i s chosen so t h a t c o n d i t i o n (3) i s s a t i s f i e d . SINGULAR INTEGRAL EOUATION O p e r a t i n n on (7) w i t h ( 1 2 ) and i n t r o d u c i n g t h e more c o n v e n i e n t quant t i e s z = l/t, u = l / v ,
c = l / k , G(u)/u = F ( v )
(13)
L.M. Brock
296
y i e l d s upon s u b s t i t u t i o n i n t o ( 3 ) t h e r e l a t i o n C
du + B I
A f0C
!!& u+z
du + C1
Ic 0 BU+Z
C
du + Cm
% !ddu au+z
2
= - y m PyG(y-es) (14)
1-1
f o r O
:'2afi2, 1,
2 K2, Bl = -A 2az
2
=-
2
___
a = ~ ( 1 - 2z
c1 =
/[1+(1-m 2 ) z 2 1
(15)
Before a t t a c k i n g (14) d i r e c t l y , we remove t h e Dirac f u n c t i o n by w r i t i n g
where f o r O
Af
c Go(u) 0
T 2 du = --m PyG(y-es) 1-1
2 and i t is noted t h a t 2az A i s t h e Rayleigh f u n c t i o n , which vanishes when z reaches t h e non-dimensionalized Rayleiqh wave speed. As noted a t t h e o u t s e t , c i s subc r i t i c a l and, thus, l i e s below t h i s speed, which i t s e l f l i e s below l / m . Equation (19) i s a standard s i n g u l a r i n t e g r a l equation o f t h e Cauchy t y p e w i t h s o l u t i o n G (z) = 0
1 A-P J (c-zL ) e-z
(20)
f o r O
f o r O
where no g e n e r a l i t y i s l o s t i n t h e argument assumed f o r h. I4=c(l+u)/2 and Z = c ( l + z ) / 2 then transforms (21) i n t o
1
B
cm
S u b s t i t u t i o n o f (231,
)dW = ITD
(24)
-1 f o r -1
The 1 .h.s.
Dislocations in the Dynamic Analysis of Cracks
297
of (24) can be f o r m a l l y approximated by Chebyshev-Gauss quadrature as n
W1. = cos JL 2n ( 2 i - 1 )
(26)
where t h e M . are t h e zeroes o f t h e Chebyshev polynomial o f t h e f i r s t k i n d o f o r d e r n and l i e i A t h e range -1<Wi
...,n )
(27)
where -1
With g a t hand, t h e crack opening f o r O
U = l G(U)du z
'
(28)
This f u n c t i o n i s p l o t t e d vs. z i n Figure 2 f o r v a r i o u s values o f c, where ef0.05 and m=J3. As m i g h t be expected, U grows w i t h c. The s i n g u l a r behavior occurs under t h e p o i n t f o r c e s , as i n Lamb's uroblem, see Achenbach (1973).
2.4
2 .O I .6 I .2
0.8 0.4
0.0
01
0.2 0.3 0.4 0.5 2
Figure 2 Crack Openinq
DISCUSSION The example considered here i n d i c a t e s t h a t problems i n dynamic crack propagation can be t r e a t e d by means o f d i s l o c a t i o n d i s t r i b u t i o n s o l u t i o n s , and t h a t the wave n a t u r e o f t h e problems and t h e t i m e and s p a t i a l v a r i a t i o n o f t h e d i s t r i b u t i o n need
298
L.M. Brock
n o t p r o v e t o be s t u m b l i n g b l o c k s . The f i r s t d i f f i c u l t y can be p a r t i a l l y a l l e v i a t e d by w r i t i n g s o l u t i o n s as sums o f s p e c i f i c wave s i g n a l terms. The second proves no o b s t a c l e i n cases w i t h no c h a r a c t e r i s t i c l e n g t h , s i n c e t h e a s s o c i a t e d s o l u t i o n homogeneity a l l o w s t h e d i s t r i b u t i o n t o be w i t h r e s p e c t t o a s i n g l e speed parameter. I t s h o u l d be noted, however, t h a t c a r e must be used i n making s u r e t h a t t h e deqree o f homogeneity o f t h e c r a c k and d i s l o c a t i o n problems a r e i d e n t i c a l , see f o r example t h e c o r r e c t i o n t o e a r l i e r work p r e s e n t e d by Burgers and Freund (1 981 ) . REFERENCES Read, H . T . ,
D i s l o c a t i o n s i n c r y s t a l s (McGraw H i l l , New York, 1953).
Kroner, E . , 599-606.
D i s l o c a t i o n s and continuum mechanics, Appl. Mech. Rev. 15 (1962)
Dundurs, J., E l a s t i c i n t e r a c t i o n o f d i s l o c a t i o n s w i t h i n h o m o g e n e i t i e s , i n : Mathematical Theory o f D i s l o c a t i o n s (ASME, New York, 1971). M u s k h e l i s h v i l i , N.I., Some b a s i c problems i n t h e mathematical t h e o r y o f e l a s t i c i t y ( N o o r d h o f f , Leyden, 1975). M i k l o w i t z , J., U a v e f r o n t a n a l y s i s i n t h e nonseparable elastodynamic q u a r t e r p l a n e nroblems, 1 P a r t 2: N a v e f r o n t e v e n t s i n t h e edge u n i f o r m p r e s s u r e problem, J r n l . A p p l . Mech. 49 (1982) 808-815. Brock, L.M.,
Two b a s i c wave p r o p a g a t i o n problems f o r t h e n o n - u n i f o r m m o t i o n
of displacement d i s c o n t i n u i t i e s i n a h a l f - p l a n e , I n t . J r n l . Engng. S c i 17 (1979) 1211-1223. Brock, L.M., Non-uniform edge d i s l o c a t i o n m o t i o n a l o n g an a r b i t r a r y p a t h , J r n l . Mech. Phys. S o l i d s 31 (1983) 123-132. Burgers, P. and Freund, L.B., An addendum t o t h e paper: dynamic growth o f an edge c r a c k i n a h a l f - s p a c e , I n t . J r n l . S o l i d s S t r u c t . 17 (1981) 721-727. Erdogan, F. and Gupta, G.D., On t h e n u m e r i c a l s o l u t i o n o f s i n g u l a r i n t e g r a l e q u a t i o n s , Quar. J r n l . Apol. Math. 30 (1972) 525-534. Achenbach, J.D., Mave p r o p a g a t i o n i n e l a s t i c s o l i d s ( N o r t h - H o l l a n d , Amsterdam, 1973).
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland),1984
299
A BACKLUND TRANSFORMATION FOR A NONLINEAR TELEGRAPH EQUATION Eric Varley
Brian Seymour University of British Columbia Vancouver, B C Canada
and
..
Lehigh University Bethlehem, PA U.S.A.
A general result for the transformation of a set of quasilinear dissipative partial differential equations into a similar system is used to linearize two systems of physical interest. One example yields a new Backlund transformation which linearizes a nonlinear telegraph equation.
1.
INTRODUCTION
There are very few known transformations which can be used to map nonlinear partial differential equations that arise naturally in physical problems into linear equations. Well known examples include the hodograph transformation for systems of two quasilinear nondissipative equations (see Courant and Hilbert [ 11 ) , the Hopf-Cole transformation for Burger’s equation (see Hopf [ 2 ] and Cole [31) and the inverse scattering transformation for the Korteweg-devries equation (see Miura [ 4 ] ) . For some classes of nonlinear partial differential equations it is possible to determine whether there exists a transformation which will map the equation into a linear equation (see Kumei and Bluman [ 5 ] ) . Nevertheless, very few equations have been linearized using these techniques. Here we will show, by using a general transformation on a system of first order quasi1 near partial differential equations, that the nonlinear telegraph equation
Q,,- (l+Qt
(1.1)
is transformed into the linear telegraph equation @--
tt
- @--xx +
by the Backlund transformation
@-
t
= 0
(1.2)
8.Seymourand E. Varley
300
and
Equation (1.1) arises in several contexts when considering wave phenomena in nonlinear dissipative systems.
The particular
example which motivated this study is the propagation of stress waves through a nonlinear viscoelastic material with an equation of state of the form
where ( o , X ) are stress and strain measures and
,$ ,
2.
JI0 and k are constants. THE BASIC TRANSFORMATION The transformations that are described in this paper are
obtained by an application of the following general result. If u(x,t) and v(x,t) satisfy a system of two, first order, quasilinear, partial differential equations of the form a u l x
+
blut
+ clvx + dlVt + e1
a u 2 x
+
b2ut
+
= 0
and
(2.1)
c2vx + d2vt + e2 = 0
where a l,a2,bl, etc. are functions of (u,v,x,t), and if the relations
x
=
X(U,V),
t = T(u,v)
(2.2)
represent particular solution to (2.11, then, in terms of the new independent variables
equations (2.1) can be written
301
A Backlund Transformation for a Nonlinear Telegraph Equation
a l u-x
+
-
a l = al
-
+ C l v-x +
+ i;2 u-t + C 2 v-x
a u2 x
where
L I u-t
+
+
e
+ a2v-t +
e
vI t
-
+
bl = bl
elXu,
-
a n d al, a 2 e t c . a r e f u n c t i o n s o f
1
2
=
o
=
o etc.
elTU,
(u,v,x,f).
T h i s r e s u l t is e a s i l y e s t a b l i s h e d b y w r i t i n g t h e f i r s t
d e r i v a t i v e s i n t e r m s of t h e t h r e e J a c o b i a n s
Thus,
U
X
+
= (U;
TVJ2)/J1
U
t
t
=
-
(U-
t
X J )/J1
v 2
(2.7) V
=
-
(V;
TUJ2)/J1
Vt
t
+
= (V:
XUJ2)/Jlr
a n d h e n c e t h e f i r s t of ( 2 . 1 ) becomes a (u1 x
+
T J ) v 2
+
b (u1 t
-
-
XvJ2)
+
cl(v;
+
X J ) u 2
+
e l J l = 0.
TUJ2) (2.8)
t d
(v1 t
Using the i d e n t i t y J1 = 1
+
X u U-x
+
X v V-x
+
+
T u U-t
T~ V- t
'
~
2
~
3
'( 2 . 9 )
( 2 . 8 ) c a n be r e w r i t t e n i n t h e f o r m
( a1+X u e l ) u x -
-
_- -
+
( b +T e )U: 1 u l
{alTv - b l X v {alux
+
blut
+
( c1+X v e l )v-x
+
( d +T e ) v c 1 v l
-
clTu
+
dlXu
+
+
clvx
+
dlvt
+ ellJ2J3
+
el (2.10)
J3el}J2
= 0.
E q u a t i o n ( 2 . 1 0 ) i s of t h e s a m e f o r m a s ( 2 . 1 ) w i t h a l r e p l a c e d by al
+
elXu,
bl replaced by bl
When e l
E 0
and e2
+
elTU,
5
0 equations (2.4)
v ( x , f ) are i d e n t i c a l to e q u a t i o n s
etc.
(2.1)
f o r u(x,E) and
for u(x,t)
and v ( x , t ) .
This
302
B. Seymour and E. Varley
f a c t i s an e x p r e s s i o n of t h e w e l l known r e s u l t t h a t i f
(2.1) a r e
w r i t t e n i n terms of h o d o g r a p h v a r i a b l e s t h e n t h e r e s u l t i n g e q u a t i o n s
are l i n e a r .
W e now show t h a t e v e n when t h e el a n d e 2 t e r m s a r e
p r e s e n t , t h e procedure d e s c r i b e d by ( 2 . 2 ) - ( 2 . 5 )
c a n be u s e d t o
t r a n s f o r m c e r t a i n n o n l i n e a r e q u a t i o n s which are of the form ( 2 . 1 ) , and which a r e of c u r r e n t p r a c t i c a l i n t e r e s t , i n t o l i n e a r e q u a t i o n s o f t h e form ( 2 . 4 ) .
3.
A SIMPLE EXAMPLE:
LONG WAVES ON A SLOPING BEACH
T h e t w o d i m e n s i o n a l n o n l i n e a r shallow-water
equations
d e s c r i b i n g l o n g waves on a c o n s t a n t s l o p i n g b e a c h c a n be w r i t t e n i n t h e form
nt
4-
C u ( ~ + u x ) l x= 0
ut
+
uu
13.1)
and
+
grlx
=o
(3.2)
I
where u ( x , t ) i s t h e h o r i z o n t a l v e l o c i t y component,
t h e bottom i s
g i v e n by y = -ux a n d t h e f r e e s u r f a c e b y y = n ( x , t ) . E q u a t i o n s ( 3 . 1 ) and ( 3 . 2 ) a r e of t h e form ( 2 . 1 ) w i t h e l =
As e 2 = 0 ,
au and e 2 = 0.
( 3 . 2 ) w i l l be l e f t unchanged b y a n y
t r a n s f o r m a t i o n of t h e i n d e p e n d e n t v a r i a b l e s o f t h e form ( 2 . 3 ) .
Thus
i n order t o l i n e a r i z e ( 3 . 2 ) w e d e f i n e
n
=
4-
u 2/ 2 9 .
(3.3)
when e q u a t i o n ( 3 . 2 ) becomes Ut
+
-
gn* = 0 .
(3.4)
While t h i s s i m p l i f i e s e q u a t i o n ( 3 . 2 ) , c o n s i d e r a b l y more c o m p l i c a t e d :
it makes e q u a t i o n ( 3 . 1 )
w e w r i t e it i n t h e form of ( 2 . 1 ) a s
W e now s e e k a p a r t i c u l a r s o l u t i o n o f e q u a t i o n s ( 3 . 4 )
and (3.5).
This
i s o b t a i n e d b y o b s e r v i n g t h a t e q u a t i o n s ( 3 . 1 ) and ( 3 . 2 ) p o s s e s s t h e simple solution
n I n t e r m s of
n
and u = g a t . u the s o l u t i o n ( 3 . 6 ) becomes
= -ax
!and -
= -ax
4-
which inverts to give
1 2
g a 2 t 2 and
u = gat
(3.6)
(3.7)
A Backlund Transformation for a Nonlinear Telegraph Equation
x = x(u,;)
2 u /2ga
+
= -;/a
and t = T ( U ) = u/ga. D e f i n i n g new i n d e p e n d e n t v a r i a b l e s
-
x = x
-
~(u,:)
E
-
T(u) = t
+
= x
-
:/a
(3.10)
u2/2ga
and = t
-
(3.11)
u/ga,
e q u a t i o n s ( 3 . 4 ) a n d ( 3 . 5 ) become
+
ut
-
grl- = 0
(3.12)
X
and
-
TIT + a ( x u ) x
.
o
(3.13)
The f i n a l system o f e q u a t i o n s is l i n e a r
1
.
T h i s example of t h e u s e o f a t r a n s f o r m a t i o n of t h e f o r m
( 2 . 3 ) t o l i n e a r i z e the shallow-water
derived w i t h hindsight.
e q u a t i o n s w a s , of c o u r s e ,
A transformation equivalent to (3.31,
a n d ( 3 . 1 1 ) w a s p r e v i o u s l y d e r i v e d by Tuck and Hwang [ 6 ] .
(3.10)
In section
4 w e d e r i v e a new t r a n s f o r m a t i o n t o l i n e a r i z e a n o n l i n e a r t e l e g r a p h
equation. 4.
A NONLINEAR TELEGRAPH EQUATION The general r e s u l t described i n s e c t i o n 2 was obtained
w h i l e s t u d y i n g the p r o p a g a t i o n of l a r g e a m p l i t u d e stress waves i n v i s c o e l a s t i c materials.
Here w e s h o w h o w t h e e q u a t i o n s which g o v e r n
s u c h waves i n a s p e c i a l c l a s s of m a t e r i a l s c a n be t r a n s f o r m e d i n t o a l i n e a r system. I n t h e i r s i m p l e s t form, t h e e q u a t i o n s c a n be w r i t t e n
u
- aX
= o
-
d(a)at
-
(4.1)
and u
e(a) = 0 ;
( x , t ) a r e d i s t a n c e and t i m e m e a s u r e s , measures.
(4.2)
( u , u ) are v e l o c i t y and s t r e s s
The f u n c t i o n s d and e c h a r a c t e r i z e t h e dynamic r e s p o n s e of
the material.
If
(4.1)
i s u s e d to w r i t e u = $x a n d a = $t,
(4.2)
implies that $xx
-
a ( $ , ) $tt
-
e ( $t) = 0 ,
(4.3)
304
B. Seymour and E. Varley
which has the form of a nonlinear telegraph equation. Equations ( 4 . 1 ) and ( 4 . 2 ) always have simple solutions describing stress-relaxation at constant strain-rate. For these solutions
x
=
X(u)
t = T(a)
and
(4.4)
where dT - kd(o) (4.5) do-=’ It then follows from the results described in section 2 that if
X(u) = ku
and
x
= x - X(u) and ? = t - T(o) (4.6) rather than x and t, are used as independent variables, equations ( 4 . 1 ) and ( 4 . 2 ) transform into
ut
-
(4.7)
u-x = 0 ,
and d e 2 o E - m = 0 . (1-ke) The functions d(o) and e(o) are as yet unspecified. They are determined by requiring that equation ( 4 . 8 ) be linear; i.e. we set
-
u-
-=
1
e ke
-
eou
(1
where (e d ) are constants. 0'
0
d
and
-
-
ke)
2 - do
Thus
e0 o e(o) = l+keo u and
(4.9)
d(o) = ( l+keoo)
(4.10)
2
(4.11)
It then follows that under the transformation (4.12)
the system of nonlinear equations ut - ox = 0
(4.13)
and
(4.14)
tranforms into the system of linear equations u- - (J- = 0 t x
(4.15)
305
A Backlund Transformation for a Nonlinear Telegraph Equation
and (4.16) A c o n s e q u e n c e of
order e q u a t i o n ( 4 . 3 ) ,
is t h a t the s e c o n d -
equations (4.10)-(4.12)
w i t h d and e g i v e n b y ( 4 . 1 0 ) and ( 4 . 1 1 ) ,
transforms i n t o the l i n e a r t e l e g r a p h equation Q--
xx
-
d 0-0 tt
-
e
Q-
0
t
= 0
(4.17)
u n d e r t h e Backlund t r a n s f o r m a t i o n
-
x = x -
(4.18)
@X
d (4.19) and
Q = $
-
Tk$: 1
-
1
do{$t-
(4.20)
fnll+keo$tl].
keO
Depending on w h e t h e r d
is n e g a t i v e , p o s i t i v e or zero t h e
0
e q u a t i o n ( 4 . 3 ) i s e i t h e r h y p e r b o l i c , e l l i p t i c or p a r a b o l i c . e x a m p l e , when do = 0 a n d eo = l / u e q u a t i o n s ( 4 . 1 7 ) - ( 4 . 2 0 )
For
imply t h a t
the n o n l i n e a r d i f f u s i o n e q u a t i o n vz
i s mapped
xx
= z
-2
(4.21)
t
n t o the l i n e a r d i f f u s i o n equation uz-- = xx
under the
z
2
(4.22)
t
ransformation
t The transformation
= t,
x
= x
-
(4.23)
vfz dt. X
( 4 . 2 3 ) f o r the n o n l i n e a r d i f f u s i o n e q u a t i o n ( 4 . 2 1 )
w a s o b t a i n e d b y Bluman and Kumei [ 7 ] b y u s i n g t h e p r o p e r t i e s o f t h e a s s o c i a t e d Lie-Backlund groups. A s i n most a p p l i c a t i o n s o f t r a n s f o r m a t i o n s of t h e form
(2.3), s o l u t i o n s t h a t a r e r e q u i r e d t o s a t i s f y b o u n d a r y c o n d i t i o n s a t f i x e d c u r v e s i n the
(x,;)
plane usually transform i n t o
v a l u e problems i n the ( x , t ) p l a n e .
free
boundary
However, w e n o t e t h a t ( 4 . 1 2 )
i m p l y t h a t s o l u t i o n s i n t h e ( x , t ) p l a n e t h a t describe waves progressing i n t o a region w h e r e u = u = 0 correspond to s i m i l a r s o l u t i o n s i n the
(G,c) p l a n e .
Also,
s o l u t i o n s t o ( 4 . 1 3 ) and ( 4 . 1 4 )
which a r e e i t h e r p e r i o d i c i n x , or i n t , c o r r e s p o n d t o s o l u t i o n s t o ( 4 . 1 5 ) a n d ( 4 . 1 6 ) which are p e r i o d i c i n
x
or i n
c.
The s o l u t i o n s o f
6.Seymour and E. Varley
306
some simple b o u n d a r y v a l u e p r o b l e m s a r e g i v e n i n V a r l e y and Seymour
C81. 1 There a r e o t h e r ways t o l i n e a r i z e e q u a t i o n s ( 3 . 1 ) and ( 3 . 2 ) .
For
example, b y u s i n g t h e h o d o g r a p h t r a n s f o r m a t i o n i n c o n s i d e r i n g v = u -gat and c = $[g(u+ax)] 1/2
a s new i n d e p e n d e n t v a r i a b l e s and x and t as new d e p e n d e n t v a r i a b l e s y i e l d s a l i n e a r s y s t e m of e q u a t i o n s f o r x and t . these r e s u l t s i n t h e l i n e a r s e c o n d - o r d e r
c(tvv-tcc)
-
E l i m i n a t i n g x from
e q u a t i o n f o r t:
3 t c = 0.
T h i s f o r m of l i n e a r i z a t i o n , u s i n g t h e hodograph t r a n s f o r m a t i o n , w a s i n t r o d u c e d by C a r r i e r and G r e e n s p a n [9]. Acknowledgements T h i s work w a s s u p p o r t e d i n p a r t by t h e U.S.
c o n t r a c t DAAG29-82K-0027
Army u n d e r
a n d i n p a r t b y t h e N a t u r a l S c i e n c e s and
E n g i n e e r i n g Research C o u n c i l of Canada u n d e r g r a n t A9117. References R. a n d D. H i l b e r t , Methods of Mathematical P h y s i c s , v o l . 2, I n t e r s c i e n c e , N e w York, 1 9 6 2 .
c11 C o u r a n t ,
c 21
Hopf, Corn.
" T h e p a r t i a l d i f f e r e n t i a l e q u a t i o n u t + uux = u x x , " P u r e Appl. Math., 2 ( 1 9 5 0 ) , p p . 201-230.
E.,
J.D., "On a q u a s i l i n e a r p a r a b o l i c e q u a t i o n o c c u r r i n g i n a e r o d y n a m i c s , " Q u a r t . Appl. Math., 9 ( 1 9 5 1 ) pp. 225-236.
c31
Cole,
C41
Miura, R.M., "The Korteweg-deVries e q u a t i o n : r e s u l t s , " SIAM Rev., 2,( 1 9 7 6 ) , p p . 412-459.
c51
Kumei, S . , a n d G.W. Bluman, "When n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s are e q u i v a l e n t to l i n e a r d i f f e r e n t i a l e q u a t i o n s , J. Appl. Math., 42 ( 1 9 8 2 ) . p p . 1157-1173.
A s u r v e y of
"
"Long wave g e n e r a t i o n on a s l o p i n g ( 1 9 7 2 ) . pp. 449-461.
C6l
Tuck, E . O . and L. Hwang, b e a c h , " J . F l u i d Mech.,
c71
Bluman, G.W. and S . Kumei, "On t h r e m a r k a b l e n o n l i n e a r d i f f u s i o n equation t a / a x ) ~ a ( u + b ) - ~ ( a u / a x ) l a u / a t = 0," Math. P h y s . , 21 ( 1 9 8 0 ) , p p . 1019-1023.
-
J.
C 81
V a r l e y , E. and B . R . Seymour, " E x a c t s o l u t i o n s f o r l a r g e Technical a m p l i t u d e waves i n d i s s i p a t i v e s y s t e m s , " I . A . M . S . R e p o r t N o . 83-27, ( 1 9 8 3 ) .
Cgl
C a r r i e r , G.F.
a n d H.P. G r e e n s p a n , "Water waves of f i n i t e a m p l i t u d e on a s l o p i n g beach," J. F l u i d Mech., 4 ( 1 9 5 8 ) , pp. 97-109.
SIAM
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
307
ON LINEAR D I S P E R S I V E WAVES WITH D I S S I P A T I O N
Francesco M a i n a r d i Department o f Physics , U n i v e r s i t y o f Bologna 1-40126 B o l o g n a , I t a l y
The aim o f t h i s paper i s t o d i s c u s s t h e c o n c e p t o f s i g n a l v e l o c i t y f o r s i n u s o i d a l waves i n l i n e a r d i s p e r s i v e media. When damping cannot be n e g l e c t e d we a r e l e d t o d i s t i n g u i s h two d i f f e r e n t types o f d i s s i p a t i o n w i t h r e s p e c t t o t h e d i s p e r s i v e p r o p e r t i e s o f t h e s i g n a l . The t h e o r e t i c a l r e s u l t s , p r e v i o u s l y o b t a i n e d by t h e a u t h o r , a r e h e r e i n t e r p r e t e d i n view o f t h e numerical d a t a f o r a t y p i c a l s i g n a l l i n g problem a s s o c i a t e d w i t h t h e 1 i n e a r Klein-Gordon e q u a t i o n . 1 . INTRODUCTION The concept o f s i g n a l v e l o c i t y f o r l i n e a r d i s p e r s i v e waves can be g i v e n u s i n g t h e s t e e p e s t descent method f o l l o w i n g t h e c l a s s i c a l argument o f B r i l l o u i n [l] f o r e l e c t r o m a g n e t i c waves. T h i s v e l o c i t y , which i s always s m a l l e r than w a v e - f r o n t v e l o c i t y , i s o f t e n i d e n t i f i e d w i t h group v e l o c i t y , a l t h o u g h t h e r e a r e cases, p a r t i c u l a r l y those i n which t h e r e i s d i s s i p a t i o n , where such an i n t e r p r e t a t i o n i s n o t a1 t o g e t h e r c o r r e c t . I n a r e c e n t paper t h e a u t h o r [ 2 1 c o n s i d e r e d t h e f o l l o w i n g one-dimensional wave equation (1.1)
Ytt
where
Y
+ 2aYt + B 2 Y = a2 Y x x =
, and
Y(x,t)
(KZein-Gordon e q u a t i o n w i t h d i s s i p a t i o n )
a,x,tbO
,
w i t h the conditions : Y(x,O)
= Yt(x,O)
Y(0,t)
= Y(t)
Y(t) =
sin(Rt)
=
0
for x
> 0,
f o r t >, 0,
, R>O.
E q u a t i o n ( 1 . l ) has many o f t h e f e a t u r e s of more general systems o f d i s p e r s i v e h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s and occurs i n a v a r i e t y o f wave propagat i o n phenomena such as waves on an e l a s t i c a l l y s u p p o r t e d s t r i n g damped by a i r f r i c t i o n , waves on e l e c t r i c a l t r a n s m i s s i o n l i n e s , e l e c t r o m a g n e t i c waves i n plasmas I n t h e above s i g n a l l i n g problem t h e i n i t i a l d i s c o n t i n u i t y o f Y ( t ) propagates f r o m x = t = 0 i n t o x>O, t > O w i t h speed c a l o n g t h e c h a r a c t e r i s t i c x = c t , where c i s t h e w a v e - f r o n t v e l o c i t y . I t s h o u l d be n o t e d t h a t some a u t h o r s (see e.g. [ 3 , 41) r e f e r . t o c a s s i g n a l v e l o c i t y b u t we p r e f e r t o use t h i s t e r m f o r t h e speed w i t h which an a p p r e c i a b l e s i g n a l i s propagated, t h u s f o l l o w i n g t h e mathematical c r i t e r i o n g i v e n by B r i l l o u i n [ 1 1
.
E q u a t i o n (1.1) can be assumed as a model e q u a t i o n i n t h e i n v e s t i g a t i o n o f l i n e a r
F. Mainardi
308
d i s p e r s i ye waves w i t h d i s s i p a t i o n . The non-dimensional parameter m = a / @ ( o < m s m ) w i l l be shown t o p l a y a fundamental r o l e i n t h e c h a r a c t e r i z a t i o n o f t h e d i s p e r s i o n p r o p e r t i e s o f t h e s o l u t i o n . A t m = 1 a t r a n s i t i o n between t h e two d i f f e r e n t e f f e c t s o f d i s s i p a t i o n i s f o u n d , r e c a l l i n g what occurs i n t h e case o f t h e l i n e a r damped o s c i l l a t o r . U s i n g t h e Laplace t r a n s f o r m technique (see,e.g. [ 5 ] ) , t h e s o l u t i o n of e q u a t i o n (1.1) may be o b t a i n e d i n a c l o s e d form By i n t r o d u c i n g t h e f o l l o w i n g v a r i a b l e s
.
6
(1 *3)
,
= x/c
~ = t - x / e
and
where J and I , denote, r e s p e c t i v e l y , t h e o r d i n a r y and t h e m o d i f i e d Bessel f u n c t i o i o f o r d e r 1. 2 2 H e n c e f o r t h , t h e cases (6 - a 2 0 ) w i l l be r e f e r r e d t o as cases ( ) r e s p e c t i v e l y . S o l u t i o n ( 1 - 4 ) c o n s i s t s o f two terms : t h e f i r s t r e p r e s e n t s t h e s i g n a l l i n g d a t a p r o p a g a t i n g a t v e l o c i t y c and s p a t i a l l y a t t e n u a t e d by a f a c t o r e x p ( - a x / c ) , w h i l e t h e second i s r e s p o n s i b l e f o r t h e d i s t o r s i o n o f t h e s i g n a l , which depends on t h e p o s i t i o n and on t h e t i m e e l a p s e d from t h e w a v e - f r o n t . The s t r e n g t h o f d i s t o r s i o n w h i c h vanishes when m = 1. Such a case i s s a i d may be measured by t h e parameter t o be undistcrted
x
In t h e n e x t s e c t i o n s we s h a l l i n v e s t i g a t e t h e d i s p e r s i o n p r o p e r t i e s o f t h e s o l u t i o n and t h e r e l a t e d concept o f s i g n a l v e l o c i t y a c c o r d i n g t o B r i l l o u i n . The numerical r e s u l t s o f two t y p i c a l cases w i l l be quoted, f r o m w h i c h t h e f i n a l c o n c l u s i o n s w i l l be drawn.
2. DISPERSION PROPERTIES The d i s p e r s i o n r e l a t i o n s h i p f o r wave-equation ( 1 .l)i s o b t a i n e d by s e t t i n g (2.1)
Y(x,t;w)
=
exp{i[k(w)x - w t l )
,
where w denotes t h e r a d i a n frequency and k ( w ) t h e complex wave number k(w) = k(w) + i 6 ( w )
(2.2)
.
F u n c t i o n s k ( w ) and 6 ( ~ )c h a r a c t e r i z e d i s p e r s i o n and a t t e n u a t i o n r e s p e c t i v e l y and may be o b t a i n e d by s o l v i n g t h e system
1
c;s(;)k(w)
c [ k (w)
=
-
ctw
S2(0)]
=
w
2
- 62
.
309
On Linear Dispersive Waves with Dissipation
U(w) a r e d e f i n e d by
Phase v e l o c i t y V(W) and group v e l o c i t y (2.4) I n absence of d i s s i p a t i o n (
i
(2.5) a # 0
For
= 0) a c u t - o f f o c c u r s a t w
ci
[ I
6 ( w ) = B2-m2
0 <w
i
6(w) =aV(w)/c w
2
2
2
2
-
[ c / V (w)
# f3
2
propagation ;
~ ( w =) c w [ w 2 - ~ 2 ] - '
.
w
11
=
2 a =
v2 (w)/c2
-
B
2
1) c o n s t a n t a t t e n u a t i o n ( 6 =
2
a/c )
( m # 1 ) we o b t a i n t h e f o l l o w i n g i n t e r e s t i n g r e l a t i o n s h i p between V
and U :
u = v
(2.7) When
.namely
,
2
2
,
no a t t e n u a t i o n
B < U < "
I n the u n d i s t o r t e d case ci = B2 ( m and no d i s p e r s i o n (V = U = c ) a r e o b t a i n e d . c1
, no
'/c
t h e f o l l o w i n g r e l a t i o n s a r e o b t a i n e d f o r any
(2.6)
If
= f3
2 2 2 2 a (1-V / c ) + 6 ( 1 - e / V ) + 2 a2 - B
[ l
9
D = 0 e q u a t i o n ( 2 . 7 ) reduces t o 3
U = 2 V - V / c
which i s quoted i n C 6 j
, while
2
when
, 01
= 0 and w > B
i t reduces t o t h e known r e l a t i o n
L
U V = c .
As a m a t t e r o f f a c t we o b t a i n t h e f o l l o w i n g i n e q u a l i t i e s : (2.8)
1
Oda
o
<6
(I< m
<
<
1 \< V/c
(O<m<1) m)
@/a< V/c
B/a
,U
6 V
1
,U
>/ V (anomaZous d i s p e r s i o n ) .
(norma2 dispersion)
From t h e known r e l a t i o n (2.9)
v/u
=
1
- wT&dV
i t may be deduced t h a t i n t h e case o f normal (anomalous) d i s p e r s i o n , Dhase v e l o c i t y i s a d e c r e a s i n g ( i n c r e a s i n g ) f u n c t i o n o f frequency.
I n F i g u r e s 1-4 we show t h e curves 6 , V , U a g a i n s t w f o r t h e f o l l o w i n g values o f t h e parameter m : ( I ) r n = 0 ; (11) m = 1/42 ; (111) m = J2 ; ( I v ) m = m Nondimensional v a r i a b l e s were used f o r convenience i n t r o d u c i n g c h a r a c t e r i s t i c t i m e scales : t T = 1/B f o r t h e cases+(+) ; +T- = l / a f o r t h e cases ( - ) ; and c h a r a c t e r i s t i c space s c a l e s X- = cT-
.
may assume .values g r e a t e r t h a n w a v e - f r o n t Except i n case ( I ) , group v e l o c i t y v e l o c i t y and t h e r e f o r e U i s expected t o l o s e t h e s i g n i f i c a n c e o f s i g n a l v e l o c i t y (speed o f t h e group) a t t r i b u t e d t o i t on t h e b a s i s of R a y l e i g h ' s c l a s s i c a l argument.
F. Mainardi
310
I
I
w
0
1 2 F i g u r e 2 : m = 1/42
3
U
1
w I
0
1 Figure 3 : m
3
2 =
0
J2
1 2 F i g u r e 4 : rn = m
J
3
3. BRILLOUIN'S SIGNAL VELOCITY I n h i s c l a s s i c a l paper about t h e p r o p a g a t i o n o f l i g h t i n d i s p e r s i v e m e d i a , B r i l l o u i n
[ l ]used t h e method o f s t e e p e s t descent i n t h e complex frequency p l a n e t o d e f i n e t h e a r r i v a l titw o f a s i g n a l a t a g i v e n d i s t a n c e f r o m t h e boundary. The r a t i o o f d i s t a n c e t o t i m e was c a l l e d signaZ ueZocity. I t was n o t g i v e n as a c l o s e d e x p r e s s i o n b u t as a computational p r e s c r i p t i o n . W i t h a n u m e r i c a l e v a l u a t i o n f o r a g i v e n complex d i s p e r s i o n r e l a t i o n s h i p k = k ( w ) ( L o r e n t z - L o r e n z r e l a t i o n ) , B r i l l o u i n showed t h a t t h e s i g n a l v e l o c i t y remained always s m a l l e r t h a n c ,the speed o f l i g h t i n t h e vacuum, i n c o n t r a s t w i t h t h e group v e l o c i t y which c o i n c i d e s w i t h t h e s i g n a l v e l o c i t y f o r weak a b s o r b t i o n . The a n a l y s i s o f B r i l l o u i n may be extended t o any one-dimensional s i g n a l l i n g problem where a F o u r i e r - L a p l a c e r e p r e s e n t a t i o n o f t h e s o l u t i o n i s a v a i l a b l e , see [ 2), l41, l6 11, [ 8 1 f o r d e t a i l s . A p p l i c a t i o n o f t h e method of s t e e p e s t descent i n t h e i n v e s t i g a t i o n o f wave-packets i n two and t h r e e dimensions may be f o u n d i n papers by Gaster [ 9 - 1 0 ] and Suchy [ll12 1 r e s p e c t i v e l y . As f a r as o u r s i g n a l l i n g problem (see ( l . l ) ,
( 1 . 2 ) ) i s concerned, t h e Laplace
31 1
On Linear Dispersive Waves with Dissipation
r e p r e s e n t a t i o n o f t h e s o l u t i o n i s , see e.g. [ Z ] , exp
where
is ( t -xn(s)/c)) s2+ n2
s denotes t h e complex Laplace parameter, B r t h e Bromwich p a t h , and n(s)
(3.2)
=
[I
.
+ B'/s']
2a/s
t
The complex f u n c t i o n n ( s ) p r o v i d e s t h e complex i n d e x o f r e f r a c t i o n when s according t o : (3.3)
n(+iu) = nr(o)
7 ini(.)
c E(w)/w
=
=
+iw
.
BY means o f t h e Cauchy theorem, t h e r e p r e s e n t a t i o n (3.1)
o f f e r s an a l t e r n a t i v e i n t e r p r e t a t i o n o f wave m o t i o n , which i s more i n t e r e s t i n g than t h e c l o s e d s o l u t i o n ( 1 . 4 ) . The wave m o t i o n now c o n s i s t s o f two p a r t s , t h e t r a n s i e n t s t a t e and t h e i t e a d y s t a t e . The t r a n s i e n t s t a t e i s t h e c o n t r i b u t i o n due t o t h e branch c u t o f n ( s ) , w h i l e t h e steady s t a t e i s t h e sum c o n t r i b u t i o n f r o m t h e p o l e s s = +in and reads (3.4)
YS(x,t) =
exp(-6x) s i n [
n( t - x / V ) ]
where (3.5)
6 = nni(n)/e
,
V = c/nr(n)
denote t h e a t t e n u a t i o n f a c t o r and t h e phase v e l o c i t y , r e s p e c t i v e l y , a t t h e i m p a c t frequency n . I t s h o u l d be n o t e d t h a t t h e two+cases (+) a r e c h a r a c t e r i z e d by a d i f f e r e n t l o c a t i o n o f t h e branch p o i n t s o f n ( s ) , s - , i n t h e complex s - p l a n e :
(+)
U<m < 1
(-)
l<m
<
m
+
(normal d i s p e r s i o n ) (anomalous d i s p e r s i o n )
s- =
-
~1
fix
(complexconjugate) ;
s+
-
~1
+X
( r e a l negative)
=
.
Another i n t e r p r e t a t i o n o f t h e s o l u t i o n can be o f f e r e d by u s i n g t h e method o f t h e s t e e p e s t descent, which l e a d s t o B r i l l o u i n ' s c o n c e p t o f s i g n a l v e l o c i t y . F o r t h i s purpose (3.1) i s r e w r i t t e n by changing t h e v a r i a b l e s x , t i n t o E = X / C , Q = c t / x > l , so t h a t
Then, t h e o r i g i n a l f i x e d p a t h B r should be deformed i n t o t h e new, moving p a t h L ( 0 ) which i s o f s t e e p e s t descent f o r t h e r e a l p a r t o f F(s,O). The i n t e g r a l on L(0) t u r n s o u t t o be equal t o t h e Bromwich i n t e g r a l , o r d i f f e r from i t by t h e steady (n) o r o > o (n), r e s p e c t i v e l y , where 8 (0,)i s s t a t e (3.4) according t o l(O
(3.7)
S(n)
= c/
OS(n)
.
W h i l e V , U can a p r i o r i be g r e a t e r t h a n c , S i s always l e s s than w a v e - f r o n t v e l o c i t y . However, t h e e v a l u a t i o n o f S depends, i n g e n e r a l , on t h e e x p l i c i t knowledge o f n ( s ) i n t h e complex s-plane and n o t o n l y i n t h e i m a g i n a r y a x i s , as f o r V and U.
312
F. M a i n a d
The e v o l u t i o n o f p a t h L ( 0 ) has been analysedby t h e a u t h o r i n [ Z ] and i s sketched i n F i g u r e s 5 , 6 f o r cases (+), ( - ) , r e s p e c t i v e l y . The arrows show t h e d i r e c t i o n of ascent through t h e s a d d l e p o i n t s ; F o r O = B C , L i s t a n g e n t t o t h e i m a g i n a r y a x i s . I n t h i s paper, we o n l y r e p o r t t h e main r e s u l t s . When a = 0 ( m = 0) i n case (+), t h e saddle p o i n t s move a l o n g t h e i m a g i n a r y a x i s s = + i w w i t h w > B .This i m p l i e s that
(3.8)
m = O
S(n)
=
U(Q)
f o r R > 6.
The i d e n t i f i c a t i o n o f t h e group v e l o c i t y w i t h t h e s i g n a l v e l o c i t y may h o l d p r o v i d e d t h a t 0 < m << 1 s o t h a t t h e saddle p o i n t s move a l o n g a l i n e which i s v e r y c l o s e t o t h e i m a g i n a r y a x i s , thus i n agreement w i t h t h e a n a l y s i s by L i g h t h i l l [ 7 ] c a r r i e d o u t by t h e method o f s t a t i o n a r y phase.
Re s
F i g u r e 5 : E v o l u t i o n i n t i m e o f t h e p a t h L ( 0 ) f o r case
(+).
I
I
1
w = Ims
F i g u r e 6 : E v o l u t i o n i n t i m e o f t h e p a t h L ( 0 ) f o r case
(-).
313
On Linear Dispersive Waves with Dissipation
An i n t e r e s t i n g r e s u l t o f t h e a n a l y s i s , which may appear unusual, occurs i n t h e case ( - ) where t h e saddle p o i n t s moves a l o n g t h e r e a l a x i s on which F(s,O) a t t a i n s o n l y r e a l v a l u e s . T h i s implges t h a t t h e s i g n a l v e l o c i t y f o r a n y R c o i n c i d e s w i t h t h e c o r r e s p o n d i n g phase v e l o c i t y . I n f a c t , when p a t h L(O) i n t e r s e c t s t h e i m a g i n a r y a x i s a t t h e p o l e s s = t i n we o b t a i n t h e f o l l o w i n g r e s u l t (3.9) from
T h i s r e s u l t has r e c e n t l y been extended by t h e a u t h o r [ 1 3 ] t o s i n u s o i d a l s i g n a l l i n g problems which e x h i b i t f u l l anomalous d i s p e r s i o n , i . e . U > V f o r 0 < w <
.
4 . NUMERICAL RESULTS AND CONCLUSIONS H e r e a f t e r , we w i l l i n v e s t i g a t e t h e s i g n i f i c a n c e o f t h e p r e v i o u s t h e o r e t i c a l s t a t e ments showing t h e numerical r e s u l t s o f o u r s i g n a l l i n g problem o b t a i n e d f r o m equat i o n s (1.1 - 6 ) i n t h e f o l l o w i n g cases : (4.1
1
(t)
a = o ,
a = 1
;
(-)
a - 1 ,
B = O .
F o r convenience we r e p o r t t h e s p l u t i o n a t a f i x e d p o s i t i o n elapsed from the wave-front , T , re-scaled according t o (4.2)
T =
TITo
,
T
0
=
2n/Q
x against the time
,
and we compare i t w i t h t h e steady s t a t e ( 3 . 4 ) , which i s p l o t t e d i n a d o t t e d l i n e tile times T,, T" , c o r r e s p o n d i n g t o velocities V and U r e We a l s o g i v e s p e c t i v e l y , d e f i n e d by 3
(4.3) I n F i g u r e s 7 and 8 we c o n s i d e r t h e case ( + ) when R = 2 and x = 5 , x = 20 respect i v e l y . It i s p o s s i b l e t o r e c o g n i z e t h e i d e n t i f i c a t i o n o f s i g n a l v e l o c i t y w i t h group v e l o c i t y ; t h i s i d e n t i f i c a t i o n improves a t g r e a t e r d i s t a n c e s . The s i g n a l , s t a r t i n g f r o m z e r o a t t h e w a v e - f r o n t , develops as a damped f o r e r u n n e r t h e a m p l i t u d e o f which i s i n c r e a s e d u n t i l i t matches t h a t o f t h e s t a t i o n a r y s t a t e a p p r o x i m a t i v e l y when T = T U' I n F i g u r e s 9 and 1 0 we c o n s i d e r t h e case ( - ) when x = 5 and ?i = 0.5 , 0 = 2 respect i v e l y . The a m p l i t u d e o f t h e steady s t a t e i s now a t t e n u a t e d by a f a c t o r A = exp ( - a ( n ) x ) =
4.39
1 1 . 0 6 lo-'
when when
R R
= 0.5 = 2
,
.
I n t h e p l o t s t h e r e s u l t s f o r t h e e x a c t s o l u t i o n and t h e s t a t i o n a r y s t a t e have b o t h been s c a l e d by m u l t i p l y i n g them by t h e f a c t o r 1/A. I n t h e s e cases t h e s i g n a l , s t a r t i n g f r o m z e r o a t t h e wave f r o n t , a t t a i n s an a m p l i t u d e of o r d e r 1 when -C = T,,. The f o r e r u n n e r i s now l e s s damped t h a n t h e steady s t a t e and t h e y match o n l y a f t e r a s u f f i c i e n t l y l o n g t i m e . The i d e n t i f i c a t i o n o f t h e s i g n a l v e l o c i t y w i t h t h e phase v e l o c i t y i n t h e p r e s e n t case i s t o be i n t e r p r e t e d i n a d i f f e r e n t manner f r o m t h e i d e n t i f i c a t i o n w i t h t h e group v e l o c i t y o f t h e p r e v i o u s case. Here we observe t h e e x i s t e n c e o f a s u b s t a n t i a l f o r e r u n n e r which i s t h e s o u r c e o f g r e a t d i s t u r b a n c e f o r t r a n s m i s s i o n s . In o t h e r words,the r o l e o f t h e main s i g n a l i s no l o n g e r p l a y e d b y t h e s t a t i o n a r y s t a t e b u t by t h i s f o r e r u n n e r w h i c h becomes a p p r e c i a b l e , and, o f t h e same o r d e r o f t h e steady s t a t e , when T = T".
314
F. Mainardi
2
0
-2 1
0
2
4
3
F i g u r e 7 : Comparison between e x a c t s o l u t i o n ( c o n t i n u o u s l i n e ) and steady s t a t e ( d o t t e d l i n e ) f o r case ( + ) o f ( 4 . 1 ) w i t h x = 5 and R = 2 .
2
I
Y(XY
7)
x=2O,R=2
I
;
-
T"’0,
I
T"=1.
0
-T
-2 0
I.
1
I
1
2
3
F i g u r e 8 : Comparison between e x a c t s o l u t i o n ( c o n t i n u o u s l i n e ) and steady s t a t e ( d o t t e d line) f o r case (+) of (4.1) w i t h x=20 and 0 = 2.
4
31 5
On Linear Dispersive Waves with Dissipation
2
x = 5 , R = 0.5
; T" = 0.24
,
T~
<
I
0 T
0 Figure
I -2 0
I
I
1
1
2
3
4
9 : Comparison between e x a c t s o l u t i o n (continuous l i n e ) and s'teady s t a t e ( d o t t e d l i n e ) for case ( - ) of (4.1) w i t h x = 5 and R = 0.5.
x
=
5
,
fi = 2 .
; T" = 0.16
,
T~
<
t
I
1
1
2
3
0 T
F i g u r e 10 : C o n a r i s ' ? n hetween e x a c t s o l u t i o n (continuous l i n e ) and steady s t a t e ( d o t t e d l i n e ) f o r case ( - ) o f (4.1) w i t h x = 5 and R = 2.
4
F. Mainardi
316
The d i f f e r e n c e s between t h e two cases may be f o r e c a s t by o b s e r v i n g t h e d i f f e r e n t n a t u r e o f t h e branch p o i n t s o f t h e complex f u n c t i o n n ( 5 ) ,which embodies t h e d i s p e r s i o n p r o p e r t i e s o f t h e e q u a t i o n , as p o i n t e d o u t i n s e c t i o n 3. The t r a n s i e n t s t a t e , d e r i v e d from t h e i n t e g r a t i o n a l o n g t h e branch c u t , i s expected t o e x h i b i t damped o s c i l l a t i o n s i n case ( + ) , and e x p o n e n t i a l decay i n case ( - ) . These two d i f f e r e n t e f f e c t s a l s o appear f r o m t h e c l o s e d s o l u t i o n (1.4) because o f I1 entering the convolution t h e d i f f e r e n t b e h a v i o u r of t h e Bessel f u n c t i o n s J, integral. Although t h e e x p l i c i t e v a l u a t i o n o f t h e t r a n s i e n t s t a t e may be h a r d t o o b t a i n , an a s y m p t o t i c e s t i m a t i o n by means o f t h e saddle p o i n t method i s f e a s i b l e t h u s confirming our conclusions. L a s t l y i t i s w o r t h w h i l e t o remark on t h e p o s s i b l e i d e n t i f i c a t i o n o f s i g n a l v e l o c i t y w i t h energy v e l o c i t y f o r quasi-monochromatic waves. Such i d e n t i f i c a t i o n i s w e l l known i n t h e case of p u r e l y d i s p e r s i v e waves where group v e l o c i t y p l a y s b o t h r o l e s . A f t e r h a v i n g d e f i n e d , w i t h due care, t h e n o t i o n o f energy v e l o c i t y i n presence o f d i s s i p a t i o n , i t i s p o s s i b l e t o demonstrate t h a t , i n t h e case o f h y p e r b o l i c waves w i t h f u l l anomalous d i s p e r s i o n , t h e energy v e l o c i t y c o i n c i d e s w i t h t h e phase v e l o c i t y [14] , t h u s e x t e n d i n g t h e s p e c i f i c r e s u l t s by Bland [ 1 5 ] f o r v i s c o e l a s t i c waves. ACKNOWLEDGMENT T h i s work was s u p p o r t e d i n p a r t by C.N.R. t h r o u g h t h e “Gruppo N a z i o n a l e p e r l a F i s i c a Matematica” and by I.N.F.N., Sezione d i Bologna. REFERENCES B r i l l o u i n , L., Uber d i e F o r t p f l a n z u n g des L i c h t e s i n d i s p e r g i e r e n d e n Medien, Ann.Pliys.Leipzig, 44 (1914) 203-240. ( E n g l i s h t r a n s l a t i o n i n Wave P r o p a g a t i o n and Group V e l o c i t y (Academic Press, New-York, 1960), pp.43-83. M a i n a r d i , F , , S i g n a l V e l o c i t y f o r T r a n s i e n t Waves i n L i n e a r D i s s i p a t i v e Media, Wave M o t i o n , 5 (1983) 33-41. Reiss, E., The Impact Problem f o r t h e Klein-Gordon Equation, S I A M J.Appl.Math., 17 (1969) 526-542. Prasad, P.,
Group V e l o c i t y , Jour. Nath. Phy. S c i . ( I n d i a ) , 14 (1980) 71-82,
Doetsch, G., I n t r o d u c t i o n t o t h e Theory and A p p l i c a t i o n o f t h e Laplace Transform ( S p r i n g e r Verlag, B e r l i n , 1974). C a r r i e r , G.F., ( McGraw-Hill,
Krook, M. and Pearson, C.E., New-York, 1966) , Ch.6.
L i g h t h i l l , M.J.,
F u n c t i o n s o f a Complex V a r i a b l e
Group V e l o c i t y , J.Inst.Maths.App1.
1 (1965) 1-28.
Thau, S.A., L i n e a r D i s p e r s i v e Waves, i n : L e i b o v i c h , S. and Seebass, A.R. Non L i n e a r Waves, ( C o r n e l l U n i v e r s i t y Pres, I t h a c a , 1974) ,pp.44-81. Gaster, M., P r o p a g a t i o n o f L i n e a r Wave Packets i n Laminar AIM J . 19 (1981) 419-423.
(eds),
Boundary Layers,
[lOIGaster, M., Estimates o f t h e E r r o r s Occurred i n V a r i o u s A s y m p t o t i c Representat i o n s o f Wave Packets, J . F l u i d Mech. 121 (1982) 365-377.
On Linear Dispersive Waves with Dissipation
317
[l
[12]
Suchy, K., Real Hamilton Equations of Geometric O p t i c s f o r Media w i t h Moderate A b s o r p t i o n , Radio S c i e n c e , 16 (1981) 1179-1182.
[ l a M a i n a r d i , F . , O n S i g n a l V e l o c i t y f o r Anomalous D i s p e r s i v e Waves, Nuovo Cimento, 74 B (1983) 52-58. [141 M a i n a r d i , F . , Energy V e l o c i t y f o r D i s p e r s i v e Hyperbolic Waves, P r e - p r i n t , Dept. o f P h y s i c s , Bologna Univ. (December 1 9 8 3 ) , s u b m i t t e d t o Nuovo Cimento.
n5] Bland, D.R., The Theory o f L i n e a r V i s c o e l a s t i c i t y (Pergamon P r e s s , Oxford, 1 9 6 1 ) , Ch.3.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) (North-Holland), 1984
319
0 Elsevier Science Publishers B.V.
LIMITING GASDYNAMIC THEORY AND ITS APPLICATIONS W . H . Hui a n d H.J. Van Roessel
Department of Applied Mathematics University of Waterloo Waterloo, Ontario N2L 3G1
An important area of hyperbolic waves i s t h a t of a n unsteady f l u i d flow with shock waves past a s o l i d body a s governed by the gasdynamic equations. This paper considers the l i m i t i n g case of such a flow when the shock wave i s extremely s t r o n g . I t i s shown t h a t in the l i m i t as t h e f r e e stream Mach number M_ approaches i n f i n i t y and independently a s the r a t i o of s p e c i f i c heats of the gas y approaches unity, the gasdynamic theory reduces t o unsteady Newton-Busemann flow theory in which the surface pressure c o n s i s t s of the Newtonian impact pressure plus a centrifugal c o r r e c t i o n . With the use of the Lagrangian method, the ll'mitlng gasdynamic equations a r e solved a n a l y t i c a l l y a n d applied to many practical problems a r i s i n g from the study of aerodynamic properties o f hypersonic vehicles. INTRODUCTION
According t o Newton's original impact theory 1 , f l u i d p a r t i c l e s do not i n t e r a c t with each o t h e r b u t , upon s t r i k i n g a body, l o s e t h e i r normal component of velocity r e l a t i v e to the body s u r f a c e , r e s u l t i n g i n an impact pressure force pimpact . After impact the p a r t i c l e s continue along t3e f r i c t i o n l e s s surface with zero tangential a c c e l e r a t i o n . However, Busemann pointed o u t t h a t the Newtonian impact pressure represents t h a t a t the t o p of t h e shock layer whereas the surface pressure must contain a centrifugal correction due t o t h e curved p a t h followed by the fluid particles. We shall f i r s t show t h a t in the most general case, unsteady Newton-Busemann flow m theory i s derivable from modern gasdynamic theory i n t h e double l i m i t as M, and y 1 , where M, i s the f r e e stream Mach number a n d y the r a t i o of s p e c i f i c heats of t h e gas. The governing equations of motion of a p e r f e c t gas are -t DD tP + p v . v = o (la) -+
-+
[
-+
where,
-
V denote the pressure, density and v e l o c i t y of the p a r t i c l e a n d denotes time d e r i v a t i v e following a p a r t i c l e . Now, i n the l i m i t as the p,
p,
-
Dt , which i s equivalent t o the shock density of the gas behind the shock p l a y e r becoming i n f i n i t e s i m a l l y t h i n (as required by conservation o f mass across the shock), the momentum equation ( l b ) may be s i m p l i f i e d . Thus when ( l b ) i s -+
W. H. Hui and H.J. Van Roessel
320
+ -+ resolved along t h e tangential d i r e c t i o n s y1 and y 2 and along the normal of the body surface ( F i g . 1 ) i t reduces to* , i n the l i m i t direction
+
-
,
(2a)
P
an
*
This i s s o , because the pressure gradient in the tangential d i r e c t i o n i s determined by the shock pressure j u m p a n d must remain f i n i t e , while in the normal direction the pressure gradient may approach i n f i n i t y as the shock l a y e r becomes infinitesimal l y t h i n . We thus see from above t h a t in the l i m i t
p + m the momentum equation of gas dynamics s t a t e s t h a t the f l u i d p a r t i c l e s move independently on the body surface with zero acceleration (Eqs. ( 2 a ) and ( 2 b ) ) in agreement with Newton's assumption, b u t the pressure change across the shock layer may be f i n i t e and i s due t o the centrifugal e f f e c t as seen from E q . ( Z c ) , as a n t i c i p a t e d by Busemann. Since the l i m i t i n g process as density p -+ i s the equivalent t o the l i m i t i n g process as Mm -+ m a n d , independently as y -+ 1 , we conclude t h a t unsteady Newton-Busemann flow model, a n d n o t the Newtonian impact model, i s derivable i n the most general case from modern gas dynamic theory i n t h a t double l i m i t . , The imp0 t n e o f the centrifugal correction i n unsteady flow has been we1 1 demonstrated.
-
5,w
From a mathematical point of view the problem of c a l c u l a t i n g hypersonic flow past a given body i s greatly simplified in the Newtonian l i m i t . T h u s , i n t h e gas dynamic theory one needs t o solve the f i v e coupled non-linear p a r t i a l d i f f e r e n t i a l equations (1) subject to boundqry conditions o n the unknown shock position f o r the f i v e unknowns p , p a n d V , each of which i s a function of three space v a r i a b l e s and t h e time v a r i a b l e . By c o n t r a s t , i n the Newtonian l i m i t , the tangential momentum equations ( 2 a ) a n d ( 2 b ) are decoupled from the normal momentum equation (2c) and the continuity equation ( l a ) . One can t h u s c a l c u l a t e t h e flow by ( a ) f i r s t determining the p a r t i c l e t r a j e c t o r i e s o n the two-dimensional surface from ( 2 a ) a n d (2b) which i s only a problem of solving two ordinary d i f f e r e n t i a l equations, ( b ) then finding t h e density v a r i a t i o n across the shock l a y e r by solving the f i r s t order p a r t i a l d i f f e r e n t i a l equation ( l a ) which i s a l s o equivalent t o solving ordinary d i f f e r e n t i a l equations, and ( c ) f i n a l l y obtaining t h e pressure change across the shock l a y e r by i n t e g r a t i o n . We see then t h a t the problem of c a l c u l a t i n g a flow i n the Newtonian l i m i t i s reduced t o t h a t of solvi n g ordinary d i f f e r e n t i a l equations. The success of t h i s a p roach i s well demonstrat d in the e a r l i e r s t u d i e s f o r o s c i l l a t i n g a i r f o i l s ' , bodies of revolution and higher frequency e f f e c t s and indicia1 response6.
s
I t should be noted t h a t the g r e a t mathematical s i m p l i f i c a t i o n s mentioned above a r e achievable only when Lagrangian method of description of f l u i d m o t i o n i s used i n solving the tangential momentum equations to find t h e p a r t i c l e t r a j e c t o r i e s . If the Eulerian method of description were used we have aa =
~
at
+
...
+ v
n av"
an
, a=l,2
(3)
f o r the tangential a c c e l e r a t i o n . For steady flow, v n = 0 , a n d the number of independent s p a t i a l variables i s reduced by 1 . This i s , however, not the case f o r unsteady flow such a s t h a t a r i s i n g from t h e motion of a s o l i d body i n a uniform hypersonic flow as v n # 0 i n general. I n t h a t case no such s i m p l i f i c a t i o n s *
Superscripts 1 , 2 , components.
and
n
of velocity and acceleration denote t h e i r respective
Limiting Gasdynamic Theory and its Applications
321
can be made even i n the Newtonian l i m i t . We thus encounter here a r a r e example in f l u i d mechanics f o r which the Lagrangian method i s superior to Eulerianmethodof description.
x3
J
Fig. 1 Three-dimensi ona 1 shapes showing no t a t i on
S p e c i f i c a l l y we consider a uniform hypersonic flow p a s t a three-dimensional body o f a r b i t r a r y shape performing a combined pitching plunging motion. We b r i q f l y i n d i c a t e how the s t a b i l i t y d e r i v a t i v e s may be c a l c u l a t e d e x h i b i t i n g r e s u l t s for two p a r t i c u l a r examples.
322
W. H. Hui and H.J. Van Roessel
NEWTONIAN IMPACT PRESSURE Consider a body 62 o f a r b i t r a r y shape p e r f o r m i n g a s m a l l a m p l i t u d e p i t c h i n g m o t i o n about an a x i s t h r o u g h t h e c e n t e r o f g r a v i t y C i n a u n i f o r m hypersonic flow 3 , ( F i g . 1 ) . S i m u l t a n e o u s l y t h e p i v o t a x i s undergoes a p l u n g i n g m o t i o n . Let
OX1X2X3
be an i n e r t i a l r e f e r e n c e frame o f C a r t e s i a n c o o r d i n a t e s , w i t h
rl, t2,t3,
corresponding u n i t vectors
i n which
a x i s and o r t h o g o n a l t o t h e f r e e stream v e l o c i t y the plungino motion.
Then
9- = U_eiEi
where
OX3
i s p a r a l l e l t o the p i v o t
fl, , and
OX2
i1 = C O S ~,
i s parallel to = sinn
L'
t h e a n q l e o f a t t a c k a b e i n g t h e a n g l e t h a t fim makes w i t h OX1 l a t i n i n d i c e s denote a sum f r o m 1 t o 3 and Greek f r o m 1 t o 2.
.
,
i3=
0
,
Here, repeated
0 1 x 1 x 2 x 3 be a b o d y - f i x e d system o f C a r t e s i a n c o o r d i n a t e s w i t h c o r r e s p o n d i n g + + + u n i t v e c t o r s el, e2, e3, i n which O ' x 3 i s k e p t p a r a l l e l t o OX3 and t h e Let
c e n t e r o f g r a v i t y C l i e s on O ' x ' a d i s t a n c e o f h u n i t s f r o m 0 ' . A l l l e n g t h s a r e s c a l e d by t h e body l e n g t h a. , v e l o c i t i e s by U, , d e n s i t y by p, , p r e s s u r e by pmU2 and t i m e b y i/U, The p i t c h i n g m o t i o n o f t h e body may be
.
r e p r e s e n t e d by t h e d i s p l a c e m e n t a n g l e 8 ( t ) which i s t h e a n g l e between O'xl and 1 2 3 OX1 . L e t (Xc,Xc,Xc) be t h e c o o r d i n a t e s o f C Then t h e p l u n g i n g m o t i o n may
.
be r e p r e s e n t e d by that
O(e,6,yEPc.{)
1 Xc = Xc = c o n s t e t c . are a l l
2
, Xc3
Xc = Y c ( t ) <<
1
=
.
0
I t w i l l be assumed
so t h a t t h e i r q u a d r a t i c terms and h i g h e r
may be n e g l e c t e d . The v e l o c i t y and a c c e l e r a t i o n o f a f l u i d o a r t i c l e r e l a t i v e r e f e r e n c e frame a r e g i v e n by
r e s p e c t i v e l y , where
W:
= +1
,
U'
2
= -1
,
and
0
i j
= 0
L e t t h e e q u a t i o n s d e f i n i n g t h e body s u r f a c e be g i v e n by where
c1c2
to
the i n e r t i a l
otherwise. x
i
=
'
1
2
H3 ( 5 ,c ) , i
i s some c u r v i l i n e a r c o o r d i n a t e system o f t h e body s u r f a c e .
corresponding vectors tangent t o t h e surface then aret) + + -f '1 X T 2 i+ w h i l e t h e normal i s g i v e n by n = ? = n e . . + 1 '21
: = Hi a ,a
'
= 1,2,3,
The
a = 1,2,
We now c a l c u l a t e t h e Newtonian impact p r e s s u r e . By s u b s t i t u t i n g t h e e q u a t i o n f o r + t h e s u r f a c e i n t o ( 4 ) and ( 5 ) and r e s o l v i n g fim , v and i n t o components + e t c . ) we g e t normal and t a n g e n t i a l t o t h e s u r f a c e ( i . e . a = aa? t an;
where
baB
surface, ra
=
i i n H
,“B
i s t h e i n v e r s e o f t h e m e t r i c t e n s o r (gaB), and
(4“’)
= gaaHi
Hi
BY ,BY i s given by ’impact
a r e t h e components o f t h e second f u n d a m e n t a l f o r m o f t h e
,o
a r e t h e C h r i s t o f f e l symbols.
=
cu; -
=
ninjeiI?j
The N e w t o n i a n i m p a c t p r e s s u r e
vnI2
-
i j j i k 2n n o I? I? 8 k
-
i i
2n R (njw$lk
-
n2 h ) 6
-
2n i n2 Q i y
.
(3)
The p a r t i c l e m o t i o n i s d e t e r m i n e d f r o m ( 2 a ) a n d ( 2 b ) , i . e .
For t h e s p e c i a l case o f steady f l o w
(6
: y :0 ) , t h i s r e d u c e s t o
i m p l y i n g t h a t t h e p a r t i c l e t r a j e c t o r i e s a r e t h e geodesics o f t h e surface. Since o u r u l t i m a t e o b j e c t i v e i s the determination o f t h e s t a b i l i t y d e r i v a t i v e s f o r a t h r e e - d i m e n s i o n a l body u n d e r g o i n g a p i t c h i n g - p l u n g i n g m o t i o n , we o u t l i n e a method b e l o w f o r t h e e v a l u a t i o n o f t h e s e s t a b i l i t y d e r i v a t i v e s w h i c h does n o t e x p l i c i t l y r e q u i r e d e t a i l e d c a l c u l a t i o n o f t h e unsteady pressure.
A s o l u t i o n o f e q u a t i o n (10)
i s sought o f t h e form Sa = Ea(t,s,CQ,SQ) 1 2
324
where
W. H. Hui and H.J. Van Roessel
1;
s =
+ -d t , vr
being t h e f l u i d ve loc ity r e l a t i v e to t h e body
Q -f
(i.e.
vr
=
*a+
5
T~),
layer a t
(5
1
2
since
and
y
9
Q’ 5 Q ) .
and
t
Since
the time when t h e f l u i d p a r t i c l e s e n t e r s t h e shock
Q t
appears i n ( 1 0 ) only through
e and
y
, and
a r e s mal l , the s o l u t i o n may be expressed a s
The scaled r e l a t i v e speed i s expressed as
together with i n i t i a l conditions obtained by demanding c o n t i n u i t y of ta nge ntia l v e l o c i t y a t t h e shock s u r f ace. I n s e r t i n g (13) a n d ( 1 4 ) i n t o ( 1 0 ) r e s u l t s i n the following system of o . d . e . ' s t o be solved successively
ds d2E:
ds2
(16a )
+
dE1 dEA
(E*ra ) -- = 0 0 By ds ds
w i t h s i m i l a r expressions f o r
F yj
and
En yj
,
325
Limiting Gasdynamic Theory and its Applications
where
we
and
f
a r e known f u n c t i o n s , 7
r:
=
1
,(-I
j
j
0 and E$I'&
= igy(EA,Ei).
Equations (16a) and (16c) a r e e a s i l y s o l v e d t o g i v e
1 2 I i s d e f i n e d as I [ f ] = f ( u , t Q , c Q ) d u and where 1 2 1 " a r e determined f r o m t h e i n i t i a l c o n d i t i o n s . ( E ~ , $ ) = F, (0,~Q,~4)
1:
where t h e o p e r a t o r
F, j
The
j
s t r e a m l i n e s a r e determined b y s o l v i n g ( 1 6 b ) f o r t h e geodesics and (16d) f o r s u c c e s s f u l l y ?or a s i m i l a r way
.
.
j = 0,1,2,...
:E
j The c o n t i n u i t y e q u a t i o n may a l s o be s o l v e d i n
To c a l c u l a t e t h e s t a b i l i t y d e r i v a t i v e s t h e p i t c h i n g moment f o r a f l u i d element i s i n t e g r a t e d a l o n g a s t r e a m l i n e t o determine t h e c o n t r i b u t i o n t o t h e t o t a l p i t c h i n g moment o f t h a t s t r e a m l i n e , and t h e n one i n t e g r a t e s o v e r t h e s u r f a c e t o t a k e i n t o a c c o u n t t h e e f f e c t s o f a l l t h e s t r e a m l i n e s . The p i t c h i n g moment coefficient
1 Cm = M/2
i s d e f i n e d as
P
where
U S P. m-b
and
Sb
c h a r a c t e r i s t i c area and l e n g t h i n t h e body r e s p e c t i v e l y and W r i t i n g t h e p i t c h i n g moment as p i t c h i n g moment a b o u t C
.
k
a r e some
M i s the t o t a l
m
we g i v e examples below o f
-Cmi
f o r various bodies.
I n f i g . 2 we p l o t t e d d a m p i n g - i n - p i t c h d e r i v a t i v e
-Cmi
versus a n g l e o f a t t a c k
a ( r a n g i n g form -120' t o 80") o f a r o c k e t whose c r o s s s e c t i o n i s an e q u i l a t e r a l t r i a n g l e . A t h i g h a n g l e o f a t t a c k o n l y t h e l o w e r s u r f a c e has a c o n t r i b u t i o n and i s proportional t o s i n u A t l a r g e n e g a t i v e a , on t h e o t h e r a c c o r d i n g l y -C
.
mi hand, o n l y t h e two upper s u r f a c e s have a c o n t r i b u t i o n . For moderate a n g l e o f i s thus continuous b u t a t t a c k a l l t h r e e s u r f a c e s c o n t r i b u t e . The c u r v e o f -Cmi n o t smooth as shown i n t h e f i g u r e . I n f i g . 3 i s p l o t t e d the damping-in-pitch d e r i v a t i v e
-C
"i
versus a n g l e o f a t t a c k
a o f a 20" s h a r p cone t o g e t h e r w i t h experiments' u s i n g a i r ( y = 1.4, h e r e y i s t h e r a t i o o f s p e c i f i c h e a t s o f t h e gas) a t M_ = 8 Good agreement i s seen between t h e o r e t i c a l p r e d i c t i o n and e x p e r i m e n t s when due allowance i s g i v e n t8 t h e f a c t t h a t t h e experiments correspond t o y = 1.4 I t has been shown b y Hui
.
.
W. H. Hui and H.J. Van Roessel
326
6-
2-
01 -120 -95 -70 -45
I
-20 Q
F i g . 2.
5
30
55
80
(degrees)
V a r i a t i o n o f damping d e r i v a t i v e o f a body w i t h t r i a n g u l a r c r o s s - s e c t i o n versus a n g l e of a t t a c k .
0’
4
8
1
,
I
12
16
20
(I(degrees)
F i g . 3.
V a r i a t i o n o f damping d e r i v a t i v e o f a 20' cone versus a n g l e o f attack.
327
Limiting Gasdynamic Theory and its Applications
t h a t d e c r e a s i n g y tends t o i n c r e a s e dynamic s t a b i l i t y . Thus Newton-Busemann flow t h e o r y t e n d s t o o v e r e s t i m a t e -Cmi , t y p i c a l l y by a b o u t 1 0 % . The remaining d i f f e r e n c e may be a t t r i b u t e d t o flow s e p a r a t i o n f o r l a r g e a n g l e o f a t t a c h which r e q u i r e s s p e c i a l t r e a t m e n t and has n o t been accounted f o r i n t h e p r e s e n t calculation. As shown by H u i and Tobak", the s t a b i l i t y d e r i v a t i v e s given i n thispaper f o r small a m p l i t u d e o s c i l l a t i o n c o n t a i n a l l t h e i n f o r m a t i o n f o r l a r g e a m p l i t u d e slow o s c i l l a t i o n o f the body. On the o t h e r hand, the t h e o r y depends c r i t i c a l l y on t h e shock l a y e r being v e r y t h i n . For flows f o r which t h e shock l a y e r i s n o t t h i n , t h e t h e o r y i s n o t e x p e c t e d t o g i v e a good a p p r o x i m a t i o n
Acknowledgement T h i s work was s u p p o r t e d by the Natural S c i e n c e s a n d o f Canada.
Engineer ng Research Council
Re f e r e nces Newton, I . , Mathematical P r i n c i p l e s o f Natural Philosophy, T r a n s l . by A . Motte ( 1 9 2 9 ) , r e v i s e d by A . C a j o r i , U n i v . o f C a l i f o r n i a Press, BerkBley, 1936, r e p r i n t e d 1946. Busemann, A . , " F l U s s i g k e i t s and Gasbewegung", HandwGrterbuch d e r N a t u r w i s s e n s c h a f t e n , Vol. IV, 2nd e d . , Gustav Fisher, J e n a , 1933, p p 244279. Hui, W . H . and Tobak, M . , "Unsteady Newton-Busemann Flow Theory, P a r t I : A i r f o i l s " , AIAA J o u r n a l , Vol. 1 9 , p p 311-318, 1 9 8 1 , H u i , W . H . and Tobak, M . , "Unsteady Newton-Busemann Flow Theory, P a r t 11: B o d i e s o f R e v o l u t i o n " , AIAA J o u r n a l , Vol. 1 9 , p p 1272-73, 1981, a l s o NASA TM-80459.
Mahood, 6 . E . and H u i , W . H . , "Remarks on Unsteady Newtonian Flow Theory", A e r o n a u t i c a l Q u a r t e r l y , Vol. 29, 1 9 7 6 , pp 66-74. Hui, W . H . , "Unsteady Newton-Busemann Flow Theory, P a r t 111: Frequency Dependence and I n d i c i a 1 Response", A e r o n a u t i c a l Q u a r t e r l y , Vol 33, November 1 9 8 2 .
.
H u i , W . H . and van R o e s s e l , H . J . , Three-Dimensional Unsteady NewtonBusemann Flow Theory", AIAA ( t o a p p e a r ) .
AEDC Report TDR-62-11
.
H u i , W . H., "An A n a l y t i c Theory o f S u p e r s o n i c / H y p e r s o n i c P i t c h i n g S t a b i l i t y " WTA AdARD Symposiun on Dynamic S t a b i l i t y P a r a m e t e r s " , CP-235, p a p e r #22, 1 9 j 8
[lo]
Hui, W . H . and Tobak, M . , " S i f u r c a t i o n A n a l y s i s o f A i r c r a f t P i t c h i n g Motions a b c u t Large Mean Angles o f A t t a c k " , J o u r n a l o f Guidance, C o n t r o l , and Dynamics, Vol. 7 , p p . 113-122, 1984.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
329
SCATTERING OF ACOUSTIC WAVES BY IMPEDANCE SURFACES T. S. Angel1 and R. E. Kleinman Applied Mathematics I n s t i t u t e University of Delaware Newark, Delaware U.S.A.
In t h i s paper we summarize r e c e n t r e s u l t s f o r t h e Robin problem f o r t h e Helmholtz equation i n e x t e r i o r domains which i s t h e mathematical model f o r s c a t t e r i n g of small amplitude a c o u s t i c waves by a surface with a known a c o u s t i c impedance. W e f i r s t develop e x i s t e n c e and uniqueness r e s u l t s f o r boundary d a t a i n L2 and use t h e s e t o e s t a b l i s h t h e completeness of a family of f u n c t i o n s . These a r e then used t o obtain t h e main r e s u l t which i s t h e c o n s t r u c t i o n of t h e T-matrix whose elements a r e shown t o be c o e f f i c i e n t s of a r e p r e s e n t a t i o n of t h e Green's f u n c t i o n f o r t h e problem.
1.
INTRODUCTION
The p r e s e n t paper i s an o u t l i n e of r e s u l t s obtained by t h e authors i n [l]. S p e c i f i c a l l y , we wish t o describe r e s u l t s which employ modified Green's f u n c t i o n s i n a boundary i n t e g r a l equation t o e s t a b l i s h t h e e x i s t e n c e and uniqueness of s o l u t i o n s of t h e e x t e r i o r boundary value problem of t h e t h i r d kind ( t h e Robin problem) f o r t h e Helmholtz equation with boundary d a t a merely i n L2. Thus, we t r e a t t h e boundary condition i n t h e "generalized L -sense" which i s e s s e n t i a l l y 2 1 a more general n o t i o n of s o l u t i o n than t h a t of s o l u t i o n s i n t h e space HRoc. (For a r e c e n t d i s c u s s i o n of such s o l u t i o n s t h e i n t e r e s t e d r e a d e r i s r e f e r r e d t o t h e papers [2] and [31 of V. P. Mikhailov.) We then t u r n t o results r e l a t e d t o t h e so-called "Null-Field Method" of Waterman and demonstrate t h e use of c e r t a i n complete i i n e a r l y i n d e p e n d e n t f a m i l i e s of f u n c t i o n s , defined i n terms of s p h e r i c a l harmonics, t o d e r i v e t h e T-matrix f o r t h e Robin s c a t t e r i n g problem.
2.
NOTATION AND STATEMEXT OF THE PROBLEM
Let D - denote a bounded domain i n
P3, c o n t a i n i n g t h e o r i g i n and with boundary aD
which i s a c l o s e d Lyapunov s u r f a c e with index 1. This l a t t e r c o n d i t i o n i m p l i e s , among o t h e r t h i n g s , t h a t t h e u n i t normal i s L i p s c h i t z continuous on aD. We denote
3
t h e e x t e r i o r of a D ( i . e . t h e complement of D-u a D i n IB ) by D+.
.
3 We w i l l Let R=R(p,q) denote t h e d i s t a n c e between two t y p i c a l p o i n t s p and q i n P concern ouselves i n t h i s work with t h e following boundary value problem f o r t h e Helmholtz equation
330 (2.3)
T.S. Angel1 and R. E. Kleinman lim r* P
r
( p ) -iku(p)) = 0
~
P(p::
,
where fEL2( a D ) , UEL-( a D ) and ImkzO, I&uzO. t o be t a k e n on i n t h e g e n e r a l i z e d L
sense.
h o l d almost everywhere on aD ( f o r d e t a i l s s e e
Here we c o n s i d e r t h e boundary v a l u e s Thus e q u a t i o n ( 2 . 2 ) i s understood t o
[4] and [5]).
We i n d i c a t e d i f f e r e n t i a t i o n i n t h e d i r e c t i o n o f t h e u n i t normal I; a t t h e p o i n t P a pEaD by - where i s d i r e c t e d outward, from aD t o D+. Furthermore we w r i t e an P P a t o d e n o t e t h e normal d e r i v a t i v e when p approaches aD from D and D+ a and anan+ P P
respectively. P o i n t s p and q are assumed t o have s p h e r i c a l p o l a r c o o r d i n a t e s r e l a t i v e t o a C a r t e s i a n c o o r d i n a t e system c e n t e r e d p , $ p ) and ( r 8 (rp, 9' 9' + q ) i n D-. It i s w e l l known ( s e e e . g . Leis [ 6 1 ) t h a t t h e c l a s s i c a l boundary v a l u e problem h a s a t most one s o l u t i o n . I n t h e p r e s e n t s e t t i n g , simple m o d i f i c a t i o n s of t h e u s u a l proof l e a d t o t h e f o l l o w i n g uniqueness theorem ( s e e [4]). L e t aD be a Lyapunov s u r f a c e o f i n d e x 1, l e t Re kzO, I m k l O and U E L , ( ~ D ) -Theorem: w i t h Imku>O. Then t h e boundary v a l u e problem (2.1)-(2.3) h a s a t most one s o l u t i o n .
A fundamental s o l u t i o n of ( 2 . 1 ) i s a two p o i n t f u n c t i o n o f p o s i t i o n y ( p , q ) which we w r i t e as ikR (2.4) y o ( p , q ) : = -%--~ITR * The n o r m a l i z a t i o n o f t h e fundamental s o l u t i o n i s chosen s o t h a t G r e e n ' s theorem f o r s o l u t i o n s of t h e Helmholtz e q u a t i o n i n D+ s a t i s f y i n g a r a d i a t i o n c o n d i t i o n ( 2 . 3 ) t a k e s t h e form.
That Green's theorem and t h e d i v e r g e n c e theorem may be employed f o r t h e c l a s s of s u r f a c e s c o n s i d e r e d h e r e i s proven i n [ b ] . The fundamental s o l u t i o n h a s a w e l l known expansion i n s p h e r i c a l wave f u n c t i o n s
wNere ' i ( k r ) P t ( c o s 8 ) { (1-j ) c o s m$+j s i n me),
zz(h) =
hp)(kr)
p > = p i f r > r P 9 = q i f r > r 9 P and t h e multi-index
,
Zi(kr) = j n ( k r )
,
p< = p i f r < r P 9 = q i f r < r 9 P’
EOsll, m=2 f o r m>O
9
33 1
Scattering of Acoustic Waves by Impedance Surfaces
= n+m+j. The f u n c t i o n s i n t h e s e r i e s ( 2 . 6 ) could be r e o r d e r e d s o t h a t t h e summation i s c a r r i e d o u t over a s i n g l e i n d e x , s e e ( 7 1 f o r e x p l i c i t d e t a i l s . Furthermore we d e f i n e a modified G r e e n ' s f u n c t i o n as
is employed w i t h
where N may be +m and Ll i s again a multi-index a s i n ( 2 . 8 ) . S i n g l e and double l a y e r d i s t r i b u t i o n s w i t h a modified k e r n e l may be d e f i n e d which have t h e u s u a l p r o p e r t i e s ; t h a t is, l e t
an N
(2.11)
N
(D.LI)(P):= J
u(q)
a?.. 2
(p,q)dsq,
P
C
3
\{(I), ~
9
an
and
where j=O,l. W e w i l l d e a l a t each s t a g e w i t h m o d i f i c a t i o n s i n v o l v i n g o n l y T i n i t e l y many terms a l t h o u g h we w i l l comment l a t e r on t h e asymptotic b e h a v i o r o f t h e expansion c o e f f i c i e n t s ( s e e below). We w i l l t h e r e f o r e drop t h e s u p e r s c r i p t N from t h e s e o p e r a t o r s . We remark t h a t K. and S. a r e completely c o n t i n u o u s J J o p e r a t o r s on L ( a n ) p r o v i d e d of c o u r s e t h a t t h e s e r i e s o c c u r r i n g i n ( 2 . 9 )
([a])
2
converge if N-.
L 2 (D - \ B a )
The o p e r a t o r s S1 and D1 a r e c o n t i n u o u s o p e r a t o r s from L
2( a n ) t o
where Ba i s a b a l l of r a d i u s a>O, and t h e jump c o n d i t i o n s a r e
an? (2.14)
lim D . p = Tp + @ u , J
3
P+P?
pCaD ,
where K)f i s t h e L ( a n ) a d j o i n t of K. and - denotes complex c o n j u g a t e . J 2 J t h e s e r e l a t i o n s h o l d p o i n t w i s e almost everywhere on an.
Note t h a t
Using t h e s e jump r e l a t i o n s , one may proceed by s t a n d a r d arguments t o d e r i v e boundary i n t e g r a l e q u a t i o n s whose s o l u t i o n s may t h e n b e u s e d t o r e p r e s e n t a s o l u t i o n t o t h e given boundary v a l u e problem. If one employs t h e unmodified fundamental s i n g u l a r i t y ( 2 . 6 ) , t h e r e s u l t i n g i n t e g r a l e q u a t i o n p o s s e s s e s a m u l t i p l i c i t y of s o l u t i o n s a t a c e r t a i n d i s c r e t e set o f v a l u e s o f t h e number k. A s h a s been p o i n t e d out many t i m e s , t h i s l a c k of uniqueness i s a n a r t i f a c t of t h e i n t e g r a l e q u a t i o n method,for t h e boundary v a l u e problem i t s e l f e x h i b i t s unique s o l u t i o n s . This s i t u a t i o n i s i n d e e d t h e m o t i v a t i o n for s t u d y i n g m o d i f i c a t i o n s . For t h e boundary v a l u e problem ( 2 . 1 ) - ( 2 . 3 ) t h e r e s u l t i n g boundary i n t e g r a l equation i s
(2.15)
( I + K +oS.)v = f j
J
.
T: S. Angel1 and R. E. Kleinman
332
The r e s u l t s of [ 4 ] show t h a t , for j = O , one must i n t r o d u c e a second i n t e g r a l e q u a t i o n , i n v o l v i n g t h e normal d e r i v a t i v e of t h e o p e r a t o r Do, i n o r d e r t o g u a r a n t e e uniqueness of s o l u t i o n s for all v a l u e s of t h e wave number k . By u s i n g a modified k e r n e l , we may a v o i d t h e c o n s i d e r a t i o n of t h i s second e q u a t i o n . Indeed, w e proved t h e f o l l o w i n g theorem i n [9]. Theorem: k#k'
I f Ink20 and Imko>O. and i f la
e9.
+:I<;
for a l l k , ) L / > O ,and aee,=O,
t h e n (I+K +US )w=O if and only i f w=O. 1 1
We may t h e n use t h e unique s o l u t i o n v of t h e i n t e g r a l e q u a t i o n ( 2 . 1 5 ) t o r e p r e s e n t a s o l u t i o n o f t h e o r i g i n a l boundary v a l u e problem i n D i n t h e form o f a s i n g l e layer u = S v inD+ 1 and t h e r e b y o b t a i n t h e b a s i c e x i s t e n c e and uniqueness theorem. Theorem:
L e t aD be a c l o s e d Lyapunov s u r f a c e of index 1 and l e t I m k >-O , Re k>O.
Then for e v e r y fcL ( a n ) t h e r e e x i s t s a s o l u t i o n utC (D )nL ( a D ) of t h e Helmholtz 2 2 + 2 e q u a t i o n ( 2 . 1 ) and s a t i s f y i n g t h e Robin boundary c o n d i t i o n ( 2 . 2 ) i n t h e g e n e r a l i z e d L - s e n s e , and t h e r a d i a t i o n c o n d i t i o n (2.3), provided 1600.
-
2
3.
COMPLETE FAMILIES AND THE NULL FIELD EQUATIONS
Having e s t a b l i s h e d t h e e x i s t e n c e and uniqueness theorem, we may t u r n t o a
IRI=O d e f i n e d i n ( 2 . 7 ) .
d i s c u s s i o n of t h e f a m i l y of t h e r a d i a t i n g f u n c t i o n s {veIm 9,
It i s w e l l known ( s e e e.g. Vekua [lo]) t h a t t h e s e f u n c t i o n s form a complete family i n L2(aD). Moreover it f o l l o w s e a s i l y from t h e o r t h o g o n a l i t y of s p h e r i c a l harmonics on s p h e r e s t h a t any f i n i t e s u b s e t i s a l i n e a r l y independent s e t . t h e e x i s t e n c e and uniqueness theorem we can e s t a b l i s h t h e f o l l o w i n g r w h i c h g e n e r a l i z e s known r e s u l t s (see e . g . [111 and [12]). meorem:
me
family
1s
+
L
Using
cmzie I=o i s a complete and l i n e a r l y independent J
s u b s e t of L ( aD) p r o v i d e d I m k l O , otLm( aD), I&o_>O and Re k>O. 2 The s o - c a l l e d " n u l l f i e l d e q u a t i o n s " f o r t h e Robin problem, w r i t t e n i n t e r m s of t h e complete l i n e a r l y independent f a m i l y (2.7), can be d e r i v e d e a s i l y from t h e Helmholtz r e p r e s e n t a t i o n . Indeed, i f u i s t h e unique s o l u t i o n of t h e Robin problem ( 2 . 1 )- ( 2.3), t h e n
(3.1) 0 = l(u aD
ai,
and t h e r e f o r e
ai, This l a t t e r s e t o f e q u a t i o n s i s r e f e r r e d t o a s t h e null f i e l d e q u a t i o n s , t h e u s u a l t r e a t m e n t of which i n v o l v e s approximating u on aD w i t h a ( f i n i t e ) l i n e a r combination of elements of a complete family and s o l v i n g t h e r e s u l t i n g a l g e b r a i c ave i l e For s i m p l i c i t y , we w i l l use t h e f a m i l i e s {vi] and {- an + ma}.W e s h o u l d emphasize t h a t t h e s e f a m i l i e s do n o t , i n g e n e r a l , form a b a s i s for L2(aD).
system.
333
Scattering of Acoustic Waves by impedance Surfaces
A s f a r a s t h e a u t h o r s a r e aware, t h e r e are no g e n e r a l c o n d i t i o n s on aD which w i l l g u a r a n t e e such b a s i c i t y ; c e r t a i n l y such r e s u l t s w i l l i n v o l v e s p e c i f i c geometric c o n d i t i o n s on t h e boundary of t h e r e g i o n ( s e e e . g . [13]). N e v e r t h e l e s s , we c o n s i d e r , f o r e a c h N=0,1,2,..
., the
subspaces o f L2( a n ) ,
ave
generated, respectively,
and
choose f o r each N , f u n c t i o n s u r e s p e c t i v e l y i n L (an). Thus 2
N
e N L by {vLl,el=o and {-an
~ i band
fN,%,
%
and
N
+ ~ ev ~ } ~ and ~ Iwe= may ~ ,
which b e s t approximate u and f
and
The completeness o f t h e f a m i l i e s of expansion f u n c t i o n s i n s u r e t h a t , w i t h t h e s e choices f o r c o e f f i c i e n t s ,
and
We
now
extend
N t h e d e f i n i t i o n ( 3 . 3 ) of u t o a l l o f D+ and d e f i n e a f u n c t i o n
9N by
so that
N (3.8)
?&+ an
u $ = ~ fN ,
We may t h e n u s e G r e e n ' s f u n c t i o n s for t h e D i s i c h l e t and Robin problems, denoted by y D and yR r e s p e c t i v e l y , t o o b t a i n t h e r e p r e s e n t a t i o n s
and
s p h e r e of r a d i u s A chosen s u f f i c i e n t l y l a r g e I n t e g r a t i n g t h e s e r e l a t i o n s over t o c o m p l e t e l y c o n t a i n t h e r e g i o n D- and r e c a l l i n g t h e o r t h o g o n a l i t y p r o p e r t y of
334
TS. Angel1 and R. E. Kleinman
em t h e f u n c t i o n s {vQ)
lz0
for some c o n s t a n t
( a similar estimate holding f o r t h e d i f f e r e n c e I c Q - d i N i l ) ,
c1
on s p h e r e s , we a r e l e d t o t h e e s t i m a t e , u s i n g (2.81,
m
where u ( p ) =
(3.12)
c , v i ( p ) , r =A.
C
IQ]=O
dr)
l i m ciN) = lirn N-ro N-ro
We a r e t h e n l e d t o t h e c o n c l u s i o n t h a t
P
= cL
.
F i n a l l y , w e c o n s i d e r t h e c a s e when t h e d a t a f i s of t h e form
(3.13) f = -
au' ~
an
i
-
UU
where u1 r e p r e s e n t s a n i n c i d e n t f i e l d . This i s t h e form i n which t h e b o u n d a r y . d a t a o f t e n a p p e a r s i n s c a t t e r i n g problems. It i s u s u a l l y convenient t o r e p r e s e n t u1 i n t e r m s of incoming waves a,
where A i s t h e r a d i u s of a c i r c u m s c r i b i n g s p h e r e which c o n t a i n s no s o u r c e s . Then t h e n u l l f i e l d e q u a t i o n s ( 3 . 2 ) become
where, for n o t a t i o n a l convenience, we have r e p l a c e d t h e multi-index i n t r o d u c e t h e approximations
2 with
t.
Now
N
where, a g a i n , t h e a Q L ,a r e chosen s o t h a t t h e r i g h t hand s i d e r e p r e s e n t s t h e b e s t approximation i n
q, t o
t h e g i v e n l e f t hand s i d e .
=
N
c
The n u l l f i e l d e q u a t i o n s t h e n a r e
N
c
( R I = O IL’)=O
where we use t h e r e l a t i o n
aD
aD
N
aeIaEQ,
J
aD
ave ($
+ av:)veds
Scattering of Acoustic Waves by Impedance Surfaces
335
Using m a t r i x n o t a t i o n , (3.18) i s o f t h e form
(3.19) Q c
= Q n
N a '
This e q u a t i o n h a s t h e obvious s o l u t i o n
(3.20) c =
nN a
i n which form nN i s o f t e n r e f e r r e d t o as t h e T-matrix o r t r a n s i t i o n m a t r i x f o r t h e Robin problem s i n c e it t r a n s f o r m s t h e c o e f f i c i e n t s a = ( a , , ) of t h e i n c i d e n t wave (3.14) i n t o t h e c o e f f i c i e n t s c = ( c k y ' ) of t h e s c a t t e r e d f i e l d (3.16). W e n o t e an a d d i t i o n a l p r o p e r t y enjoyed by t h e elements aNQ R ,of t h e
T-matrix. I n [l] we show t h a t f o r f i x e d Q and R ' t h e sequence of approximations has a l i m i t i . e .
Furthermore t h e Green's f u n c t i o n f o r t h e Robin problem h a s t h e f o l l o w i n g r e p r e s e n t a t ion
when b o t h p and q l i e o u t s i d e any s p h e r e c o n t a i n i n g t h e boundary an. Thus we have e s t a b l i s h e d t h e c o n n e c t i o n between t h e T-matrix and t h e G r e e n ' s f u n c t i o n .
ACKNOWLEDGMENT T h i s work w a s s u p p o r t e d ur,der NSF Grant No. MCS-82-02-033.
336
T.S. Angell and R.E. Kleinman
REFERENCES A n g e l l , T.S. and Kleinman, R . E . ,
The Helmholtz Equation w i t h L
Boundary 2 Values, U n i v e r s i t y of Delaware Applied Mathematics I n s t i t u t e Technical Report 136A. Mikhailov, V.P., On t h e Boundary Values of t h e S o l u t i o n s of E l l i p t i c E q u a t i o n s , Appl Math Optim, 6 ( 1 9 8 0 ) 193-199. Mikhailov, V . P . , On D i r i c h l e t ' s Problem f o r E l l i p t i c Equations of t h e Second Order, D i f f E q u a t i o n s , 1 2 (1976) 1877-1891. A n g e l l , T . S . and Kleinman, R . E . , Boundary I n t e g r a l Equations f o r t h e Helmholtz Equation: The Third Boundary Value Problem, Math Meth i n t h e Appl s c i , 4 ( 1 9 8 2 ) 164-193. Miranda, C . , P a r t i a l D i f f e r e n t i a l Equations of E l l i p t i c Type, 2nd Rev. Ed. (Springer-Verlag, New York, H e i d e l b e r g , B e r l i n , 1970 ). Leis, R . ,h e r d a s Neumannsche Raudwert problem fer d i e Helmholtzsche Schwingungsgleichung, Arch R a t i o n a l Mech Anal, 2 (1958) 101-113. Kleinman, R.E., Roach, G.F., and StrEm, S., The Null F i e l d Method and Modified Green's F u n c t i o n s , t o a p p e a r . Mikhlin, S.G., Mathematical P h y s i c s : An Advanced Course (North Holland Pub. Co., Amsterdam 1970). A n g e l l , T.S. and Kleinman, R . E . , Modified G r e e n ' s F u n c t i o n s and t h e T h i r d Boundary Value Problem for t h e Helmholtz Equation, J. Math Anal and A p p l i c s , 97 (1) ( 1 9 8 3 ) 81-94. Vekua, I.N., On Completeness of a System of Metaharmonic F u n c t i o n s , Dokl. Akad Nauk. SSSR 90 (1953) 715-178. M i l l a r , R . F . , On t h e Completeness of S o l u t i o n s t o t h e Helmholtz Equation, M I A J . Appl. Math., 30 (1) (1983) 27-38. Colton, D . , Far F i e l d P a t t e r n s f o r t h e Impedance Boundary Value Problem i n Acoustic S c a t t e r i n g , Applic Anal, 16 ( 1 9 8 3 ) 131-139. K r i s t e n s s o n , G . , Ram, A . G . , and Strljm, S . , Convergence of t h e T-Matrix Approach i n S c a t t e r i n g Theory 11, J. Math. Phys, 24 (11) (1983) 2619-2631.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) Q Elsevier Science Publishers B.V. (North-Holland). 1984
337
INTERACTING NONLINEAR WAVES I N A NEURAL CONTINUUM MODEL: ASSOCIATIVE MEMORY AND PATTERN RECOGNITION Marcus S. Cohen Department o f Mathematics U n i v e r s i t y o f Kentucky L e x i n g t o n , Kentucky 40506 U.S.A.
Coupled wave e q u a t i o n s a r e d e r i v e d f o r t h e i n t e r a c t i o n o f graded p o t e n t i a l s i n a n e u r a l continuum. Waves d i v e r g i n g from two d i f f e r e n t sensory i n p u t p a t t e r n s i n t e r f e r e t h r o u g h a q u a d r a t i c memory n o n l i n e a r i t y , i m p r i n t i n g a parameter g r a t i n g . A r e p l i c a o f one o f t h e s e i n p u t waves i s Bragg-ref l e c t e d a t t h e g r a t i n g , r e c o n s t r u c t i n g t h e a s s o c i a t e d wave. Chains and c y c l e s of a s s o c i a t e d p a t t e r n s a r e r e c o n s t r u c t e d by m u l t i p l e Bragg r e f l e c t i o n , s u g g e s t i n g mechanisms f o r ass o c i a t i v e memory and p a t t e r n r e c o g n i t i o n . INTRODUCTION U n t i l t h e l a t e 1 9 6 0 ' s , most o f n e u r o p h y s i o l o g y c o n s i s t e d o f e x p e r i m e n t a l and theor e t i c a l i n v e s t i g a t i o n o f t h e " a l l o r none" p r o p a g a t i o n of a c t i o n p o t e n t i a l s i n axons.
There i s a g r o w i n g body o f o p i n i o n , however, t h a t these r e p r e s e n t t h e l o n g
d i s t a n c e " t e l e g r a p h y " o f sensory and motor i n f o r m a t i o n i n a d i g i t i z e d form, and t h a t t h e i n t e g r a t i v e f u n c t i o n s o f t h e b r a i n must i n v o l v e t h e spread and n o n l i n e a r i n t e r a c t i o n o f graded p o t e n t i a l s i n d e n d r i t i c t r e e s ( 1 , Z ) d r i t i c o r c o l a t e r a l axo-somatic synapses ( 3 ) .
and across dendro-den-
Employing a s i m p l e continuum model
f o r a n e u r a l t i s s u e c o n s i s t i n g o f c e l l s w i t h Hodgkin-Huxley t y p e membrane conductances connected b y r e c i p r o c a l l a t e r a l synapses (such as i s found i n t h e r e t i n a and sensory p r o j e c t i o n a r e a s ) we analyze t h e n o n l i n e a r i n t e r a c t i o n s o f s u b t h r e s h o l d graded p o t e n t i a l s , i.e. s m a l l a m p l i t u d e d e p a r t u r e s f r o m t h e r e s t i n g p o t e n t i a l Vr. Since Vr
i t s e l f changes w i t h i n t e g r a t e d a c t i v i t y on a s l o w t i m e s c a l e T , we d e f i n e
a q u i e s c e n t r e s t i n g p o t e n t i a l 1, t h e " b i a s p o i n t " i n t h e p r o l o n g e d absence o f exc i t a t i o n , which p l a y s t h e r o l e of a c o n t r o l parameter i n t h e model. f e a t u r e f o r o u r purposes i s t h a t , as v a l u e ,A,
The e s s e n t i a l
X i s r a i s e d past t h e threshold ( c r i t i c a l )
t h e homogeneous r e s t s t a t e becomes u n s t a b l e , and s m a l l a m p l i t u d e s t a b l e
waves b i f u r c a t e a t f r e q u e n c y wc and wavenumber kcgo. t i a l wayes."
We c a l l these "graded poten-
The s i t u a t i o n i s s i m i l a r t o a r e a c t i o n - d i f f u s i o n system whose homo-
geneous f o r m a d m i t s a Hopf b i f u r c a t i o n (4,5); however, t h e s y n a p t i c c o u p l i n g i s much more c o m p l i c a t e d t h a n d i f f u s i o n , p r o c e e d i n g as i t does t h r o u g h two i n t e r m e d i a t e v a r i a b l e s , t h e p r e - and p o s t s y n a p t i c t r a n s m i t t e r c o n c e n t r a t i o n s , w i t h d e l a y s .
338
M.S. Cohen
NONLINEAR SYNAPTIC TRANSFER F i r s t , we develop a s i m p l e model o f a chemical synapase which embodies i n a s k e l e t a l f o r m t h e known p h y s i o l o g y ( 3 ) : 1.
) Q t h a t depends on p r e Pre s y n a p t i c v o l t a g e and t h e s t a n d i n g pool o f t r a n s m i t t e r Q i s p r e s y n a p t i c P r e s y n a p t i c t r a n s m i t t e r Q i s r e l e a s e d a t r a t e K(V
vesicles. 2.
Q i s r e p l e n i s h e d a t a c o n s t a n t r a t e b.
3.
T r a n s m i t t e r r e l e a s e d f r o m c e l l s m-1 and m + l q u i c k l y d i f f u s e s a c r o s s t h e syna p t i c c l e f t s and i s bound by t h e l e f t and r i g h t p o s t s y n a p t i c membranes o f c e l l Bound p o s t s y n a p t i c t r a n s m i t t e r P o r R i s r e l e a s e d
m r e s p e c t i v e l y ( F i g u r e 1).
and degraded a t a r a t e p r o p o r t i o n a l t o i t s c o n c e n t r a t i o n .
Thus
1)
m- 1 m-1 m-1 m+l m+l m+l Qt = -K( V ) Q + b , Qt = -K( V ) Q + b
2)
m m-1 m-1 m Pt = K( V ) 4 - LP,
4.
P o s t s y n a p t i c conductance f o r some i o n i c s p e c i e s B ( n o m i n a l l y , C h l o r i d e ) w i t h
m Rt
=
m + l rn+l m -K( V ) Q - RR
r e v e r s a l p o t e n t i a l VB i s p r o p o r t i o n a l t o bound t r a n s m i t t e r . The p o s t s y n a p t i c g e n e r a t o r c u r r e n t s a r e t h u s
m 3)
m m
I = oP(V
m J =
- VB)
m m R(V - V B )
Note t h a t Ipost c o n t a i n s an " e s s e n t i a l " q u a d r a t i c n o n l i n e a r i t y : t h e p r o d u c t o f a f u n c t i o n o f v o l t a g e a t one p o i n t and t h e v o l t a g e a t a n o t h e r . L a t e r a l membrane r e s i s t a n c e i s i g n o r e d i n t h i s lumped model.
Departures f r o m t h e
m a
r e s t i n g l e v e l o f t h e n e t p o s t s y n a p t i c g e n e r a t o r c u r r e n t i n c e l l m, i + j m m ( I - I r ) + ( J - J r ) , a r e t h u s s i m p l y r e t u r n e d t h r o u g h t h e transmembrane conductance Z m m m c a u s i n g a s h i f t away f r o m t h e r e s t i n g p o t e n t i a l v :V-Vr expressed by t h e f u n c t i o n a l r e 1a t i o n
m t ~('){i(tl)
where Z
m
+j(tl),
m m i(t11) + j ( t ' l ) ,
m i(tg'1)
L i s t h e l i n e a r i z e d membrane conductance, 2'')
in
+j(t"l)}+
...
i s a quadratic nonlinearity
r e p r e s e n t i n g "delayed r e c t i f i c a t i o n " and Z ( 3 ) i s a c u b i c n o n l i n e a r i t y r e p r e s e n t i n g s a t u r a t i o n o f some i o n i c channel. t h e r e s t i n g p o t e n t i a l Vr.
The phenomenological conductances Z depend on
I n b o t h t h e Hodgkin-Huxley (6,7)
and Fitzhugh-Nagumo
models, Z ( L ) becomes u n s t a b l e v i a a growing o s c i l l a t i o n a t frequency urn (Hopf b i f u r c a t i o n ) as Vr i s r a i s e d beyond " t h r e s h o l d " Am;
t h e a m p l i t u d e i s l i m i t e d by t h e
interacting Waves in a Neural Continuum Model
339
Figure 1 A Line of Cells With Reciprocal Lateral Synapses cubic nonlinearity Z ( 3 ) ( 8 ) .
V r i t s e l f changes over a slow time scale T i f these
osci 1 1a t i ons pers i st: m m m vr = x + D ( * ) { ~ ( T )I , ~ ( T I I;TI ) 5)
m where 6 ( * ) i s a quadratic functional of v giving the e f f e c t of integrated a c t i v i t y in c e l l m upon i t s r e s t i n g potential and i s t h e quiescent value of the r e s t i n g p o t e n t i a l . This e f f e c t a r i s e s from the " r e c t i f i c a t i o n " properties of t h e quadratic synaptic and membrane n o n l i n e a r i t i e s ; o s c i l l a t o r y a c t i v i t y separates charge which changes the ionic balances across the membrane (Nernst p o t e n t i a l s ) and thus the r e s t i n g potential ( s e e ( 9 ) f o r a d e t a i l e d d i s c u s s i o n ) . As we s h a l l see below, i t i s t h i s "memory" nonlinearity b ( ' ) t h a t i s responsible f o r imprinting t h e shortterm memory t r a c e in our model. PROPAGATION EQUATIONS FOR HARMONIC WAVES W e give here the basic equations governing t h e spread of graded potential waves, an o u t l i n e of the perturbation method, and a statement of the main r e s u l t s ; the reader i s r e f e r r e d t o ( 9 , l O ) f o r a more d e t a i l e d account. The method follows t h a t of Hopf (11). The parameter deviation from c r i t i c a l i t y A-Xc and a l l t h e dynamical variables in each c e l l m a r e expanded in powers of a small parameter E , which s c a l e s the amplitude of b i f u r c a t i n g waves. Instead of the usual expansion of the
2
frequency W = W + E W ~ + E w 2 . - - with w1=0. however, we define a slow time s c a l e C d a z-,a and derive modulation T = E2t , so the time d e r i v a t i v e operator -becomes -+E dt at aT equations f o r the r a t e o f change o f the amplitude o f waves a t t h e fundamental f r e quency w with slow time T . Since we a r e looking f o r quasi-periodic s o l u t i o n s , we C f i r s t expand each v a r i a b l e in each c e l l as a Fourier s e r i e s
M.S. Cohen
340
and t h e n expand each F o u r i e r c o e f f i c i e n t as a power s e r i e s i n
E:
i
7)
m
v r- A
=
E
A-Ac
=
€1 + E 2 f 1 2
v 1+ E 2
m V2t."
x
We a l s o t r a n s f o r m t h e phenomenological r e l a t i o n 4) t o t h e frequency domain, obt a i n i n g f o r each harmonic n a t each o r d e r ( k ) ,
where we sum o v e r each i n d e x such t h a t mtr=s+u+w=n and a + b = c+d+e= k . Through t h e expansions 6 ) , 7 ) , and 8 ) , we may o b t a i n s o l u t i o n s t o 1 ) - 5 ) a t each o r d e r ( k ) f o r each harmonic n.
W e may t h e n reassemble t h e c o n t r i b u t i o n s a t t h e
f i r s t few o r d e r s i n a d j a c e n t c e l l s t o o b t a i n approximate coupled e q u a t i o n s f o r t h e p r o p a g a t i o n o f o s c i l l a t i o n s vneinWt
a t frequency n
from c e l l t o c e l l .
Finally,
we pass t o t h e continuum l i m i t t o o b t a i n an approximate p r o p a g a t i o n e q u a t i o n f o r m For example, we o b t a i n a t O ( E ) V1=O, A1=O, and v(x,t,T).
L
where
r
x(s)
= "(vy.-vB)c
J 7
and
g=aP
C
are evaluated a t
Vr=Ac
Recognizing
as t h e d i s c r e t e f o r m o f t h e L a p l a c i a n , t h i s becomes i n t h e continuum l i m i t
where
I
341
Interacting Waves in a Neural Continuum Model
and h i s t h e s p a c i n g between t h e c e l l s . O(E)
9)
C2vJi1('x,T) + k 2 ( X C , i i w c ) v
For N s p a t i a l dimensions, we o b t a i n a t
= 0
k1
A t 0 ( c L ) , we o b t a i n no c o n t r i b u t i o n t o t h e fundamental a m p l i t u d e ; however, we do o b t a i n D.C.
and second harmonic terms
11) vh2) = F ( i l o ) v i l ) v i i )
,
F(iw) = F*(iw)
= H(iw)vkl( l ) v +1 (l)
12) ~1.)
f r o m t h e m i x i n g o f terms a t + i w and a t - i w t h r o u g h t h e q u a d r a t i c n o n l i n e a r i t i e s . The importance o f t h e D.C.
t e r m i s t h a t i t d r i v e s charge s e p a r a t i o n across t h e
membrane, l e a d i n g t h r o u g h ( 5 ) t o a d r i f t slow t i m e T ( s e e ( 9 ) ) .
13)
D(2){~l:)(;,T’)~i:)(jt,T’);T}
i n Vr w i t h
2
Thus, t o O ( E ) ,
v ~ ( z , T ) - A :E 2 v 2 = E 2 ~ ( 2 ) {1 v ( 1 -) 1 v ( 1 ) ; ~.~
We a l s o f i n d h2 > O f o r t h e c l a s s o f models t h a t g i v e s m a l l a m p l i t u d e b i f u r c a t i n g waves, c o r r e s p o n d i n g t o a s u p e r c r i t i c a l b i f u r c a t i o n ; t h u s E2
X;l(X-Xc)
f o r small
X-Xc
.
3 A t O ( E ) we o b t a i n a new c o n t r i b u t i o n t o t h e fundamental t h r o u g h t h e c u b i c n o n l i n e a r i t y , gvil)vIi)vil),
where g i s complex, and A 3 = V 3 = 0 .
No new c o n t r i b u t i o n s t o
4
t h e fundamental appear a t O ( E ) . 4 Reassembling a l l t h e c o n t r i b u t i o n s t o v1 up t o O ( E ), we f i n a l l y o b t a i n a t r u n c a t e d m o d u l a t i o n e q u a t i o n f o r s l o w l y v a r y i n g waves vl(;.T)eiWt quency w 14)
a t t h e fundamental f r e -
=w : C
a
i p h aT v +
where v - v l , U = X 2 ( ak 2
v2 v + Ckc2 + v(U + F{v*v;TI + g v * v l v
v*-vel,
p=c2=Xi1(X-Ac),
= 0
h = ( a k2
and FIv*v;T) = ( a k 2/ a A ) c D ( 2 ) { ~ * ~ ; T 1 .
k 2 = k 2 (Xc,iwc), C
342
M.S. Cohen
f Irn
k2
Re k 2
A C
Figure 2 D i s p e r s i o n r e l a t i o n k2(A,iw). k: k 2 ( h c , i u c ) and (ak2/aw)c a r e b o t h r e a l . (Decaying waves broken, growing waves sol i d 1 i n e s . ) 2 k (i,i+w)
i s t h e l i n e a r d i s p e r s i o n r e l a t i o n 10) c h a r a c t e r i z i n g p l a n e waves
z-%=k 2 and
-
To a s c e r t a i n t h e s t a b i l i t y o f propaC -t ++ g a t i n g waves, we i n t r o d u c e a l o n g space s c a l e X = v z and reexpand v(x,X,T) =
ei(k'xtwt),
where
v(')(z,?,T)
+pv(')(z,;,T)
15)
+ k;v(O)
$v(o)
(x,y,z).
=;
We o b t a i n a t O ( 1 ) :
+O(p2).
= 0 ;
t h i s has planewave s o l u t i o n s v'')(;,?,T)
++ =A(?,T)eikax, where
llKll 2 = kc .
For X=X
C'
waves a t wC n e i t h e r grow n o r decay i n t h e i r d i r e c t i o n s o f propagation; t h u s , k c i s 2 r e a l . Since we r e q u i r e waves a t W Z W ~t o decay, (ak / a w ) c = 2 k c ( a k / a w ) c must be r e a l ; t h u s , t h e group v e l o c i t y V = - ( a o i / a k ) c i s r e a l . g o f growing waves about wc,kc (see F i g u r e 2 ) .
For X > X
C'
t h e r e i s a band
A t O(u) we o b t a i n an inhomogeneous e q u a t i o n f o r v(') in t h e s e l f - a d j o i n t ope:a$or 2 V2 + kc; t h e Fredholm a l t e r n a t i v e t e l l s us t h a t t h e terms on t h e R.H.S. i n eikSx 2 for k must vanish, g i v i n g a n e t m o d u l a t i o n e q u a t i o n C
16)
Ac :AT
V
-V
E - v - ~ A = -i 3 ( U g
x
+ F{A*A;T} + gA*A)A
2 kc
i s t h e r a y c o o r d i n a t e i n t h e frame moving a t V i n t h e where 5 E T - V-'^k.? 9 g t i o n . The "energy" e q u a t i o n
^k
direc-
t h e n t e l l s us t h a t a ) waves w i l l grow i n 5 f r o m since
I A l2 = O
if Im U > O ,
A2 > O ( c o n v e c t i v e i n s t a b i l i t y ) .
t h e n o n l i n e a r terms, i f
i.e.
2 i f ( a I m k /aA)c>O,
The a m p l i t u d e w i l l be l i m i t e d b y
343
Interacting Waves in a Neural Continuum Model
b ) lm g < 0 ( h a r d l i m i t t h r o u g h t h e c u b i c n o n l i n e a r i t y ) , and/or
2
c ) t h e d e l a y f u n c t i o n a l F{ / A \ ;T] accumulates a s u f f i c i e n t n e g a t i v e i m a g i n a r y 2 p a r t as \ A ] p e r s i s t s i n t i m e ( A u t o m a t i c Gain C o n t r o l , o r A.G.C.). a ) and b ) o r c ) a r e f u l f i l l e d by t h e c l a s s o f models o f i n t e r e s t t o us here; we conclude t h a t bifurcating waves are stabZe (see ( 9 ) f o r a s p e c i f i c example based on t h e Hodgkin-Huxley scheme). Now a l i n e a r i z e d model o f t h e charge d i s p l a c e m e n t c u r r e n t s across t h e membrane i n response t o t h e D.C.
'I
i n t e r f e r e n c e " term v o - p l v ( o ) 1 2
r e s t i n g p o t e n t i a l V,-x
where A,
a
(9) predicts a s h i f t i n the
w i t h i n t e g r a t e d a c t i v i t y g i v e n by
z ~ D ( ~ ) {
F o r C
i s t h e r e l a x a t i o n r a t e o f Vr.
( h y p e r p o l a r i z e d ) by c o n t i n u e d e x c i t a t i o n and t h e a m p l i t u d e d r i v e n down v i a t h e
A.G.C.
effect i n c).
Our model t h u s e x h i b i t s desensitization; t h i s i s an observed
neurophysiological e f f e c t (12). WAVE-WAVE INTERACTIONS The r e a l i n t e r e s t l i e s , however, i n t h e i n t e r a c t i o n o f two o r more waves t h r o u g h t h e n o n l i n e a r i t i e s F and g.
We g i v e o n l y t h e main r e s u l t s here; see (10) f o r a
The 0 ( 1 ) e q u a t i o n 1 5 ) has s o l u t i o n s c o n s i s t i n g o f two super-
d e t a i l e d discussion.
posed modulated p l a n e waves, o r writing waves
++ 18)
v(O)(;,I,T)
= A(d,T)eib"+
where A = IAleiay
B = IB[ei8,
B(?,T)eibaX Say these e x i s t f r o m T ' = O t o TI.
and a * = b 2 = k:.
They i n t e r f e r e t h r o u g h t h e q u a d r a t i c n o n l i n e a r i t y t o produce a D . C . p r i n t s an i n t e r f e r e n c e g r a t i n g p D ( 2 ) { ~ ( o ) * ~ ( o f ; T ) i n Vr;
t e r m vo which
t h i s appears as an
index grating F { ~ V ( ~ ) ~ * ;=T K[cos(z-?+n(?)) } +b] + i n 14), where 6:;-b i s t h e grating vector, 19)
K = Me-RT(ak2/aA)c.
A ( 1 ) = a(%)-B(T)i t s
phase, and
We may c a l c u l a t e t h e s c a t t e r i n g o f newly i n p u t reading waves ++
20) v(O)
= S(x,T)ei"'+R($,T)eir.X
;
S = Isleiu
and
R = /RleiP
o f f t h i s g r a t i n g b y assuming i t fades l i t t l e d u r i n g t h e s c a t t e r i n g process and t r e a t i n g F as an inhomogeneous t e r m i n t h e O(p) e q u a t i o n . t i v e t h e n y i e l d s t h e Bragg conditions 21)
++ s-r =
d
++
z +(a-b)
;
s 2 = r 2 = k2 c '
The Fredholm a l t e r n a -
344
M.S.Cohen -t
which i m p y s =
R where
n
E
td
and
f
+
= +b, and t h e
(aT - V 9?.v.j;)R
coupled wave equations
- R*R)lR
- i A [ U +g(2E
- iite-iAS
,
A = V /2kc, E = S*S + R*R, and K = KV /4kc i s the modulation depth. (We have 9 9 + s i g n i n 2 1 ) . ) From 22) we o b t a i n t h e phase-matching condition
chosen t h e 23)
a(?) - p ( ? ) = A ( x )
a(X) -
B(?)
.
I f t h e boundary c o n d i t i o n s i n 22) consis:
$f an i n p u t wave I s l e
i o iT-2 e , there w i l l
be a s t r o n g (O(1)) r e f l e c t e d wave ReiPeir’x
i f and o n l y i f t h e i n p u t wave has t h e + + same wave v e c t o r as t h e o r i g i n a l w r i t i n g wave o r i t s c o n j u g a t e , s = ? a ; f u r t h e r m o r e ,
i f t h e phase m o d u l a t i o n s o f t h e i n p u t wave S reproduce t h o s e o f t h e w r i t i n g wave + A, o(X) t h e phase m o d u l a t i o n s o f t h e r e f l e c t e d wave R reproduce t h o s e o f
=+,(x),
t h e a s s o c i a t e d w r i t i n g wave B, p ( x ) =
+@(St).
Since phase m o d u l a t i o n s on a w a v e f r o n t
encode t h e images o f p a t t e r n s , t h e system e x h i b i t s a s s o c i a t i v e p a t t e r n memory: I n p u t o f a r e p l i c a o f p a t t e r n A evokes a v i r t u a l image o f p a t t e r n B w i t h which i t was a s s o c i a t i v e l y s t o r e d .
Many p a i r s o f a s s o c i a t e d images (A.,B.) J
can be m u l t i -
.J
p l e x e d i n t h e medium w i t h o u t c r o s s t a l k by f o r m i n g t h e i r r e s p e c t i v e i n t e r f e r e n c e g r a t i n g s K . = A *B p a i r w i s e ; each i n p u t wave S . t h a t r e p l i c a t e s a p r e v i o u s w r i t i n g J j j J wave A . w i l l evoke a r e f l e c t e d wave R . o f f t h e g r a t i n g K . t h a t r e c o n s t r u c t s t h e J J 3 i . e . a v i r t u a l image o f t h e a s s o c i a t e d i n p u t p a t t e r n . a s s o c i a t e d w r i t i n g wave B j’ The same c o n c l u s i o n h o l d s f o r p a i r s o f i n p u t p a t t e r n s t h a t a r e each t h e s u p e r p o s i t i o n o f many 0 ( 1 ) modulated p l a n e waves, i n p a r t i c u l a r f o r p a t t e r n s composed o f s p h e r i c a l waves d i v e r g i n g f r o m source p o i n t s (e.g.
p1 and p,)
i n each p a t t e r n
(Figure 3). The analogy t o volume h o l o g r a p h y ( 1 3 ) i s e v i d e n t ; a b a s i c d i f f e r e n c e i s t h a t t h e r e c o r d i n g and g a i n f u n c t i o n s have been i n c o r p o r a t e d i n t o t h e same medium.
I n the
n e u r a l model t h e i n p u t waves have t h e i r sources i n t h e t e r m i n a t i o n s o f a f f e r e n t axons on t h e bodies o f c e l l s i n t h e sensory p r o j e c t i o n areas ( 1 4 ) .
An a f f e r e n t
v o l l e y causes d e p o l a r i z a t i o n i n t h e d e n d r i t i c t r e e o f i t s t a r g e t c e l l which has t h e same e f f e c t as r a i s i n g Vr
(and t h u s F) l o c a l l y : t h e t a r g e t c e l l becomes a
s c a t t e r i n g c e n t e r f o r b i f u r c a t i n g waves a t wc w h i c h r a d i a t e i n t o t h e s u r r o u n d i n g medium t o i n t e r f e r e w i t h waves s c a t t e r e d from o t h e r sensory i n p u t l o c i .
There i s
no n e c e s s i t y f o r an e x t e r n a l c o h e r e n t source; t h e g l o b a l b i f u r c a t i n g wave a t wc i s generated by t h e c o n v e c t i v e g a i n I m U w i t h i n t h e medium i t s e l f . I n t h e absence o f sensory i n p u t , r a i s i n g h beyond A, spontaneously r a d i a t e f r o m t h e s t o r e d g r a t i n g s
K
j’
causes b i f u r c a t i n g waves t o
f o r m i n g b o t h v i r t u a l and r e a l
Interacting Waves in a Neural Continuum Model
345
Figure 3 S p h e r i c a l waves d i v e r g i n g f r o m source p o i n t s p1 and p 2 form an h y p e r b o l o i d a l i n t e r f e r e n c e g r a t i n g (broken l i n e s ) w i t h f o c i p1,p2. A s p h e r i c a l wave from p 1 i s t h e n i n c i d e n t on t h e g r a t i n g , producing+a+reflected wave app a r e n t l y d i v e r g i n g f r o m p 2 . The Bragg c o n d i t i o n s s - r = G h o l d l o c a l l y : i n c i d e n t wave-vector - r e f l e c t e d wave-vector = g r a t i n g v e c t o r . images o f t h e a s s o c i a t e d p a t t e r n s A. and B . t h a t had i n t e r f e r e d t o c o n s t r u c t them J J i n t h e f i r s t p l a c e , R a d i a t i o n f r o m t h e most r e c e n t l y formed g r a t i n g s w i l l t e n d t o predominate, s i n c e t h e s e have t h e g r e a t e s t d e p t h K~
K
j*
R a d i a t i o n f r o m any g r a t i n g
may be s t i m u l a t e d by a s i n g l e newly i n p u t p a t t e r n Cm t h a t b e a r s a s i g n i f i c a n t
c r o s s - c o r r e l a t i o n t o e i t h e r o f t h e s t o r e d p a t t e r n s Am o r Bm t h a t had i n t e r f e r e d t o form
K ~ .
New g r a t i n g s CGA,
and CiB,
B i n a r y a s s o c i a t i o n s ATA2,A;A3,...,A;-1An
chains.
a s s o c i a t i n g Cm w i t h Am and Bm a r e t h e n s t o r e d . may be composed t o form associative
M u l t i p l e Bragg r e f l e c t i o n o f f t h e r e s u l t i n g s t a c k o f g r a t i n g s
K ~ ~ K ~ ~ , . . . , K ~t h-e~n Kr e~c o n s t r u c t s
f i r s t member.
t h e whole c h a i n s e q u e n t i a l l y upon i n p u t of t h e
Chains may be composed t o form c y c l e s ; i n p u t o f any member o f t h e
c y c l e A . w i l l evoke r e v e r b e r a t i n g a c t i v i t y i n t h a t c y c l e , r e c o n s t r u c t i n g a l l i t s J o t h e r members s e q u e n t i a l l y . I n p u t o f a p a t t e r n B . d i s j o i n t from the A c y c l e w i l l J evoke r e v e r b e r a t i n g a c t i v i t y i n t h e B c y c l e , l e a v i n g t h e A c y c l e q u i e s c e n t . The i m p l i c a t i o n s f o r p a t t e r n r e c o g n i t i o n by n e u r a l networks o r t h e i r o p t i c a l analogues a r e t a n t a l i z i n g : we c o n j e c t u r e t h a t recognition o f any newly i n p u t p a t t e r n as be-
longing t o a group o f previously associated patterns i s impZemented by evocation o f reverberating a c t i v i t y i n t h e corresponding associative c y c l e . F o r t h i s mechanism t o be e f f e c t i v e , t h e r e must be some means o f c o n s o l i d a t i n g t h e s h o r t - t e r m memory t r a c e K (which r e l a x e s a t a r a t e p ) i n t o a l o n g - t e r m change i n j t h e l o c a l c o n d u c t i o n p r o p e r t i e s o f t h e d e n d r i t i c o r s y n a p t i c membranes f o r graded
MS. Cohen
346 p o t e n t i a l waves.
S t r u c t u r a l and biochemical changes i n t h e d e n d r i t e s a r e indeed
imp1 i c a t e d i n t h e psychophysiology o f memory ( 1 5 ) . OPEN QUESTIONS
There remain many i n t e r e s t i n g q u e s t i o n s about t h e b e h a v i o r o f t h e model and i t s r e l e v a n c e t o s h o r t - t e r m memory.
Though N-wave s o l u t i o n s t o 14) a r e s t a b l e i n t h e
sense o f bounded energy, t h e medium may break up i n t o e n t r a i n m e n t domains about "leading centers," i . e .
sensory i n p u t l o c i w i t h d i f f e r e n t frequency s h i f t s w2,
separated by shock f r o n t s ( 1 6 ) b e f o r e any i n t e r f e r e n c e g r a t i n g s can be p r i n t e d . We c o n j e c t u r e t h a t t h i s occurs i f a m p l i t u d e s a r e " h a r d l i m i t e d " t h r o u g h t h e i n stantaneous c u b i c n o n l i n e a r i t y 7m g, b u t t h a t N wave s u p e r p o s i t i o n s a r e s t a b l e i f amplitudes a r e l i m i t e d t h r o u g h t h e d e l a y (A.G.C.)
n o n l i n e a r i t y lm F C l v l
2 ;TI;
Re F t h e n a c t s as a phase-lock l o o p c o n t r o l t o e n t r a i n c e n t e r s t o t h e same f r e quency.
A f t e r a g r a t i n g has been p r i n t e d , s u p e r p o s i t i o n s a r e s t a b i l i z e d by coup-
l i n g o f waves t h r o u g h t h e g r a t i n g . Whether o s c i l l a t o r y graded p o t e n t i a l waves do indeed propagate t h r o u g h n e u r a l n e t works w i t h l a t e r a l i n h i b i t i o n i s s t i l l an open q u e s t i o n ; h o p e f u l l y , e x p e r i m e n t a l i n v e s t i g a t i o n w i l l p r o v i d e t h e answer i n t h e n e a r f u t u r e .
'Assuming ( K " ) ~ , ( K ' ) $< E ( K ' Q ) ~i n 1) o r 2'') v * ( o v ) ~ appears.
-
2 c d 3 ) i n 4); otherwise, a term
REFERENCES S c h m i t t , F.O., Dev, P., and Smith, B.H., E l e c t r o t o n i c p r o c e s s i n g o f i n f o r m a t i o n by b r a i n c e l l s , Science 193 (1976) 114-120. Poggio, T. and T o r r e , V., A new approach t o s y n a p t i c i n t e r a c t i o n s , i n : Heim, R. and Palm, P. ( e d s . ) , L e c t u r e Notes i n Biomathematics 21 ( S p r i n g e r V e r l ag , B e r l i n , 1977). Katz, B. and M i l e d i , R., A s t u d y o f s y n a p t i c t r a n s m i s s i o n i n t h e absence o f n e r v e impulses, J. P h y s i o l . (Lond.) 192 (1967) 407-436. Ermentrout, B., S t a b l e s m a l l a m p l i t u d e s o l u t i o n s i n r e a c t i o n - d i f f u s i o n equat i o n s , Q u a r t . Appl. Math., A p r i l (1981) 61-86. K o p e l l , N. and Howard, L.N., Plane wave s o l u t i o n s t o r e a c t i o n - d i f f u s i o n e q u a t i o n s , Stud. Appl. Math. 52 (1973) 291-328. Sabah, N.H. and L e i b o v i c , K.N., S u b t h r e s h o l d o s c i l l a t o r y responses o f t h e Hodgkin-Huxley c a b l e model f o r t h e Squid G i a n t Axon, Biophys. J o u r . 9 (1969) 1206-1222. Mauro, A., C o n t i , F., Dodge, F., and Schor, R., S u b t h r e s h o l d b e h a v i o r and phenomenological impedance o f t h e Squid G i a n t Axon, J o u r . Gen. P h y s i o l . 55 (1970) 497-523. Hassard, B.D., K a z a r i n o f f , N.D., and Wan, Y.H., Theory and a p p l i c a t i o n s o f Hopf b i f u r c a t i o n , London Mathematical S o c i e t y L e c t u r e Note S e r i e s 41 (Camb r i d g e U n i v e r s i t y Press, Cambridge, 1981).
Interacting Waves in a Neural Continuum Model
347
(9)
Cohen, M., S t a b l e s u b t h r e s h o l d waves i n a n e u r a l continuum model w i t h l a t e r a l r e c i p r o c a l s y n a p s e s , preprint; s u b m i t t e d J o u r . Theor. B i o l . ( 1 0 ) Cohen, M . , A s s o c i a t i v e p a i r s , c h a i n s , and c y c l e s i n a n e u r a l continuum model, p r e p r i n t ; s u b m i t t e d J o u r . Theor. B i o l . (11) Marsden, J.E. and McCracken, M., The Hopf b i f u r c a t i o n and i t s a p p l i c a t i o n s , Applied Math. S c i e n c e s , Vol.19 ( S p r i n g e r - V e r l a g , New York, 1976). ( 1 2 ) Connor, J.A., Mechanisms of pacemaker d i s c h a r g e i n i n v e r t e b r a t e n e u r o n s , i n : C a r p e n t e r ( e d . ) , C e l l u l a r Pacemakers, Vol . I (Wiley, New York, 1982). (13) Kogelnik, H., Coupled wave t h e o r y f o r t h i c k hologram g r a t i n g s , Bell System Technical J o u r n a l 48 (1969) 2909-2947. ( 1 4 ) Anderson, P . and Andersson, S . A . , P h y s i o l o g i c a l b a s i s f o r the a l p h a rhythm ( A p p l e t o n - C e n t u r y - C r o f t s , New York, 1 9 6 8 ) . ( 1 5 ) R i n z e l , J . , Neuronal p l a s t i c i t y ( l e a r n i n g ) , i n : Miura, R. ( e d . ) , Lectures on Mathematics i n t h e L i f e S c i e n c e s , Vol.15 (American Math. S o c i e t y , P r o v i d e n c e , 1982). ( 1 6 ) Howard, L . N . and Kopell, N . , Slowly v a r y i n g waves and shock s t r u c t u r e s in r e a c t i o n - d i f f u s i o n e q u a t i o n s , S t u d . Appl. Math. 56 (1977) 95-145.
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.)
349
@ Elsevier Science Publishers B.V. (North.Holland), 1984
SOLITON DYNAMIC STATES IN THE SINE-GORDON SYSTEM Peter Leth Christiansen Laboratory of Applied Mathematical Physics The Technical University of Denmark DK-2800 Lyngby Denmark
The sine-Gordon system that models the Josephson tunnel junction oscillator is presented. This system is particularly representative of modern non-linear dynamics because it exhibits both soliton dynamic states and chaotic intermittency between such states. The paper discusses some recent computational results for the perturbed sine-Gordon equation with homogeneous and inhomogeneous boundary conditions. The results are compared with experimental measurements on the Josephson oscillator. MODELLING THE JOSEPHSON OSCILLATOR A Josephson tunnel junction consists of two superconductivity metal layers separated by a thin insulating oxide layer of uniform thickness (tax) that i s small enough to permit quantum-mechanical tunnelling of electrons. The geometry of an overlap junction is shown in Figure 1 .
//Superconductor
Figure 1 Josephson tunnel junction of overlap geometry [ 11. The region where the two superconducting layers overlap has the length L in the X-direction and the width W in the Y-direction. Typical values in experimental Josephson junctions are L 6 XJ and W 0.8 XJ where the Josephson length XJ 1.56 x m. Thus the overlap region is approximately 1 -dimensional.
-
-
-
P. L. Christiansen
350
The tunnelling supercurrent is described by the two basic Josephson equations j(X,T)
=
jo sin$
(1)
and
Here $ = $(X,T) is the difference between the phases of the order parameter of the two superconductors, T is laboratory time, and j(X,T) i s the Josephson current crossing the barrier per unit length in the X-direction, j o being the maximum current. The voltage drop across the insulating barrier is V = V ( X , T ) . Combination of Eqs. (1) and (2) with Maxwell s equations yields the following partial differential equation
Details of the derivation are given in Ref. [ I ] , which also gives references to the original works. In Eq. ( 3 ) Lp and Rp are inductance and resistance per length unit of the oscillator as illustrated in the equivalent circuit diagram for the junction in Figure 2 .
Figure 2 Element of lumped transmission line equivalent circuit representing the Josephson oscillator 111. The capacitance and the effective normal resistance per length unit are represented by C and G-l respectively. The externally applied bias current per length unit is j,, while Qo = h/2e = 2 . 0 6 4 ~ Wb is the magnetic flux quantum. Introducing the normalized coordinates, x = X / X J and t = Two, we obtain t h e perturbed sine-Gordon equation
Here the Josephson length and the Josephson plasma frequency are given by AJ = (Q0/2nj oLp) band wo = (2nj0/Q0 C)/, and the coefficients in E q . ( 4 ) are given j B / j o . Typical values of XJ and w0 are AJ = by a = G / w C D = wo Lp/Rp, and 1 . 5 6 ~10-40m and wo = 5 . 8 ~10 - such that the propagation velocity for electromagnetic signals (i.e. solutions of the linear wave equation, I # J ~ -~ @tt = 0 ) becomes c = A J W ~ = 9 . 0 5 ~l o 6 m/s in laboratory coordinates along a typical Josephson oscillator. In the normalized coordinates, x and t, this velocity is of course equal to unity.
2
At the ends of the junction, X boundary conditions:
=
0 and X
=
L , we apply the following two sets of
Sofiton Dynamic States in the Sine-Gordon System
351
(i) When no external magnetic field is applied (Hex = 0 in Figure 1 ) we approximate the physical situation by an open-end condition, i.e. zero current on the junctions at the ends. Since the current is proportional to $x we get @x(o,t)
=
@x(a,t)
(5a)
= 0
in normalized units with R = L / X J . Eq. (5a) is o n l y an approximation since the coupling between the junction and the surrounding microwave circuit is neglected in the equation. However, Eq. (5a) leads to reasonable agreement between computational results obtained with this boundary condition and the experimental measurements as we shall see. (ii) When an external magnetic field is applied (Hex + 0 in Figure 1 ) in the negative Y-direction the boundary condition becomes
where 17 is the normalized magnetic field 17
= (-
W/jo XJ)H,,.
The initial conditions for the computational modelling of the oscillator are @(x,O) = F(x)
and
@,(x,O)
=
G(x)
where the functions F and G are chosen such that the stationary soliton dynamic states of the oscillator are obtained in the numerical computations without too distributions from a nearby stationlong transients. In practice the final 4) and ary dynamic state in parameter space ( a , 13, y, R , and q ) are often used as initial values. The influence of small perturbations on the stationary soliton dynamic states is illustrated in Ref. [2].
DYNAMICS OF THE JOSEPHSON OSCILLATOR In this section we give a brief review of the soliton dynamic states which we have found by computational solution of Eqs. ( 4 - 6 ) . In the absence of an external magnetic field (i.e. boundary conditions (5a)) and with one soliton in the initial condition (6) the computer picture shown in Figure 3 results.
Figure 3 Computer solution to the perturbed sine-Gordon equation ( 4 ) with a = 0.05, B = 0 . 0 2 , y = 0.35. Boundary condition ( 5 ) with R = 6, T- = 0. One soliton in the initial conditions ( 6 ) . The inset shows ’$ (a,t) for 50 time units [ l ] . t
P. L. Christiansen
352
SOLITON TRAJECTORIES SOUTON
-
ANTISOLITON
-- -
ZFS 1
ZFS 2 Symmetric
ZFS 3 3- Bunch
ZFS 3 2 -Bunch
PLASMA OSCILLATION
..........
ZFS 2 2 -Bunch
-
1 Free
FSI
-
FS14
FS2 Symmetric
FS 2 2- Bunch
FS 3
Figure 4 S o l i t o n dynamic s t a t e s . ZFS = z e r o f i e l d s t e p . FS = F i s k e s t e p .
Soliton Dynamic States in the Sine-Gordon System
353
Initially, the soliton travels in the negative x-direction. Then it is reflected at the left end of the oscillator into an anti-soliton which travels in the positive x-direction until it hits the right end of the oscillator. At this boundary the anti-soliton is reflected into a soliton. Thus one cycle of the soliton motion is completed. The motion continues periodically and creates the electromagnetic radiation from the oscillator. In this stationary state the soliton and anti-soliton travel back and forth in the oscillator with a velocity determined by the energy balance in the system between the input energy (due to bias y) and the losses (due to the terms a@t and -B$,,t in E q . ( 4 ) ) . This stationary velocity can be determined by means of Hamiltonian perturbation theory as it is done in Ref. [31. When the soliton velocity implied by the initial conditions (6) deviates from the stationary velocity there will be a transient period in which the soliton velocity is adjusted from the initial velocity to the stationary velocity. Such a period is not visible in Figure 3 . A s we shall see in the following section the soliton dynamic states described above
give rise to a branch of the current-voltage characteristic for the oscillator designated the first zero field Step (ZFS 1 ) in the literature. With N solitons present on the oscillator ( N = 1,2, ) we get the N’th zero field step (ZFS N ) . In Figure 4 we have illustrated the various soliton dynamic states schematically by drawing the trajectories in the xt-plane between x = 0 and x = R. F u l l curves correspond to solitons, dashed curves to anti-solitons, and dotted curves to linear waves (often called plasma oscillations in the literature). Thus in the diagram marked ZFS 1 we see the schematic representation of Figure 3: Soliton trajectory in negative x-direction followed by anti-soliton trajectory in positive x-direction etc.
...
With two solitons in the initial conditions (6) we may get a different computer picture - shown in Figure 5 a and b - for different values of the parameters. For small values of y (Figure 5a) the two solitons travel in a symmetric configuration - i.e. soliton and anti-soliton in opposite directions as shown schematically in Figure 4 in the diagram marked ZFS 2 symmetric. For higher values of y (Figure 5b) the solitons travel in a bunched state - i . e . two solitons in negative x-direction followed by two anti-solitons in positive x-direction as shown in Figure 4 in the diagram marked ZFS 2 2-bunch. There is a gradual transition from the symmetric state to the 2-bunch state as the bias current y is increased and vice versa as y is decreased. Ref. [ l ] reports on a hysteresis phenomenon in connection with this transition. The computer picture for lower values of y (Figure 5a) resembles linear cavity oscillations. For higher values of y (Figure 5 b ) the non-linear character of the model becomes more dominant and as a consequence the bunched soliton state occurs. States with bunches of three solitons have also been found in certain parameter regions [ l ] and are illustrated schematically in Figure 4 as ZFS 3 3-bunch. However, in narrow intervals of the bias current y a different configuration occurs. In this soliton dynamic state two solitons travel in a bunch while the third soliton is "free" and travels with a slightly different velocity. The velocity difference in this state is exaggerated in the schematic diagram marked ZFS 3 2-bunch 1-free in Figure 4 . The fundamental frequency of the resulting electromagnetic radiation is given by f
=2Rv
(7)
where u is the soliton velocity in normalized units. In the soliton dynamic state just described we have two adjacent velocities, u and u + Au, present. Consequently, two frequency components, f and f + af, should be found in the frequency spectrum for the corresponding electromagnetic radiation. In the following section we shall see that this is indeed the case. In Refs. [ 2 ] and [ 4 1 we have recently investigated the case where an external (constant) magnetic field is present. Thus we get the boundary conditions (5b) with q + 0. The technical motivation for this research is to study the frequency tuning
P. L. Christiansen
354
a
.
Figure 5 Computer solutions to the perturbed sine-Gordon equation ( 4 ) with a = 0.05, B = 0.2. Boundary condition (5) with 8 = 6, q = 0 . Two solitons in the initial conditions ( 6 ) . (a): y = 0.125, ( b ) : y = 0.3.
of the oscillator caused by the external magnetic field. The tuning results from a change in the soliton dynamic state a s we shall see. For a relatively small value of Q the computer picture seen in Figure 6 i s obtained. Analysis of this picture shows a soliton travelling in the negative x-direction. = q at the left end of the oscillator absorbs energy The boundary condition @,(O,t) from the incident soliton. A s a result the minimum energy for the sine-Gordon s o l i ton (= 8 in normalized energy units) is no longer available. Therefore no anti-soliton i s reflected. Instead we observe the reflection of plasma oscillations which contain lessenergy. The plasma oscillations travel in the positive x-direction and excite a soliton at the right end of the junction. Here the boundary condition &(!Z,t) = q pumps energy into the system and makes the creation of this soliton possible. This i s the first cycle of a stationary state giving rise to a branch in
Soliton Dynamic States in the Sine-GordonSystem
355
Figure 6 Computer solution to the perturbed sine-Gordon equation ( 4 ) with a = 0.252, 6 = 0 , y = 0 . 5 4 . Boundary condition (5) with L = 5 , il = 0.75. One Soliton in the initial conditions (6). the current-voltage characteristic for the oscillator designated the first Fiske step (FS 1 ) in the literature. Again with N solitons present on the oscillaror ( N = 1,2,...) we get the N’th Fiske step (FS N ) . In Figure 4 we have illustrated the soliton dynamic state of Figure 6 schematically in the diagram marked FS 1 . For higher values of il we may have two solitons present on the oscillator as illustrated in the computer picture in Figure 7.
PHIX 5.13
5.00
-2 . E)6 12
Figure 7 Computer solution to the perturbed sine-Gordon equation ( 4 ) with a = 0.252, B = 0 , y = 0.71. Boundary condition (5) with 1 = 5, rl = 1.35. Two solitons in the initial conditions (6). Here the solitons and the plasma oscillation travel in opposite directions at the same time giving rise to the second Fiske step. In Figure 4 this soliton dynamic state is shown schematically as FS 2 symmetric. Our recent work [5] has demonstrated
356
P. L. Christiansen
that the solitons on FS 2 may also be bunc-hed as shown schematically in Figure 4 in the diagram marked FS 2 2-bunch. This shift from symmetric to bunched state on FS 2 is similar to the shift from symmetric to bunched state observed on ZFS 2. We have, however, only found bunched states when surface l o s s is present in the model, i.e. for D 9 0. In our recent Computations we have also found the third Fiske step indicated schematically in Figure 4 in the diagram FS 3 . In a certain small portion of parameter space ( a = 0.252, 6 = 0, y = 0 . 4 5 6 - 0 . 4 9 0 , J? = 5 , n = 1.25) we have found[4] that the oscillator shifts between the two soliton dynamic states denoted by FS 1 and FS 2. This is a new result. So far chaos phenomena have only been found in the literature for small and large Josephson junctions driven by an oscillating bias current (i.e. in the sine-Gordon system in the ordinary and partial differential equation cases with a sinusoidal perturbing term). Relevant references are given in [ 4 1 . By performing very long computer simulations for sets of parameter values in the region mentioned above we could study the statistics of the switching between FS 1 and FS 2. There is a chaotic intermittency between the two soliton dynamic states. Our results indicate that the switching can be described statistically by a Poisson process. There is an interesting asymmetry in the switching probability: On FS 1 the probability that the oscillator switches to FS 2 increases linearly with the time spent on FS 1 . In contrast, on FS 2 the probability that the oscillator switches t o FS 1 develops discontinuously as a function of the time spent on FS 2. As a consequence the oscillator has a preference to switch from FS 2 t o FS 1 after six oscillation cycles. The reason for this asymmetry is not understood at present. The phenomenon gives rise to a particular branch in the current-voltage characteristic for which we have proposed the name Fiske step 1 4 ” (FS 1 4 ) [41. A similar branch has been found experimentally [61 and may be interpreted as FS 14. COMPARISON BETWEEN COMPUTATIONAL RESULTS AND EXPERIMENTAL RESULTS From the computational results we may extract data which can be compared with experimental results. In particular we have compared
i) the computed current-voltage ( I - V ) characteristic and the experimentally measured I-V characteristic and ii) the power spectrum computed for the voltage at one end of the oscillator and the experimental measurements of frequency components of the electromagnetic radiation from the oscillator. The computer results may also be compared with direct dynamic measurements at the single soliton level which are beginning to appear in the literature. First we discuss the I-V characteristic. According to Eq. ( 2 ) the voltage on the oscillator (V) is proportional to &. The time-average value, <@t(O,t)> or <&(J?,t)>, therefore represents the dc-voltage at the end of the oscillator. The applied dccurrent (I) is the term y in Eq. (4) in our model. Figure 8 illustrates the agreementbetween computational results from the model and experimental measurements on the first three zero field steps. With applied magnetic field we find the Fiske step branches of the I-V characteristic both computationally and experimentally I S \ . It turns out that FS 2 overlaps ZFS 1, FS 4 overlaps ZFS 2, etc. while FS 1 lies between the axis (V = 0) and ZFS 1 , FS 3 lies between ZFS 1 and ZFS 2. Also various kinds of substructure of the FS s have been discovered [2 and 41. The power spectra of the radiation emitted from the overlapping ZFS 1 and FS 2 are, however, quite different as we shall see.
357
Soliton Dynamic States in the Sine-GordonSystem
.laQhso ZFS 1
31/--"^"^"" 2.5
0.0
1( 0
75
5.0
I,,(mA) Figure 8 DC-voltage v e r s u s a p p l i e d b i a s c u r r e n t showing t h e f i r s t t h r e e z e r o f i e l d s t e p s (ZFS 1 , ZFS 2 , ZFS 3 ) . For t h e n u m e r i c a l r e s u l t s we have used a = 0.05, = 0 . 0 2 i n E q . ( 4 ) , 9. = 6 , rl = 0 i n E q . ( 5 ) , and one, two, and t h r e e s o l i t o n s i n E q . ( 6 ) on ZFS 1 , ZFS 2 , and ZFS 3 r e s p e c t i v e l y . Furthermore y N I d c and < $ t ( O , t ) > vdc. C i r c l e s i n d i c a t e t h e n u m e r i c a l r e s u l t s and s o l i d c u r v e s r e p r e s e n t the experimental r e s u l t s [ I ] .
-
-1st
5 0 harm.
40
I
0.0
,
I
I
1 0.5
'
I
,
,
1.(I
Dc-Bias
Figure 9 F i r s t harmonic of power v e r s u s t h e d c - b i a s y on ZFS 3 o b t a i n e d by n u m e r i c a l s o l u t i o n of E q . ( 4 ) w i t h a = 0.05, 6 = 0 . 0 2 , E q . (5) w i t h II = 6 , II = 0 , and t h r e e s o l i t o n s i n Eq. ( 6 ) [ I ] . F i g u r e 9 i l l u s t r a t e s t h e f i r s t harmonic of t h e r a d i a t e d power from t h e o s c i l l a t o r v e r s u s t h e dc-bias on t h e t h i r d z e r o f i e l d s t e p . I n t h e range of r a p i d o s c i l l a t i o n s (y = 0.34 - 0.41) two a d j a c e n t f r e q u e n c y components a r e p r e s e n t i n t h e s i g n a l . I n t h i s i n t e r v a l t h e computer p i c t u r e s show t h a t t h e s o l i t o n dynamic s t a t e c o n s i s t s of
358
P. L. Christiansen
two bunched solitons and one "free" soliton as mentioned in the previous section. I n Figure 10 we have computed the power of the first three harmonics of bt(0,t) as a function of time for a y-value ( 0 . 4 0 ) in the oscillatory interval in Figure 9. The oscillations in Figure 10 indicate the presence of two adjacent frequency components separated by A f = 50 MHz.
1st horrn
v
- _ _2nd
20/-
-’OI -20
4
harm
1 0
2000
4000
6000
8000 10000
TIME Figure 10 Harmonics in power versus time on ZFS 3 by numerical solution of E q . (4) with a = 0.05, R = 0.02, y = 0 . 4 0 , Eq. (5) with R = 6 , q = 0, and three solitons in Eq. ( 6 ) [ l ] .
6
I
Frequency, GHz
Figure 1 1 Frequency display of experimentally observed microwave radiation from niobium oxid-lead Josephson junction. For this junction the l o s s parameters were c1 = 0.05, 13 = 0.02. The length was !L = 6 in normalized units. No external magnetic field was applied. Traces a and b were obtained for bias currents (in normalized units) y = 0 . 1 4 and 0.115 respectively [ 7 ] . Figure 1 1 shows corresponding experimental results. I n the frequency sweep, marked a, there is a single frequency component at 8.86 GHz. This peak corresponds to three bunched solitons on ZFS 3 (Figure 4: ZFS 3 3-bunch). I n the frequency sweep, marked b, there are two frequency components at 8.76 GHz and 8.81 GHz correspond-
359
Soliton Dynamic States in the Sine-Gordon System
-120
F
D
a
I
v
/2
40
-0.2 Time - t
Frequency
-
f
a
-120 m a v
5 80 P,
40
Time
-
t
b
0 -0.2 .O .2 .4 .6 .8 1.0 Frequency - f
Figure 12 Time series for Qt(0,t) and corresponding power spectra (normalized to = 1 . 0 ~10-l’). Parameter values a = 0.05, B = 0.02 in Eq. ( 4 ) , .I = 2 in Eq. ( 5 ) , suitable initial conditions. (a): y = 0.15, ri = 0 . ( b ) : y = 0 . 1 6 , r \ = 2.5.
@:
P. L. Christiansen
360
ing to two bunched solitons and one "free" soliton on ZFS 3 (Figure 4 : ZFS 3 2-bunch 1-free). The distance between the two peaks in the frequency spectrum is approximately 50 MHz. This experimental result thus agrees with the computational results of Figure 10. It is noteworthy that the soliton dynamic computational model of the Josephson oscillator can reproduce even such subtle details in the power spectrum for the electromagnetic radiation. We wish to remark that Josephson oscillators with radiation frequencies around and above 100 GHz have also been designed. (This can be achieved for s h o r t e r junctions with f a s t e r solitons according to Eq. (7)). The reason why a junction emitting radiation around 10 GHz was chosen for the reported studies is simply limitations of the microwave circuits in the experimental set-up available for the frequency measurements. Clearly, it is important to have programs available which can immediately compute the power spectrum for &(O,t) resulting from the numerical simulation of the oscillator. Figure 12 thus shows time series and corresponding power spectra obtained by such programs in two different cases. In Figure 12a ( q = 0) and Figure 12b ( q = 2.5) the oscillator is operating on ZFS 1 and FS 2 respectively. As mentioned the two steps overlap each other in the current-voltage diagram. Nevertheless the corresponding soliton dynamic states (Figure 4: ZFS 1 and FS 2 symmetric) are quite different. As a consequence the power spectra for the radiation in the two cases are also very different. On ZFS 1 we see strong frequency components at fl = 0.2 and f2 = 0 . 4 4 in Figure 12a. On FS 2 we only get a frequency component at f2 = 0 . 4 4 in Figure 12b. That the frequency component fl must vanish as we pass from ZFS 1 to FS 2 (turning on the external magnetic field) is obvious since the soliton number per time unit passing through the system is doubled. At present further studies of the power spectra of the electromagnetic radiation on the different steps are carried out and compared with experimental measurements. ACKNOWLEDGEMENT The financial support of the European Research Office of the United States Army through contract No. DAJA 37-82-C 0057 is gratefully acknowledged. REFERENCES Lorndahl, P.S., Soerensen, O.H., and Christiansen, P.L., "Soliton excitations in Josephson tunnel junctions", Phys. Rev. B (1982) 5337-5748. Soerensen, M.P., Christiansen, P.L., Parmentier, R.D., and Skovgaard, O., "Subharmonic generation in Josephson junction fluxon oscillators biased on Fiske steps", Appl. Phys. Rev. Lett. (1983) 739-741. McLaughlin, D.W. and Scott, A.C., "Perturbation analysis of fluxon dynamics", Phys. Rev. A (1978) 1652-1680. Soerensen, M.P., Arley, N., Christiansen, P.L., Parmentier, R.D., and Skovgaard, O., "Intermittent Switching between Soliton Dynamic States in a Perturbed SineGordon Model", Phys. Rev. Lett. (1983) 1919-1922. Soerensen, M.P., Parmentier, R.D., Christiansen, P.L., Skovgaard, O., Dueholm, B., Joergensen, E., Koshelets, V.P., Levring, O.A., Monaco, R., Mygind, J., Pedersen, N.F., and Samuelsen, M.R., "Magnetic field dependence of microwave radiation in intermediate length Josephson junctions" (to appear). Cirillo, M., Costabile, G., and Parmentier, R.D., "Possible observation of chaotic intermittency effect in a long Josephson junction in a magnetic field" (to appear). Lomdahl, P.S., Soerensen, O.H., Christiansen, P.L., Scott, A.C., and Eilbeck, J.C., "Multiple frequency generation by bunched solitons in Josephson tunnel junctions", Phys. Rev. B (1981) 7460-7462.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.E. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
36 1
AN ASYMPTOTIC ANALYSIS OF ACOUSTIC SCATTERING BY NEARLY RIGID OR SOFT TARGETS
Gregory A. Kriegsmann, Andrew Norris’
and Edward L. Reiss
Department of Engineering Sciences and Applied Mathematics The Technological Institute Northwest ern Univer s ity Evanston IL 60201
1.
Introduction
The qualitative features of the field scattered by a penetrable acoustic target, insonified by a time periodic incident wave, depends on a number of factors. Among them are:
the frequency of the incident wave: the densities and indicies
of refraction of the target (pt and nt respectively) and the surrounding acoustic fluid (pa and n , respectively); and the shape of the target. ratio
E-P
a
/p
t
is small (large) then the target
is
If the density
nearly rigid (nearly soft).
When the incident frequency is hounded away from all the "in-vacuo" or resonant frequencies of the target, then the scattered field is given essentially by the field scattered either by the rigid or by the soft target, depending on whether
E
is small or large.
We refer to either the rigidly or the softly
scattered fields as the background fields.
However, when the incident frequency
is nearly resonant, then the target oscillates in response to the incident wave with relatively large amplitudes, resulting in a scattered field comparable in amplitude to the background field.
In previous publications [ 1 , 2 ] , we have employed the method of matched asymptotic expansions
(MMAE)
[ 3 ] to obtain asymptotic approximations of the scattered fields
that are uniformaly valid in the incident field frequency as E+O nearly rigid (soft) targets.
(m)
for the
The outer expansion of the MMAE is a small pertur-
bation to the background field.
It is invalid, that is, it becomes unbounded,
as the incident frequency approaches a resonant frequency of the target.
The
inner expansion is valid when the incident frequency is near a resonant frequency. The composite expansion of the MMAE gives the desired uniform asymptotic approximation.
We emphasize that in the usual applications of the MMAE to acoustics
and other problems, see e.g. [3,4], the singularities, or boundary layers, develop as the space variables approach special values.
In the present analysis
the singularities develop as a parameter, which is proportional to the incident frequency, approaches special values, namely the resonant frequencies of the target.
362
G.A. Kriegsrnann et al.
The classical resonant scattering approximation of nuclear scattering theory, which has been extended and applied to acoustical scattering problems 151, corresponds to our inner expansion.
However, to obtain the classical resonant
scattering approximation, which does not require large density mismatches between the target and its surrounding fluid, it
is
necessary to first explicitly
solve the scattering problem, e.g., by partial wave expansions, and then expand these solutions near the resonant frequencies of the target.
Thus, resonant
scattering theory is applicable to essentially separable geometries and n !constant.
In our method, which is applicable to problems with large density
t
mismatches, but for arbitrary targets, it is not necessary to first solve the scattering problem since the structure of the scattered field and its dependence on the incident frequency is given explicitly by the asymptotic approximation. Furthermore, the asymptotic approximation is uniformly valid in the incident frequency.
Moreover, the analysis suggests more efficient methods for numer-
ically solving such scattering problems, as we have discussed previously [1,2], since the "singularity" that arises near resonance is analytically resolved.
In this paper we give a brief exposition of the method for the elementary problem of one dimensional scattering by a uniform, nearly rigid acoustic layer.
Since
this problem can be solved explicitly we then compare our asymptotic result with this solution.
A similiar analysis is applicable to the nearly soft acoustic
layer, but we do
not present these results.
The asymptotic method is applicable
to other scattering problems such as for elastic and electromagnetic waves, as we shall demonstrate in future publications.
2. A
Formulation uniform layer (the target) of thickness L lies in the interval OjXjL in an
infinite acoustic medium with constant density p tion n
.
by pt and nt. parameter
and constant index of refrac-
The constant density and index of refraction of the layer are denoted
E
We assume the layer is nearly rigid so that we define a small
by
We consider the one dimensional scattering problem of a time periodic wave proportional to e-iwt that is propagating from the left in the surrounding fluid and is normally incident on the surface X=O of the layer. assumed to be independent of Y and - iut
X, upon omitting the factor e
.
2
The incident wave is
so that the scattered field depends only on
We denote the acoustic potentials in the surrounding fluid and in the target by Ga(x;k) and G t (x;k) respectively. They
363
Acoustic Scattering by nearly Rigid or Soft Targets
s o l v e t h e s c a t t e r i n g problem
(2.2a)
OaXx
+
k20a
(2.2b)
@:x
+
k2n20t
(2.2c)
:O
=
:O
=
0
=
, ot
, 0
f o r x < 0 and x > 1 ;
,
for 0 < x < 1
,
=
;
f o r x = 0,1
;
a n d t h e r e s u l t i n g s c a t t e r e d f i e l d s a r e o u t g o i n g f r o m t h e t a r g e t a s Ixl+m.
In
( 2 . 2 ) , t h e s u b s c r i p t s x d e n o t e d i f f e r e n t i a t i o n , a n d we h a v e u s e d t h e f o l l o w i n g notation,
where C
and C
respectively.
t
a r e t h e sound v e l o c i t i e s o f t h e s u r r o u n d i n g f l u i d and t a r g e t , The i n t e r f a c e c o n d i t i o n s ( 2 . 2 )
a r e c o n t i n u i t y c o n d i t i o n s on t h e
a c o u s t i c p r e s s u r e and v e l o c i t y .
A wave 0 ' .
which i s d e f i n e d b y ,
i s i n c i d e n t from
x=-m
on t h e t a r g e t .
The r e s u l t i n g s c a t t e r e d f i e l d c o n s i s t s o f
a r e f l e c t e d wave i n x l which
is o u t g o i n g a s x+=. i n Section 4.
The e x a c t s o l u t i o n o f t h i s s c a t t e r i n g p r o b l e m i s p r e s e n t e d
I t is e a s i l y o b t a i n e d f o r any v a l u e o f E .
However,
f o r exposi-
t o r y p u r p o s e s we assume t h a t i t i s " s i m p l e r " t o s o l v e ( 2 . 2 ) f o r t h e r i g i d t a r g e t . T h u s , f o r E=O t h e s o l u t i o n o f
( 2 . 2 ) t h a t i s i n r e s p o n s e t o t h e i n c i d e n t wave
( 2 . 4 ) a n d t h a t s a t i s f i e s t h e o u t g o i n g wave c o n d i t i o n s a s Ixl+m i s a wave r i g i d l y r e f l e c t e d a t x=O a n d p r o p a g a t i n g t o x=--m.
(2.5)
I t is g i v e n by,
I
o
,
x > l .
H e r e , t h e b a c k g r o u n d f i e l d O a is t h e sum of t h e i n c i d e n t a n d r l g i d l y r e f l e c t e d B -ikx ) f o r xO, b e c a u s e t h e t a r g e t i s r i g i d s o t h a t waves ( e t h e r e is no t r a n s m i t t e d f i e l d .
W e e x p r e s s t h e s o l u t i o n of t h e n e a r l y r i g i d s c a t t e r i n g p r o b l e m E # O
(2.2)
as
364
G.A. Kriegsrnann eta/.
(2.6)
Then,by inserting (2.6) into (2.2) and using (2.5), we deduce that the scattered potentials $a and
Ot
must satisfy the Helmholtz equations (2.2a) and (2.2b),
respectively, the boundary conditions:
a
+x
(2.7b)
=
t
t
+x ,
= E + ~
,
for x
1
=
and the appropriate radiation conditions on Qa as
;
Ixl+m.
We now reformulate this scattering problem for $a and t problem for $ . To do this we express the potential
and the outgoing radiation conditions as Ixl-,
- ikx Rse $a
i
and T
t @I
it is given by
,
for x 5 0 ,
,
for x
=
! T elkx
where R
as a boundary value in terms of values of
Since $a satisfies the Helmholtz equation (2.2a)
on the target boundaries x=@,1.
(2.8)
Ot
s
>
1
,
are the scattered reflection and transmission coefficients, to
be determined. The total reflection and transmission coefficients are related to R
and Ts by
(2.9)
R
1 + Rs
,
T
Z
T
By inserting (2.8) into the boundary conditions (2.7) and eliminating R
and T
from the resulting four equations, we obtain the required boundary value problem t for Q as
(2.10b)
When (2.10 is solved then R
and T
are given by
Acoustic Scattering b y nearly Rgid or Soft Targets
(2.11)
R
l t
s
k x (0) ,
=-$
T
=-
and hence the scattered potential ’$
3.1
365
is given by (2.8)
The Outer Expansion
To solve (2.10), we first seek an asymptotic expansion of $ t in the form
Then by inserting (3.1) into (2.10) and equating coefficients of the same powers t of E , we find, in the usual way, that satisfies
The solution of (3.2) is given by
(3.3)
where for u=1,2, ..., the functions JI ( X I
the constants k
are defined by
are given by V
(3.4b)
k,, = v n / n ,
and the parameters B
are defined by
The functions $ (x) and parameters k the "in vacuo" target problem
are the eiqenfunctions and eigenvalues of
G.A. Kriegsmann et a/.
366 (3.5.3)
2 2 Jixx+knJi
=
0 ,
O < x < l
The eigenfunctions are normalized by the condition ($u,@u)
=I, where we use the
inner product,
for any two functions f and g.
All the eiqenvalues of (3.5) are simple.
For
more complex scattering problems the Corresponding eiqenvalues of the target may be multiple, see e.g. [l].
Higher order terms in (3.1) can similarly be computed. In particular, t t satisfies (3.2a) and the boundary conditions Q ( O ) = i Q (O)/k and 2 1.x t t t Q2(1)=-iQ (l)/k. Thus, Q, is proportional to l/k and our outer expansion l,x becomes, to leading order
(3.6a)
Q t (x,k;E)
=
t
E $ ~ ( x , + ~ )O ( E
2 /k)
It is clear from (3.3) that the outer expansion (3.6.3) is invalid as k+k
u=1,2,3... because then
Qi+-
as k-k
.
v ' Furthermore, it i s invalid as k+k EO 0 2
because the second term in (3.6a) is O ( E /k).
The corresponding leading order outer expansions for R
and T
which follow from S'
(2.11), (3.4) and (3.6a), are
(3.6b)
These functions become singular as k+k , u=0,1,2, ...
3.2
.
The Inner Expansion
We now obtain asymptotrc representations of Qt which are valid for k near k fc,r each fixed u=1,2,...
.
Accordlngly, we define the "inner variable" a by
367
Acoustic Scattering by nearly Rigid or Soft Targets
k = k (1+Ea)
(3.7)
This Lmplies that a=O(1) when k-k = O ( E ) .
Then we seek an asymptotic expansion
for $t, valid for a=0(1),in the form
Inserting (3.7) and (3.8) into (2.10) and equating coefficients of the same powers of
E,
we find, for example, that Uo and U1 satisfy:
(3.9a)
U
(3.10a)
Ulxx
(3.10b)
U1(0)
Since k
+ k 2n2U
v
oxx
+
0
0
=
-2k2an2U
2 2 k n U v 1 2
=
+
,
=
' 1u k
Ox
is an eigenvalue of (3.51,
f o r O < x < l ,
,
(0)
it
f o r O < x < l ,
U](l)
=
- 1.
UOX(l)
.
V
follows from (3.9) that
is to be determined.
where the constant A
,
When (3.11) is inserted into (3.10)
an inhomogeneous eigenvalue problem must be solved to determine U (x). To 1 insure the existence of U (x) a solvability condition must be satisfied. Appli1 cation of this condition leads to a linear equation for A . Omitting all details we finally obtain
(3.12)
Thus, to leading order, and for each fixed v=1,2 ...
(3.13a)
$t
=
A
$J
v v
(x)
+
O(E)
, we have the inner expansion
.
The corresponding inner expansions, to leading order, for R (2.,11) with k=k
V’
(3.4a), (3.7), (3.12), and (3.13a).
and T
They are
follow from
G.A. Kriegsrnann et al.
368 (3.13b)
R~
=
i f i A"
(3.13~)
T
=
-ifi
+
O(E)
.
+ o(E)
Finally, we investigate the solution of (2.10) when k is near k
0
. We intro-
duce the new stretched variable a by
(3.14)
k
a~
=
and seek an asymptotic expansion for Q
t
valid for a = 0 ( 1 ) , of the form
InsertLng (3.14) and ( 3 . 1 5 ) into (2.10) and equating the coefficients of the same powers of
(3.16)
we find that U1 satisfies
E,
Ulxx = 0, O < x < l ,
U (O)+iaU (0)= 2ia lx 1
,
Ulx(l)-icuU (1) = 0 1
.
The solution of (3.16) is
and thus, to leading order, (3.15) gives (3.17b)
Qt
.
eQ0(x,a)+O ( E 3 )
=
The corresponding inner expansions, to leading order, for R
and T
valid near k=O, follow from (2.11), (3.14), ( 3 . 1 7 ) , and (3.18a).
(3.18)
3.3
= --
R S
2i~ + k+2i~
2
O ( E ) ,
T
=
~
2 1 ~ e-lk + k+2ie
which are They are
2
O(E )
.
The Matching Conditions
In the method of matched asymptotic expansions it is assumed that there exists an overlap interval in which the outer and inner expansions are valid asymptotic approximations o f the solution of the scattering problem.
This implies that in
this interval the expansion coefficients must satisfy matching conditions.
They
are obtained by first expressing the outer expansions in terms of the stretched variable rr by using (3.7) or (3.14) and then reexpanding the result as a new
Acoustic Scattering by nearly Rigid or Soft Targets
power serLes in
E
to give
at
(3.19)
369
m
=
1,
-t $j(XtU)E1
.
i=O
The series (3.19), which we call the residual expansion, is the outer expansion in terms of the stretched variable in the overlap interval.
The matching
conditions are then given by
1im
(3.20)
, j
1
=
0,1,..
Omitting all details, we can show that the conditions of (3.20) are identically satisfied.
3.4
and T
Similar calculations and results follow for R
.
The Uniform Expansion
The composite expansion of the method of matched asymptotic expansions is defined as the sum of the outer and inner expansions minus the residual expansion.
Thus,
by employing (3.3). (3.6)-(3.8), (3.11), and (3.12) we obtain to leading order
(3.21.3)
(3.21b)
These expressions are uniformly valid in an interval containing the simple eigenvalues k
or ko, respectively.
the k=k. ( j # v ) . 3
It is valid for any other k bounded away from
We obtain an asymptotic expansion that is uniformly valid for
all k by summing the last term in (3.21) over all values of v .
We find that
at
becomes
(3.22a)
$t
=
2~ [k(1-X)+IE k+21~
1 k2-k2
m=l
Similar considerations and calculations for R
m
(3.22b)
R
=
-2i~-+21 [k+iie
m=l
{
+
and T
- -k k2-k2
(k-k
m
21fiE ) (k-k
give the uniform results
ie (k-k ) (k-k +2ie) m m
il
370
G.A. Kriegsrnann et al.
(3.22~)
A uniform asymptotic expansion of the scattered potential now follows by
inserting (3.16)-(3.17), (3.22b) and ( 3 . 2 2 ~ )into (2.8).
4.
A Comparison of Exact and Asymptotic Results
The exact solution of the scattering problem (2.10) is
0,
(4.1)
for any value of
(4.2)
t =
~
a(l+an) eikn(x-l) A
+
(an-1)
A
-ikn(x-l)
e
20, where 2 2 2an cos(kn) - i(l+a n ) sin(kn)
A
.
By inserting these results into (2.11) we obtain the exact scattered reflection and transmission coefficients, respectively as
(4.3)
RE S
(4.4)
TS
E
=
=
-2an cos(kn)
+
2ia2n2 sin(kn)
A
2ane
-ik
/n
.
We now show the relationships between the exact results (4.1)-(4.3) and the uniform asymptotic results by taking various limits.
We first consider the outer limit in (4.3) where kfk m is fixed and a+O. gives
(4.5)
R~
=
This
.
-2ia cot(kn) + O(E 2)
S
In the same limit the uniform asymptotic approximation (3.22b) is reduced to the outer expansion (3.6b).
The equivalence between these two results is demon-
strated by inserting the Mittag-Leffler expansion for cot(kn) ,
(4.6)
into ( 4 . 5 ) .
cotikn)
=
2
['
+ 2
m=l
4 1 k -k
E
A similar agreement can be shown in this limit between TS and ( 3 . 6 ~ ) .
37 1
Acoustic Scattering b y nearly Rigid or Soft Targets
The outer limit of t r$E
(4.7)
r$g from
(4.1) i s
- 2 ~ sin kn(x-1) + sin (kn)
=
2
O(E )
.
To demonstrate the equivalence between (4.7) and the outer limit (3.6ai of (3.22a) we first rewrite (4.7) as
where g(x) is defined by,
sin kn(x-1) sin(kn)
-2(1-x) - 2
z
g(x)
(4.8b)
Since g(x) is a smooth function and q(l)=g(O)=O, it can be expanded in an eigenfunction series using the IJ (x). A straightforward calculation shows that the m 2 2 /(k -k ) . Inserting this result in (4.8) shows
expansion coefficients are
m
that (4.7) and (3.6) are identical.
We now consider the inner limit where a
(k-k )/(Ek ) and v > 0 are fixed, and
=
V
E + 0.
In this limit the exact scattered reflection coefficient (4.3) is
(4.9)
which agrees exactly with the inner limit of RS given by (3.13b).
Similar
agreement is obtained for the transmission coefficient.
I n the inner limit the exact potential given by (4.1) is
t ’ E
(4.10)
=
-2 sin VTX n[kua+21)
f
O(E)
.
This is identical to the inner limit given by (3.12) and (3.13) of the uniiorm asymptotic approximation (3.22~1). Similarly, we can show that the inner limit of the exact result and of the uniform asymptotic approximation are identical when a
=
k/E O ( 1 ) and
E
+
0. E
Finally, we have made graphs of RE and RS, and of TS and TS, to numerically evalS uate the accuracy of the uniform asymptotic approximations. For E < . 1 and k in the interval O
372
G.A. Kriegsmann et a/.
indistinquishable. Footnotes ’This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 80-0016A, the National Science Foundation under Grant No. MSC 8300578 and the Office of Naval Research under Grant N00014-76-C-0063 and Contract N o . N00014-83-C-0518. 2Now at Exxon Research and Engineering Corp., Clinton, N . J .
Re ferences
[l] Kriegsmann, G.A., Norris, A.N. and Reiss, E.L., Acoustic scattering by baffled membranes, J. Acoust. Soc. of Amer., in press. 121 Kriegsmann, G.A., Norris, A.N., and Reiss, E.L., Scattering by penetrable acoustic targets, Wave Motion, in press. [31 Cole, J.D., Perturbation Methods in Applied Mathematics (Blaisdell, Waltham, MA, 1968). [4] Kriegsmann, G.A. and Reiss, E.L., Low frequency scattering by local inhomogeneities, SIAM J. Appl. Math. 43,923 (19831. [5] Flax, L., Gaunaurd, G. and Uberall, H., The resonance scattering theory, in Physical Acoustics, ed. by W.P. Mason and R.W. Thurston, 15,Academic, NY (1981), p. 193.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers 8.V. (North-Holland), 1984
373
SYMMETRY REDUCTION FOR NONLINEAR WAVE EQUATIONS I N RIEMANNIAN AND PSEUDO-RIEMANNIAN SPACES A.M.Grundland,
J.Harnad, and P . W i n t e r n i t z
C e n t r e de recherche de mathematiques a p p l i q u e e s U n i v e r s i t e de M o n t r e a l M o n t r e a l , Quebec Canada H3C 357
N o n l i n e a r e q u a t i o n s o f t h e f o r m H(Au, ( V U ) ~ , u ) = 0 a r e discussed, where H i s an a r b i t r a r y s u f f i c i e n t l y smooth f u n c t i o n o f t h r e e v a r i a b l e s , u =U(xl, xn) i s an unknown
...,
s c a l a r f u n c t i o n and A and (V.)2 a r e t h e Laplace and g r a d i e n t square o p e r a t o r s on some n-dimensional Riemannian o r pseudoRiemannian m a n i f o l d M. I t i s shown t h a t t h i s e q u a t i o n can be reduced t o a d i f f e r e n t i a l e q u a t i o n i n k ( 1 5 k 5 n ) v a r i a b l e s by making t h e Ansatz t h a t u s h o u l d depend o n l y on k ) . These s h o u l d some new v a r i a b l e s 5, (xl ,..., xn) ( a = l ,
...,
be chosen t o be k independent i n v a r i a n t s o f a subgroup G o c G h a v i n g g e n e r i c o r b i t s o f codimension k i n M . A c l a s s i f i c a t i o n o f subgroups o f G t h u s p r o v i d e s a c l a s s i f i c a t i o n o f a l l p o s s i b l e symmetry v a r i a b l e s . The examples o f n + l dimensional Minkowski space and o f t h e h y p e r b o l o i d xf+x$+x$-xz-xf = 1 :re t r e a t e d i n d e t a i l . S p e c i a l a t t e n t i o n i s devoted t o degenerate symmetry v a r i a b l e s " i n v o l v i n g a r b i t r a r y f u n c t i o n s o f n u l l v a r i a b l e s i n pseudo-Riemannian spaces. (The group G i s t h e group o f i s o m e t r i e s o f M . ) 1. INTRODUCTION The purpose o f t h i s p r e s e n t a t i o n i s t o show how group t h e o r y can be s y s t e m a t i c a l l y employed t o reduce n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n n independent v a r i a b l e s t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n fewer v a r i a b l e s and, i n p a r t i c u l a r , t o ordinary d i f f e r e n t i a l equations. More s p e c i F i c a l l y , we c o n s i d e r n o n l i n e a r wave e q u a t i o n s i n an n-dimensional Riemannian o r pseudo-Riemannian space M w i t h m e t r i c
(throughout, summation o v e r r e p e a t e d i n d i c e s i s t o be u n d e r s t o o d ) . t i o n s considered a r e o f t h e type H ( A ~ (, o u f ,
u ) = 0,
where H i s some g i v e n s u f f i c i e n t l y smooth f u n c t i o n , s c a l a r f u n c t i o n and
(real)
The equa(1.2)
u(xl,
...,xn)
a r e t h e L a p l a c e and g r a d i e n t square o p e r a t o r s i n M, r e s p e c t i v e l y .
i s an unknown
Our aim i s t o
A.M. Grundland et a/.
374
reduce t h e dimension o f e q u a t i o n ( 1 . 2 ) by i n t r o d u c i n g a new s e t o f independent variables Ea(xl
,...,x k ) ,
i n such a manner t h a t t h e Ansatz
E~
,...,Ek
only.
u = u(c
a = 1
,...,k <
l,... , E ~ )
n
(1.4)
reduces ( 1 . 2 ) t o a PDE i n
F o r k = 1 we o b t a i n an ODE as a s p e c i a l case.
The m o t i v a t i o n f o r p e r f o r m i n g t h i s t y p e o f d i m e n s i o n a l r e d u c t i o n i s an o b v i o u s one, namely t o o b t a i n p h y s i c a l l y i n t e r e s t i n g p a r t i c u l a r s o l u t i o n s o f e q u a t i o n Lower dimensional e q u a t i o n s a r e e a s i e r t o t r e a t t h a n h i g h e r dimensional (1.2). ones; O D E ' S are,at l e a s t i n p r i n c i p l e , e a s i e r t o s o l v e than PDE's. I n a d d i t i o n , many techniques f o r i n t e g r a t i n g c e r t a i n c l a s s e s o f PDE’s, such as t h e i n v e r s e s c a t t e r i n g t r a n s f o r m , 62cklund t r a n s f o r m a t i o n s , e t c . , a r e e s s e n t i a l l y r e s t r i c t e d t o e q u a t i o n s w i t h 2, o r e x c e p t i o n a l l y 3, independent v a r i a b l e s ( A b l o w i t z and Segur (1981), Fokas and A b l o w i t z ( 1 9 8 3 ) ) . The r e d u c t i o n t o n=1,2, o r a t most 3 v a r i a b l e s i s t h u s o f p a r t i c u l a r i n t e r e s t and we can, i n combination w i t h o t h e r techniques, p r o v i d e whole c l a s s e s o f s o l u t i o n s o f e q u a t i o n (1.2). Equations o f t y p e (1.2) a r e o f c o n s i d e r a b l e i n t e r e s t i n a p p l i c a t i o n s . c l u d e wave e q u a t i o n s o f t h e t y p e
They i n -
Au = F ( u ) , (1.5) such as t h e sine-Gordon e q u a t i o n ( F ( u ) = s i n u ) , t h e L i o u v i l l e e q u a t i o n ( F ( u)=eU) and many o t h e r s ( u s u a l l y s t u d i e d i n f l a t space). They a l s o n c l ude t h e e i konal e q u a t i o n ( o r Hamilton-Jacobi e q u a t i o n w i t h a " p o t e n t i a l " V(u depending on Hamilton's c h a r a c t e r i s t i c f u n c t i o n u) ( 0 ~ +) V(u) ~
=
E*
f i g u r i n g i n c l a s s i c a l and quantum mechanics as w e l l as i n general r e l a t i v i t y theory. The main r e s u l t t h a t we w i s h t o emphasize i s t h a t t h e t h e o r y o f L i e groups makes i t p o s s i b l e t o s y s t e m a t i c a l l y g e n e r a t e "symnetry v a r i a b l e s " E~ t h a t p r o v i d e a r e d u c t i o n . Indeed, l e t t h e m e t r i c ds2 on t h e space M have an i s o m e t r y group G. T h i s group w i l l ' l e a v e e q u a t i o n (1.2) i n v a r i a n t . A subgroup Go c G, h a v i n g g e n e r i c o r b i t s o f codimension k i n M, can be used t o o b t a i n k symnetry v a r i a b l e s , namely t h e k f u n c t i o n a l l y independent i n v a r i a n t s o f G o , whose l e v e l s e t s a r e these o r b i t s . A c l a s s i f i c a t i o n o f subgroups o f G t h u s p r o v i d e s a c l a s s i f i c a t i o n o f s e t s of symmetry v a r i a b l e s . Furthermore, t r a n s f o r m a t i o n s b e l o n g i n g t o t h e group G make i t p o s s i b l e t o i n t r o d u c e symmetry v a r i a b l e s t h a t depend on a r b i t r a r y c o n s t a n t s and i n some cases on a r b i t r a r y f u n c t i o n s .
The i d e a o f symmetry r e d u c t i o n i s an o l d one - as o l d as t h e t h e o r y o f L i e groups i t s e l f ( L i e (1891), ( 1 8 8 5 ) ) . More r e c e n t l y , a p p l i c a t i o n s o f group t h e o r y t o t h e s t u d y o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s have been t r e a t e d i n s e v e r a l books (Ames (1965), Bluman and Cole (1974), I b r a g i m o v (1967), Ovsiannikov (1982), and W i n t e r n i t z ( 1 9 8 3 ) ) . Our main c o n t r i b u t i o n i s t h e s y s t e m a t i c use o f t h e subgroup s t r u c t u r e o f t h e i n v a r i a n c e group G, t o generate, i n a w e l l d e f i n e d sense, a l l p o s s i b l e symmetry v a r i a b l e s f o r a g i v e n e q u a t i o n . Methods f o r c l a s s i f y i n g c l o s e d connected subgroups of L i e groups i n t o conjugacy c l a s s e s and f o r c o n s t r u c t i n g r e p r e s e n t a t i v e s of each conjugacy c l a s s have o n l y become a v a i l a b l e q u i t e r e c e n t l y (Patera, W i n t e r n i t z and Zassenhaus (1975a,b), Patera, Sharp, W i n t e r n i t z and Zassenhaus (1976, 1 9 7 7 ) ) . Symnetry r e d u c t i o n i s a s t a n d a r d method f o r o b t a i n i n g s o l u t i o n s o f t h e E i n s t e i n e q u a t i o n s o f g e n e r a l r e l a t i v i t y , whenever t h e y a d m i t n o n t r i v i a l K i l l i n g v e c t o r s (Misner, Thorne and Wheeler (1973), K i n n e r s l e y (1980), and E r n s t ( 1 9 6 8 ) ) . Recentl y we have used i t t o o b t a i n new s o l u t i o n s of t h e sine-Gordon e q u a t i o n (Grundland, Harnad and W i n t e r n i t z ( 1 9 8 2 ) ) i n n + l dimensional Minkowski space M(n,l) and a l s o
375
Symmetry Reduction for Nonlinear Wave Equations
t o s t u d y e q u a t i o n (1.2) i n M ( n , l ) (Grundland, Harnad and W i n t e r n i t z (7984)). S i m i l a r s t u d i e s have been performed f o r t h e n o n l i n e a r S c h r d d i n g e r e q u a t i o n u + i u t = F(u,u*,x,t) (Boyer, Sharp and W i n t e r n i t z (1976), and Harnad and xx W i n t e r n i t z ( 1 9 8 2 ) ) and t h e Y a n g - M i l l s e q u a t i o n s i n t h e c o m p a c t i f i e d Minkowski space (Harnad, S h n i d e r and V i n e t (1979a,b)). I n S e c t i o n 2 we p r e s e n t some r e s u l t s on symmetry r e d u c t i o n i n general and on soc a l l e d "degenerate symmetry v a r i a b l e s " i n p a r t i c u l a r . S e c t i o n 3 i s devoted t o examples, namely n + l dimensional Minkowski space and t h e 4-dimensional c u r v a t u r e space r e p r e s e n t e d by t h e 0(3,2) h y p e r b o l o i d X ? + X $ + X ~ - X $ - X=$ 1 . As an a p p l i c a t i o n of t h e method o f symmetry r e d u c t i o n we d i s c u s s i n S e c t i o n 4, a " g e n e r a l i z e d hyperbolic L i o u v i l l e equation" U
Au = ee +x. E = i1, x = const. (1.7) and show t h a t f o r x # 0 t h i s e q u a t i o n does n o t s a t i s f t h e " P a i n l e v @ c r i t e r i o n " ( A b l o w i t z , Ramani and Segur (1Y80)). Indeed, we show t i a t i n g e n e r a l , s o l u t i o n s o f (1.7) o b t a i n e d by symmetry r e d u c t i o n w i l l have moving c r i t i c a l p o i n t s . 2.
GENERAL RESULTS ON SYMMETRY REDUCTION
Our f i r s t r e s u l t s a r e t h e necessary and s u f f i c i e n t c o n d i t i o n s f o r a s e t o f k v a r i a b l e s t o p r o v i d e a dimensional r e d u c t i o n o f e q u a t i o n ( 1 . 2 ) . We b e g i n by formu l a t i n g t h e r e d u c t i o n procedure i n an i n v a r i a n t g e o m e t r i c a l manner. A l t h o u g h t h e f o r m u l a t i o n w i l l be g l o b a l , i n a p p l i c a t i o n s t h e r e s u l t s a r e o f t e n v a l i d o n l y on s u i t a b l y d e f i n e d " l a r g e " open s e t s , which t y p i c a l l y i n v o l v e t h e complements o f c e r t a i n l o w e r dimensional m a n i f o l d s . I n t e r p r e t i n g t h e v a r i a b l e s Ea as c o o r d i n a t e s on a m a n i f o l d N, such t h a t t h e map n:M
n(x
1
+
,...,xn) *
N
(E
1
,...,E, k )
i s a submersion, we s h a l l seek c o n d i t i o n s on n a s s u r i n g t h a t a l l e q u a t i o n s o f t h e t y p e ( 1 . 2 ) reduce t o d i f f e r e n t i a l e q u a t i o n s on N f o r f u n c t i o n s u which a r e const a n t on t h e f i b r e s o f n ( i . e . t h o s e which a r e p u l l b a c k s o f f u n c t i o n s on N). It i s c l e a r t h a t , i n p a r t i c u l a r , t h e L a p l a c i a n must p r o j e c t t o a s u i t a b l e second o r d e r d i f f e r e n t i a l o p e r a t o r on N (when r e s t r i c t e d t o such f u n c t i o n s ) . I t i s easy t o v e r i f y t h a t t h i s c o n d i t i o n i s a c t u a l l y necessary and s u f f i c i e n t . T h i s i s t h e c o n t e n t o f o u r f i r s t theorem. Theorem 1: The necessary and s u f f i c i e n t c o n d i t i o n f o r a l l e q u a t i o n s o f t y p e (1.2) t o p r o j e c t t o second o r d e r d i f f e r e n t i a l e q u a t i o n s on N (when r e s t r i c t e d t o f u n c t i o n s u which a r e t h e p u l l b a c k s of f u n c t i o n s on N) i s t h a t t h e L a p l a c i a n on M p r o j e c t s t o a second o r d e r d i f f e r e n t i a l o p e r a t o r on N. Before p r o v i n g Theorem 1, we r e f o r m u l a t e i t i n terms o f c o o r d i n a t e s as f o l l o w s . Theorem 1 ' : 1 n x ), a = l , k 5 N , be a s e t o f f u n c t i o n a l l y L e t Ea(x which we i n t e r p r e t as c o o r d i n a t e s on M. The Ansatz 1 k u ( x I . . . , x n ) = $ ( E l ,...,E )
,...,
...,
independent v a r i a b l e s
(2.2)
w i l l reduce t h e e q u a t i o n s (1.2) t o a d i f f e r e n t i a l e q u a t i o n i n t h e v a r i a b l e s c1 , . . . , E ~ , if and o n l y i f these new v a r i a b l e s s a t i s f y a
( 8 5 ,VE
b
1
=
p
Aga = a a ( E
a b 1
(E ,.-.,E
1
,...,E k 1,
k
1
(2.3a) (2.3b)
376
A.M. Grundland et al.
where pab, a a a r e some ( a r b i t r a r y ) f u n c t i o n s o f t h e i n d i c a t e d v a r i a b l e s . reduced e q u a t i o n t h e n i s
The
(2.4) Comment: The e q u i v a l e n c e o f t h o s e two f o r m u l a t i o n s i s c o n t a i n e d i n t h e f a c t t h a t a r b i t r a r y l i n e a r second-order d i f f e r e n t i a l o p e r a t o r s on N can be l o c a l l y expressed as: n
-
I n p a r t i c u l a r , assuming t h e c o n d i t i o n s of Theorem 1, we have, f o r u =
don
and choosing + t o be t h e c o o r d i n a t e f u n c t i o n s I S a } o r t h e p r o d u c t s ( c a c b ) , we o b t a i n r e l a t i o n s (2.3a,b). The converse i s immediate. Proof: s a t i s f y (2.2), i . e .
be a p u l l b a c k o f a f u n c t i o n on N and l e t us a p p l y t h e
Laplace and g r a d i e n t square o p e r a t o r s t o u =
+on
( w i t h 4 c Cm(N)).We o b t a i n
<
9
AU
( A E a ) a+
('Vga,VEb) *+
=
aE ac
(2.5a)
ac (2.5b)
where (oca,,gb)
= gPJ
a
ax' &a
=
1_
b
a5: ag , ax"
a d i SF" a 5a*
fi axCL
axv
We see t h a t (2.5a) and (2.5b), and hence a l s o (2.4) N, i f and o n l y i f (2.3a) and (2.3b) a r e s a t i s f i e d .
a r e we1
d e f i n e d e q u a t i o n s on
Q.E.D.
Theorem 1 p e r t a i n s t o dimensional r e d u c t i o n as such. Group t h e o r y i s used t o generate s o l u t i o n s o f e q u a t i o n s (2.3) i n t h e f o l l o w i n g way: Theorem 2: L e t G be t h e L i e group o f i s o m e t r i e s o f t h e Riemannian o r pseudo-Riemannian space M w i t h m e t r i c g as i n ( 1 . 1 ) and l e t L be i t s L i e a l g e b r a , i . e . t h e L i e a l g e b r a o f K i l l i n g vector f i e l d s x^ = x q x ) , x = ( x 1 , . . . I xn ) , (2.6) ax' L e t Go c G be a s a t i s f y i n g L X g = 0 ( % i s t h e L i e d e r i v a t i v e w i t h respect t o X).
a
.,
L i e subgroup o f G, h a v i n g g e n e r i c o r b i t s o f codimension k i n M. Assume t h a t t h e space o f o r b i t s G,\M s N may be g i v e n a m a n i f o l d s t r u c t u r e , such t h a t t h e p r o j e c t i o n n:M + N t a k i n g a p o i n t i n t o i t s o r b i t i s a submersion. Such a submersion 1 k s a t i s f i e s t h e c o n d i t i o n s o f Theorem 1. The f u n c t i o n s 15 ( x ) , . . . , s ( x ) ) on N a r e any s e t o f smooth f u n c t i o n a l l y independent i n v a r i a n t s o f G. They can be o b t a i n e d by s o l v i n g t h e e q u a t i o n s A
xis(x)
= 0,
i = 1,
...,r ,
(2.7)
377
Symmetry Reduction for Nonlinear Wave Equations 1
where {X,, variants t o (2.4).
... % Xr }
ca,
i s a basis o f L
.
(the X . ' s are K i l l i n g vector f i e l d s ) .
o
1
The i n -
a = l ,. , ,k s a t i s f y e q u a t i o n s ( 2 . 3 ) and hence reduce e q u a t i o n s (1.2)
Proof: Since t h e Xi's A
rator A.
a r e K i l l i n g v e c t o r s , t h e y commute w i t h t h e Laplace B e l t r a m i ope-
We hence have ii(AEa)
= A(iiEa)
= 0,
i=1,
...,r ,
a=l
,..., k.
(2.8)
Since Go has p r e c i s e l y k f u n c t i o n a l l y independent i n v a r i a n t s (codimension k ) , and k 5 , i . e . (2.3a) h o l d s . aga i s a s o l u t i o n o f ( 2 . 7 ) , i t must be a f u n c t i o n o f 5 ' ) Furthermore we have
...,
ii(VEEa,VEb) = Xitg(vEa,vEb)l = L A d v E a ,vcb;+g[9(iiEa)
,vEb!+g(vEa,v(Xicb)
xi
= o
1
(2.9)
n
( t h e f i r s t t e r m vanishes because Xi i s a K i l l i n g v e c t o r , t h e l a s t two because o f b k E , so ( 2 . 7 ) ) . Thus, (vga,vE ) s a t i s f i e s ( 2 . 7 ) and must be a f u n c t i o n o f E l , t h a t (2.3a) h o l d s . Q.E.D.
...,
D e f i n i t i o n 1: 1 We c a l l t h e v a r i a b l e s 5 ables".
,... ,gk,
s a t i s f y i n g (2.7) "codimension-k s y m e t v y vari-
Theorem 2 p r o v i d e s a s y s t e m a t i c procedure, which we c a l l symmetry reduction, f o r g e n e r a t i n g t h e v a r i a b l e s E~ o f ( 2 . 2 ) - ( 2 . 4 ) . c o n s t i t u t i n g an a l g o r i t h m .
I t consists o f the f o l l o w i n g steps,
1. Given e q u a t i o n ( 1 . 2 ) , f i n d t h e i s o m e t r y group G o f t h e u n d e r l y i n g space c o n s t r u c t a b a s i s o f K i l l i n g v e c t o r s ( 2 . 6 ) f o r t h e i s o m e t r y a l g e b r a L.
M and
2. C l a s s i f y a l l subalgebras L o o f L i n t o conjugacy c l a s s e s under t h e a c t i o n o f t h e group G and choose a r e p r e s e n t a t i v e o f each conjugacy c l a s s .
3. Determine t h e codimension o f t h e
generic o r b i t s i n
t h e r e p r e s e n t a t i v e s e t o f subalgebras. presentative algebra.
M f o r each a l g e b r a Lo i n 1 k cR,...,5 o f each r e -
Find the invariants
R
..
1 k 4. Extend t h e " r e p r e s e n t a t i v e i n v a r i a n t s " E ~ . . ,cR i n t o general i n v a r i a n t s k e l , . . . , ~ by c o n j u g a t i o n under t h e group G. T h i s w i l l t a k e t h e r e p r e s e n t a t i v e a l g e b r a L O i n t o a g e n e r a l a l g e b r a i n t h e same conjugacy c l a s s and w i l l i n t r o d u c e a c e r t a i n number o f f r e e parameters i n t o t h e v e c t o r f i e l d s and hence i n t o t h e i n v a r i a n t s , i.e., t h e general codimension k symmetry variables. 5. S u b s t i t u t e t h e symmetry v a r i a b l e s i n t o ( 2 . 1 ) and o b t a i n t h e reduced d i f f e r e n t i a l equation. B e f o r e c o n s i d e r i n g some examples i n t h e n e x t s e c t i o n , we s h a l l d i s c u s s a n o t h e r phenomenon, namely t h a t of n u l l v a r i a b l e s and degenerate symmetry v a r i a b l e s i n pseudo-Riemannian spaces. Consider a space M w i t h m e t r i c g o f s i g n a t u r e ( p , q ) , p+q = n, p 2 q 2 1. D e f i n i t i o n 2: We s h a l l c a l l a s e t o f v a r i a b l e s
q
1
,. . . ,qt
" n u l l variables" i f t h e y s a t i s f y
A.M. Grundland et al.
378
i ' ( v q ,vqJ) = 0,
A?'
i , j = 1 ,..., G.
= 0,
A t most q independent n u l l v a r i a b l e s e x i s t i n M.
orthogonal n u l l variables
f o r p r o j e c t i o n n:M
k C'+',...,E }, i f i n a d d i t i o n t o (2.101, o f the coordinate functions s a t i s f y :
i (Vq ,vCa) = 0, Now l e t
{c',
...,E k }
+
(2.10)
We s h a l l c a l l t h e v a r i a b l e s qi
N t o a space N={q1 ,...,qG ,
t h e p u l l - b a c k s q a ( x ) = q a w , t;i =
i = 1 ,...,C ,
a = C+1,
... ,k.
(2.11)
be a s e t of codimension k symmetry v a r i a b l e s , c o n t a i n i n g 1 c ,q We p u t
,. ..
orthogonal n u l l variables q
c
.
Ea P Ca ,
,...,C ,
i = l
a=G+l
,...,k.
(2.12)
I n view o f r e l a t i o n s (2.10) and ( 2 . 1 1 ) , a l l d e r i v a t i v e s w i t h r e s p e c t t o t h e n u l l v a r i a b l e s drop o u t o f t h e reduced e q u a t i o n s and we have k 2 k Au(qi,Ea) = C pab 2 aa a,b=8 aCaazb a=8 aza (2.13)
au,
au
Thus, a s e t o f codimension k symmetry v a r i a b l e s , i n v o l v i n g 8 o r t h o g o n a l n u l l v a r i a b l e s , reduces e q u a t i o n (1.2) t o a PDE i n k-8 v a r i a b l e s , r a t h e r t h a n k v a r i ables. The n u l l v a r i a b l e s ql,...,qC may s t i l l f i g u r e i n t h e reduced e q u a t i o n , ab i b u t o n l y as parameters i n t h e c o e f f i c i e n t s p (q ,<') and II~(~~,E~). Since t h e r e d u c t i o n procedure i n v o l v e s t h e q u o t i e n t i n g o f t h e space M by a subgroup Go of t h e i s o m e t r y group G, i t i s r e l e v a n t t o determine t h e c o n d i t i o n s l e a d i n g t o o r t h o g o n a l n u l l v a r i a b l e s qi ( s a t i s f y i n g (2.10) and ( 2 . 1 1 ) ) i n terms o f t h e L i e a l g e b r a L o . T h i s w i l l be done r i g o r o u s l y in a subsequent p u b l i c a t i o n ; here we s i m p l y l i s t t h e c o n d i t i o n s on L o t h a t we use i n o r d e r t o o b t a i n a t l e a s t one n u l l v a r i a b l e q . L e t a b a s i s f o r t h e L i e a l g e b r a Lo c o n s i s t o f t h e K i l l i n g v e c t o r s {Y,X1, We r e q u i r e t h a t a t l e a s t one o f t h e s e v e c t o r s , say Y, t o a l l o t h e r s i n Lo, 1.e. A . .
(Y,Y)
..A
0,
=
(Y,Xi)
= 0,
i = 1
...,X,}.
be i s o t r o p i c and o r t h o g o n a l
,...,r.
(2.14)
I n a d d i t i o n , we r e q u i r e t h a t Y be an i d e a l i n Lo: A
,
.
[x.,Y] 1
A
= a.Y, 1
i = 1
,...,r.
(2.15)
The importance o f n u l l v a r i a b l e s i s t h a t t h e y make p o s s i b l e t h e i n t r o d u c t i o n o f a new t y p e o f symmetry v a r i a b l e , namely "degenerate symmetry variabZes", i n v o l v i n g a r b i t r a r y f u n c t i o n s o f t h e n u l l v a r i a b l e s . These a r b i t r a r y f u n c t i o n s i n t h e new independent v a r i a b l e s make i t p o s s i b l e t o o b t a i n more general p a r t i c u l a r s o l u t i o n s o f t h e o r i g i n a l PDE t h a n t h e usual symmetry v a r i a b l e s , s i n c e t h e l a t t e r o n l y i n volve a r b i t r a r y constants. The a l g o r i t h m t h a t one uses f o r o b t a i n i n g a s e t o f s y n e t r y v a r i a b l e s i n v o l v i n g a t l e a s t one degenerate symmetry v a r i a b l e , i s t h e f o l l o w i n g :
379
Symmetry Reduction for Nonlinear Wave Equations
1. Y
c
Given t h e i s o m e t r y a l g e b r a L o f t h e space M, f i n d an i s o t r o p i c K i l l i n g v e c t o r L.
2. Q u o t i e n t M by
?,
t h a t i s , determine t h e p r o j e c t i o n ny:M
dimension n-1, such t h a t t h e p u l l - b a c k o f f u n c t i o n s a n n i h i l a t e d by
?.
I n o t h e r words, s o l v e t h e e q u a t i o n ?F(x
(oq)'
=
,,...,xn)
..
I n t r o d u c e c o o r d i n a t e s (q,tl,.
= 0.
(2.16)
on No s a t i s f y i n g
( o q , v g i ) = 0,
A? = 0,
0,
6 c Cm(NO) c o n s i s t s
o f functions
{F=$on}
t o a space o f
No
-+
i=7
,..., n-2,
(2.17)
so t h a t q i s a n u l l v a r i a b l e . T h i s can always be done s i n c e t h e n u l l v a r i a b l e q i s determined by t h e c o n d i t i o n t h a t dq be p r o p o r t i o n a l t o t h e c o v e c t o r o b t a i n e d by l o w e r i n g t h e i n d i c e s on Y. n
n
n
Lo = {Y,Z l,...,Zr}
3. C o n s t r u c t a subalgebra
o f t h e i s o m e t r y a l g e b r a L, s a t i s -
fying
(v",?)
=
0,
(?,?i)
[ii,?] = ui?,
0,
=
and hence SEi = 0,
2.q = 0, 1
i
n-2
n
?q = 0,
ziga
c
=
...,r,
i=l,
p
b= 1
i 5 +a q, ab b a
(2.18)
...
i=l,, r .
(2.19)
4. C l a s s i f y t h e subalgebras o f L O c o n t a i n i n g Y i n t o conjugacy c l a s s e s under t h e a c t i o n o f G, choose a r e p r e s e n t a t i v e o f each conjugacy c l a s s and f i n d i t s i n v a r 1 S i i i a n t s iq,zR,...,zR}, 1 5 s .= n-2. We have ZR = ZR(Cl ,...,5n-2). 5. L e t t h e group G o , c o r r e s p o n d i n g t o L o , a c t on N o and l e t t h e elements g 6 G o i be f u n c t i o n s of ?,i.e. g=g(q).The v a r i a b l e s ER, i=l s, w i l l be t r a n s f o r m e d
,...,
.
i n t o new v a r i a b l e s (c',. , ,rS), i n v o l v i n g a r b i t r a r y f u n c t i o n s o f q. t h e degenerate symmetry v a r i a b l e s . 6. Use t h e v a r i a b l e s (cl PDE i n s v a r i a b l e s .
Example;
,...,c S ) t o
These a r e
reduce t h e o r i g i n a l PDE i n n v a r i a b l e s , t o a
o f t h i s procedure a r e g i v e n i n S e c t i o n 3 below.
3. EXAMPLES OF SYMMETRY REDUCTION INVOLVING ORDINARY AND DEGENERATE SYMMETRY VARIABLES
3.1 Symmetry Reduction t o O D E ' S i n Minkowski Space M(n,l) The symmetry group o f n i l dimensional Minkowski space, i . e . 2 2 ds2 = dx -dx 0 1
-. . .-dxn
2
f l a t space w i t h m e t r i c (3.1)
i s t h e P a i n c a r 6 group ( t h e inhomogeneous L o r e n t z group) P ( n , l ) . I t s L i e algebra p ( n , l ) i s spanned by i n f i n i t e s i m a l homogeneous L o r e n t z t r a n s f o r m a t i o n s M and vv ' satisfying translations P P'
[M
P.v
,M
P5
1=
M
PP vcJ
-9
M
+gwMpv-gvDMpli.
VP I-La
(3.2a)
A.M. Grundland et al.
380 [M
P ] = g
pv’ u
[P
p,v=O,l
,. . . ,n,
w’
gaa = -1
goo = 1,
P-g
pD v
(3.2b)
P
vu p
P ] = 0,
(3.2~)
v
,
a=l
,. . . ,n,
V9,
The c o r r e s p o n d i n g K i 11 ing v e c t o r s a r e rotations)
= xaab-xbaa
MOa = -x a -x a Oa a 0
boosts)
P
trans1a t i o n s )
K
= a
P
,...,n,
0,l
p =
a,b = 1
L e t us now t u r n t o e q u a t i o n (1.2) o f S e c t i o n 2.
,... ,n .
f o r ds
Step 1 has j u s t been performed: the isometry algebra L = p(n,l).
2
as i n (3.1)
and f o l l o w t h e p r e s c r i p t i o n
t h e K i l l i n g v e c t o r s (3.3) p r o v i d e a b a s i s f o r
Step 2 c o n s i s t s o f f i n d i n g a l l subalgebras o f p ( n , l ) . Since we a r e i n t e r e s t e d i n r e d u c t i o n s t o O D E ’ S , t h e c o r r e s p o n d i n g subgroups o f G = P ( n , l ) must have g e n e r i c F o r a d e t a i l e d d i s c u s s i o n we o r b i t s o f codimension 1 ( h y p e r s u r f a c e s i n M ( n , l ) ) . r e f e r t o a p r e v i o u s a r t i c l e (Grundland, Harnad and W i n t e r n i t z ( 1 9 8 4 ) ) , here we c o n s i d e r j u s t one example, namely t h e a l g e b r a : a,b=3,.
{Mab3MOa-M1a ,pa ,Mo1+pP2 ’Po-P1 1 , Find t h e i n v a r i a n t s o f (3.4).
Step 3.
,..., x n )
PaF(xo,xl we f i n d ally,
F = F(x +x ,x ) ;
0
1
2
M
ab
F
= 0
aF ( M ~ , + ~ P ~ =) F -?-+
arl
implies
(3.4)
Imposing
(P 0 -P 1 )F(xo,xl
= 0,
.. .n.
and
,...,xn)
= 0
(MOa-Mla)F
aF ax2
p -
=
= 0,
is
a=3
,...,n
s a t i s f i e d automatic-
q = x + x 0 1
0,
F = F(ER),
ER
=
x2 + p l n ( x 0+x1 ) .
(3.5)
Step 4. Apply a g e n e r a l P ( n , l ) t r a n s f o r m a t i o n t o t h e space M(n,l) and t h e v a r i a b l e cR, We o b t a i n t h e general codimension 1 symmetry v a r i a b l e 5 a s s o c i a t e d w i t h t h e c l a s s o f L i e a l g e b r a s r e p r e s e n t e d by ( 3 . 4 ) , namely =
(_x.+~,!)
and
3
5 where
A,
Step 5.
Put
2 (05)
+ p,ln(?+h,fi),
=
-1,
are constant vectors i n M(n,l),
N~ = p
<
0,
($,PI)
=
o
(3.6)
IR.
u = u ( ~ ) ; we have
= -1,
A5 = 0,
= -us, H(-u ,-u2,u) = 0. ( 3 . 7 ) 55’ 55 5 Dropping a l l d e t a i l s , we summarize t h e r e s u l t s on symmetry r e d u c t i o n t o an ODE f o r e q u a t i o n (1.2) i n M(n,l) as a theorem, proven elsewhere (Grundland, Harnad and Winternitz (1984)).
AU =
-U
381
Symmetry Reduction for Nonlinear Wave Equations
Theorem 3: be a L i e group w i t h g e n e r i c o r b i t s o f codimension 1 i n M ( n , l ) , L e t G c P(n,l) ( h y p e r s u r f a c e s i n M ( n , l ) ) . Then G has p r e c i s e l y one i n v a r i a n t t h a t can be obt a i n e d by a P o i n c a r 6 t r a n s f o r m a t i o n f r o m one o f t h e f o l l o w i n g r e p r e s e n t a t i v e i n v a r i a n t s cR: xl,
xO
q=xo+xl,
p=x t g ( x +x 12, u = x 2 + a l n ( x +x 1 , (o+a 2 0 1 0 1
2 2 2 1 r = ( x t x +...+ x ~ + ~ ) ' (,k = l , k 1 2 The Ansatz
..., n - l ) ,
T
k
2 2 = ( x - x -...-x:)', 0 1
c
(k=l,
IR)
(3.8a)
...,n ) .
(3.8b)
u = u ( ~ ) reduces e q u a t i o n ( 1 . 2 ) t o t h e ODE H(x[uEt;t pk c l , x u c 2, u ) = 0.
(3.9)
( i . e . q i s a n u l l v a r i a b l e and (3.9) r e The v a l u e o f x i s x = 0 f o r q = xO+xl duces t o an a l g e b r a i c e q u a t i o n ) , x = 1 f o r x o and T ~ x, = -1 i n a l l o t h e r cases. The v a l u e of k i n (3.9) i s i n d i c a t e d by t h e s u b s c r i p t f o r rk and T and we have k k=O i n a l l o t h e r cases. Comnent : A general P o i n c a r 6 t r a n s f o r m a t i o n takes t h e s e t o f v a r i a b l e s (3.8) i n t o t h e s e t
where
k2 =
4, Fa, 0,
PJ and =
are constant vectors i n M(n,l),
0, (Ba,f-
=
01, ( k a , j b )
=
jt
=
g2 =
-1,
-
T2
= 1,
-6ab*
A general v a r i a b l e i n (3.10) reduces eq. ( 1 . 2 ) t o t h e same ODE ( 3 . 9 ) as t h e c o r responding representative v a r i a b l e (3.8). Degenerate symmetry v a r i a b l e s i n M ( n , l ) a r e d i s c u s s e d below i n S e c t i o n 3.3.
3.2 Symmetry Reduction t o O D E ' S i n a Space o f Nonzero Constant C u r v a t u r e L e t us c o n s i d e r a f o u r - d i m e n s i o n a l h y p e r b o l o i d x2 +x2 +X 2 - x 2 -x 2 1 2 3 0 4 = l '
(3.11)
The group o f i s o r n e t r i e s i s t h e de S i t t e r group 0(3,2) and t h e h y p e r b o l o i d can be i d e n t i f i e d w i t h t h e q u o t i e n t space 0 ( 3 , 2 ) / 0 ( 2 , 2 ) . We i n t r o d u c e h o r o s D h e r i c coo r d i n a t e s ( V i l e n k i n ( 1 968)) :
xo = re-Ysha,
x1 = c h y - f r 2 e - Y ,
x2 = re-Ychasi n$, (3.12)
x3 = re-Ychacos$,
x 4 = -sh++r 2 e -Y
i n which t h e m e t r i c , L a p l a c e o p e r a t o r and g r a d i e n t square a r e ds2 = gabdxadxb = -(dy)2-r2e-2y(da)2+e-2y(dr)2+r2e-2Ych2a(d$)2, 2 2 2 2Y 2 1 a + . A= -- a + 3 a - q g - + t h a "1 + e z y ( C + 2 2 2 -7) r2 aa aY ar2 ar r ch a a$ ay2
-
(3.13)
A.M. Grundland et al.
382
respectively. We s h a l l use a b a s i s f o r t h e L i e a l g e b r a o f 0 ( 3 , 2 ) i n which t h i s group i s i d e n t i f i e d as t h e group o f conformal t r a n s f o r m a t i o n s o f a 2+1 dimensional Minkowski space (Beckers, Harnad, Perroud and W i n t e r n i t z ( 1 9 7 8 ) ) . U s i n g t h e same n o t a t i o n as i n (3.2a), b u t w i t h
MPv
gcLv = diag(1,1,1,-1,-1) we c a l c u l a t e i n h o r o s p h e r i c c o o r d i n a t e s :
=E+r--a
D s M 14
L
3
--M
23
=
ar
ay = _ -a
a4 a
K1
z
M30 = -cosb-
Kp
t
M20 = -sin4-
Po
z
M10-M40
aa
a
aa
+ thasinb-
a
a+
-
thacos4-
a
a$
a
= shaF -
cha a -r aa
(3.15)
P1
-M31+M34 = -chacos$- a + p1 h a c o s 6 - a + s i n + ar aa
P2
M12+M24
= -chasin+d
+ +hasin&! 1
ar C C
0
1
= Zrsha-a + (r2+eZy)sha&+ +M 10 40 aY
=
M
=
-M
31
-M
34
= -2rchacos$- a
aY
-
+ ( - r 2+e 2 y )-s i n $
?-
aa
r
a
r c h a a$
-(r cha r
2 -e 2 y )-a aa
)r a
(r2+e2')chacos6T a + ( - r2+e2 y shacos$ aa
a
r c h a a$
cp
M12-M24
= - 2 r c h a ~ i n $- ~(r2+e 2Y )chasin$$
ay
2 2 y cos9 + ( r -e
1%
a
+
2 2v ) shasin9 a r aa
( - r +e
a6
We a g a i n f o l l o w t h e p r e s c r i p t i o n o f S e c t i o n 2. Step 1. We have j u s t c o n s t r u c t e d a c o n v e n i e n t b a s i s o f K i l l i n g v e c t o r s o f 0(3,2), namely ( 3 . 1 5 ) . Step 2. A l l subalgebras o f 0(3,2) a r e known (Patera, Sharp, W i n t e r n i t z and Zassenhaus ( 1 9 7 7 ) ) . We again l i m i t o u r s e l v e s t o t h e case when t h e c o r r e s p o n d i n g subgroups o f 0(3,2) have g e n e r i c o r b i t s o f codimension 1 on t h e h y p e r b o l o i d (3.11). Step 3. F i n d t h e i n v a r i a n t of each subalgebra i n t h e l i s t . Here we s h a l l r e s t r i c t o u r s e l v e s t o one n o n t r i v i a l example and t h e n summarize t h e r e s u l t s i n T a b l e 1. Consider t h e L i e a l g e b r a : I D + K ~ + E ( P O + P>-Ki+L33P1 ~) ,Po-P2). Using e x p r e s s i o n s (3.15) and imposing
E
=
t1.
(3.16)
383
Symmetry Reduction for Nonlinear Wave Equations
Plf(y,a,r,+)
=
0, (Po-P2)f(v,a,r,+)
=
0
we f i n d
q = r(sha+chasin+)
f = f(y,q),
[(-K1+L3)f(y,q)
= 0 i s satisfied automatically].
Imposing
( F+ K2+ e ( P O+P2 ) 1f (Y.TI = 0 we f i n a l l y o b t a i n f
=
f(<,) (3.17)
Step 4. Apply a general 0(3,2) t r a n s f o r m a t i o n t o t h e o b t a i n e d i n v a r i a n t s ; i n t h i s example t o ER. We o b t a i n
< Step 5.
+
-1n(A,x)
=
[_A,!
C
IR 5 ,
(A,&)
(3.18) =
A1B1+A2B2+A3B3-AOBO-A4B4].
Au(<) = -'G+3ti, and hence
We p u t u = u ( z )
Perform t h e dimensional r e d u c t i o n .
H ( ~ u , ( V u ) ~ , u )= 0
( V U ) ~=
reduces t o t h e H(-;+3;,-h2,u)
and f i n d
-i2
ODE (3.19)
= 0.
I n Table 1 we p r e s e n t the codimension 1 symmetry v a r i a b l e s f o r t h e c o n s i d e r e d e q u a t i o n . I n column 2 we g i v e t h e maximal symmetry a l g e b r a y i e l d i n g t h e c o r r e s p o n d i n g i n v a r i a n t . In column 3 we p r e s e n t t h e r e p r e s e n t a t i v e symmetry v a r i a b l e E~ (we d r o p t h e R ) . The reduced e q u a t i o n i s o b t a i n e d by s u b s t i t u t i n g Au and (Vu) 2 f r o m t h e l a s t 2 columns o f Table 1 i n t o t h e e q u a t i o n H(Au,(VU) 2 ,u)=O. space we do n o t g i v e t h e general v a r i a b l e s 5 , o b t a i n e d f r o m 5 by a R general O(3,Z) t r a n s f o r m a t i o n ( i .e. t h e r e s u l t s o f s t e p 4 o f t h e a l g o r i t h m have been o m i t t e d i n t h e T a b l e ) .
To save
3.3 Examples o f Symmetry Reduction t o O D E ' S U s i n g Degenerate Symmetry V a r i a b l e s A . Degenerate Symmetry VariabZes in Minkowski Space Mln, 11
We proceed a c c o r d i n g t o t h e a l g o r i t h m o f S e c t i o n 2 . 1. The i s o m e t r y a l g e b r a i n t h i s case i s p ( n , l ) (see ( 3 . 2 ) ) . Up t o conjugacy under t h e P o i n c a r 6 group P ( n , l ) t h i s a l g e b r a c o n t a i n s a s i n g l e i s o t r o p i c K i l l i n g v e c t o r , namely
v^ 2 . The e q u a t i o n YF(xo,xl c o o r d i n a t e s on N
0
,... ,xn)
(3.20)
= Po-P1.
= 0 determines t h e p r o j e c t i o n ny:M
are
(q =
XO+X1,X2,X3,.
Here q i s a n u l l v a r i a b l e s a t i s f y i n g ( 2 . 1 7 ) .
. . ,xn).
+
No,
where
384
A.M. Grundland et al.
3. The subalgebra Lo o f p ( n , l ) , (2.19) i s
?,
orthogonal t o
i.e.
Lo = {Mab,MOa-M1 a ,Pa ,Po-P1 1, On N o ,
s a t i s f y i n g (2.18) and
a, b=2,.
. . ,n .
(3.21)
P -P
a c t s t r i v i a l l y and MOa-Mla i s p r o p o r t i o n a l t o Pa. Thus, Lo a c t s 1 l i k e t h e E u c l i d e a n a l g e b r a e ( n - 1 ) i n an n-1 dimensional E u c l i d e a n space { x 2 , * . ,xn). 0
.
4. The subalgebras o f e ( n - 1 ) a r e known and so a r e t h e i r
codimension 1 i n v a r i a n t s namely (Grundland, Harnad and W i n t e r n i t z ( 1 9 8 4 ) ) k 2 ZR = (x2+. ..+x2 )', k=l, ,n-1 (3.22) k+l k 5. A c t upon ZR b y t h e E u c l i d e a n group E ( n - l ) , a l l o w i n g t h e general group element
...
t o depend upon t h e n u l l v a r i a b l e
(A,&), A2
=
o r more g e n e r a l l y
qR = xo+xl,
- 2 2 2 - A -A -...-An = 0. 0 1
(3.23)
The degenerate symnetry v a r i a b l e s r e d u c i n g (1.2) t o an ODE a r e k k 5 = I .z (i$+&(d&jh)) 2 3.4 1=1 k=l,
q
...,n-1,
(Ei(q),i4(q))
as i n (3.23).
=
...,k ,
i,G=l,
-€jiC,
(&i(q),b)
(3.24) = 0,
A and
with
We have k 2
(v5
1
= -1
k I
s o t h a t e q u a t i o n (1.2) i s reduced t o t h e ODE k-1 2 ~ 3 - = u0, H(-u E5 - ~
k-1
=-7 ~
k = ~l , .
. . ,n-1. ~
(3.25)
)
N o t i c e t h a t t h e d e f i n i t i o n o f q i n v o l v e s one c o n s t a n t i s o t r o p i c v e c t o r i n M ( n , l ) , t h e d e f i n i t i o n o f gk i n v o l v e s k + l v e c t o r f u n c t i o n s o f r) (one a r b i t r a r y one, n a m e l y ki (q)) namely L(q),p l u s k m u t u a l l y o r t h o g o n a l ones , o r t h o g o n a l t o
A,
.
B . Degenerate Variables on an 0 (3,2) Hyperboloid
L e t us again c o n s i d e r t h e h y p e r b o l o i d (3.11) and use t h e a l g o r i t h m f o r o b t a i n i n g degenerate symnetry v a r i a b l e s . 1 . The i s o m e t r y a l g e b r a 0(3,2) c o n t a i n s ( u p t o conjugacy under t h e de S i t t e r group 0(3,2)) p r e c i s e l y one i s o t r o p i c K i l l i n g v e c t o r w h i c h we choose t o i d e n t i f y with . a 1 a cow a (3.26) Y = P -P = (sha+chasin$)-(cha+shasin&)+ 0 2 ar r aa r c h a a& * 2. Equation
-
,.
YF(y,a,r,+)=
0
d e f i n e s t h e space
N 0 w i t h coordinates
{ q = r ( sha+chasin&), E=rchacos$,y)
(3.27)
satisfying 2
(Vq) = 0,
AT = 0,
(Vq,Vy) = 0,
(Vrl,VE)
(3.28)
= 0
so t h a t -q i s t h e r e q u i r e d n u l l v a r i a b l e ( t h e o p e r a t o r s A and (V.) (3.13) and ( 3 . 1 4 ) ) .
2
a r e as i n
Symmetry Reduction for Nonlinear Wave Equations
385
3. The s u b a l g e b r a Lo s a t i s f y i n g (2.18) and (2.19) i n t h i s case i s
{D+K2,-L
+K P P -P } . 3 1 ' 1' 0 2
(3.29)
4. Subalgebras of (3.29) c o n t a i n i n g t h e i s o t r o p i c element Po-P2 a r e e a s i l y const r u c t e d , o r can be r e a d o f f from Patera, Sharp, W i n t e r n i t z and Zassenhaus (1977). I f we r e s t r i c t o u r s e l v e s t o a l g e b r a s t h a t have two i n v a r i a n t s i n No. namely q and one more, we a r e l e f t w i t h two such subalgebras and t h e i r i n v a r i a n t s :
(3.30)
? = -
xo+x2
,
re-Ychacos+ = x3
'1+'4
.
(3.31)
5. L e t t h e L i e group G o generated by (3.29) a c t on t h e h y p e r b o l o i d ( 3 . 1 1 ) and c o n s i d e r t h e a c t i o n i n d u c e d on t h e above i n v a r i a n t s . L e t t h e elements o f G o depend on t h e n u l l v a r i a b l e q. The i n v a r i a n t s y and x 3 , r e s p e c t i v e l y , g i v e r i s e t o t h e f o l l o w i n g degenerate symmetry v a r i a b l e s x +x p = y + f ( q ) = -1n(xl+x4)+f(-) 0 2 (3.32) R '1+'4 u R= e-Y{ rchacos$+r(sha+chasin+)fl ( q ) + f 2 ( q ) }
(3.33)
(3.34) (3.35) where
A,
and
are constant vectors, s a t i s f y i n g A* =
N
,$ =
0,
k2 =
1,
(5,EJ
(:,&).
(3.36)
Au = -4u,
(3.37)
(&,C)
=
=
6. We have (Vp)
2
=
-1,
Ap = 3,
(00)'
2 f1,
= -u
so t h a t e q u a t i o n ( 1 . 2 ) i n these cases reduces t o t h e O O E ' s
H(-u and
+3~~,-u;,O) PP
=
(3.38)
0
H ( ( 1 - u 2 ) u u ~ - 4 ~ u O , ( l - u 2 ) u ~ , U =) 0, respectively. N o t i c e t h e occurrence o f a r b i t r a r y f u n c t i o n s i n p and
CJ.
(3.39)
386
4.
A.M. Grundland et al.
COMMENTS ON THE REDUCED EQUATIONS
I n t h i s p r e s e n t a t i o n we have c o n c e n t r a t e d m a i n l y on symmetry r e d u c t i o n f r o m p a r t i a l d i f f e r e n t i a l e q u a t i o n s t o o r d i n a r y ones. The t e c h n i q u e s f o r r e d u c i n g t o l o w e r dimensional PDE's a r e i d e n t i c a l t o t h e ones used here. As an example o f t h e t y p e o f i n s i g h t t h a t symmetry r e d u c t i o n can g i v e , l e t us consider the equation
-...-u
U
~ = t l , x=const. (4.1) xlxl 'nXn T h i s i s a s p e c i a l case o f ( 1 . 2 ) i n n + l dimensional Minkowski space. For x = 0 (4.1) i s t h e L i o u v i l l e e q u a t i o n which i s o f c o n s i d e r a b l e i n t e r e s t i n p h y s i c s . For x # 0 an e l l i p t i c v e r s i o n o f t h i s e q u a t i o n i n two dimensions o c c u r s i n t h e s t u d y o f v o r t i c e s (e.g. i n t h e t h e o r y o f s u p e r c o n d u c t i v i t y and i n t h e a b e l i a n Higgs model (Jacobs and Rebbi ( 1 9 7 9 ) ) ) . -u
= Ee + x ,
uxoxo
I n t r o d u c i n g t; as one o f t h e codimension 1 symmetry v a r i a b l e s (3.8) o r ( 3 . 1 0 ) , o r one o f t h e degenerate v a r i a b l e s ( 3 . 2 4 ) , we o b t a i n t h e ODE u" +
E u'
=
Eeu +
X,
k=0,1
,..., n.
(4.2)
L e t us now a p p l y a known a l g o r i t h m ( A b l o w i t z , Ramani and Segur (1980)) t o t e s t whether t h i s i s an e q u a t i o n o f t h e Painlev; t y p e . We r e c a l l t h a t s o l u t i o n s o f a P a i n l e v e t y p e e q u a t i o n have, by d e f i n i t i o n , no moving c r i t i c a l p o i n t s , i . e . no branch p o i n t s o r e s s e n t i a l s i n g u l a r i t i e s , t h e p o s i t i o n s o f which depend on t h e i n i t i a l c o n d i t i o n s ( I n c e ( 1 9 5 6 ) ) . We f i r s t t r a n s f o r m (4.2) i n t o an e q u a t i o n w i t h a r a t i o n a l n o n l i n e a r i t y by p u t t i n g y = eu y ~ -;
Next, we p u t
T
=
c - ~ where ~ , 5
i2 + rk yy*
so t h a t =
Ey3
+ x y2 .
(4.3)
i s an a r b i t r a r y c o n s t a n t , and l o o k f o r a s o l u t i o n
t o (4.3) i n the form y =
E
a=O
a -a y a ~ ,
u = positive integer.
(4.4)
A necessary c o n d i t i o n f o r (4.3) t o have t h e Painleve' p r o p e r t y i s t h a t a solut i o n o f t h e f o r m (4.4) e x i s t s and depends on two f r e e parameters, e0 and one o f t h e c o e f f i c i e n t s ya.
We d r o p a l l d e t a i l s and only s t a t e t h a t t h e s u b s t i t u t i o n
o f ( 4 . 4 ) i n t o ( 4 . 3 ) l e a d s t o t h e e q u a t i o n a = 2 and ko a r e c u r s i o n r e l a t i o n f o r t h e c o e f f i c i e n t s y, t h a t breaks down t t h e "resonance v a l u e a=2. The c o e f f i c i e n t yp i s hence e i t h e r undetermined so we o b t a i n t h e r e q u i r e d f r e e parameter, o r we o b t a i n a c o n t r a d i c t i o n f o r a=2. A c l o s e r i n v e s t i g a t i o n , performed on a computer u s i n g an ALTRAN program t h a t does t h e necessary symbolic m a n i p u l a t i o n s , lea.ds t o t h e compati b i lit y c o n d i t i o n s 2k(k-1
+
X5;
= 0.
(4.5)
For x=O we f i n d t h a t e q u a t i o n (4.3) s a t i s f i e s t h e necessary c o n d i t i o n f o r b e i n g o f t h e P a i n l e v 6 t y p e i f and o n l y i f k=O o r k = l . For x#O, e q u a t i o n ( 4 . 5 ) cannot be s a t i s f i e d f o r i n t e g e r values o f k, s i n c e co i s an a r b i t r a r y c o n s t a n t . The e q u a t i o n f o r x#O i s hence n o t o f t h e P a i n l e v 6 t y p e ( f o r any v a l u e o f k ) . A c c o r d i n g t o t h e "Painleve' c o n j e c t u r e " ( A b l o w i t z and Segur (1981), and A b l o w i t z , Ramani and Segur (1980)), a PDE i s i n t e g r a b l e by i n v e r s e s c a t t e r i n g t e c h n i q u e s
Symmetry Reduction for Nonlinear Wave Equations
o n l y i f a l l O D E ' S o b t a i n e d by A c c e p t i n g t h i s c o n j e c t u r e , we v e r s e s c a t t e r i n g o n l y f o r n-1 n 3 2 t h i s equation w i l l , in
387
dimensional r e d u c t i o n a r e o f t h e Painlev; t y p e . conclude t h a t e q u a t i o n ( 4 . 2 ) can b e s o l v e d by i n and x=O. T h i s a l s o i n d i c a t e s t h a t f o r rfO and/or g e n e r a l , n o t have s o l i t o n t y p e s o l u t i o n s .
REFERENCES
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Boyer, C.P., Sharp, R.T. and W i n t e r n i t z , P., Symmetry b r e a k i n g i n t e r a c t i o n s f o r t h e t i m e dependent SchrUdinger e q u a t i o n , J.Math.Phys. 17 (1976) 74391451. E r n s t , F.J., New f o r m u l a t i o n o f t h e a x i a l l y symmetric g r a v i t a t i o n a l f i e l d problem, Phys.Rev. 167 (1968) 1175-1178. Fokas, A.S. and A b l o w i t z , M.J., The i n v e r s e s c a t t e r i n g t r a n s f o r m f o r m u l t i dimensional ( 2+1) problems, i n : K.B. Wolf ( e d . ) , N o n l i n e a r Phenomena ( L e c t u r e Notes i n P h y s i c s 189, S p r i n g e r , 1983). Grundland, A.M., Harnad, J . , and W i n t e r n i t z , P . , S o l u t i o n s o f t h e m u l t i dimensional s i n e Gordon e q u a t i o n o b t a i n e d by symmetry r e d u c t i o n , KINAM, Rev. de F i s i c a 4 (1982) 333-344. Grundland, A.M., Harnad, J . and W i n t e r n i t z , P . , Symmetry r e d u c t i o n f o r nonl i n e a r r e l a t i v i s t i c a l l y i n v a r i a n t e q u a t i o n s , J.Math.Phys. 25 (1984). Harnad, J., Shnider, S . and V i n e t , L., The Y a n g - M i l l s system i n compacti f i e d Minkowski space; i n v a r i a n c e c o n d i t i o n s and S U ( 2 ) i n v a r i a n t s o l u t i o n s , J.Math.Phys. 20 (1979,a) 931-942. Harnad, J.,
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and V i n e t ,
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S o l u t i o n s t o Yang-Mills e q u a t i o n s
on M4 i n v a r i a n t under subgroups o f 0(4,2), i n : Complex M a n i f o l d Techniques i n T h e o r e t i c a l P h y s i c s (Pitman Research Notes i n Mathematics 32 (1979,b) 21 9-230). Harnad, J., and W i n t e r n i t z , P., P s e u d o p o t e n t i a l s and L i e symmetries f o r t h e g e n e r a l i z e d n o n l i n e a r S c h r 6 d i n g e r e q u a t i o n , J.Math.Phys. 23 (1982) 517-525. Ibragimov, N. Kh., Gruppovye s v o i s t v a n e k o t o r y k h d i f f e r e n t s i a l ' n y k h u r a v n e n i i [Group t h e o r e t i c a l p r o p e r t i e s o f some d i f f e r e n t i a l e q u a t i o n s ] (Nauka, N o v o s i b i r s k , 1967) I n c e , E.L.,
O r d i n a r y D i f f e r e n t i a l D i f f e r e n t i a l Equations (Dover, 1956).
Jacobs, L. and Rebbi, C . , I n t e r a c t i o n energy o f superconducting v o r t i c e s , Phys.Rev. B19 (1979) 4486-4494. K i n n e r s l e y , W.M., The Geroch group and s o l i t o n s o l u t i o n s o f t h e s t a t i o n a r y a x i a l l y symmetric E i n s t e i n ' s e q u a t i o n s , i n : Group T h e o r e t i c a l Methods i n P h y s i c s ( L e c t u r e Notes i n P h y s i c s 135, S p r i n g e r , New York, 1980).
388
A.M. Grundland et at.
L i e , S., Vorlesungen Uber O i f f e r e n t i a l g l e i c h u n g e n m i t bekannten i n f i n i t e simal en T r a n s f o r m a t i onen (Teubner, L e i p z i g, 1891 ) . L i e , S., Allgemeine Untersuchungen Ober D i f f e r e n t i a l g l e i c h u n g e n d i e e i n e c o n t i n u i r l i c h e e n d l i c h e Gruppe g e s t a t t e n , Math.Ann. 25 (1885) 71-151. Misner, C.W., Thorne, K.S. San F r a n c i s c o , 1973).
and Wheeler, J.A.,
G r a v i t a t i o n (Freeman,
Ovsiannikov, L.V., Group A n a l y s i s o f D i f f e r e n t i a l Equations (Academic Press, New York, 1982). P a t e r a , J . , W i n t e r n i t z , P. and Zassenhaus, H., Continuous subgroups o f t h e fundamental groups o f Physics. I . General method and t h e P o i n c a r 6 group, J .Math.Phys. 16 ( 1 975 ,a) 1597-1 614. P a t e r a , J . , W i n t e r n i t z , P., and Zassenhaus, H . , Continuous subgroups o f t h e fundamental groups o f Physics. 11. The s i m i l i t u d e group, J.Math.Phys. 16 ( 1 975,b) 1615-1624. c241
Patera, 3.. Sharp, R.T., W i n t e r n i t z , P. and Zassenhaus, H . , Continuous subgroups o f t h e fundamental groups o f P h y s i c s . 111. The de S i t t e r groups. J.Math.Phys. 18 (1977) 2259-2288. Patera, J . , Sharp, R.T., W i n t e r n i t z , P. and Zassenhaus, H., Subgroups o f t h e Poincare' group and t h e i r i n v a r i a n t s , J.Math.Phys. 17 (1976) 1439-1451. V i l e n k i n , N.J., S p e c i a l F u n c t i o n s and Group R e p r e s e n t a t i o n Theory, T r a n s l . o f Math.Monographs V01.22 (Amer.Math.Soc., Providence, R . I . , 1968). W i n t e r n i t z , P., L i e groups and s o l u t i o n s o f n o n l i n e a r d i f f e r e n t i a l equat i o n s , i n : Wolf, K.B. ( e d . ) , N o n l i n e a r Phenomena ( L e c t u r e Notes i n P h y s i c s 189, S p r i n g e r , New York, 1983).
WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
390
W
a
3
v
h
-4N
+X
N m N N X I 4 I 1
v
r"
A.M. Grundland eta/.
I/
b
II
a L ,
?-
c
v
N
N
X I
Im
+X
-3.7 N X N
-
C
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
391
INTEGRAL REPRESENTATIONS FOR SOLUTIONS TO SOME DIFFERENTIAL EQUATIONS THAT ARISE I N WAVE THEORY
Robert F. M i l l a r Department o f Mathematics U n i v e r s i t y o f A1 b e r t a Edmonton, A1 b e r t a Canada T6G 2G1 I n t e g r a l r e p r e s e n t a t i o n s a r e g i v e n f o r s o l u t i o n s t o some l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n two independent v a r i a b l e s , s e v e r a l o f which a r i s e i n wave theory.
These p r o v i d e s o l u t i o n s t o a n a l y t i c Cauchy problems
f o r t h e e q u a t i o n s i n terms o f t h e complex Riernann f u n c t i o n . R e s u l t s a r e g i v e n f o r a c l a s s o f second-order equations, and f o r s e v e r a l f o u r t h - o r d e r equations.
For t h e Helmholtz
equation, a p p l i c a t i o n s t o special functions,
analytic
c o n t i n u a t i o n , and an i n v e r s e s c a t t e r i n g problem, a r e described. INTRODUCTION Complex v a r i a b l e methods p r o v i d e an e f f e c t i v e means f o r i n v e s t i g a t i n g some They form t h e b a s i s f o r t h e so-
classes o f p a r t i a l d i f f e r e n t i a l equations.
c a l l e d f u n c t i o n t h e o r e t i c methods (see, f o r example,
[l])
which, though used
c h i e f l y i n t h e s t u d y o f e q u a t i o n s w i t h two independent v a r i a b l e s , can be a p p l i e d t o some cases w i t h s e v e r a l independent v a r i a b l e s .
I n t h e p r e s e n t paper, a t t e n -
t i o n w i l l be c o n f i n e d t o e l l i p t i c e q u a t i o n s i n two independent v a r i a b l e s . It i s w e l l known t h a t s o l u t i o n s t o e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s w i t h
a n a l y t i c c o e f f i c i e n t s a r e a n a l y t i c f u n c t i o n s o f t h e independent v a r i a b l e s .
Such
s o l u t i o n s may t h u s be extended a n a l y t i c a l l y i n t o t h e complex domain o f t h e s e variables.
The method o f i n t e g r a l o p e r a t o r s t h a t e x p l o i t s t h i s i d e a was d e v e l -
oped e x t e n s i v e l y by Bergman [ Z ] . I n t h e case o f two independent v a r i a b l e s
x
and
y, where t h e h i g h e s t o r d e r
d e r i v a t i v e s appear as t h e L a p l a c i a n , i n t e g r a t i o n o f s u i t a b l e combinations o f a s o l u t i o n and t h e complex Riemann f u n c t i o n o f Vekua [3] a l o n g a p p r o p r i a t e paths i n t h e space o f two complex v a r i a b l e s
z
:= x
+ i y and z* := x -
iy
leads t o
f o r m u l a s f o r a g e n e r a l s o l u t i o n o f t h e e q u a t i o n i n terms o f d a t a on t h e complex
R.F. Millar
392 characteristics
[l].
z = c o n s t a n t and z* = c o n s t a n t ; see, f o r example [3]-[8],
and
Such methods have been used t o s o l v e boundary v a l u e problems [ 6 ] , (91, t o
o b t a i n r e s u l t s about s p e c i a l f u n c t i o n s [4],
[7],
[8], and t o s o l v e Cauchy prob-
lems w i t h d a t a g i v e n on an a n a l y t i c a r c i n t h e r e a l
x,y-plane
[5].
Vekua [31
has a l s o s t u d i e d c l a s s e s o f h i g h e r o r d e r e q u a t i o n s . Recently, and i n a manner d i f f e r e n t from [5],
a r e p r e s e n t a t i o n was found f o r t h e
s o l u t i o n t o t h e a n a l y t i c Cauchy problem f o r an e l l i p t i c e q u a t i o n o f secondorder, w i t h d a t a g i v e n on an a n a l y t i c a r c i n t h e p l a n e [lo].
T h i s a n a l y s i s has
been extended subsequently t o a x i a l l y symmetric problems [ll], and t o a c l a s s o f f o u r t h - o r d e r e q u a t i o n s [12].
U n l i k e Vekua's r e p r e s e n t a t i o n s , which d e t e r m i n e
s o l u t i o n s t h r o u g h o u t a g i v e n domain and a r e t h u s a p p r o p r i a t e f o r t r e a t i n g bounda r y v a l u e problems, s o l u t i o n s t o Cauchy problems a r e s u i t e d t o a n a l y t i c c o n t i n u ation.
Although t h e problems u s u a l l y a s s o c i a t e d w i t h e l l i p t i c e q u a t i o n s a r e
boundary v a l u e problems, f o r second o r d e r e q u a t i o n s a t l e a s t a number o f a p p l i c a t i o n s o f a n a l y t i c Cauchy problems have a r i s e n and i t i s expected t h a t s i m i l a r uses w i l l be found i n c o n n e c t i o n w i t h h i g h e r o r d e r e q u a t i o n s .
One purpose o f t h e p r e s e n t paper i s t o e x h i b i t some o f t h e r e p r e s e n t a t i o n s d e s c r i b e d i n t h e p r e c e d i n g paragraph, and t o extend t h e r e s u l t s t o two new equat i o n s o f f o u r t h order.
Applications i n t h e contexts o f special functions,
l y t i c c o n t i n u a t i o n , and i n v e r s e problems a l s o w i l l be discussed.
ana-
None o f t h e s e
w i l l be t r e a t e d i n depth, t h e i n t e n t b e i n g t o g i v e an i n d i c a t i o n o f t h e r a t h e r wide range o f a p p l i c a t i o n o f t h e s e r e p r e s e n t a t i o n s . wave phenomena i s i n d i r e c t :
Thus t h e c o n n e c t i o n w i t h
t h e paper i s concerned w i t h e x h i b i t i n g r e s u l t s t h a t
m i g h t be u s e f u l i n t h e study o f wave (and o t h e r ) phenomena. A method f o r o b t a i n i n g t h e r e p r e s e n t a t i o n f o r m u l a s i s d e s c r i b e d v e r y b r i e f l y i n t h e n e x t s e c t i o n , and t h e e a r l i e r r e s u l t s f o r c l a s s e s o f second-and f o u r t h - o r d e r equations are stated.
Corresponding f o r m u l a s f o r s o l u t i o n s t o t h e v i b r a t i n g
p l a t e equation, and t o an e q u a t i o n t h a t a r i s e s i n p l a t e b u c k l i n g t h e o r y , a r e obtained.
Some a p p l i c a t i o n s a r e t h e n discussed, a l l i n t h e c o n t e x t o f t h e Helm-
h o l t z equation.
T h i s keeps t h e d i s c u s s i o n about as s i m p l e as p o s s i b l e and,
moreover, t h e H e l m h o l t z e q u a t i o n has many a p p l i c a t i o n s i n a v a r i e t y o f areas. The a p p l i c a t i o n s deal w i t h s p e c i a l f u n c t i o n s (Bessel f u n c t i o n r e p r e s e n t a t i o n s , Mathieu f u n c t i o n r e s u l t s ) , a n a l y t i c c o n t i n u a t i o n ( r e f l e c t i o n f o r m u l a s ) , and i n v e r s e problems.
I n t h i s l a t t e r connection,
i t i s shown t h a t t h e problem o f
d e t e r m i n i n g p r o p e r t i e s o f t h e i n t e r i o r o f a medium f r o m s c a t t e r i n g measurements on and o u t s i d e a p o r t i o n o f i t s boundary can be f o r m u l a t e d w i t h t h e a i d o f t h e
Integral Representations
393
r e p r e s e n t a t i o n f o r t h e s o l u t i o n t o Cauchy's problem.
T h i s problem i s u n s t a b l e
w i t h r e g a r d t o measurement e r r o r , and a method t h a t may s t a b i l i z e i t i s o u t 1 ined. REPRESENTATION FORMULAS S i n c e d e r i v a t i o n s o f t h e s e f o r m u l a s have been g i v e n p r e v i o u s l y f o r some secondo r d e r [lo]
and f o u r t h - o r d e r [12]
e q u a t i o n s , d e t a i l s w i l l be o m i t t e d .
For t h e
second-order case, we quote t h e b a s i c r e s u l t and d e s c r i b e t h e n o t a t i o n .
We
s t a t e t h e c o r r e s p o n d i n g f o r m u l a f o r s o l u t i o n s t o a c l a s s o f f o u r t h - o r d e r equations.
To conclude t h e s e c t i o n , we show how analogous r e s u l t s can be found f o r
s o l u t i o n s t o two o t h e r f o u r t h - o r d e r equations.
I n a l l cases, o n l y t h e homogene-
ous d i f f e r e n t i a l e q u a t i o n i s considered; t h e nonhomogeneous e q u a t i o n can be reduced t o t h i s by s u b t r a c t i n g o u t an a n a l y t i c p a r t i c u l a r s o l u t i o n . Second-Order E q u a t i o n s L [ u l = 0, where
Consider second-order e q u a t i o n s o f t h e f o r m L[u]
:= u
EE
+
u
r)V
+
au
5
+ bu + cu. r)
The c o e f f i c i e n t s and s o l u t i o n a r e assumed t o be a n a l y t i c f u n c t i o n s o f i n a neighbourhood o f a simple, a n a l y t i c a r c s e n t a t i o n f o r m u l a t h e n i s expressed ((4.2)
C
i n t h e F;,q-plane.
o f [lo])
E and
q
The r e p r e -
as
a r e s u l t t h a t i s r e m i n i s c e n t o f Riemann's s o l u t i o n f o r a h y p e r b o l i c e q u a t i o n . Here
s
i s a r e a l parameter on
C; ~ ( s ) and
~ ( s ) a r e a n a l y t i c f u n c t i o n s , and
( ~ ( s ) , q ( s ) ) a r e r e c t a n g u l a r c o o r d i n a t e s o f a p o i n t on
2
r := [ ( x - e ) + ( y - ~ - , ) ~ ] ~ ' ~and , if
A(~;,n;x,y)
c
=
and
E +
iT, c* = t;
A(t;,q;x,y)
+
-
iq,
A ( C , ~ ; E , ~ )= 1. z = x + i y , z* = x
R(c,c*;z,z*).
Also
If
denotes t h e c o e f f i c i e n t o f l o g ( l / r )
in t h e a d j o i n t fundamental s o l u t i o n , t h e n A ( c , ~ ; x , y ) i t i s n o r m a l i z e d so t h a t
C.
i s t h e Riemann f u n c t i o n ;
Under t h e change o f v a r i a b l e s
- i y , we have u(x,y) + U(z,z*), u ( s ) := u ( ~ ( s ) , q ( s ) ) , and s i m i l a r l y f o r
R. F. Millar
394
t h e n s = S(z) i s t h e s o l u t i o n t o t ( s ) := ~ ( s )- i q ( s ) t h e n s = s ( z * ) i s t h e s o l u t i o n t o
If Z(s) := ~ ( s )f i?-,(s),
(au/av)(s).
Z(s) = z, w h i l e i f
f ( s ) = z*. R( s;z,z*)
The i n t e g r a l i s a c o n t o u r i n t e g r a l i n t h e complex s-plane. := R(Z( S )
,?( S ) ;z,z*)
U f o r (z,z*)
The r e p r e s e n t a t i o n (2) d e t e r m i n e s
denoting t h e arc t h a t i s conjugate t o u(s)
and
(au/av)(s). L[u] = 0.
i n a neighbourhood o f
Cxc
(c
i n terms o f t h e a n a l y t i c Cauchy d a t a
C)
Since i t depends on two a r b i t r a r y f u n c t i o n s o f one
v a r i a b l e , l i k e t h e s o l u t i o n s o f Vekua [3] solution of
Finally,
.
i t may be regarded as a general
Vekua's s o l u t i o n , however, i s a n a l y t i c i n a s p e c i f i e d
domain whereas (2) g i v e s t h e s o l u t i o n t o t h e Cauchy problem; i t i s p r e c i s e l y t h e r e s u l t t h a t i s o b t a i n e d when t h e c l a s s i c a l r e p r e s e n t a t i o n ( e x p r e s s i n g domain bordered i n p a r t by
i n terms o f
Cxc
s o l u t i o n ) i s c o n t i n u e d across
U
in a
u, au/av, and t h e fundamental
CxE.
S p e c i a l cases of ( 2 ) a r e d e s c r i b e d and discussed i n
[lo]
and [ll]. These
i n c l u d e formulas f o r s o l u t i o n s t o t h e Laplace, Helmholtz, and g e n e r a l i z e d a x i a l l y symmetric Helmholtz (GASH) equations.
We s h a l l make use o f t h e r e s u l t
f o r t h e Helmholtz e q u a t i o n l a t e r . F o u r t h-Order Equations F o r f o u r t h - o r d e r e q u a t i o n s o f t h e form
L2[u] = 0
with
L
r e p r e s e n t a t i o n formulas a r e found i n a s i m i l a r manner [12].
We have
+ U(S(Z*))L*[RI)
U(Z,Z*) = -x{u(S(z))L*[R]
(3)
d e f i n e d by ( l ) ,
- ua L*[R])ds, av where terms i n (2.12) and
R(S(z*);z,z*)
( 3 ) , t h e argument
o f [12]
i n v o l v i n g t h e vanishing functions
have been d e l e t e d . s
R(S(z);z,z*)
I n the f i r s t bracketed expression i n
i s evaluated a t S(z)
or at
S(z*),
as t h e case may be.
395
Integral Representations
The Riemann f u n c t i o n fundamental s o l u t i o n .
R
again i s the c o e f f i c i e n t o f l o g ( l / r ) i n the a d j o i n t 2 The formula ( 3 ) g i v e s a g e n e r a l s o l u t i o n f o r L [u] = 0
which, again, d i f f e r s from t h e t y p e o b t a i n e d by Vekua [3]. I n c l u d e d a r e formulas f o r
Some s p e c i a l cases o f ( 3 ) a r e discussed i n [12].
s o l u t i o n s t o t h e b i h a r m o n i c , g e n e r a l i z e d a x i a l l y symmetric biharmonic, and i t e r a t e d GASH equations. I t i s convenient t o m e n t i o n h e r e a f u r t h e r r e s u l t i n c o n n e c t i o n w i t h
c o n t i n u a t i o n o f a s o l u t i o n t o the biharmonic equation. t h e E-axis,
(4)
t h e n ( s e e (3.2)
u(z,z*)
1 2
= -
CU(Z,O)
f o r solutions t o
C
i s an a r c o f
+ u(z*
From t h i s t h e r e f l e c t i o n r e s u l t (3.28) (5)
When
o f [12])
U(X,-y)
= -u X,Y)
2
u = u
A u = 0,
Y
o f P o r i t s k y [13],
+ 2YUy(X,Y)
on
= 0
-
namely
2
Y AU(X,Y),
y = 0, i s r e a d i l y o b t a i n e d .
We s h a l l conclude t h i s s e c t i o n by o b t a i n i n g some r e s u l t s f o r two o t h e r f o u r t h o r d e r e q u a t i o n s , one o f which a r i s e s i n wave t h e o r y .
These a r e t h e e q u a t i o n f o r
t h e v i b r a t i n g p l a t e , and one t h a t a r i s e s i n t h e t h e o r y o f bending o r b u c k l i n g o f a plate.
A l t h o u g h e i g e n v a l u e problems a r e a p p r o p r i a t e f o r t h e homogeneous
e q u a t i o n s , o t h e r v a l u e s o f t h e parameters a r e s i g n i f i c a n t when t h e o r i g i n a l differential
e q u a t i o n i s nonhomogeneous.
D i f f e r e n t general s o l u t i o n s f o r t h e s e
two e q u a t i o n have been g i v e n by Vekua C31, $40, $554, 55. V i b r a t i n g p l a t e equation:
2
A u
This equation i s o f t h e form
-
K
2
2 L [u]
u
= 0
-
K
2
u = 0, r a t h e r t h a n
2 L [u] = 0, b u t
f o r m u l a ( 3 ) s t i l l a p p l i e s , w i t h t h e a p p r o p r i a t e c h o i c e o f R; f o r 2 2 2 2 2 2 u(L* Cv1-K v ) = VL [ u l - uL* [v], so t h e d e r i v a t i o n f o r V ( L [ u ] - K u)
K
f o r m a l l y t h e same as f o r
may be
-
K
= 0 c121.
I n t h e p r e s e n t circumstances, R
found by s p e c i a l i z i n g r e s u l t s of Vekua 131, p p . 339-347. r e g u l a r s o l u t i o n , t h e most general fundamental s o l u t i o n i s
+
C2Yo(iJzr).
Then
The a p p r o p r i a t e n o r m a l i z a t i o n imp1 i e s t h a t
#
0 is
To w i t h i n an a d d i t i v e C Y
(&r)
1 0
C1 =
-c2
=
-(8K)
-1
.
R.F. Millar
396
i n which
I.
i s t h e m o d i f i e d Bessel f u n c t i o n .
c a l c u l a t i o n s show t h a t
Then, s i n c e
L = L* = A , s i m p l e
= L*[R](S(z*);z,z*)
L*[R](S(z);z,z*)
= -(2~)-'.
Thus
(3) yields
Bending p l a t e e q u a t i o n :
L
A u
+
AAU = 0
The nonhomogenous form o f t h i s e q u a t i o n a r i s e s i n t h e t h e o r y o f bending o f a t h i n p l a t e ; see, f o r example, [3],
5554, 55.
The homogeneous form occurs i n
s t a b i l i t y problems i n l i n e a r p l a t e t h e o r y , when a u n i f o r m compression i n t h e p l a n e o f t h e p l a t e i s a p p l i e d a l o n g t h e normal t o i t s edge; t h e n
i s an
e i g e n v a l u e which, depending on t h e boundary c o n d i t i o n s on t h e edge, may be d i f f i c u l t t o obtain.
The case o f a square p l a t e w it h clamped edges i s
considered i n [14], where i t i s shown t h a t
A > 0.
We s h a l l g i v e a
r e p r e s e n t a t i o n f o r t h e s o l u t i o n t o t h e Cauchy problem f o r
Since t h e o p e r a t o r i n ( 8 ) i s n e i t h e r
L2
nor
L2
-
K'
, the
p r e v i o u s methods o f
d e r i v a t i o n must be m o d i f i e d . Because t h e r e i s a Lagrange-Green i d e n t i t y 2 2 unv, by a d d i t i o n we i n v o l v i n g VL [u] - uL* [v], and one i n v o l v i n g vhu
-
obtain
Here
5 and
D
i s a domain c o n t a i n e d w i t h i n one i n which 0, and p a r t o f
aD
consists o f the analytic
u
and a r c C.
a p p r o p r i a t e fundamental s o l u t i o n may b e o b t a i n e d from [3], c o r r e s p o n d i n g Riemann f u n c t i o n i s
v
are a n a l y t i c i n Once again, t h e
pp. 339-347.
The
397
Integral Representations
R(~,~*;z,z*)
(10)
= ('2x1)
-1
U(z,z*)
=
T
[u(S(z)) + u(s(z*))] + n i j
3Z*) (R(s;z,z*) S( z )
-
AU(S)
aR
On c a r r y i n g o u t
i t i s found t h a t
1
(11)
11,
h + 0.
which reduces t o t h a t f o r t h e b i h a r m o n i c e q u a t i o n when an a n a l y s i s s i m i l a r t o t h a t i n [12],
-
[J0(Jh(C-z)(C*-z*))
(S;Z,Z*) + (AR+hR)
au av ( S ) -
u(S)
a ay
av
(s)
(AR+hR))ds
.
APPLICATIONS The r e p r e s e n t a t i o n f o r m u l a s d e s c r i b e d p r e v i o u s l y p r o v i d e c l a s s e s o f g e n e r a l s o l u t i o n s o f t h e d i f f e r e n t i a l e q u a t i o n s , t h a t depend on t h e a n a l y t i c a r c C. Given such s o l u t i o n s , s e v e r a l p o s s i b l e a p p l i c a t i o n s come t o mind, such as: 1.
t h e d e t e r m i n a t i o n o f p r o p e r t i e s o f known s o l u t i o n s t o p a r t i a l d i f f e r e n t i a l
e q u a t i o n s and, i n p a r t i c u l a r , o f t h e s p e c i a l f u n c t i o n s t h a t a r e separated s o l u t i o n s t o a d i f f e r e n t i a l e q u a t i o n ; f o r example, p r o p e r t i e s o f Bessel, Mathieu, and Lam; 2.
f u n c t i o n s may be found.
the analytic continuation,
or r e f l e c t i o n , o f a s o l u t i o n t o a boundary v a l u e
problem across an a n a l y t i c boundary a r c , and t h e l o c a t i o n o f s i n g u l a r i t i e s o f t h e solution.
T h i s , i n f a c t , was a p r i m a r y m o t i v a t i o n f o r t h e p r e s e n t , and
e a r l i e r , work.
Whereas Vekua's f o r m u l a s seem t o be more s u i t a b l e f o r t h e
t r e a t m e n t o f boundary v a l u e problems, t h e s e a r e b e t t e r adapted t o c o n t i n u a t i o n or reflection.
I n p a r t i c u l a r , t h e y a r e w e l l s u i t e d t o examining t h e v a l i d i t y o f
t h e s o - c a l l e d R a y l e i g h h y p o t h e s i s [15].
3.
t h e s o l u t i o n o f i l l - p o s e d ( i n c o n t r a s t t o a n a l y t i c ) Cauchy problems f o r
e l l i p t i c e q u a t i o n s ; f o r example, u s e f u l r e s u l t s f o r some i n v e r s e s c a t t e r i n g problems may be o b t a i n e d i f t h e s o l u t i o n can be s t a b i l i z e d .
4.
t h e s o l u t i o n o f boundary v a l u e problems by t h e use o f c o m p a t i b l e Cauchy
d a t a on t h e boundary; t h a t i s by e n s u r i n g t h a t no unwanted s i n g u l a r i t i e s o c c u r i n t h e g i v e n domain.
I n t h e case o f t h e D i r i c h l e t problem f o r t h e Laplace
e q u a t i o n , t h i s p o i n t of view was adopted many y e a r s ago by Fantappi; o b t a i n e d t h e Poisson f o r m u l a f o r t h e c i r c u l a r boundary. Amerio [17] lemniscate.
[16],
who
Shortly thereafter,
a l s o c o n s i d e r e d boundaries o f t h e form o f an e l l i p s e and a It does n o t appear t h a t t h e s e i d e a s have been a p p l i e d subsequently
t o boundary v a l u e problems, a l t h o u g h r e l a t e d f u n c t i o n t h e o r e t i c methods have been used; see, f o r example, [9].
R. F. Millar
398
I n t h i s s e c t i o n , a t t e n t i o n w i l l be c o n f i n e d t o t h e f i r s t and t h i r d o f these a p p l i c a t i o n s , w i t h a few comments on t h e second. w i l l be considered.
Only t h e Helmholtz e q u a t i o n
I t i s a r i c h source o f a p p l i c a t i o n ,
and t h e r e s u l t s o f a
thorough t r e a t m e n t o f t h i s e q u a t i o n should be o f v a l u e i n t h e study o f t h e more d i f f i c u l t h i g h e r o r d e r equations. Speci a1 F u n c t i o n s Several r e s u l t s f o r s p e c i a l f u n c t i o n s a s s o c i a t e d w i t h s o l u t i o n s t o t h e H e l m h o l t z e q u a t i o n , and based on t h e s p e c i a l i z a t i o n o f ( 2 ) t o t h i s case, namely
were g i v e n e a r l i e r
[lo].
Some a p p a r e n t l y new Bessel f u n c t i o n r e p r e s e n t a t i o n s
were obtained, b o t h f o r Cauchy d a t a on a s t r a i g h t l i n e , and f o r d a t a on a c i r c u l a r arc.
Several o t h e r r e s u l t s have been o b t a i n e d subsequently, and t h e y
w i l l be presented here. In
[lo], a f u n c t i o n Kp(p,a)
K : !J
:=
J
Jo(kJ(a-px)(a-plx))x-p-ldx
(a,p > 0 )
Pla was i n t r o d u c e d , b u t n o t i d e n t f i e d . a c y l i n d e r f u n c t i o n o f order Kp(p,a)
(14)
p;
It i s not d i f f i c u l t t o v e r i f y t h a t
K
more p r e c i s e l y ,
v
is
=
Y i s t h e Neumann f u n c t i o n , which can be r e p l a c e d by a Hankel f u n c t i o n i f P d e s i r e d . T h i s r e s u l t seems t o be new.
Here
The c r o s s p r o d u c t o f Bessel f u n c t i o n s t h a t appears i n (14) a r i s e s i n t h e e x p r e s s i o n f o r t h e Green's f u n c t i o n f o r a c i r c u l a r c y l i n d e r .
A t t h i s time i t i s
u n c l e a r whether (14) would be o f use i n summing t h i s , o r o t h e r , s e r i e s . i s such t h a t
J (ka) = 0 o r P
representations f o r
J (kp) &
If
Y ( k a ) = 0, t h e n (13) and (14) p r o v i d e i n t e g r a l
P
and
Y (kp), r e s p e c t i v e l y . P
Other Bessel f u n c t i o n r e l a t i o n s can be based on a d i f f e r e n t r e s u l t from
[lo]:
k
399
Integral Representations
i f t h e Cauchy d a t a f o r t h e c y l i n d r i c a l wave f u n c t i o n
the radial line
a r e g i v e n on
> 0, i t i s found t h a t
8 = 0, 0
Cp(kP)
(15)
Cp(kp)eipe
=
eie
-
C (ks)
iJp -i0
2 2 J o ( k A + p -2sp cos e)ds.
Pe
V a r i o u s r e s u l t s can be found by i n c r e a s i n g 1 e =7 ,one o b t a i n s 2 1 s i n ( 7 pn) (16)
CJkd
=
-7
c o n t i n u o u s l y from
0
0.
If
-
C (ikt) t J o ( k m ) d t ,
p
iJ -P
where t h e p a t h o f i n t e g r a t i o n i s i n d e n t e d below t h e p o s s i b l e s i n g u l a r i t y a t t = 0.
C
By s e t t i n g
Re p > 0), we f i n d
= J
C L P
,
I.r > 0 ( o r even
k = 1, and assuming t h a t
which corresponds t o a r e s u l t g i v e n by Watson “1, p. 374, ( 4 ) ; t h e omission o f 1 a factor of on t h e r i g h t - h a n d s i d e i n Watson’s r e s u l t has been noted by H e n r i c i [4] p. 292, and o t h e r s .
I f we s e t
C
P
= J
P
and
k = i i n ( 1 6 ) , we
get
which may be regarded as a r e l a t i o n i n v e r s e t o ( 1 7 ) .
s = 0
( 1 5 ) , t h e i n t e g r a t i o n p a t h becomes a l o o p around On l e t t i n g
C
P
= J
P
( p > 0)
n e g a t i v e r e a l a x i s from
0
from pe-in
in
0 =
TC
to
pein.
and c o l l a p s i n g t h i s l o o p o n t o t h e segment o f t h e to
-p
t h i s i s a s p e c i a l case o f f o r m u l a
(p
> 0 ) , one o b t a i n s t h e r e s u l t
11.3.40
As a f i n a l Bessel f u n c t i o n example, l e t Then
When one s e t s
i n [19]. C
P
= J
,
@
p > 0, 8 = 3n/4 i n ( 1 5 ) .
400
R. F. Millar
p (ber t+ibei t )
(20)
sin(3d4) = Im
JJP)
which expresses
Jo
,
) I Jo(J(p-t)(p+it))dt
t
i n terms o f t h e K e l v i n f u n c t i o n s g i v e n by
J (p)
P
b e r t + i b e i t := J (tei3r/4). P P P R e s u l t s f o r Mathieu f u n c t i o n s can a l s o be o b t a i n e d from ( 1 2 ) . r e l a t i o n s g i v e n by Volkmer [7]
I n particular,
and by Meixner e t a1 c20] a r e r e a d i l y found.
It
i s o n l y necessary t o p r e s c r i b e Cauchy d a t a f o r a separated s o l u t i o n t o t h e H e l m h o l t z e q u a t i o n on a c o o r d i n a t e a r c i n an e l l i p t i c c o o r d i n a t e system. rectangular coordinates by
f =
( c , ~ )o f
c cosh p cos t ~ , v = c s i n h
s i n 0, t h e n a separated s o l u t i o n i s a
product o f a s o l u t i o n t o Mathieu's equation i n m o d i f i e d Mathieu e q u a t i o n i n
If the
a p o i n t i n t h e p l a n e a r e g i v e n i n t h i s system
On s e t t i n g
p,.
0 and a s o l u t i o n t o t h e p = is
where
0
i s real, it i s
seen t h a t t h e p r o d u c t c o n s i s t s e n t i r e l y o f s o l u t i o n s t o t h e Mathieu e q u a t i o n
+
(b-
f o r t h e same p a i r o f values o f
kc
Y"(x)
(21)
1 2 2
7k c
COS
~ x ) Y ( x )= 0
,
and t h e s e p a r a t i o n parameter
b.
The f o r m
of t h e f i n a l r e s u l t does n o t depend on whether a h y p e r b o l i c a r c o r an e l l i p t i c a r c i s chosen f o r t h e choice o f
0
C. = 0
I n p a r t i c u l a r , i f v and w a r e any s o l u t i o n s t o ( 2 1 ) , C w i t h t h e r e p r e s e n t a t i o n (12) l e a d s e v e n t u a l l y t o
for
the relation
i n which
Z := k c d c o s 6-cos( t ~ + ~ ) ) ( c o e-cos( s +-a)).
By d i r e c t computation, i t may be v e r i f i e d t h a t any p a i r o f t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s t h a t s a t i s f y (22) a r e s o l u t i o n s t o (21) f o r t h e same parameter p a i r
kc, b.
I f one i n t e g r a t e s t h e f i r s t i n t e g r a l i n ( 2 2 ) by p a r t s , i t i s seen t h a t
40 1
Integral Representations
V(~)W(@ = ) v ( o ) ~ ( $ - ~+)
(23)
1
J
$+0
Cv'(O)w(e) + v(o)w'(e)lJo(Z)de
@-
0
p. 72, (10). On t h e o t h e r hand, i f t h e f i r s t and t h i r d i n t e g r a l s i n (22) a r e combined, t h e r e r e s u l t s
a r e l a t i o n g i v e n e a r l i e r i n [20],
a r e s u l t obtained previously i n
(71.
B o t h Volkmer and Meixner e t a1 c o n s i d e r t h e s p e c i a l case i n which yl(0)
= 0, y i ( 0 ) = 1.
By f i x i n g
where
Then
i n ( 2 5 ) , t h e y show t h a t two homogeneous i n t e g r a l e q u a t i o n s may b e
w.
o b t a i n e d f o r c e r t a i n p e r i o d i c Mathieu f u n c t i o n s i s known;
v = yl,
The form o f t h e s e e q u a t i o n s
what seems t o be new i s t h a t t h e e i g e n v a l u e parameters a r e g i v e n
e x p l i c i t l y i n terms o f values o f t h e f u n c t i o n
yl.
The p r e s e n t w r i t e r i s n o t aware o f any f u r t h e r s t u d i e s o f ( 2 3 ) o r ( 2 4 ) . Clearly,
w
i f a solution
any o t h e r s o l u t i o n
v
i s known, each g i v e s an e x p l i c i t r e p r e s e n t a t i o n f o r
i n terms o f
v(0)
r e l a t i o n s a r e o b t a i n e d by i n t e r c h a n g i n g
and v
and
v'(0).
w
Evidently the inverse
i n ( 2 3 ) or ( 2 4 ) .
i t i s not d i f f i c u l t t o obtain nonlinear equations f o r solutions.
by t a k i n g
i n which
v = w
and
Zo := Z ( 4 = o
=
.
Then
i n ( 2 3 ) , we f i n d
Moreover,
F o r example,
R. F. Miiiar
402
and, i f
y2
i s a s o l u t i o n t o (21) w i t h
y 2 ( 0 ) = 1, y;(O)
= 0,
It i s n o t known whether a s t u d y o f t h e s e n o n l i n e a r e q u a t i o n s has been made.
But
c e r t a i n l y i t i s t e m p t i n g t o s p e c u l a t e t h a t t h e work o f Volkmer [ 7 ] and o f Meixner e t a1 [20]
does n o t exhaust a l l t h e p o s s i b i l i t i e s f o r (23) o r ( 2 4 ) .
I n p r i n c i p l e , by s e p a r a t i n g a l l b u t two o f t h e v a r i a b l e s , t h e methods d e s c r i b e d above can be a p p l i e d t o e q u a t i o n s i n more t h a n two independent v a r i a b l e s .
More
s p e c i f i c a l l y , Volkmer [8] has r e c e n t l y a p p l i e d h i s t e c h n i q u e t o study c e r t a i n p r o p e r t i e s o f Lam;
f u n c t i o n s , which a r i s e when t h e H e l m h o l t z e q u a t i o n i s
separated i n e l l i p s o i d a l c o o r d i n a t e s . A n a l y t i c Continuation I t i s o f t e n u s e f u l t o know t h e e x t e n t t o which a n a l y t i c c o n t i n u a t i o n i s p o s s i b l e
f o r t h e s o l u t i o n t o an e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n .
For example, t h e
p o s i t i o n o f s i n g u l a r i t i e s determines t h e convergence p r o p e r t i e s o f some s e r i e s representations.
T h i s i s i n t i m a t e l y r e l a t e d t o t h e R a y l e i g h h y p o t h e s i s , which
has a c o n s i d e r a b l e l i t e r a t u r e C15-j.
Methods based on (12) have been used i n
t h i s connection, and a r e o u t l i n e d i n [21]. Another area i n which knowledge o f p o i n t s o r r e g i o n s o f n o n a n a l y t i c i t y i s o f i n t e r e s t i s t h a t o f i n v e r s e problems.
Here we r e f e r s p e c i f i c a l l y t o t h o s e t h a t
can be f o r m u l a t e d as problems f o r t h e H e l m h o l t z e q u a t i o n w i t h i n a c c u r a t e Cauchy data.
A problem o f t h i s s o r t w i l l be d i s c u s s e d l a t e r , b u t f i r s t we s h a l l o b t a i n
from ( 1 2 ) a few u s e f u l f o r m u l a s under t h e assumption t h a t a t l e a s t some o f t h e Cauchy d a t a a r e known p r e c i s e l y .
I n essence, these r e s u l t s w i l l p r o v i d e r u l e s
f o r r e f l e c t i n g a s o l u t i o n across an a n a l y t i c boundary c o n d i t i o n . With r e g a r d t o ( 1 2 ) , we make t h e f o l l o w i n g o b s e r v a t i o n s : ( i ) The v a r i a b l e l i m i t s o f i n t e g r a t i o n suggest V o l t e r r a i n t e g r a l s . ( i i ) S i n c e r = 0 a t each l i m i t o f i n t e g r a t i o n , i t may be p o s s i b l e t o choose a p a t h i n t h e s-plane on which
r
i s real.
403
Integral Representations
I n fact,
i n t h e s i t u a t i o n t h a t we c o n s i d e r , we s h a l l be a b l e t o choose s p e c i f i c
paths o f i n t e g r a t i o n t h a t l e a d t o r e a l i n t e g r a l e q u a t i o n s t h a t can be i n v e r t e d expl i c it l y
.
The s i m p l e s t case occurs when t h e a r c C s h a l l confine a t t e n t i o n t o t h i s .
i s p a r t o f a s t r a i g h t l i n e , and we
( A n o t h e r f a i r l y s i m p l e case a r i s e s when
C
is
a c i r c u l a r arc; b u t we have been unable t o o b t a i n e x p l i c i t l y i n v e r t i b l e e q u a t i o n s i n t h e s e circumstances.) x
and
= V(x,y)
+ J
t h e o r i g i n , and i f
y
u(x,y)
V,W
Here
0
V(x,a)
v(x,y)
:= $u(x+iy,o)
(31)
w(x,y)
:= $uI1(xtiy,o
Y
&
+ J
J0(k@?)da
0
W(x,o)Jo(kAq)da.
even i n y, t h a t a r e d e f i n e d b y
a r e harmonic f u n c t i o n s
(30)
i s an a r c o f t h e c - a x i s t h a t c o n t a i n s
a r e r e a l , t h e n ( 1 2 ) may be reduced t o
Y
(29)
C
If
u(x-iy,O)l
t
+
u (x-iy,O)] r)
,
V(x,O)
,
= u(x,O)
w(x,O)
,
= u (x,O) rl
.
Although we now have t o deal w i t h f a i r l y s i m p l e i n t e g r a l s , t h e r e i s a p r i c e t o pay: we have l o s t t h e f l e x i b i l i t y i n d e f o r m i n g paths o f i n t e g r a t i o n t h a t we had i n ( 1 2 ) . Consequently, ( 2 9 ) w i l l n o t g e n e r a l l y be s u i t a b l e f o r mapping o u t t h e detailed singularity structure.
I f Cauchy d a t a
are prescribed a n a l y t i c a l l y , then (29) u(x,O), u (x,O) r) determines u ( x , y ) , a t l e a s t i n a neighbourhood o f t h e z - a x i s . Suppose, however, y > 0
that
b u t s u f f i c i e n t l y s m a l l , and t h a t e i t h e r
prescribed analytically. either on or
0 V.
V(X,O)
5
0
5y
or
-
or
W(X,U) u(x,-y)
0 5 U’Y -, ( 2 9 ) can be
i s known f o r
y = 0.
is
( y > 0)
and, i f
More s p e c i f i c a l l y ,
if
Thus
u ( x , ~ ) i s known
s o l v e d f o r t h e unknown
W
t h e n can be found from
which f o l l o w s f r o m ( 2 9 ) , and which may be regarded as a r u l e f o r across
u (x,O) r)
even by measurement
The v a l u e o f
u(x,O)
(Impedance-type c o n d i t i o n s can a l s o be handled.)
W
i s known, t h e n
e f lection
Jr/k ) / J?
v ( t ) := V ( x ,
s a t i s f i e s t h e V o l t e r r a i n t e g r a l e q u a t i o n o f t h e second k i n d
R. F. Millar
404
(33) i n which
F
i s known and i n v o l v e s
u(x,a)
(0
5 0 5y)
and
W.
The s o l u t i o n
by Laplace t r a n s f o r m methods i s
(34) so
V
i s determined.
If
V
i s known, t h e n d i f f e r e n t i a t i o n o f ( 2 9 ) shows t h a t
w ( t ) := W(x,JF/k)/(PkJf-) involving
U ~ ( X , ~ ()0
s a t i s f i e s ( 3 3 ) w i t h given right-hand side
5
5 y)
and
V.
Thus
W
G(c)
may be determined.
When an a n a l y t i c D i r i c h l e t o r Neumann c o n d i t i o n i s p r e s c r i b e d on an a r c o f t h e c - a x i s t o g e t h e r w i t h a p p r o p r i a t e values o f
u
in
s t a b l e c o n t i n u a t i o n across t h i s a r c i s f e a s i b l e .
y > 0, i t i s c l e a r t h a t
Unfortunately, t h i s i s not t h e
s t a t e o f a f f a i r s f o r i n v e r s e problems w i t h i n a c c u r a t e Cauchy d a t a , t o which we now d i r e c t our a t t e n t i o n .
I n v e r s e Problems I n v e r s e problems f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s f r e q u e n t l y r e q u i r e t h e s o l u t i o n t o Cauchy problems w i t h i n a c c u r a t e d a t a .
As examples, we may mention
t h e c o n t i n u a t i o n o f an a n a l y t i c f u n c t i o n o u t o f some domain, i n v e r s e g r a v i t a t i o n a l problems, o r i n v e r s e c l a s s i c a l s c a t t e r i n g problems.
Further
c o n s i d e r a t i o n i s r e s t r i c t e d t o problems where d a t a a r e known a p p r o x i m a t e l y i n some l i m i t e d r e g i o n i n c l u d i n g o n l y p a r t o f i t s boundary.
The r e p r e s e n t a t i o n
( 1 2 ) appears t o be w e l l s u i t e d f o r f o r m u l a t i n g i n v e r s e problems o f t h i s k i n d i n t w o independent v a r i a b l e s .
We c o n s i d e r now an i n v e r s e s c a t t e r i n g problem i n
which Cauchy d a t a a r e known only a p p r o x i m a t e l y on a segment T h i s i s presumably t h e case i n
C
o f the c-axis.
many p r a c t i c a l s i t u a t i o n s where t h e c h o i c e o f
data surface i s a t t h e d i s c r e t i o n o f t h e i n v e s t i g a t o r . Suppose t h a t d i f f e r e n t media occupy n o t fill a l l o f medium i n
< 0
q < 0.)
q
> 0 and q < 0.
The wave number t h r o u g h o u t
(The l o w e r medium need
11 > 0
c o n s i s t s o f m a t e r i a l w i t h wave number
is
K(> 0),
k ( > 0).
The
i n which t h e r e
may be imbedded i n t e r n a l i n h o m o g e n e i t i e s o r v o i d s , cracks, o r o t h e r sources o f s c a t t e r i n g a t unknown l o c a t i o n s .
I t may have f i n i t e b u t unknown depth.
A
405
Integral Representations
> 0 and, from measurements on
s c a l a r f i e l d i s i n c i d e n t from
tl =
O+ (and,
> 0), we wish t o deduce i n f o r m a t i o n about t h e medium i n
perhaps, elsewhere i n
rl < 0. D e p a r t u r e s from homogeneity w i l l generate s i n g u l a r i t i e s i n t h e r e l e v a n t
s o l u t i o n t o t h e H e l m h o l t z e q u a t i o n w i t h wave number
K
in
T
< 0.
Our goal i s
t o l o c a t e these s i n g u l a r i t i e s . Let
u(')
be t h e t o t a l s o l u t i o n i n
w i t h wave number K.
In
> 0; i t s a t i s f i e s t h e Helmholtz e q u a t i o n
rl
< 0, t h e t o t a l s o l u t i o n i s u ( ~ ) ,w i t h wave number
It i s assumed t h a t t h e t r a n s i t i o n c o n d i t i o n s on
i n which into
k.
a
and
11 < 0, and
p
are constants.
u ( ~ )i n t o
hen
and
W(l)
can be c o n t i n u e d a n a l y t i c a l l y
i n terms o f even harmonic f u n c t i o n s
t h a t a r e d e f i n e d i n accordance w i t h ( 3 0 ) and ( 3 1 ) :
A representation f o r
u ( ~ ) i s o b t a i n e d from ( 3 6 )
s u p e r s c r i p t s ( 1 ) by ( 2 ) , and
k
by
K.
and
u ( ~ ) i s expressed i n terms o f
V(l)
(37)
by replacing
I n accordance w i t h ( 3 5 ) , t h e harmonic
f u n c t i o n s a r e r e l a t e d by
and
a r e o f t h e form
> 0.
A t t h i s stage, we use ( 2 9 ) t o express V(l)
u(l)
n = 0
and
W( 1 )
.
406
R. F. Millar
I t i s e v i d e n t t h a t t h e problem has been reduced t o t h a t o f f i n d i n g two harmonic
functions
V(l)
and
W(l)
t h a t a r e even i n
known a p p r o x i m a t e l y on a segment o f
and which, t h r o u g h ( 3 7 ) , a r e
11
f r o m measurements o f
= O+
u
and
u
1’
There a r e methods t h a t use a p r i o r i bounds t o s t a b i l i z e s o l u t i o n s t o such Cauchy problems f o r t h e L a p l a c e e q u a t i o n i n a s p e c i f i e d domain [ 2 2 ] ; here, however, t h e domains o f e x i s t e n c e o f Although
V(')
and
and
W(')
W(')
a r e unknown.
i n d i v i d u a l l y may be s i n g u l a r , t h e combination o f them
i n (36) i s n o t s i n g u l a r i n
y > 0.
( I t i s assumed t h a t t h e i n c i d e n t e x c i t a t i o n i s a n a l y t i c ; i f i t were s i n g u l a r anywhere i n 11 > 0, we c o u l d deal w i t h t h e nons i n g u l a r s c a t t e r e d component.)
We c o n j e c t u r e t h a t t h i s p r o p e r t y may be u s e f u l
f o r s t a b i l i z i n g t h e s o l u t i o n , t h e i d e a b e i n g t o use measurements on
to
11 = 0
g i v e a f i r s t a p p r o x i m a t i o n t o t h e Cauchy data, and t o employ ( 3 6 ) w i t h measured ( y > 0)
u(')(x,y)
t o improve t h e a p p r o x i m a t i o n .
To t h i s end, t h e f o l l o w i n g
procedure i s suggested: 1.
Represent
even i n
0
W ( 1 ) ( x , ~ ) as s e r i e s o f harmonic f u n c t i o n s t h a t a r e
V(')(X,O),
and t h a t form a s e t t h a t i s complete on
c o e f f i c i e n t s be { v }, {wn}, r e s p e c t i v e l y . n ( = u ( l ) ( x , O ) ) and W ( l ) ( x , O ) Measure V ( l ) ( x , O ) 2. a p p r o x i m a t i o n s ?,,, Gn t o t h e t r u e c o e f f i c i e n t s vn e s t i m a t e s o f t h e form 3.
Measure
u(')(x,y)
iivn
-
inii
< 6, liwn
-
(= u ( l ) ( x , O ) ) and
T
wn,
t o give
and e r r o r
< 6.
Gnii
on l i n e s o f c o n s t a n t
L e t t h e undetermined
C.
y
( y > 0)
t o give constraints
on t h e t r u e c o e f f i c i e n t s , t h r o u g h ( 3 6 ) .
4.
Attempt t o choose o p t i m a l values f o r t h e c o e f f i c i e n t s , f o r example by
l i n e a r programming o r l e a s t squares methods. 5.
Use t h e V(')
and
W(l)
so o b t a i n e d t o f i n d
~ ( ~ ) ( x , y )i n y < 0
by
means o f ( 3 9 ) , and t h u s determine t h e approximate l o c a t i o n o f s i n g u l a r i t i e s i n y < 0. I t has n o t y e t been a s c e r t a i n e d whether t h i s procedure w i l l l e a d t o a u s e f u l
r e s u l t , b u t i t i s planned i n v e s t i g a t e t h i s aspect o f t h e i n v e r s e problem soon. CONCLUDING REMARKS I n t h i s paper, we have g i v e n r e p r e s e n t a t i o n s f o r s o l u t i o n s t o some second- and f o u r t h - o r d e r d i f f e r e n t i a l equations.
For t h e Helmholtz equation, various
p o s s i b l e a p p l i c a t i o n s have been d e s c r i b e d i n areas l i k e s p e c i a l f u n c t i o n s , a n a l y t i c continuation,
and i n v e r s e s c a t t e r i n g t h e o r y .
It i s expected t h a t
s i m i l a r a p p l i c a t i o n s f o r o t h e r e q u a t i o n s can be t r e a t e d i n t h e same manner.
407
Integral Representations
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Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
409
NONLINEAR WAVE PHENOMENA I N THE BRAIN
R o b e r t M.
Miura
Departments of Mathematics and Pharmacology and T h e r a p e u t i c s I n s t i t u t e of Applied Mathematics U n i v e r s i t y of B r i t i s h Columbia Vancouver, B.C. V6T 1Y4 Canada
U n d e r s t a n d i n g t h e m u l t i t u d e of phenomena a s s o c i a t e d w i t h t h e b r a i n remains one of man's g r e a t e s t s c i e n t i f i c and i n t e l l e c t u a l challenges. Among t h e s e phenomena are n o n l i n e a r waves s u c h as n e r v e i m p u l s e s which p r o p a g a t e r a p i d l y down an i n d i v i d u a l n e r v e c e l l axon and s p r e a d i n g c o r t i c a l d e p r e s s i o n which p r o p a g a t e s s l o w l y through a p o p u l a t i o n of n e r v e cells. Nerve impulse p r o p a g a t i o n h a s been s t u d i e d e x t e n s i v e l y w h i l e s p r e a d i n g d e p r e s s i o n h a s o n l y r e c e n t l y been modelled matheW e d i s c u s s v a r i o u s m a t h e m a t i c a l , n u m e r i c a l , and matically. I t i s our hope m o d e l l i n g a s p e c t s of t h e s e wave phenomena. t h a t t h i s knowledge c a n be p u t t o a u s e f u l p u r p o s e , e.9.. c o n t r o l l i n g or a n n i h i l a t i n g t h e s e waves. INTRODUCTION I
N o n l i n e a r "wave" phenomena o c c u r t h r o u g h o u t n a t u r e and, a s s c i e n t i s t s , our j o b i s t o d i s c o v e r t h e i r o c c u r r e n c e , t h e mechanisms by which t h e waves a r e g e n e r a t e d and p r o p a g a t e d , and, i n g e n e r a l , t h e i r s i g n i f i c a n c e and q u a l i t a t i v e and q u a n t i t a t i v e b e h a v i o r under v a r i o u s c o n d i t i o n s . The approach t o t h e s e problems by t h e m a t h e m a t i c a l m o d e l l e r is t o p r o p o s e a r e p r e s e n t a t i o n of t h e p h y s i c a l , c h e m i c a l , o r b i o l o g i c a l s y s t e m i n terms of some s u i t a b l e m a t h e m a t i c a l f o r m u l a t i o n . Ideall y , one t h e n s t u d i e s t h e model u s i n g a v a r i e t y of m a t h e m a t i c a l , a p p r o x i m a t i o n , and n u m e r i c a l t e c h n i q u e s , v a l i d a t e s t h e model e q u a t i o n s by q u a l i t a t i v e and quant i t a t i v e comparisons w i t h e x p e r i m e n t a l d a t a , and f i n a l l y a t t e m p t s t o p r e d i c t new b e h a v i o r p r e v i o u s l y unseen or u n n o t i c e d . A l l of t h e waves we see i n n a t u r e are t r a n s i e n t phenomena; however, t h e r e a r e many which c a n be viewed a p p r o x i m a t e l y a s s t e a d y p r o g r e s s i n g waves, i.e., as waves which move a t a c o n s t a n t speed w i t h unchanging waveform. Among t h e s t e a d y p r o g r e s s i n g waves are s o l i t a r y waves which a r e e s s e n t i a l l y s p a t i a l l y l o c a l i z e d p u l s e s moving on a c o n s t a n t background, w a v e f r o n t s which have a s p a t i a l l y l o c a l i z e d f r o n t and have a s y m p t o t i c a l l y d i f f e r e n t v a l u e s as x + f m , p e r i o d i c waves which have f i n i t e s p a t i a l and t e m p o r a l p e r i o d s , and m u l t i p l e waves which c a n be viewed as a g r o u p of s o l i t a r y - l i k e waves moving t o g e t h e r . ~
2
I n t h i s p a p e r , w e d e s c r i b e t w o t y p e s of n o n l i n e a r wave phenomena i n t h e b r a i n However, b e f o r e d o i n g t h i s , w e f i r s t make some g e n e r a l remarks on t h e b r a i n .
. A
f u n d a m e n t a l u n d e r s t a n d i n g of t h e b r a i n remains one of man's g r e a t e s t i n t e l l e c t u a l and s c i e n t i f i c c h a l l e n g e s . As t h e " c e n t e r " of t h e "mind, " " c o n s c i o u s n e s s , " and "memory," the b r a i n c o n t i n u e s t o confound p h i l o s o p h e r s and s c i e n t i s t s a l i k e . On t h i s t o p i c , t h e world famous n e u r o s u r g e o n , W i l d e r P e n f i e l d ( 1 9 7 5 ) w r o t e : "--consciousness and t h e r e l a t i o n s h i p of mind to b r a i n are probl e m s d i f f i c u l t to s t u d y i n a n i m a l s . C l i n i c a l physicians, on t h e o t h e r hand, i n t h e i r approach t o man, may hope w i t h r e a s o n t o push toward a n u n d e r s t a n d i n g of t h e p h y s i o l o g y of memory and t h e p h y s i o l o g i c a l b a s i s of t h e mind and c o n s c i o u s n e s s . "
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Our immediate g o a l i s more modest b u t s t i l l e x t r e m e l y d i f f i c u l t - t o u n d e r s t a n d t h e mechanics of t h e b r a i n . To t h i s end, i t i s noted t h a t t h e e x p e r i m e n t a l lite r a t u r e c o n t a i n s a w e a l t h of i n f o r m a t i o n and d a t a which have n o t been examined by theoreticians. A s y n e r g y of e x p e r i m e n t a l f i n d i n g s and t h e o r e t i c a l s t u d i e s w i l l g r e a t l y enhance our b a s i c u n d e r s t a n d i n g of t h e b r a i n . The b a s i c u n i t i n t h e b r a i n i s t h e n e r v e c e l l , o r neuron as i t i s a l s o c a l l e d . T h e r e are a b o u t 1 0 l 1 n e u r o n s i n t h e human b r a i n ( H u b e l , 1 9 7 9 ) . I n a d d i t i o n , t h e s e neurons are s u r r o u n d e d by n e u r o g l i a l c e l l s of which t h e r e are e s t i m a t e d t o be a t l e a s t 10 t i m e s as many as neurons and which make up a b o u t h a l f of t h e volume of t h e b r a i n ( K u f f l e r and N i c h o l l s , 1 9 7 6 ) . However, t h e primary f u n c t i o n of t h e s e n e u r o g l i a l c e l l s i s n o t known, and hence t h e y are g e n e r a l l y i g n o r e d i n any d i s c u s s i o n of b r a i n f u n c t i o n - undoubtedly t h i s w i l l t u r n o u t t o be e r r o n e o u s as more d a t a on g l i a l c e l l s a r e accumulated. There are many d i f f e r e n t t y p e s of n e u r o n s , b u t i n g e n e r a l t h e y c o n s i s t of a c e l l body o r from which e x t e n d "processes" o r branches. Most of t h e s e b r a n c h e s are c a l l e d d e n d r i t e s and s e r v e t o r e c e i v e s i g n a l s from o t h e r n e r v e c e l l s and t o i n t e g r a t e t h e s e s i g n a l s a t t h e c e l l body b e f o r e p a s s i n g on s i g n a l s t o o t h e r neurons. One d i s t i n g u i s h e d branch i s called the a x ~ n and t h r o u g h i t s many b i f u r c a t i o n s s e r v e s a s t h e o u t p u t branch S i g n a l s are s e n t o u t a l o n g t h i s branch a t v e r y high s p e e d s f o r t h e nerve c e l l . t o o t h e r c e l l s , e.g., n e i g h b o r i n g neurons or muscle c e l l s . The geuron boundary i s d e f i n e d by a t h i n l i p i d b i l a y e r membrane ( a p p r o x i m a t e l y 50 A i n t h i c k n e s s ) embedded w i t h v a r i o u s p r o t e i n s and t h i s membrane i s c a p a b l e of p e r m i t t i n g t h e movement of i o n s a c r o s s it and a l s o of s e p a r a t i n g e l e c t r i c a l c h a r g e s . I n the normal r e s t i n g s t a t e , t h e r e i s a nonzero e l e c t r i c a l p o t e n t . i a 1 a c r o s s t h i s memb r a n e c a l l e d t h e membrane p o t e n t i a l w i t h a t y p i c a l v a l u e of a b o u t -70 mV ( m i l l i v o l t s ) , negative i n s i d e r e l a t i v e t o t h e outside. The r e g i o n o u t s i d e t h e neurons i s c a l l e d t h e e x t r a c e l l u l a r space and t h e i n s i d e r e g i o n s of neurons are c a l l e d the i n t r a c e l l u l a r space. These neurons a r e c o l l e c t e d t o g e t h e r i n the major c o m p o n e n t s of t h e b r a i n , t h e r i g h t and l e f t hemispheres of t h e cerebrum, which a l m o s t c o m p l e t e l y f i l l s t h e s k u l l , and t h e c e r e b e l l u m , which i s a much smaller p a r t of t h e b r a i n l o c a t e d under t h e back p a r t of t h e cerebrum. F i n a l l y there i s t h e b r a i n s t e m which e x t e n d s between t h e cerebrum and t h e s p i n a l c o r d . The cerebrum and t h e c e r e b e l l u m both have a c o r t e x , i.e., a t h i n l a y e r near t h e s u r f a c e ( l i k e t h e p e e l of an o r a n g e ) . T h i s c o r t e x i s c a l l e d g r e y matter and c o n s i s t s mainly of t h e nerve c e l l b o d i e s and d e n d r i t i c trees ( a n d , of c o u r s e , g l i a l cells). Beneath t h e c o r t e x i s t h e w h i t e matter c o n s i s t i n g of axons from t h e nerve cells. These axons are u s u a l l y covered w i t h a f a t t y s u b s t a n c e c a l l e d myelin from which t h e name w h i t e matter a r i s e s . The two t y p e s of n o n l i n e a r wave phenomena to be d e s c r i b e d h e r e a r e t h e a c t i o n The p o t e n t i a l o r n e r v e i m p u l s e and s p r e a d i n g c o r t i c a l d e p r e s s i o n (SD f o r s h o r t ) . a c t i o n p o t e n t i a l c o r r r s p w i s ko r\ membrane p o t e n t i a l c b r i ~ r ewhi <::I -p r o p a q.a t e s i n axons of s i n g l e c e l l s a t h i g h s p e e d s v a r y i n g from 1 t o o v e r 120 meters/second (Hodgkin, 1964; K u f f l e r and N i c h o l l s , 1 9 7 6 ) . One c e l l is t h u s a b l e t o d i r e c t l y a f f e c t a n e i g h b o r i n g c e l l by f i r i n g an a c t i o n p o t e n t i a l a l o n g its axon which b i f u r c a t e s i n t o many b r a n c h e s , some of which end n e x t t o t h e d e n d r i t e of t h e neighboring c e l l . The d e p o l a r i z a t i o n of t h e p r e s y n a p t i c t e r m i n a l a t t h e end p e r m i t s t h e release of c h e m i c a l s , c a l l e d n e u r o t r a n s m i t t e r s , which d i f f u s e across a narrow gap, c a l l e d t h e s y n a p t i c c l e f t , and a t t a c h t o r e c e p t o r sites on t h e p o s t s y n a p t i c medirane. Conductance changes of the membrane are e f f e c t e d by t h e s e r e c e p t o r molecules. A s e q u e n c e of a c t i o n p o t e n t i a l s transmits information w i t h i n , t o , and from the normal b r a i n . On t h e o t h e r hand, SD i s a s l o w l y propag a t i n g wave phenomenon i n v o l v i n g l a r g e i o n c o n c e n t r a t i o n changes w i t h s p e e d s between 1 t o 15 m i l l i m e t e r s / m i n u t e ( K r a i g and N i c h o l s o n , 1978; N i c h o l s o n e t a l . , I t is o b s e r v e d i n p o p u l a t i o n s of neurons i n t h e c o r t e x of v a r i o u s b r a i n 1978). s t r u c t u r e s under p a t h o l o g i c a l c o n d i t i o n s i n e x p e r i m e n t a l a n i m a l s and i t s f u n c t i o n is n o t known. However, because S D m a n i f e s t s i t s e l f a s a moving l e s i o n , i.e., t h e
41 1
Nonlinear Wave Phenomena in the Brain
n e r v e c e l l s i n t h e r e g i o n a f f e c t e d a c t l i k e dead c e l l s , one m i g h t c o n j e c t u r e t h a t i t s f u n c t i o n is t o t e m p o r a r i l y " s h u t down" p a r t of t h e b r a i n . W e say m o r e about t h i s l a t e r . The n o n l i n e a r i t y of t h e s e waves arises from t h e f a c t t h a t t h e medium t h r o u g h which t h e y p r o p a g a t e is " e x c i t a b l e . " Q u a l i t a t i v e l y , a n e x c i t a b l e medium, s u c h a s n e r v e c e l l membrane, i s one i n which a s u b t h r e s h o l d s t i m u l u s , s a y i n t h e membrane p o t e n t i a l , r e s u l t s i n a r e s p o n s e which i m m e d i a t e l y d e c r e a s e s t o t h e t h e membrane p o t e n t i a l g o e s t o t h e e q u i l i b r i u m s t a t e . HOWr e s t i n g s t a t e , e.g., 3
e v e r , w i t h a s u p r a t h r e s h o l d s t i m u l u s , t h e r e s p o n s e f i r s t undergoes a r e l a t i v e l y l a r g e e x c u r s i o n away from t h e e q u i l i b r i u m b e f o r e r e t u r n i n g t o t h e r e s t s t a t e , e.g., an a c t i o n p o t e n t i a l . I n t h e n e r v e c e l l membrane case, t h i s e x c i t a b i l i t y comes a b o u t b e c a u s e of t h e a c t i v e ( n o n l i n e a r ) p r o p e r t i e s of t h e membrane, i . e . , t h e electrical p r o p e r t i e s of t h e membrane depend on i t s e l e c t r i c a l state. A p a s s i v e membrane h a s c o n s t a n t e l e c t r i c a l p r o p e r t i e s . Both of t h e s e phenomena i n v o l v e t h e movement of i o n s a c r o s s n e r v e c e l l membrane. I n t h e n e r v e i m p u l s e problem, t h e r e i s v e r y l i t t l e movement of i o n s a c r o s s t h e membrane and hence changes i n e x t r a c e l l u l a r and i n t r a c e l l u l a r i o n c o n c e n t r a t i o n s c a n be i g n o r e d . However, t h e r e s u l t i n g changes i n t h e membrane p o t e n t i a l due t o t h e c a p a c i t i v e e f f e c t of t h e l i p i d memhrane are l a r g e and c h a r a c t e r i z e t h e t r a n s i e n t e l e c t r i c a l s t a t e o f t h e membrane a t a g i v e n p o s i t i o n and t i m e . I t is the p r o p a g a t i o n a l o n g t h e axon of t h i s change i n t h e membrane p o t e n t i a l which cons t i t u t e s t h e nerve impulse. The n e r v e i m p u l s e is b a s i c a l l y one d i m e n s i o n a l a l though t h e membrane a l o n g which i t p r o p a g a t e s is a c y l i n d r i c a l s u r f a c e which s u r r o u n d s t h e i n t r a c e l l u l a r f l u i d (axoplasm) On t h e o t h e r hand, S D i n v o l v e s m a s s i v e movements of i o n s a c r o s s n e u r o n a l membrane and c o n c o m i t a n t l a r g e changes i n e x t r a c e l l u l a r ion concentrations occur. Thus t h e i o n c o n c e n t r a t i o n s p l a y a p r i m a r y r o l e i n S D whereas t h e i n t i m a t e l y r e l a t e d membrane p o t e n t i a l p l a y s a secondary r o l e .
.
NERVE IMPULSE PROPAGATION
Modern t h e o r e t i c a l n e u r o b i o l o g y can be s a i d t o have begun i n 1952 w i t h t h e p u b l i c a t i o n by Hodgkin and Huxley ( 1 9 5 2 ) of t h e i r N o b e l P r i z e winning work on t h e a c t i o n p o t e n t i a l i n t h e s q u i d g i a n t axon. T h e i r work which d e t a i l e d t h e mechan i s m s l e a d i n g to t h e a c t i o n p o t e n t i a l c o n s t i t u t e s o u r p r e s e n t u n d e r s t a n d i n g of Although t h e s q u i d n e r v o u s how t h e a c t i o n p o t e n t i a l i s c r e a t e d and p r o p a g a t e d . s y s t e m is a long way from t h e human b r a i n , one of t h e s t a n d a r d working h y p o t h e s e s o f p h y s i o l o g i s t s is t h a t many of t h e mechanisms o c c u r r i n g i n lower a n i m a l s q u a l i t a t i v e l y c o n t i n u e t o o p e r a t e i n h i g h e r animals. The i o n s which make up t h e e x t r a c e l l u l a r and i n t r a c e l l u l a r f l u i d s c o n s i s t m a i n l y o f sodium, p o t a s s i u m , and c h l o r i d e and t h e s e are r e s p o n s i b l e f o r both t h e r e s t i n g membrane p o t e n t i a l and f o r t h e a c t i o n p o t e n t i a l i t s e l f . I n t h e r e s t i n g state, t h e r e a r e e l e c t r o c h e m i c a l g r a d i e n t s s e t up a c r o s s t h e n e r v e c e l l membrane 440 r e s u l t i n g i n i o n c o n c e n t r a t i o n s i n s q u i d of a p p r o x i m a t e l y [Na’],
-
mM/l
(millimolar per l i t e r ) ,
[Na+Ii
-
50 m M / L ,
[K+Ii
..
[K+Io
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400 mM/L,
[C1-Ii
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560 m M / l ,
120 m M / l ,
[ Ca++],-
2 mM/!t,
where s u b s c r i p t s o and i
d e n o t e e x t r a c e l l u l a r and i n t r a c e l l u l a r , r e s p e c t i v e l y . The m a t h e m a t i c a l model p r o p o s e d by Hodgkin and Huxley ( 1 9 5 2 ) c o n s i s t s of a nonl i n e a r d i f f u s i o n e q u a t i o n F o r t h e membrane p o t e n t i a l , V, g i v e n by
c o u p l e d t o t h r e e o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i n t h e form o f r e l a x a t i o n equations,
412
R. M. Miura
an
_at
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Tn(V)
'
w h e r e C i s t h e membrane c a p a c i t a n c e , d i s t h e axon d i a m e t e r , Ri is the axoplasm r e s i s t i v i t y , qNa, g K , gL are f u l l y a c t i v a t e d c o n d u c t a n c e s f o r sod i u m , p o t a s s i u m , and a l e a k a g e c u r r e n t ( m a i n l y c h l o r i d e ) , t h e q u a n t i t i e s m, h, and n a l l o w v a r i a b i l i t y i n t h e s o d i u m a n d p o t a s s i u m c o n d u c t a n c e s , .b.( V ) I h,(V), n,(V) a r e e m p i r i c a l l y d e t e r m i n e d v a l u e s of m, h, n a t the steady s t a t e V , and T m ( V ) , Th(V), and T n ( V ) a r e e m p i r i c a l l y determined r e l a x a t i o n times. The d e t e r m i n a t i o n of t h e membrane p r o p e r t i e s a t d i f f e r e n t membrane pot e n t i a l s was o b t a i n e d by Hodgkin and Huxley ( 1 9 5 2 ) by a c l e v e r s e t of experiments. The l e f t s i d e of ( 1 ) c o r r e s p o n d s t o c a p a c i t i v e c u r r e n t due t o t h e a b i l i t y of t h e a x o n a l membrane t o s e p a r a t e c h a r g e s . On t h e r i g h t s i d e , t h e d i f f u s i o n t e r m g i v e s t h e c u r r e n t f l o w down t h e axoplasm and t h e remaining t h r e e terms desc r i b e t h e i o n i c c u r r e n t s t h r o u g h t h e membrane due t o sodium, p o t a s s i u m , and VK, VL a r e N e r n s t p o t e n t i a l s which are t h e chloride. The q u a n t i t i e s VNa,, membrane p o t e n t i a l s needed t o m a i n t a i n t h e i o n i c c o n c e n t r a t i o n s on the t w o s i d e s o f t h e membrane, i.e., each c o r r e s p o n d s t o t h e membrane e l e c t r i c p o t e n t i a l which i s n e c e s s a r y to p r e v e n t a n e t f l o w of t h a t i o n across t h e membrane due t o diffusion. T h u s V - VNa i s t h e d r i v i n g e l e c t r i c p o t e n t i a l f o r sodium i o n s a c r o s s t h e membrane and t h e d i r e c t i o n of i o n i c flow depends on i t s s i g n . I n s p i t e of t h e i r c o m p l e x i t y and l a c k of c o r r e l a t i o n w i t h some d a t a , t h e HodgkinHuxley ( H H ) e q u a t i o n s remain t h e b a s i c model f o r t h e a c t i o n p o t e n t i a l i n s q u i d g i a n t axons. Via t h e HH model, one c a n " e x p l a i n " t h e t r a j e c t o r y of t h e a c t i o n p o t e n t i a l d u r i n g i t s p r o p a g a t i o n a s a s o l i t a r y wave. Through t h e d i f f u s i o n t e r m t h e f r o n t of t h e wave g r a d u a l l y d e p o l a r i z e s t h e membrane ahead of i t u n t i l t h e membrane p o t e n t i a l r e a c h e s a v a l u e such t h a t t h e c u r r e n t f l o w t h r o u g h the membrane i s dominated by t h e s o d i u m i o n s , i . e . , t h e s o d i u m c o n d u c t a n c e g i v e n by qNam3h becomes l a r g e relative t o its resting level. T h e r e s u l t i n g e f f e c t of this is t o d r i v e the membrane p o t e n t i a l towards t h e sodium N e r n s t p o t e n t i a l ( a b o u t +SO mV i n s q u i d As t h e membrane p o t e n t i a l a p p r o a c h e s i t s peak, t h e sodium conductance axon). b e g i n s t o d e c r e a s e w h i l e t h e p o t a s s i u m conductance c o n t i n u e s t o i n c r e a s e . The c u r r e n t t h e n becomes dominated by the p o t a s s i u m i o n s and t h e membrane p o t e n t i a l i s d r i v e n towards t h e p o t a s s i u m N e r n s t p o t e n t i a l ( a b o u t -75 mV i n s q u i d a x o n ) . F i n a l l y , sodium and p o t a s s i u m c o n d u c t a n c e s d e c r e a s e t o t h e i r rest s t a t e v a l u e s and t h e membrane p o t e n t i a l r e t u r n s t o i t s r e s t i n g l e v e l ( a b o u t -60 mV i n s q u i d axon)
.
Models based on t h e HH e q u a t i o n s have been d e r i v e d f o r axons of o t h e r i n v e r t e b r a t e and v e r t e b r a t e c e l l s . However, t h e i r c o m p l e x i t y has s e v e r e l y l i m i t e d t h e m a t h e m a t i c a l a n a l y s e s of t h e e q u a t i o n s . T h e r e have been e x i s t e n c e r e s u l t s f o r s o l i t a r y waves of t h e HH e q u a t i o n s o b t a i n e d by H a s t i n g s (1976a) and C a r p e n t e r ( 1 9 7 7 ) who u s e d s i n g u l a r p e r t u r b a t i o n methods. S o l i t a r y wave p r o f i l e s u s i n g n u m e r i c a l c o m p u t a t i o n s have been o b t a i n e d by Cole e t a l . (1955) and Hassard (1980). The c o m p l e x i t y of f o r them g i v e n by
t h e HH e q u a t i o n s
l e d FitzHugh (1961 )
t o propose a "metaphor"
Nonlinear Wave Phenomena in the Brain
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= $ ( V ia
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bR),
where f ( V ) i s a c u b i c s h a p e d f u n c t i o n and $ , a , b are c o n s t a n t s . The quant i t y R i s r e f e r r e d t o as t h e r e c o v e r y v a r i a b l e and i s e s s e n t i a l f o r t h e t a i l of t h e s o l i t a r y wave t o r e c o v e r b a c k t o t h e r e s t s t a t e . F o r f ( V ) = V - V3/3, t h e s e e q u a t i o n s a r e c a l l e d t h e FitzHugh-Nagurno (FHN) e q u a t i o n s (Nagumo e t a l . , 1 9 6 2 ) . A l t h o u g h t h e o r i g i n a l p r o p o s a l f o r t h e FHN e q u a t i o n s w a s a d h o c a n d b a s e d on a n a l o g y w i t h t h e van d e r P o l o s c i l l a t o r , t h e y h a v e b e e n j u s t i f i e d a p o s t e r i o r i by C a s t e n e t a l . ( 1 9 7 5 ) and R i n z e l ( 1 9 7 8 ) . E x i s t e n c e o f s o l i t a r y wave s o l u t i o n s f o r t h e FHN e q u a t i o n s was o b t a i n e d by H a s t i n g s (197633) a n d C a r p e n t e r ( 1 9 7 7 ) . Asympt o t i c s o l i t a r y wave p r o f i l e s w e r e o b t a i n e d b y C a s t e n e t a l . ( 1 9 7 5 ) . Solitary wave p r o f i l e s h a v e b e e n computed u s i n g v a r i o u s n u m e r i c a l methods by F i t z H u g h and A n t o s i e w i c z ( 1 9 5 9 ) and Miura ( 1 9 8 2 ) . The most r e c e n t new r e s u l t i s t h a t t h e t r a v e l l i n g wave s o l u t i o n f o r t h e FHN e q u a t i o n s a r e s t a b l e r e l a t i v e t o t h e f u l l s y s t e m of p a r t i a l d i f f e r e n t i a l e q u a t i o n s ( J o n e s , 1983). A c a r i c a t u r e of
(7)
f(V)
t h e FHN e q u a t i o n s was g i v e n by McKean ( 1 9 7 0 ) w i t h Z
H(V
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-
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where H i s t h e h e a v i s i d e step function. T h i s is r e f e r r e d t o as t h e p i e c e w i s e l i n e a r model. D e t a i l e d s t u d i e s o f t h i s model, i n c l u d i n g e x a c t s o l u t i o n s , h a v e b e e n made by McKean ( 1 9 7 0 ) and R i n z e l a n d K e l l e r ( 1 9 7 3 ) ( c f . R i n z e l , 1 9 7 7 ) . SPREADING CORTICAL DEPRESSION L e a 0 ( 1 9 4 4 ) ' t h e B r a z i l i a n p h y s i o l o g i s t , d i s c o v e r e d t h e wave phenomenon c a l l e d s p r e a d i n g c o r t i c a l d e p r e s s i o n ( S D f o r s h o r t ) w h i l e s t u d y i n g e p i l e p s y i n t h e cort e x of t h e r a b b i t . T h e s e waves o c c u r i n v a r i o u s b r a i n s t r u c t u r e s i n a v a r i e t y of a n i m a l s and h a v e s p e e d s b e t w e e n 1 - 15 mm/min. The name " d e p r e s s i o n " comes f r o m t h e d e p r e s s i o n of t h e s p o n t a n e o u s e l e c t r o e n c e p h a l o g r a p h i c a c t i v i t y (EEG) w h i c h l a s t s f o r s e v e r a l minutes. The r e a d e r i s r e f e r r e d t o t h e book by B u r e s e t a1 ( 1 9 7 4 ) f o r f u r t h e r d e t a i l s on SD. I n d e s c r i b i n g t h e a c t i o n p o t e n t i a l v i a t h e HH e q u a t i o n s , one of t h e u n d e r l y i n g a s s u m p t i o n s i s t h a t t h e r e a r e no c h a n g e s i n t h e e x t r a c e l l u l a r and i n t r a c e l l u l a r c o n c e n t r a t i o n s of t h e major i o n s . I n t h a t case, t h e i m p o r t a n t d e p e n d e n t v a r i a b l e s w e r e t h e membrane p o t e n t i a l and t h r e e a u x i l i a r y v a r i a b l e s u s e d t o r e p r e s e n t t h e i o n i c c o n d u c t a n c e s of t h e membrane. F o r S D , t h e main m a n i f e s t a t i o n s of t h e phenomenon are l a r g e c h a n g e s i n t h e e x t r a c e l l u l a r i o n c o n c e n t r a t i o n s . T h e c h a n g e s i n i o n c o n c e n t r a t i o n s h a v e b e e n m e a s u r e d w i t h i o n - s e l e c t i v e microe l e c t r o d e s and the p e a k c h a n g e s e x c e e d t h e r a n g e of i o n c h a n g e s g e n e r a l l y obt h e e x t r a c e l l u l a r potassium s e r v e d u n d e r n o r m a l h o m e o s t a t i c c o n d i t i o n s , e.g., c o n c e n t r a t i o n g r e a t l y e x c e e d s 12 m M / t which is n e v e r e x c e e d e d u n d e r n o r m a l 1978). Recognizing t h e i mport ance of ion c o n d i t i o n s (Nicholson e t a l . , c o n c e n t r a t i o n c h a n g e s i n S D phenomenology, it becomes o b v i o u s t h a t the p r i m i t i v e d e p e n d e n t v a r i a b l e s s h o u l d b e t h e c o n c e n t r a t i o n s of t h e m a j o r i o n s i n v o l v e d i n s t e a d of t h e membrane p o t e n t i a l which i s t h e p r i m i t i v e v a r i a b l e f o r a c t i o n I n a d d i t i o n to t h e d e p r e s s i o n o f t h e EEG, t h e s e c h a n g e s p r o d u c e potentials. o t h e r o b s e r v a b l e phenomena s u c h a s d e p o l a r i z a t i o n of t h e membrane p o t e n t i a l t o w a r d s z e r o a n d s u b s e q u e n t r e c o v e r y c a u s i n g b u r s t i n g o f t h e c e l l s and a n e g a t i v e b r a i n s u r f a c e DC p o t e n t i a l wave of a b o u t 5 - 15 mV. A l t h o u g h S D i s a p a t h o l o g i c a l phenomenon which h a s b e e n o b s e r v e d o n l y i n e x p e r i m e n t a l a n i m a l s , e x p e r i m e n t a l and t h e o r e t i c a l s t u d i e s o f SD s h o u l d y i e l d v a l u a b l e One r e a s o n f o r t h i s i s t h a t SD new i n f o r m a t i o n on h o w t h e b r a i n o p e r a t e s .
R. M. Miura
414
e x p e r i m e n t s p r o v i d e a w i d e r d a t a b a s e from w h i c h we h o p e t o g l e a n more i n f o r m a t i o n o n b r a i n mechanisms t h a n c o u l d b e o b t a i n e d from t h e d a t a measured u n d e r t h e A s e c o n d r e a s o n is t h a t SD i s c o n f i n e d t o g r e y n a r r o w r a n g e o f normal o p e r a t i o n . m a t t e r which c o n s i s t s m a i n l y o f n e r v e c e l l b o d i e s and d e n d r i t e s as w e l l as g l i a l
cells. The s i m i l a r i t y b e t w e e n t h e s p a t i a l p a t t e r n s of SD waves e m e r g i n g from a p o i n t s t i m u l u s and n o n l i n e a r c h e m i c a l r e a c t i o n p a t t e r n s i n t h e Z a i k i n - Z h a b o t i n s k i i ( 1 9 7 0 ) r e a c t i o n l e d T u c k w e l l a n d t h e a u t h o r ( 1 9 7 8 ) t o p o s t u l a t e a s y s t e m o f nonl i n e a r d i f f u s i o n e q u a t i o n s f o r t h e movement o f i o n s i n t h e e x t r a c e l l u l a r s p a c e . I t s h o u l d b e n o t e d t h a t t h e r e are t w o d i s t i n c t a s p e c t s of S D waves which may r e q u i r e d i f f e r e n t models. One a s p e c t i s t h a t o f s t i m u l a t i o n , i . e . , t h e i n s t i g a t i o n or g e n e r a t i o n o f t h e SD wave, and t h e o t h e r i s t h e p r o p a g a t i o n o f SD waves. A v a r i e t y o f s t i m u l i c a n g e n e r a t e S D waves, a n d , h e n c e , e a c h would r e q u i r e a s e p a r a t e model. H e r e we c o n c e n t r a t e on t h e p r o p a g a t i o n of S D waves w h i c h i s i n d e p e n d e n t o f t h e mode of g e n e r a t i o n and f o r which a s i n g l e model suffices. T h e r e a r e p r o p e r t i e s o f S D waves which a r e s i m i l a r t o t h o s e of a c t i o n p o t e n t i a l s b u t t h e r e a r e a l s o d i f f e r e n c e s which must be a c c o u n t e d f o r i n a n y model. Simil a r i t i e s of SD waves t o a c t i o n p o t e n t i a l s i n c l u d e :
.
( i ) S o l i t a r y w a v e s , i e . , l o c a l i z e d t r a v e l l i n g p u l s e s w i t h c o n s t a n t s p e e d and shape; ( i i )A p p r o x i m a t e l y a l l or none phenomenon, i . e . , t h r e s h o l d b e h a v i o r ; ( i i i )R e f r a c t o r i n e s s , i . e . , a s e c o n d SD wave or a c t i o n p o t e n t i a l c a n n o t b e g e n e r a t e d i f t h e s e c o n d s t i m u l u s i s a p p l i e d too s o o n a f t e r t h e f i r s t one; and ( i v ) A n n i h i l a t i o n o f t w o waves when t h e y c o l l i d e h e a d on. D i f f e r e n c e s b e t w e e n SD waves a n d a c t i o n p o t e n t i a l s which n e e d t o b e t a k e n i n t o a c c o u n t f o r t h e SD model i n c l u d e : ( i )T i m e s c a l e f o r S D waves is m i n u t e s and mere m i l l i s e c o n d s f o r a c t i o n p o t e n tials; ( i i ) I o n f l u x e s o c c u r t h r o u g h somatic and d e n d r i t i c membranes, e s p e c i a l l y s y n a p t i c membrane, i n SD and t h r o u g h a x o n a l membrane f o r t h e a c t i o n p o t e n t i a l ; ( i i i )L a r g e e x t r a c e l l u l a r i o n c o n c e n t r a t i o n c h a n g e s o c c u r i n S D b u t n o t i n a c t i o n p o t e n t i a l s ; and, ( i v ) Active t r a n s p o r t mechanisms, which consume e n e r g y , a r e n e e d e d i n S D t o r e t u r n t h e i o n c o n c e n t r a t i o n s back to normal. V a r i o u s a t t e m p t s have b e e n made t o m a t h e m a t i c a l l y model SD; however, none of t h e s e a t t e m p t s h a v e a d d r e s s e d t h e q u e s t i o n s of w h a t are t h e s p e c i f i c p h y s i o l o g i c a l mechanisms u n d e r l y i n g SD. G r a f s t e i n ( 1 9 6 3 ) p r o p o s e d a model s i m i l a r t o t h e FHN e q u a t i o n s i n which p o t a s s i u m and i t s d i f f u s i o n w e r e r e s p o n s i b l e f o r t h e i n s t i g a t i o n and p r o p a g a t i o n of S D waves. H e r c o n j e c t u r e was b a s e d on t h e f a c t t h a t n e u r o n a l b u r s t i n g o c c u r s d u r i n g t h e SD wave and release of p o t a s s i u m s h o u l d result. T h i s p o t a s s i u m would t h e n d i f f u s e c a u s i n g n e u r o n a l f i r i n g s i n n e i g h b o r i n g c e l l s w i t h f u r t h e r r e l e a s e of p o t a s s i u m , etc. The release of p o t a s s i u m d u r i n g n e u r o n a l b u r s t i n g i s c e r t a i n l y a f a c t o r i n t h e i n s t i g a t i o n of S D waves b u t n e u r o E x p e r i m e n t s by n a l b u r s t i n g is n o t e s s e n t i a l f o r t h e p r o p a g a t i o n of S D waves. Sugaya e t a l . ( 1 9 7 5 ) showed t h a t a n S D wave c a n p r o p a g a t e t h r o u g h a region of c o r t e x t r e a t e d w i t h t e t r o d o t o x i n (TTX) which i s a p o i s o n from t h e p u f f e r f i s h ( f u g u ) . The e f f e c t of T T X i s t o p r e v e n t t h e sodium c h a n n e l s i n the membrane from o p e n i n g t h e r e b y p r e v e n t i n g t h e n e u r o n from f i r i n g . T h e r e f o r e , t h e r e are o t h e r mechanisms which a r e n e e d e d t o e x p l a i n the p r o p a g a t i o n of S D . Based o n o t h e r p h y s i o l o g i c a l e x p e r i m e n t s , T u c k w e l l and t h e a u t h o r ( 1 9 7 8 ) ( c f . M i u r a , 1981 ) c o n j e c t u r e d t h a t s y n a p t i c mechanisms n o t t r i g g e r e d by a c t i o n p o t e n t i a l s are l a r g e l y r e s p o n s i b l e f o r S D p r o p a g a t i o n . T h i s p r o p a g a t i o n is c a u s e d by t h e r e l e a s e of l a r g e amounts of p o t a s s i u m from t h e d e n d r i t e s a f t e r d e p o l a r i z a t i o n
41 5
Nonlinear Wave Phenomena in the Brain
of s y n a p s e s due t o a n i n c r e a s e d e x t r a c e l l u l a r pot assi um c o n c e n t r a t i o n r e s u l t i n g from d i f f u s i o n . The d e p o l a r i z e d s y n a p s e s r e l e a s e n e u r o t r a n s m i t t e r c h e m i c a l s and open c h a n n e l s ( n o t a b l y f o r p o ta ssiu m ) i n t h e p o s t s y n a p t i c neurons. Because t h e e x c i t a t o r y s y n a p s e s are l o c a t e d i n t h e a p i c a l d e n d r i t e s , i . e . , n e a r t h e s u r f a c e of t h e c o r t e x , S D waves h a v e t h e l a r g e s t a m p l i t u d e s i n t h i s r e g i o n . I n o r d e r t o model SD m a t h e m a t i c a l l y , i t i s u n n e c e s s a r y ( a n d i m p o s s i b l e ) t o a c c o u n t f o r e a c h i n d i v i d u a l n e u r o n b e c a u s e t h e scale o f the S D wave is l a r g e compared to cell size. W e , t h e r e f o r e , d i v i d e t h e c o r t e x i n t o t w o i n t e r a c t i n g cont i n u u m c o m p a r t m e n t s r e p r e s e n t i n g t h e e x t r a c e l l u l a r and i n t r a c e l l u l a r s p a c e s , a b o u t 2 0 % and 80% o f t h e b r a i n volume, r e s p e c t i v e l y ( N i c h o l s o n a n d P h i l l i p s , 1981). However, mechanisms a t t h e c e l l u l a r l e v e l a r e t a k e n i n t o a c c o u n t . Bec a u s e t h e e x t r a c e l l u l a r s p a c e i s c o m p l e t e l y c o n n e c t e d , i o n s c a n d i f f u s e from one On t h e o t h e r h a n d , i n t r a c e l l u l a r i o n s c a n n o t move s p a t i a l position to another. very far without f i r s t e n t e r i n g t h e e x t r a c e l l u l a r space. T h e r e i s , however, a t l e a s t o n e e x c e p t i o n t o t h i s ; g l i a l c e l l s , which t a k e u p a large p r o p o r t i o n of t h e b r a i n volume, c a n c o n n e c t t o g e t h e r t o e f f e c t i v e l y form l o n g i n t r a c e l l u l a r s t r u c tures. G l i a l c e l l membranes a l l o w m a i n l y p o t a s s i u m i o n s t o pass t h r o u g h them and a c t l i k e p o t a s s i u m e l e c t o d e s , i . e . , h a v e a membrane p o t e n t i a l e q u a l t o t h e l o c a l potassium Nernst p o t e n t i a l . T h e r e f o r e , a membrane p o t e n t i a l g r a d i e n t c a n b e s e t u p a l o n g t h e s e g l i a l c e l l s e f f e c t i n g r a p i d t r a n s p o r t of p o t a s s i u m i o n s i n t h e e x t r a c e l l u l a r s p a c e by t a k i n g up p o t a s s i u m i o n s a t o n e end and r e l e a s i n g o t h e r p o t a s s i u m i o n s a t t h e o t h e r e n d . We d o n o t d i s c u s s this p o s s i b i l i t y f u r t h e r h e r e 1983a,b; b u t n o t e t h a t t h i s i s c u r r e n t l y u n d e r i n v e s t i g a t i o n (Gardner-Medwin, Gardner-Medwin a n d N i c h o l s o n , 1 9 8 3 ) . The o n e - d i m e n s i o n a l
c o n t i n u u m model f o r
n
a co 3 = at
~ o
(8)
a2
D . 3 ax2
’+
i o n s t a k e s t h e form I. 3
+
P.
3’
where D . i s t h e d i f f u s i o n c o e f f i c i e n f of t h e j t h i o n i n a q u e o u s s o l u t i o n N i c h o l s o n ’ and P h i l l i p s , 1981 1, C? and C ? are t h e e x t r a c e l l u l a r and i n t r a c e l l 3 1 u l a r c o n c e n t r a t i o n s o f t h e j t h i o n ( p o s s i b l y i n c l u d i n g n e u r o t r a n s m i t t e r chemic a l s ) , a n d a a c c o u n t s f o r t h e r a t i o of e x t r a c e l l u l a r t o i n t r a c e l l u l a r volumes. T h e term I. c o r r e s p o n d s t o t h e m o v e m e n t of i o n s across the membrane d u e t o 3 g . ( V - v . ) where g j and V . m e m b r a n e e l e c t r i c a l forces, i . e . , a r e t h e con3 3 The c o n d u c t a n c e s d u c t a n c e and N e r n s t p o t e n t i a l f o r t h e j t h i o n . need n o t be constant. The q u a n t i t y P . r e p r e s e n t s t h e movement of i o n s a c r o s s t h e membrane d u e t o e n e r g y c o n s u m i n g mecAanisms, e.g., pumps.
4,
An e x p l i c i t f o r m o f t h i s model h a s b e e n g i v e n i n t h e s i m p l i f i e d case of o n l y two i o n s , p o t a s s i u m and c a lc iu m . T h e s e t w o i o n s are i n c l u d e d b e c a u s e p o t a s s i u m i s c o n j e c t u r e d t o b e t h e main d i f f u s i n g i o n and c a l c i u m e n t r y i n t o t h e p r e s y n a p t i c t e r m i n a l i s r e s p o n s i b l e f o r n e u r o t r a n s m i t t e r release. We assume t h e o t h e r t w o major i o n s , s o d i u m and c h l o r i d e , h a v e f i x e d c o n c e n t r a t i o n s , b u t , of c o u r s e , t h i s is not t r u e physiologically. Thus t h e r e a r e t w o e q u a t i o n s f o r t h e e x t r a c e l l u l a r i o n s and t w o f o r t h e i n t r a c e l l u l a r i o n s . L e t K and C r e p r e s e n t t h e p o t a s s i u m a n d c a l c i u m i o n c o n c e n t r a t i o n s , t h e n t h e s i m p l i f i e d SD model i s g i v e n by
-
(10)
- = D
(11)
K1- -a =
(12)
ax at -aa co t-
- D
:
a [ P K g K ( v ) ( v- V,)
a 2 ~ +o 2 ax
P c g c ( V ) ( V - Vc)
+ P,],
+ Pc’
416
R. M. Miura
(13) w h e r e DK and DC a r e t h e d i f f u s i o n c o e f f i c i e n t s f o r p o t a s s i u m and c a l c i u m and pK and pc a r e constants. The q u a n t i t i e s V , VK, Vc a r e r e l a t e d to t h e i o n c o n c e n t r a t i o n s through t h e e q u a t i o n s
v
(14)
KO
+ +
RT F’
y
A, := A t n [w] K1
vK =
6
A t n
KO [ TI,
K
where
F,R,T,y,6
a r e constants.
The c o n d u c t a n c e s are g i v e n by
where i t is assumed t h a t n e u r o t r a n s m i t t e r release is p r o p o r t i o n a l t o t h e c a l c i u m membrane c u r r e n t ( L l i n a s e t a l . , 1976) and gc is d e t e r m i n e d e m p i r i c a l l y w i t h f i t t i n g c o n s t a n t s g,, pg, Vg. The pump terms PK and Pc a r e assumed t o res t o r e t h e normal i o n i c s t a t e and are g i v e n by (19)
where
fK,
fC,
r K , rC,
fi,
f;
are f i t t i n g c o n s t a n t s and
Kg
and
Cft
are
Numerit h e r e s t i n g l e v e l s of e x t r a c e l l u l a r p o t a s s i u m and i n t r a c e l l u l a r calcium. c a l s o l u t i o n s of t h e s e e q u a t i o n s have i n d i c a t e d t h e f o l l o w i n g r e s u l t s (Tuckwell and Miura, 1 9 7 8 ) : ( i )S o l i t a r y waves a r e g e n e r a t e d w i t h a s u f f l c i e n t l y l a r g e s t i m u l u s , i . e . , a p p l i c a t i o n o f enough p o t a s s i u m ; and, ( i i ) The head-on c o l l i s i o n o f t w o S D waves r e s u l t s i n t h e a n n i h i l a t i o n of b o t h waves. with T h e s e r e s u l t s a r e q u a l i t a t i v e l y t h e same a s t h e e x p e r i m e n t a l r e s u l t s . s p e c i a l l y chosen p a r a m e t e r v a l u e s , t h e S D wave s p e e d w a s a b o u t 1.5 mm/min., w i t h i n t h e p h y s i o l o g i c a l r a n g e b u t u n d e r a v a r i e t y o f a s s u m p t i o n s which are i n c o n s i s t e n t with t h e physiology, e.g., c o n s t a n t sodium and c h l o r i d e concentrations. TO o b t a i n a t r u l y q u a n t i t a t i v e model, i .e ., one w i t h which wave s h a p e a s w e l l as wave speed a r e produced a c c u r a t e l y , r e q u i r e s t h e i n c o r p o r a t i o n of o t h e r major i o n s , namely sodium and c h l o r i d e , s e p a r a t i o n of p r e - and p o s t s y n a p t i c compartFor m e n t s , and t a k i n g i n t o a c c o u n t g l i a l c e l l i n f l u e n c e s on p o t a s s i u m t r a n s p o r t . e a c h a d d i t i o n a l i o n i n c l u d e d i n t h e model, t w o a d d i t i o n a l e q u a t i o n s are needed, one e a c h f o r t h e e x t r a c e l l u l a r and i n t r a c e l l u l a r compartments. Separating the
417
Nonlinear Wave Phenomena in the Brain
p r e - and p o s t s y n a p t i c compartments would r e q u i r e t h r e e e q u a t i o n s f o r each i o n i c species.
e a r l i e r , it is n o t known i f S D waves s e r v e any u s e f u l f u n c t i o n i n t h e human b r a i n under p a t h o l o g i c a l c o n d i t i o n s . Tuckwell and t h e a u t h o r (unpubl i s h e d ) c o n j e c t u r e d t h a t i f it d o e s o c c u r i n humans, i t could p l a y a r o l e i n t h e o c c u r r e n c e of some t y p e s of comatose s t a t e s . Some of t h e c i r c u m s t a n t i a l e v i d e n c e f o r t h i s c o n j e c t u r e a r e t h a t SD waves are i n s t i g a t e d by s t i m u l i which a l s o l e a d t o comatose s t a t e s , e.g., mechanical impact, e l e c t r i c a l s t i m u l i , and a p p l i c a t i o n of drugs. I n t h i s c o n n e c t i o n , i f it were p o s s i b l e f o r an SD wave t o s u s t a i n i t s e l f i n t h e b r a i n , t h e n i t c o u l d s u p p r e s s n e u r o n a l a c t i v i t y i n t h e b r a i n necesSuch s u s t a i n e d S D a c t i v i t y has been s a r y f o r maintaining a conscious state. o b s e r v e d i n l a b o r a t o r y a n i m a l s and c o n s i s t s of r e v e r b e r a t i o n of r o t a t i n g S D waves around a l e s i o n i n t h e c o r t e x ( S h i b a t a and Bures, 1972, 1 9 7 4 ) .
A s mentioned
Some q u e s t i o n s w e have asked a b o u t r e v e r b e r a t i n g SD waves have been:
( i ) Is t h e l e s i o n n e c e s s a r y f o r t h e phenomenon, i . e . , can r e v e r b e r a t i o n o c c u r w i t h o u t a c e n t r a l dead a r e a ; ( i i )A r e t h e r e any n a t u r a l p h y s i o l o g i c a l l i m i t a t i o n s on t h e number of r o t a t i o n s of t h e waves; and, ( i i i ) How can t h e s e r o t a t i n g waves be a n n i h i l a t e d once t h e y have s t a r t e d . To study these questions,
one needs t o e x t e n d t h e e q u a t i o n s t o two s p a t i a l dimens i o n s , n e g l e c t i n g t h e dependence on d e p t h . I n p l a c e of ( 8 ) ,w e o b t a i n
a c?
(21)
&=
Dj
a2c0
a%?
ax2
ay
( -++)
+
Ij
+
Pj.
I n s t e a d of s t u d y i n g t h e s e e q u a t i o n s [ ( 9 ) , ( 2 1 ) ] , w e have chosen t o s t u d y a simpler two-dimensional v e r s i o n of t h e FitzHugh-Nagumo e q u a t i o n s which r e p r e s e n t s an e x c i t a b l e medium (Miura and P l a n t , 1 9 8 1 ) . Using a f i n i t e e l e m e n t method, w e have been a b l e t o d e m o n s t r a t e s e v e r a l d i f f e r e n t f e a t u r e s of t h e two-dimensional e x c i t a b l e medium. F i r s t and m o s t i m p o r t a n t , it i s p o s s i b l e t o u s e p h y s i o l o g i c a l l y r e a l i s t i c s t i m u l i t o g e n e r a t e a r o t a t i n g wave. T o accomplish t h i s , f i r s t a oned i m e n s i o n a l SD wave i s g e n e r a t e d n e a r one s i d e of t h e r e g i o n ( i n t h i s c a s e a c i r c l e ) . A s t h e S D wave a p p r o a c h e s t h e o p p o s i t e s i d e of t h e r e g i o n and is a b o u t t o e x i t , a second s t i m u l u s i s a p p l i e d t o one s i d e of and j u s t behind t h e wave. I f t h e t i m i n g i s c o r r e c t , t h e n on t h e S D wave s i d e of t h e s t i m u l u s i t i s r e f r a c t o r y and no new S D wave i s g e n e r a t e d whereas a new wave i s g e n e r a t e d on t h e nonr e f r a c t o r y s i d e of t h e s t i m u l u s . I f t h e l o c a t i o n , a m p l i t u d e , and w i d t h of t h e s t i m u l u s is s u f f i c i e n t , t h e n a c i r c u l a t i n g wave is g e n e r a t e d . P r e l i m i n a r y tests i n d i c a t e t h a t t h e waves c o n t i n u e t o r o t a t e around i n d e f i n i t e l y . Finally, the g e n e r a t i o n of a second wave i n t h e o p p o s i t e d i r e c t i o n a n n i h i l a t e s t h e f i r s t wave. Although t h e s e r e s u l t s have been o b t a i n e d f o r an e x t r e m e l y s i m p l i f i e d system, i t i s e x p e c t e d t h a t s i m i l a r r e s u l t s w i l l be o b t a i n e d f o r t h e two-dimensional S D equations.
R.M. Miura MOTNOTES 1 I n some d i c t i o n a r i e s , t h e word "wave" has no less t h a n n i n e d i f f e r e n t meanings. None of them, I f e e l , r e a l l y g i v e s the e s s e n c e of t h e meaning of t h e word as i t i s used i n t h i s c o n f e r e n c e on Wave Phenomena: Modern Theory and A p p l i c a t i o n s . My f a v o r i t e d e s c r i p t i o n of a "wave" i s s i m i l a r t o t h a t g i v e n o r i g i n a l l y by John S c o t t Russell (1844), the S c o t t i s h s h i p b u i ld e r . A "wave" i s t h e s p a t i a l propaga t i o n of a t e m p o r a l change i n some o b s e r v a b l e , e.g., water e l e v a t i o n , e l e c t r i c p o t e n t i a l , i o n c o n c e n t r a t i o n s , etc. Thus a "water wave" is t h e p r o p a g a t i o n of Note t h a t a s t a n d i n g wave i s n o t i n t h e l o c a l e l e v a t i o n change of t h e water. cluded i n t h i s d e s c r i p t i o n .
2
The t i t l e of t h e t a l k on w h i c h t h i s p a p e r i s based was i n i t i a l l y i n c o r r e c t l y a d v e r t i s e d a s " N o n l i n e a r Wave Phenomena i n the Ocean." "Ocean" and " b r a i n " have t h e s u p e r f i c i a l s i m i l a r i t y of e a c h c o n t a i n i n g f i v e l e t t e r s i n c l u d i n g "a" and "n". However, t h e r e i s a d e e p e r more fundamental c o n n e c t i o n between them. The r a t i o s o f t h e c o n c e n t r a t i o n s of t h e t h r e e major i o n s (sodium, Na'; c h l o r i d e , C1-; poi n t h e e x t r a c e l l u l a r a n d i n t r a c e l l u l a r f l u i d s of t h e s q u i d g i a n t t a s s i u m , K') axon a r e similar t o t h a t found f o r n e r v e c e l l s i n v e r t e b r a t e s . The magnitudes of t h e i o n c o n c e n t r a t i o n s i n v e r t e b r a t e s a r e smaller by a b o u t a f a c t o r of t h r e e o r f o u r (Kandel and Schwartz, 1 9 8 1 ) . Note t h a t t h e i o n c o n c e n t r a t i o n s f o r e x t r a c e l l u l a r f l u i d around s q u i d axon are almost i d e n t i c a l t o t h a t of s e a w a t e r (Hodgkin, 1 9 6 4 ) . For i n t r o d u c t i o n s t o and more d e t a i l s of t e r m i n o l o g y i n neuroanatomy and n e u r o p h y s i o l o g y ; t h e r e a d e r i s r e f e r r e d t o t h e s t a n d a r d t e x t b o o k s by S t e v e n s (19661, M i l e s (19691, Junge ( 1 9 7 6 ) , K u f f l e r and N i c h o l l s ( 1 9 7 6 ) , Aidley ( 1 9 7 8 ) , and Kandel and Schwartz ( 1 9 8 1 ) . 3
A l t h o u g h w e h a v e i m p l i e d t h e e x i s t e n c e of a t h r e s h o l d , there does n o t e x i s t a s h a r p s p a t i a l l y l o c a l i z e d t h r e s h o l d v a l u e , i.e., the responses a r e continuously graded. W e c a n , however, t a l k a b o u t a t h r e s h o l d f o r t h e g e n e r a t i o n of an a c t i o n potential.
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t h e Hodgkin-Huxley
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423
SOLITON SOLUTIONS TO ZAKHAROV-SHABAT SYSTEMS BY THE REDUCTION METHOD J.Harnad*,
Y.Saint-Aubin**
and S.Shnider*+
I t i s shown how t h e Zakharov-Mikhailov-Shabat " d r e s s i n g method" f o r t h e c o n s t r u c t i o n o f m u l t i - s o l i t o n s o l u t i o n s t o g e n e r a l i z e d Zakharov-Shabat systems may be combined w i t h t h e " r e d u c t i o n method" o f M i k h a i l o v i n o r d e r t o o b t a i n mu1 t i - s o l it o n s o l u t i o n s f o r a1 1 a1 g e b r a i c a l l y reduced systems, The r e s u l t s a r e i l l u s t r a t e d f o r two s i m p l e examples: ( 1 ) t h e S 2 sigma model and ( 2 ) t h e T h i r r i n g model.
1. MULTI-SOLITON SOLUTIONS
TO
ZMS SYSTEMS
a ) ZMS Systems and t h e D r e s s i n g Method Numerous i n t e g r a b l e 2-dimensional systems o f P.D.E.'s a d m i t t i n g s o l i t o n i c solut i o n s may be analyzed i n a u n i f i e d framework due t o Zakharov, M i k h a i l o v and ( h e n c e f o r t h ZMS) and developed f u r t h e r by t h e p r e s e n t a u t h o r s [4-91 Shabat [1,2,3] The s t a r t i n g p o i n t i s a p a i r o f l i n e a r f i r s t o r d e r m a t r i x e q u a t i o n s o f t h e form:
with i n t e g r a b i l i t y conditions:
u? -vE +[U,V]
= 0.
Here JI, U and V a r e complex nxn m a t r i x f u n c t i o n s of two r e a l v a r i a b l e s 6 , 7 ( l i g h t - c o n e c o - o r d i n a t e s i n 2-dimensional Minkowski space), depending on a comp l e x parameter k such t h a t U(h] and V ( X ) a r e meromorphic i n X w i t h f i x e d p o l e s on t h e Riemann sphere t P and Ji i s i n v e r t i b l e where d e f i n e d . The i n t e g r a b i l i t y c o n d i t i o n s (1.2) must be s a t i s f i e d i d e n t i c a l l y i n and t h e r e s u l t i n g c o u p l e d system o f n o n l i n e a r PDE's i s t h e one under s t u d y . The " d r e s s i n g method" o f Zakharov, M i k h a i l o v and Shabat, r e s t r i c t e d t o t h e s o l i t o n i c s e c t o r , c o n s i s t s of t r a n A f r i n g a g i v e n s o l u t i o n { q , U, V } o f t h e system d e f i n e d by: (l.l(a),(b), 2) t o a new one {@,#,!)
* ** +
C e n t r e de recherche de mathematiques a p p i i q u e e s , U n i v e r s f t e de M o n t r e a l , C.P.6128, Succ."A", M o n t r e a l H3C 357, Quebec, Canada; and Department o f Mathematics, M c G i l l U n i v e r s i t y , M o n t r e a l , Quebec, Canada H3A 2T5. C e n t e r f o r T h e o r e t i c a l P h y s i c s , Massachusetts I n s t i t u t e o f Technology, Cambridge, Massachusetts, U.S.A. 02139. Department o f Mathematics, Ben-Gurion U n i v e r s i t y o f t h e Negev, P.O.Box 653, Beer-Sheva 84105, I s r a e l .
424
J. Harnad et a/.
where t h e " d r e s s i n g m a t r i x " X ( h , 5, y ) meromorphic i n X, h a v i n g t h e form:
i s an i n v e r t i b l e
nxn
matrix function,
(1.4(a)) w i t h inverse
-
i s r e l a t e d t o t h e gauge i n v a r i a n c e o f The n o r m a l i z a t i o n o f X ( h ) a t h = ( l . l ) , (1.2) ( t r a n s f o r m a t i o n s of t y p e ( 1 . 3 ) w i t h X independent o f X) and i s u s u a l l y chosen such t h a t X(m) = 1. I n o r d e r t o p r e s e r v e t h e meromorphic s t r u c t u r e o f (1.1 a,b), t h e r e s i d u e m a t r i c e s {Qi,Ri) must s a t i s f y c e r t a i n d i f f e r e n t i a l and a l g e b r a i c e q u a t i o n s whose complete s o l u t i o n may be reduced t o t h e f o l l o w i n g l i n e a r a l g e b r a i c r e s u l t . We may e x p r e s s Qi, Ri i n t h e form:
Qi R.
1
where
{Xi,
qi = r k Q i
Fi)
are rectangular
=
XiFT
(1.5(a))
=
+ H.K.
(1.5(b))
1 1
(nxqi)
dimensional m a t r i c e s o f maximal r a n k
whose columns span t h e image
the kernel ( k e r Ri)',
( k e r Qi)', respectively.
r e s p e c t i v e l y ; and s i m i l a r l y Then
Fi
and
o f a r b i t r a r y constant matrices
fi
and hi
Fi
=
Hi and
Xi,
Ki
and t h e o r t h o g o n a l a n n i h i l a t o r o f
ImQi
{Hi,
Ki }
span
ImRi
and
a r e determined e x p l i c i t l y i n terms
Hi
o f t h e same t y p e by:
J,+-'(Xi)fi =
44ki)hi
a r e determined by s o l v i n g t h e l i n e a r a l g e b r a i c system: k
z x 1. r .i j
=
H~
(1.7(a
k + C K.F.. = i=l1 1 J
F.
(1.7(b
i=l
J
where
and
(1 - 8 )
425
Soliton Solutions to Zakharov-Shabat Systems
= constant.
Ci
The f u l l d e r i v a t i o n o f t h i s r e s u l t i s g i v e n i n r e f e r e n c e s [4,5]. I t i n v o l v e s t h e i n t r o d u c t i o n of t h e nkxnk dimensional " s o l i t o n c o r r e l a t i o n m a t r i x " M ( E , ~ ) whose nxn dimensional s u b - b l o c k s a r e d e f i n e d b y : (1.10) ( i n t e g r a t i o n around c o n t o u r s c o n t a i n i n g o n l y t h e p o l e s i n d i c a t e d ) . The M-matrix s a t i s f i e s a m a t r i x R i c c a t i system which may be l i n e a r i z e d and c o m p l e t e l y s o l v e d . The r e s i d u e s {Qi,R.) are recovered from t h e r e l a t i o n s 1
k
k
(1.11) leading t o the r e s u l t s (1.5)-(1.9)
as shown i n r e f e r e n c e s [4,5].
b ) The Reduction Method The r e d u c t i o n method o f M i k h a i l o v [10,11] c o n s i s t s o f r e s t r i c t i n g t h e g e n e r a l system ( l . l ) , ( 1 . 2 ) by imposing c 0 n s t r a i n t . s i n t h e f o r m o f i n v a r i a n c e under a f i n i t e group o f automorphisrns. The automorphisms c o n s i d e r e d a r e g e n e r a l l y of t h e form
*
$(A)
/-Ti" (X))l
I
fd$(S
where : u
i s an automorphism o f t h e group GL(n,U)
o f i n v e r t i b l e nxn m a t r i c e s ,
u* i s i t s d i f f e r e n t i a l a t t h e i d e n t i t y ,
s
i s a conformal map o f 6P1 o f t h e form, ah+b
s: k+- cx+d, N
X
=
X
i f u i s holomorphic,
=
h
ifu i s a n t i - h o l o m o r p h i c ,
ad-bc
such t h a t s(?)
=
1.
(1.13)
permutes t h e p o l e s of U and V ,
and f ( 5 , q ) C GL(n,&) i s an a s s o c i a t e d gauge t r a n s f o r m a t i o n . u a r e a l l o f t h e form: Ul(d =
w-l,
f o r some f i x e d
a2(g) =
tst-',
t c GL(n,t')
and
U3(d =
T-It-I tg
g c GL(n,(C)
,
or
arbitrary.
The automorphisms
+-1 -1 04(g) = t g t (1.14)
J. Harnad et a/.
426
1
(a)
I f u i s o f t y p e ul
(b)
H . = ftHi, F . = (ft)+-'Fi, s(1) S(1) I f u i s o f t y p e u2
(c)
(d)
H 7 = ftRi, F ~ ( ? )= ( f t ) + - l F i , s(1) I f u i s o f type u 3 t-1HS(i) = f t F i , Fs(i) = ( f t ) Hi,
1
r s ( i ) s ( j )= =
rs(i)s(3) rf(j)s(i)
=
rij(cXi+d)(cp,.+d) J
?.
.(cxi+d)(ci.+d) J
1.I
* ( 1.16)
- r . . ( c i i + d ) ( c p . +Jd ) 1J
I f u i s o f t y p e u4 H
7
s(1)
= ftF., 1
F
7
s(1)
= (ft)+-lHi,
r i ( j ) s ( 7 )=
-rij(cxi+d)(cL.+d) J
,,
2. THE ONE-SOLITON SOLUTION OF THE u MODEL WITH VALUES I N S2 The u models a r e examples o f n o n l i n e a r systems o f H a m i l t o n i a n t y p e which a r e bel i e v e d t o be c o m p l e t e l y i n t e g r a b l e . There i s a u-model a s s o c i a t e d w i t h each Riemannian symmetric space and t h e problem o f c o n s t r u c t i n g and s o l v i n g m u l t i Bzcklund t r a n s f o r m a t i o n s f o r these models has been s t u d i e d by t h e a u t h o r s [4,6,7]. The ' s i m p l e s t c$se i s t h e u model w i t h values i n t h e sphere S2, d e f i n e d by a u n i t s a t i s f y i n g t h e f i e l d equations: v e c t o r f i e l d q = (ql,q2,q3)
where ( matrix g
, )
denqtes t h e s c a l a r p r o d u c t on o u t o f q by:
P 3 . One can c o n s t r u c t a u n i t a r y
Soliton Solutions to Zakharov-Shabat Systems
427
which s a t i s f i e s t h e e q u a t i o n s o f t h e s o - c a l l e d " p r i n c i p a l " o r " c h i r a l " a-model (gEg-l)ri
-1
+
(gq9
IE
=
0
w i t h the reduction constraints
As an example, t h e u n i t v e c t o r
i s mapped t o t h e f o l l o w i n g m a t r i x :
and these s a t i s f y , r e s p e c t i v e l y , t h e e q u a t i o n s ( 2 . 1 ) and ( 2 . 3 ) . We s h a l l use go i n t h e f o l l o w i n g as t h e "vacuum" s o l u t i o n ; i .e., t h e s t a r t i n g p o i n t o f t h e d r e s s i n g method. F o r t h e n o n l i n e a r system ( 2 . 3 ) , an a s s o c i a t e d ZMS l i n e a r system i s known [ l - 3 1 :
A 1+x B 1-1
I
where A and B belong t o t h e L i e a l g e b r a a s s o c i a t e d w i t h t h e L i e group where g t a k e s i t s values -1 A s g g (2.6(c)) 5 -1 B P g g (2.6(d)) T
To b e g i n w i t h , we s h a l l assume t h i s group i s SL(2,E). (The d e t e r m i n a n t i s a l ways p r e s e r v e d by t h e d r e s s i n g t r a n s f o r m a t i o n s up t o a c o n s t a n t , and hence we may c o n s i s t e n t l y s e t t r A = t r B = 0 ) . The r e d u c t i o n group i s o f o r d e r f o u r and gener a t e d by t h e two t r a n s f o r m a t i o n s T1 $(A) fl*+-l(h) ( 2 . 7 ~
$(A) where
-T2
fa-rJ,(l/h
1.
-1
(2.7(b))
-
The gauge t r a n s f o r m a t i o n s f, and f 2 a r e d e t e r m i n e d b y s e t t i n g X = i n (2.7). U s i n g t h e f a c t t h a t $(X=O) = g and t h a t $(X=m) i s a c o n s t a n t which can be s e t equal t o t h e i d e n t i t y , one a r r i v e s a t
J. Harnad et al.
428
f l = l
The f i r s t r e d u c t i o n c o n d i t i o n t h e n g i v e s T1:
f
and
SL(2,E)
-
al
= g .
2
1
3
( 9 ) = g+-l = g
SU(2)
h,
S1(X) =
w h i l e t h e second reproduces t h e c o n d i t i o n ( 2 . 4 )
rv
S2(X) =
l/X.
The combination o f t h e two c o n d i t i o n s g i v e s r i s e t o (ql.q2,q3) a r e a l u n i t vector.
g
o f t h e form (2.2), w i t h
Using t h e r e d u c t i o n group, we can proceed t o t h e c o n s t r u c t i o n o f t h e o n e - s o l i t o n s o l u t i o n . The r e d u c t i o n s T1 and T2 r e q u i r e t h e m a t r i x X(X) t o s a t i s f y t h e equations: x+-l(x) = x(X)
(2.11 ( a ) )
5
where i s t h e new s o l u t i o n . Ngte t h a t a l t h o u g h t h e r e d u c t i o n c o n d i t i o n (2.11(b)) i n v o l v e s t h e q u a n t i t y g which i s a t t h i s s t a g e unknown, t h e s e t o f e q u i v a l e n t c o n d i t i o n s ( I - 1 6 ) does n o t . The c o n d i t i o n ( 2 . 1 1 ( a ) ) r e l a t e s t h e p o l e s o f ~ ( h and ) X I(X): f o r each p o l e hi i n ~ ( x ) x-l(X) , must have a p o l e a t The c o n d i t i o n ( 2 . 1 1 ( b ) ) p u t s a c o n s t r a i n t upon t h e r e s i d u e s o f X(X) = xi. o n l y . S i n c e S2(X) = l / X , t h e p o l e s o f ~ ( 1 must ) come i n p a i r s ( k i , l / l i ) . The pi
( t h e form i n v o l v i n g t h e minimal number o f p o l e s )
s i m p l e s t f o r m f o r x(X) and X-’(X) i s then:
X A)
1
=
QT + -x-x1 +Qi x-l/xl (2.12)
X
(1) =
I + Ri - + -R j
x-l/xl
h-hl
*
From t h i s p o i n t on, t h e e x p l i c i t r e s u l t s a r e as i n d i c a t e d i n S e c t i o n 1. With t h e n o t a t i o n ( 1 . 5 ) , t h e c o n s t r a i n t s ( 2 . 1 1 ( a ) ) and ( 2 . 1 1 ( b ) a r e r e s p e c t v e l y e q u i v a l e n t t o ( 1 . 1 6 ( d ) ) and ( l . l 6 ( a ) ) , i . e .
(2.13)
In o t h e r words, Hi,
Fi
i t i s enough t o g i v e F1 t o d e t e r m i n e a l l t h e o t h e r b l o c k m a t r i c e s
and H i which e n t e r t h e c a l c u l a t i o n s .
fl t o be:
fl =
T a k i n g t h e rank-one c o n s t a n t m a t r i x
(:;)
’
(2.14)
429
Soliton Solutions to Zakharov-Shabat Systems
and s o l v i n g t h e l i n e a r system ( 2 . 6 ) f o r $ ( A ) :
one can c o n s t r u c t t h e one-sol i t o n s o l u t i o n t h r o u g h a l e n g t h y b u t s t r a i g h t f o r w a r d computation: formula ( 1 . 6 ) g i v e s F1 i n terms o f f and f o r m u l a (2.13) g i v e s 1 H i , FT and H i . Using t h e d e f i n i t i o n ( 1 . 8 ) o f I?, one can s o l v e (1.7) t o o b t a i n X1,
T h i s enables us t o o b t a i n
X i , K i and K i . IV
The new s o l u t i o n
g = X(h=O)g
, Q f , -R i
Q,
d e f i n e s t h e new v e c t o r
and R4 t h r o u g h ( 1 - 5 ) . :
(2.16)
The components o f t h i s u n i t v e c t o r -2s in i i s i n hQ
* -
q1 - cosh p i n h
Q+COS
.
d
a r e found t o be:
(cosh~sinvsinhBcosA+sinh~cosvcoshQsinA)
vsin A
(2.17) x[
2 (cosh p - c o s 2 v ) s i nhslsi nA+i (coshpsinhwsi n h Q c o s ~ - c o s v snvcoshBsi i nA)]}
where: X1ae
d E
iA+Q ,
1-lh1 i(x,-$
c
l2 )
,
.
x1-i;1
=ti----
?
Il*Xl
v
5
I
,
d(~+E-c-T)+v,
C
p
5
e iv+u ,
2
c+t;+c-q+u,
c1 A,Q,U,V,C+,~,K,V
C
IR.
(2.18)
-
T h i s s o l u t i o n i s c l o s e t o a sine-Gordon b r e a t h e r i n t h e sense t h a t i t d i s p l a y s t h e i n t e r n a l o s c i l l a t i o n mode (due t o t h e s i n and cos ) . I f v # i; + 0 (and hence d + 0 ) g i v e s back t h e i n i t i a l s o l u t i o n qo.
, the
limit
Q
A 1 - s o l i t o n s o l u t i o n was p r e v i o u s l y r e p o r t e d by one o f us i n reference [ 8 ] . Howe v e r t h i s s o l u t i o n i s n o t o b t a i n a b l e as a p a r t i c u l a r case o f t h e above, b u t r a t h e r f r o m t h e - g e n e r a l case i n w h i c h t h e d r e s s i n g m a t r i x X ( k ) has f o u r d i s t i n c t poles a t I h , X1, l / X l , l/h,l.
430
J. Harnad e t a / .
3. THE THIRRING MODEL As a second i l l u s t r a t i o n of the reduction technique, we consider the massive
Thirring model which, within a choice of normalizations may be expressed by t h e p a i r of equations
with E = ? l . The complex functions ($1,$2) a r e i n t e r p r e t e d as components of a s e l f - i n t e r a c t i n g spinor f i e l d i n two dimensions. These equations may be obtained as i n t e g r a b i l i t y conditions of a l i n e a r ZMS system o f the type ( 1 . 1 ) due t o Mikhailov [12] with U and V of the form:
;.f
ih
-
=
-€$I 1
To analyze t h i s system through the reduction procedure, we consider more genera l l y , t r a c e l e s s , complex 2 x 2 matrices U(X), V(h) with the same meromorphic
structure: (3.3)
Imposing reduction conditions of the form: (3.4)
where =
reduces the system t o t h e form:
’
( 0 -1O )
(3.5)
431
Soliton Solutions to Zakharov-Shabat Systems
F u r t h e r , making use o f t h e gauge i n v a r i a n c e : a
+
a+A
b
5’
+
b+A
(3.7)
and conformal i n v a r i a n c e : 5 -+?(5),
q +;(i-l).
(3.8)
we reduce t h e system, up t o n o r m a l i z a t i o n , t o t h e f o r m ( 3 . 2 ) . results i n A p p l y i n g t h e d r e A s i n t r a n s f o r m a t i o n ( 1 . 3 ) t o a g i v e n s o l u t i o n (J/,U,V) a new s o l u t i o n ( q , g , ) o f t h e same s t r u c t u r e p r o v i d e d t h e d r e s s i n g m a t r i x X(X) has t h e f o r m
#
The d e t a i l e d a p p l i c a t i o n o f t h e procedures o f s e c t i o n ( 1 ) t o determine t h e m u l t i - s o l i t o n s o l u t i o n s i s d e s c r i b e d i n r e f e r e n c e [9]. We l i m i t o u r s e l v e s h e r e t o c i t i n g the r e s u l t : (3.10) d2 = pe
where
u,p,B
and x a r e g i v e n by:
u
=
4i
z
i ,j = l
fiyij4j
J. Harnad eta/.
432
c
eix
z [xi/xi]
=
i=l with fi
=
-2 --2 aiexpIi[XiE+Ai El
gi
=
biexpI-i[AiE+Ai -2 --2 q]
(ai,bi
= constants)
and {y. . ) t h e e n t r i e s of t h e i n v e r s e of t h e m a t r i x 1J
r..
=
1
’
d e f i n e d by
.s . g .
2x.7 . f J.+285; J
1J
{ r1.J. )
1 J
(3.13)
2 -2 X.4. 1
J
F u l l d e t a i l s may be found i n r e f e r e n c e [9]. F o r o t h e r examples of models s o l v e d t h r o u g h t h e a p p l i c a t i o n o f t h e r e d u c t i o n method, we r e f e r t h e r e a d e r t o references [4,5,9,10,11]. REFERENCES
[l] Zakharov, V.F.
and M i k h a i l o v , A.V.,
Sov.Phys.JETP,
47 (1979) 1017.
[2]
Zakharov, V.E.
and Shabat, A.B.,
[3]
Zakharov, V.E.
and M i k h a i l o v , A.V.,
[4]
Harnad, J., S a i n t - A u b i n , Y . and Shnider,S.,
Commun.Math.Phys.
92 (1984) 329.
[5]
Harnad,J.,
Saint-Aubin,
Y . and Shnider,S.,
Commun.Math.Phys.
93 (1984) 33.
[6]
Harnad, J . , Saint-Aubin,
Y . a w l Shnider,S.,
J.Math.Phys.
[7]
Harnad, J . , Saint-Aubin, Y . and S h n i d e r S., Q u a d r a t i c p s e u d o p o t e n t i a l s f o r GL(n,C) p r i n c i p a l sigma models, Physica D (1984, i n p r e s s ) .
[8]
Saint-Aubin,Y.,
[9]
David,D.,
[lo]
Mikhailov,A.V.,
Lett.Math.Phys.
Funct.Anal.App1.
13 (1979) 166.
Commun.Math.Phys.
74 (1980) 21.
25 (1984) 368.
6 (1982) 441.
Harnad,J. and Shnider,S.,
Lett.Math.Phys.
JETP L e t t . 32 (1980) 174.
[ll]Mikhailov,A.V.,
P h y s i c a 30 (1981) 73.
[12] Mikhailov,A.V.,
JETP L e t t . 23 (1976) 321.
8 (1984) 27.
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland). 1984
433
MULTIPLE SCATTERING O F S H WAVES BY RANDOMLY DlSTRIEUTED DISSIMlLAR SCATTERERS Matthew F. McCarthy Faculty of Science, National University of Ireland, University College, Galway, Ireland and Michael M. Carroll, Department of Mechanical Engineering, University of California, Berkeley, California, U.S.A.
A scattering matrix approach is used to study the multiple scattering of elastic waves in a medium containing two families of identical, long, parallel, randomly distributed cylinders of arbitrary cross-section. In general, both the geometrical and mechanical characteristics of the two families will be different and the elastic properties of each family will be assumed to be different from those of the matrix in which they are embedded. A statistical approach is employed to obtain the average amplitudes of the scattered and exciting fields and these equations determine the propagation characteristics of coherent waves in the composite. Dynamical properties of several types of composite media are determined and the behaviour of such composites is discussed in detail in the Rayleigh limit. 1. INTRODUCTION Multiple scattering of waves is of considerable interest in many fields of science and technology. A statistical approach to multiple scattering has been employed by a number of authors (1-5). Such an approach has been employed by Bose and Ma1 (6) and Ma1 and Chatterjee (7) t o examine the scattering of SH waves by a single family of circular cylinders. In particular, detailed results in the low frequency Rayleigh limit have been given in ( 6 , 7). The method of matched asymptotic expansions has been employed by Datta (8)t o study the scattering of SH waves by cylinders of elliptic crosssection. The T matrix method of Waterman ( 9 ) has been applied in a series of papers by Varadan et a1 (10-12) to study the scattering of SH waves by a random distribution of cylinders of arbitrary cross-sections. Here we employ the T matrix method t o study the scattering of SH waves incident upon an arbitrary configuration of obstacles in the form of infinitely long cylinders of uniform crosssection, the cylinders being parallel to each other. The T matrix approach has the advantage of being applicable t o cylinders of arbitrary crosssection and, furthermore, the wavelengths considered are not restricted in our general formulation. In contrast to the assumptions made in (10--12), we do not assume that the cylinders are identical. In the interests of simplicity we confine our attention to the scattering of plane waves which are incident normally on two families of cylinders, the members of each family being identical. The orientations of the members of either of the families are not restricted in our general formulation. To begin with, we consider the problem of the scattering of a plane wave by a large number N of cylinders which are arbitrarily distributed in an infinite matrix. We assume that the N cylinders are made up of two families of identical cylinders, the first family containing N, members, the second N, = N-N, . The resulting equations are then averaged, assuming the positions of the cylinders to be random while the orientations of the scatterers of each family are assumed t o be either parallel or randomly distributed. The averaged
434
M. F. McCarthy and M.M. Carroll
equations are solved by using Lax’s (13 ) quasi-crystalline approximation to yield the propagation characteristics of the coherent waves in the composite. The particular equations which govern the propagation of SH waves are derived when (i) the orientations of each of the families are parallel, (ii) the orientations of one set are aligned while those of the other family are random and (iii) the orientations of both sets are random. When the orientations of either or both families are aligned, the composite is anisotropic and the wave propagation characteristics of the material depend on the direction of propagation. On the other hand, when the orientations of all the scatterers are randomly distributed, the composite is isotropic. Detailed results are presented in the Rayleigh limit which generalize the earlier results presented in ( 6 , 7 , 10-12). In particular, results are presented for materials containing both obstacles of finite crosssection and cracks.
Figure 1. Geometry of scatterers and circumscribing circular cylinders. 2 . B A S l C EQUATIONS A N D FORMULAE Consider N arbitrarily shaped, infinitely long parallel cylindrical scatterers each with smooth surface S which are referred t o a coordinate system as shown in Fig. 1. The centre of the smallest circle which circumscribes the ith scatterer is at Oi and has cylindrical polar coordinates (ri, B i , z). We suppose that the matrix in which the scatterers are embedded extends to infinity, that the scatterers are located within a large region A of a crosssection of the material and that the density and rigidity of the matrix are p and p, respectively. As far as the two scatterers are concerned, we assume that they consist of two families containing Nl and N, (=N-Nl ) members respectively. We further assume that the members of each of these families are identical but, for the moment, we do not specify the mechanical or geometrical characteristics of either family. Furthermore, we note that while our considerations here apply to only two families of scatterers they may be generalised without difficulty to apply to the situation which arises when the number of families is arbitrary. We suppose that a time harmonic plane SH wave of unit amplitude is generated at infinity and propagates in a direction perpendicular t o the fibres. Such an incident wave may be represented by
Randomly Distributed Dissimilar Scatterers
435
where k is the wave vector. There is no loss of generality in assuming that wave propagates in the positive . r = k x where k = w ( p / p ) t . Since the incident wave has time dependence given by exp(-iwt), all field quantities will have the same time factor which henceforth will be suppressed. The total displacement field at any point outside the scatterers is the sum of the incident displacement field and the fields scattered by all the scatterers. Thus, at the point of observation r, we have
x direction so that k
where w;"(r-ri) is the field scattered by the ith scatterer t o the point r. The field that excites the ith scatterer is denoted by wp and at any point r in the vicinity of the ith scatterer it is given by
We denote by aa the radius of the circle which just circumscribes scatterers of the type a and we assume that there is no interpenetration between the circular cylinders which circumscribe the scatterers.
of jth and The functions
$0 - ri)= Hn(klr-rjI)ein@i,$(r-q) =Jn(klr-ril)ein@'i
(2.4)
where H n ( .) and J,( .) denote the Hankel function of the first kind and Bessel function of order a , respectively, and @, denotes the angle which the vector r-ri makes withe the positive x direction [see Fig. 21, are solutions of the Helmholtz equation with wave number k . Of course, $n(r-ri) is the regular part of $n(r-ri). The functions (2.4) serve as a basis set in terms of which we may expand wf and w5 as I follows :
M.F. McCarthv and M.M. Carroll
436
Next we note, as is well known (10, 14), that a' and b' are related through the transition matrix of the
jth scatterer P ' by the formula
where The matrices @ and @ are determined by the crosssection of the jth scatterer as well as the boundary conditions prevailing on the surface of the scatterer. Expressions for the matrix Q when the scatterer is an inclusion or a void are given in the appendix. When the scatterer is a crack Q may be found from (A.8) by a limiting process as described by Weaver and Pao (1 5 ) or from the results of Van den Berg (16). On substitution from (2.5), (2.6) and (2.7) into equation (2.3), we find that if there is a scatterer of the first family at ri then (2.9) while if the scatterer at ri is one of the second type we have ~ ~ J / ~ ( c=- w(0)(r) c ~ ) +
NI
N
z zP z ~b m' i ~ i $ ~+j(c+~icQ-z~q i.a)A $ Q ( r - r i )
]=1
m
1
i NI
m
.
(2.10)
When a standard procedure, which follows closely that described in ( l o ) , is employed i t may be shown that equations (2.9) and (2.10) are equivalent t o the equations
(2.11)
and
(2.12)
Many scatterers possess cross-sections which have axes of symmetry and when these are used as reference axes the calculation of the T mamces is frequently simplified. Thus, if the natural (X,Y) axes of the scatterer make an angle o( with the x axis, it follows that (2.13) T,, = ei(m-n)cuF m n (2.13) where T is the transition matrix referred t o the preferred axes of the scatterer. Substitution from (2.13) into (2.11) and (2.12) now yields the equations
and
(2.15)
437
Randomly Distributed Dissimilar Scatterers
At any point r outside the scatterers, the total displacement is given by (2.2). On the other hand if r is a point within the ith scatterer, which we assume t o be composed of a material of shear modulus p and f density pp then the displacement is
where kf = w/c , c j = pf,/pr. In order to write a single formula for the displacement at any point in the composite we folfow establlshed practice (6, 17) and introduce the symbol
(2.17)
where Siis the crosssection of the ith scatterer. Then, since there is no interpenetration of scatterers, the expressions (2.2) and (2.16) may be combined to yield a single formula for the displacement in the form
(2.18)
3 . CONFIGURATIONAL AVERAGING We now wish to average over the positions of all the scatterers. The probability density of finding the first scatterer at r l , the second at rz and so forth will be denoted by p(rl, r2, ... r N ) . Of course, we may express the probability density as p(rl , r z , .... ’N)= p(ri)p(rI, rz , .... / ...rN Iri) (3.1) = p(ri)p(r.lr.)p(rl, r2 ... / ... / ... Ir., ri) I ' I - ..., -
where p(ri) denotes the probability density of finding a scatterer at T i , while p(r./r.)denotes the condiI ' tional probability of finding a scatterer at r . given that there is already a scatterer at ri. A prime in the 1 first of (3.1) means that ri is absent while two primes in the second mean that both ri and r . are absent. If the scatterers are randomly distributed, the positions of all scatterers are equally probable within A and hence
On the other hand, if p(S)(r,)denotes the probability density of finding a scatterer of type s at ri then
(3.3)
M. F. McCarthy and M. M. Carroll
438
and, of course, p(ri) = p ( ' ) ( r i ) + ~ ( ~ ) ( -p(l~z)(rilrl), r~) wherep(S*t)(r.lr.) denotes the conditional probaI I bility of finding a scatterer of type t a t r.,given that there is a scatterer of type s a t ri. We assume that the I parallel cylinders do not overlap and furthermore, in the interests of simplicity, effects of correlation in the position of cylinders will be ignored. Thus, we approximate the pair correlation functions as follows:
(3.4)
where at denotes the radius of the smallest circle which circumscribes a type t scatterer. In what follows we assume that a ,
(3.5)
Next, we note that, since the scatterers are not all identical, the decomposition (3.1) is not unique. We may, in particular, replace (3.1) by the formulae
p(ri, ...., rN)
... / ...rNIri)
= p(S)(ri)p(S)(r,
... ... / ... rNlrj,ri).
= p(s)(r~)p(s)(r.lr,)p(s)(rl/
I '
(3.6)
We denote the conditional expectation values of the quantity $ by ($)!') = JA...JA$p(')(r1 ... / ($)(:) = J,...
91
... rNlri)drl
... / ...drN,
J, $p(s)(rl ... / ... / ...rN\r.I' r.)dr, ' ... / ... /
(3.7)
...drN.
In the first of (3.7) the averaging takes place over all scatterers except the scatterer a t ri which is assumed t o be of type s. In (3.7), the averaging takes place over all scatterers except the scatterer of type s at ri and whatever type of scatterer may be at rj. I t follows from (3.6) and (3.7) that ($)(f) = JAp(s)(r\ri)($)(s$dri,
and when ( 3 . 5 ) is used in (3.9) we find that
(3 3)
Randomly Distributed Dissimilar Scatterers
439
(3.9)
We now assume that scatterers of a given family are indistinguishable i.e. we assume that the scatterers of a given family are either all parallel or have their orientation randomly distributed. Next we apply formulae (3.7) and ( 3 . 1 0 ) to obtain the conditional expectation values of equations ( 2 . 1 4 ) and (2.15), thus obtaining the equations (@f)
= peik.ri
+xQ 2m(-l)n+P
A
TI fii1) + -2p N 612) Qm grnn A Pm rQmn
(3.10)
where ~ rPmn P S )= J
APq
(&m-Q)aji a;\
( P J ) ~e-n(rl-ri~ri.
(3.11)
In a similar fashion, we find that, when we take the expectation value of (2.18) and use the expansion theorem for the wave function $,(r - rj), we have (v)=( I - c ) ~ ' ~ . ~ + z( nz 1 ~ i m 6 . A + n m
n2qm6:A)
(3.12)
where
(3.13)
In ( 3 . 1 8 ) c d y o t e s the fraction of the crosssection of the composite occupied by the scatterers, ni = Ni/N and is obtained by replacing k by R in (2.4), . f In order to make further progress, we must now make some assumption concerning the orientations of the scatterers. Three particular cases are of special interest to us. Firstly, the situation which arises when the preferred directions of the two families of scatterers make fixed angles cq ,a3respectively with the x axis; secondly, when the orientations of the first set of scatterers are aligned while the orientations of the
$l(r)
440
M. F. McCarthy and M.M. Carroll
second set are randomly distributed, and finalIy, when the orientations of both sets of scatterers are randomly distributed. When all the scatterers of family q are parallel and make an angle aq with the x axis we find that (ei(m-Q)a.l a ,j? ) ( pI J . q ) = ei(m-P)aq(uj,)(p,q), 11
(3.14)
On the other hand, when the orientations of the preferred directions of family q are randomly distributed, we have (3.15)
For two families of aligned scatterers equations ( 3 . 1 1 ) reduce to
(3.16)
where
Equations ( 3 . 1 6 ) give the conditional average of u ; , with the ith scatterer fixed, in terms of integral relations involving the conditional average of u6 with two scatterers fixed. In order to close the system, we follow conventional practice ( 6 , l O ) and adopt the quasi-crystalline approximation ( 1 3 ) . Thus, we write
where
When the preferred directions of the first family of scatterers are aligned and those of the second family are randomly oriented, equations ( 3 .lo) yield the equations
44 1
Randomly Distributed Dissimilar Scatterers
Finally, when the orientations of all of the scatterers are randomly distributed, equations (3.12) must be replaced by the equations =jn,ik.r, n
+ C (-l)n+m
m
i
I",
1 T
; ~ T $ ~+ ~NAA ~m m$ 1i m2 )m n TZ
2-
( 32 2 )
+ N d T $222) A m m zmmn T m m immn $21)
m
4. P R O P A G A T I O N C H A R A C T E R I S T I C S O F T H E M E D I U M In order t o solve the integral equations (3.19), we assume that the average coherent wave is a plane wave propagating with an effective complex wave number K in the direction of propagation of the incident wave. We assume that = .m 2
Xm
i $2)
iK.ri 7
(am
i
= i m y eiK.ri
(4.1)
m
where X m , Y , and K are constants. On substituting from ( 4 . 1 ) into (3.19) we find that j n X eiK.ri = jneik.ri
+
2
Pm
(-l)n+P
1"
I
1 A-L\i,,)nT$mXm + 2 N2 L $ b k ) n q m Ym
where
When the integrals in ( 4 . 3 ) are evaluated and Lax's extinction theorem (6, lo), which states that the unperturbed incident wave is extinguished within the composite by waves induced at the boundary of the system, is used, equations (4.2) reduce t o
where Z":)
= k R p q J n ( K R p q)ff’,, ( k R p4 )
- K R p qHn ( k R pq)J'n ( K R p q ) ,
(4.5)
with R p q = u p + aq, Gi = r/2-ai and we assume that both N,,N, and A are infinitely large while
442
M.F. McCarthy and M. M. Carroll
Nl/A = n l and N z / A = n2 are both finite. Observe that, as one would expect, equations (4.3)reduce t o precisely those derived by Varadan et a1 (10, 12) in their studies on the propagation of SH waves in the presence of a single family of identical scatterers of parallel orientation. Equations (4.4) form a system of simultaneous linear homogeneous equations for the unknown amplitudes X,, Y,. For a non-trivial solution, the determinant of the coefficients must vanish and this yields an equation for the effective wave number K in terms of k , n , ', n 2 , a l , a*,T1 and T 2 ,Because the two families of scatterers are aligned in different directions, the composite has two preferred directions and, because of this anisotropy, the effective wave number depends heavily on the orientations of the two preferred directions relative to the direction of propagation of the unperturbed wave. In order to assess the influence of the angles a l , az on K , we make use of the properties of the T matrices as outlined in t
(kz- Kz)
where X"=(X,"l, Y"==(Y$
x"=(x;},
YO
= (Yn")
n
= 0, 1,
...
(4.7)
with
x,"= %(Xn+ x-,), x; = %(X,-X-, = %(Y, + Y-,)
y", = M(Y,-Y-,
(4.8)
and
and
where
,
E~
= 1 and
E,
= 2 , n > 1, while
(4.10)
and, finally,
Randomly Distributed Dissimilar Scatterers
443
(4.11)
Obviously, equations (4.4) are extremely complicated when a, and az are arbitrary angles. However, when the preferred directions of the families of scatterers are in the direction of propagation or orthogonal to it, or when one is in the direction of propagation and the other is orthogonal to it, equations (4.4)uncouple, since in these cases = $B ( i j ) = 0 and we have
Since these equations have non-trivial solutions if and only if the determinants of their coefficients vanish, it appears that two waves, one corresponding to each of the values of K which follow from these equations, may propagate. However, on substitution from (4.1) into (3.12) we find that for the particular orientations under consideration, ( w ) depends only on Xe and Ye and so the root which follows from (4.11) determines the wave number of the coherent wave which may propagate in the composite. Next we turn our attention to the case when the scatterers of family one are aligned while those of the second family are randomly oriented. It now follows that equations (3.21) reduce to
(k2- K z )
(4.13)
where
.,d;jA
=
'
WCij)
R n j e m T L m . m,n,
% ( Vn m) =
Rn.E
I m
~ i ,dii) ,m , n .
(4.14)
Finally, if the preferred directions of both sets of scatterers are randomly oriented, equations (3.22) reduce to
(kZ-KZ)
[I]
=
.FJ(21)
"""] k:] d 2 2 )
(4.15)
(4.16)
Clearly equations (4.15) and (4.16) are uncoupled. An elementary calculation shows that the mean displacement in the composite is determined by Xe and Ye and is independent of Xo and Yo.Thus, the wave number of the coherent wave which may propagate in the composite is determined by the equation (4.15).
M. F. McCarthy and M. M. Carroll
444
5 . WAVE PROPAGATION I N T H E R A Y L E I G H LIMIT In the Rayleigh or low frequency limit, the size of the scatterers is assumed t o be small when compared t o the wavelength of the incident disturbance. In this limit, it suffices to take only the lower order terms in the expansions of the Bessel and Hankel functions. On adopting this procedure, i.e. retaining only the lowest order terms in kaj and Kai, i = 1,2, we find
where { = K / k . Next, in the Rayleigh limit, it follows from (2.8) and Appendix A that the only nonvanishing components of the T matrix of a scatterer whose boundary is described by the equation r = .fro), where a is the radius of the circular cylinder which just circumscribes the scatterer, are To,
i(d - 1 )
=-
2
k2a21,
TI1 = -T1,-1 =
i(l -m)k2a21 2(1 + m)(l - A 2 )
(5.2)
where (5.3)
and m = ~1 l p , d = p l p A may be regarded as a measure of the skewness of the scatterer while its area is f f ’ 212. In order t o obtain specific results for particular types of scatterers we note the following particular cases:
(i) Circular Cylinders: f ( 8 ) = 1,
I =r/2,
A =0,
(5.4)
(ii) Elliptic cylinders:
fro)= ( C o s 2 6 + a 2 / b z Sin6 \-l'z,
(1 - m)(b - a ) ab I =- , A = 2a ( 1 + m ) ( b+ a )
(5.5)
where a,b (b>a) are the semi-axes of the ellipse. (iii) Rectangular Cylinders:
I = 2b/a,
A=-
4(1 - m )
r(l + m )
tan:l
b-a b+a '
where 2a and 2 b (
~ 2 )
(5.7)
Randomly Distributed Dissimilar Scatterers
445
and it is to be noted that all of these quantities are independent of the radii of the circumscribing cylinders. We now examine in turn each of the three particular cases discussed in the previous section. In the interests of simplicity we begin by considering wave propagation in a material in which the preferred directions of both sets of scatterers are randomly oriented. It follows from (4.15) that
where (5.9)
and here d j , m i , A, are the values of d, m and A for a scatterer of type i, while ci = 2nia/li is the concentration of scatterers of type i. In what follows we only deal with scatterers which have circular crosssections or else are cracks. In either case ci = anpt , so that in the case of cracks c, is the concentration of the circular cylinders which circumscribe the cracks. Notice that if the scatterers have the same mechanical and geometrical properties so that m1 = m 2 = m, d , = d, = d , but are of different sizes so that a, # a 2 , then they behave as a single family of scatterers whose concentration is c1 + c 2 . Thus, in the Rayleigh limit, the wave does not distinguish between randomly oriented scatterers whose mechanical and geometrical properties are identical but whose sizes differ. A number of special cases which follow from (5.8) and ( 5 . 9 ) are of particular interest. If the average shear modulus of the composite is denoted by ( p ) it follows that if both families of scatterers are cracks then
(5.10)
if the first family is composed of cracks and the second of circular rods then
(5.11)
while finally, if both families are circular rods, we have
(5.12)
We have computed the shear modulus (1.1) of the composite for a variety of situations and the results are shown in Fig. 3 . The shear modulus of the composite is normalized with respect to the shear modulus of the matrix. When the scatterers are holes or cracks it is clear that the shear modulus decreases as the concentration of either increases. For a given concentration of defects, the reduction in shear modulus produced by cracks is less than that produced by holes.
M. F. McCarthy and M. M. Carroll
446
1
.5
.4 .I
0
.2
c1
Figure 3 . Normalized shear modulus versus concentration c2 for various values of c 1 : - - - - - - - - two families of randomly oriented cracks; family of holes of concentration c1 , family of cracks of concentration c2 ;- . - - . - two families of holes. When the preferred directions of one set of scatterers are aligned, while those of the other family are randomly oriented, it follows from (4.15) that, in the Rayleigh limit, we have:
(5.13)
where
(5.14)
and here a1 is the angle which the preferred direction of the scatterers of family 1 makes with the x axis.
447
Randomly Distributed Dissimilar Scatterers
When both families are cracks it follows from (5.13) that
(5.15)
and when the second family is a set of circular rods we find that
The implications of (5.15) and (5.16) are presented graphically in Figs. 4 and 5 . Fig. 4shows the influence of the concentration and shear strength of reinforcing rods on the shear modulus of a material which contains a fixed concentration of randomly distributed parallel cracks. The influence of crack concentration and orientation in the presence of a fixed distribution of randomly oriented cracks or circular holes is illustrated in Fig. 5. I t is clear that, for a given concentration of holes or randomly oriented cracks, the shear modulus of the composite decreases more rapidly with the concentration of aligned cracks as the angle which the cracks make with the direction of propagation increases. I
io
.9 1.o
'
2
90’
1 (P) P
5
0 O
.5
.8
30" 45"
.8
60' 0
rn
.05
90'
I
.05
.1
C2
Figure 4. Normalized shear modulus versus concentration cg of rods for various values of m2 and fixed concentration c 1 = 0.1 of cracks which are aligned at 60' t o direction of propagation.
.I
C1
Figure 5 . Normalized shear modulus versus concentration of aligned cracks when the second family of scatterers is either a family of cracks ( - - - - - -1 or a family of holes of concentratio or a family of holes ( -) of concentration c 2 = 0.1,
M. F. McCarthy and M. M. Carroll
448
The variation of the shear modulus with the orientation of the aligned cracks is plotted in Fig.6 for various concentrations of aligned cracks and randomly oriented cracks or holes. The shear modulus is seen to be a decreasing function of the angle of alignment and is least sensitive to changes in alignment of the cracks when rhe angle of alignment is 45'.
Figure 6. Normalized shear modulus versus angle of alignment a1 for various concentrations c 1 of aligned cracks and concentrations c 2 of randomly oriented
1 .o
*
A
= .I;-
L.
c2
0
.9
--
.1
.2
0 .8
.1
90'
45" a1
When each of the families of scatterers is a set of aligned cracks, it follows from (4.6) that, in the Rayleigh limit, we have
?!h
I.I
=
1
R / { 1 + %(cl + c 2 ) + c1 c2 Sinz(al-a,) /4
(5.17)
where R = 1 + % (cl Cos 2u1 + c2 Cos 2az
+ cf SinZ 2al + 2c1 c2 Sin2(a1+ a2) + c: Sin2 2a21 /8.
(5.18)
The effects of crack orientation on the shear modulus are examined in Figs. 7 and 8. I t is clear from Fig. 7 that there is very little difference between the manner in which the shear modulus vanes with concentration of aligned cracks when the second family of cracks is aligned at 45" to the direction of propagation or is randomly oriented. As is clear from Fig, 8 a similar result holds as far as the variation of the shear modulus with the angle of orientation of the first family is concerned.
WAVE PHENOMENA: MODERN THEORY AND APPLICATIONS
450
M.F. McCarthy and M.M. Carroll
References 1. Foldy, L. L., The Multiple Scattering of Waves, Pbys. Rev. 67 (1945), 107. 2. Twersky, V., On Scattering of Waves by Random Distributions. I. Free Space Scatterer Formalism, J . Math. Pbys. 3 (1962), 700. 3. Twersky, V., On Scattering of Waves by Random Distributions. 11. Two Space Scatterer Formalism, J. Math. Pbys. 3 (1962), 724. 4. Waterman, P. C. and R. Truell, Multiple Scattering of Waves,]. Math. Pbys. 2 (1961), 512. 5. Tsang, L. and J . A. Kong, Multiple Scattering of Acoustic Waves by Random Distributions of Discrete Scatteres with the Use of Quasicrystalline-coherent Potential Approximation, J. Appl. Pbys. 5 2 (1981), 5848. 6. Bose, S. K. and A. K. Mal, Longitudinal Shear Waves in a Fiber Reinforced Composite, Int. J Solids and Structures 9 (1973), 1075. 7. Mal, A. K. and A. K . Chatterjee, The Elastic Moduli of a Fiber Reinforced Composite, J. Appl. Mech. 44(1977), 61. 8. Datta, S. K., Propagation of SH Waves through a Fiber-reinforced Composite - Elliptic Cylindrical Fibers, J . Appl. Mech. 42 (1975), 165. 9. Waterman, P. C. New Formulation of Acoustic Scattering, J. Acoust. Sac. A m e r . 45 (1969), 1417. 10. Varadan, V. K., Varadan, V. V. and Y. H. Pao, Multiple Scattering of Elastic Waves by Cylinders of Arbitrary Cross-section. I. SH Waves,J. Acoust. SOC.A m e r . 63 (1978), 1310. 11. Varadan, V. K. and V. V. Varadan, Characterization of Dynamic Shear Modulus in Inhomogeneous Media Using Ultrasonic Waves, in Berger, H. and M. Linzer (eds.) Ultrasonic Materials Cbaracterization (U.S. National Bureau of Standards, Washington D.C., 1980). 12,Varadan, V. K. and V. V. Varadan, Frequency Dependence of Elastic (SH) Wave Velocity and Attenuation in Anisotropic Two-Phase Media, Wave Motion 1 (1979), 5 3 . 13. Lax, M., Multiple Scattering of Waves 11. Effective Field in Dense Systems, Phys. Rev. 85 (1952), 621. 14. Pao, Y. H., The Transition Matrix for the Scattering of Acoustic Waves and for Elastic Waves, in Miklowitz, J. and J. D. Achenbach (eds.) Modern Problems in Elastic Wave Propagation (Wiley, New York, 1978). 15. Weaver, R. L. and Y. H. Pao, Application of the Transition Matrix t o a Ribbon-Shaped Scatterer, J. Acoust. Sac. A m e r . 6 4 (1979), 1199. 16. van den Berg, P. M., The Transition Matrix in Acoustic Scattering by a Strip, J . Acoust. SOC. Amer. 70 (1981), 615 17. Sobczyk, K., Elastic Wave Propagation in a Discrete Random Medium, A c t u Mech. 25 (1976) 13. Appendix
The Q matrix for an inclusion is given by the expression
where B is the boundary of the inclusion. Whenever a superscript appears, rhe corresponding wave function is evaluated within B , otherwise i t is evaluated i n the matrix. The matrix 0 follows from (A.1) when $-, is replaced by . When the boundary B is described by the equation r = r(0) and has the symmetry properties
6-,
*
r(0) = d2r - e ) = ~ ( ne ) (A.1) reduces t o
(A.2)
Randomly Distributed Dissimiiar Scatterers when rn + n is even, Q,,
= 0 when m
451
+ n is odd. In (A.3) we have
I t follow from (A.3) and (A.4) that
When the inclusion is replaced by a cavity, the elements of the Q matrix follow from ( A . l ) on setting = 0 and replacing $: by J n so that
fif
Q,,
1 = -Sg$,n.V4-,dB 4
64.7)
When the boundary has the symmrtry properties described by (A.2) it follows that -
(-1yn n
Q m n - F
I
k J , ( k r ) { u(;) (kr) de) Cos [(rn-n)e] (A.8)
Wave Phenomena: Modern Theory and Applications C. Rogers and T.B. Moodie (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
453
AN INTEGRAL EQUATION A R I S I N G I N DIFFRACTION THEORY
R.A. Ross Department o f Mat hemat ics U n i v e r s i t v o f Toronto Toronto, g n t a r i o , Canada
INTRODUCTION Many mixed boundary v a l u e problems i n d i f f r a c t i o n t h e o r y can be reduced t o t h e s o l u t i o n o f dual i n t e g r a l e q u a t i o n s o f t h e t y p e
,
uhf(u)J (ru)du = g ( r ) lJ
0
(11
r < 1
<
0
(2) where g ( r ) , h ( r ) a r e g i v e n f u n c t i o n s and f ( u ) i s a f u n c t i o n t o be found. There i s an e x t e n s i v e l i t e r a t u r e d e a l i n q w i t h t h e case a = 0 and a f u l l account o f t h i s case can be found i n Sneddon (1966). I f a z 0 t h e problem o f s o l v i n g (1) and ( 2 ) i s much h a r d e r . I n t h e paper o f S c h m e l t z e r and Lewin (1963) a procedure i s g i v e n f o r r e d u c i n g ( 1 ) and ( 2 ) t o dual i n t e g r a l e q u a t i o n s o f t h e f o r m
1
m
$ ( u ) s i n (ux)du
=
f(x)
,0
<
x
<
1
(3)
0
where f ( x ) i s a known f u n c t i o n and $ ( u ) i s t o be found. Schmeltzer and Lewin g i v e a method f o r s o l v i n g ( 3 ) and ( 4 ) b y showing t h a t t h e y can be r e p l a c e d by t h e s i n g l e s i n g u l a r i n t e g r a l equation
,/
1 ;
$ ( t ) g ( x , t ) d t = -E(x)
,x
> 1
(5)
1 where g ( x , t ) = K o ( a l x - t l ) - K o ( a l x + t l ) o r d e r z e r o and E(x) =
,
’S
; 0
KO(x)
i s t h e Macdonald f u n c t i o n of
f ( t ) g(x,t)dt
,
i s a known f u n c t i o n . The f u n c t i o n $ ( t ) = f ( t ) , 0 < t < 1 , and i s t o be found when t ' >1 From $ ( t ) , $ ( u ) i n ( 3 ) and ( 4 ) can be determined by
.
m
$ ( u ) = ,/ $ ( t ) s i n ( u t ) d t
0
(6)
454
R.A. Ross
The s o l u t i o n o f ( 5 ) can be found by a t e c h n i q u e o r i g i n a l l y due t o Carleman (1922) i n which ( 5 ) i s r e p l a c e d by an e q u i v a l e n t Riemann-Hilbert problem. T h i s method i s discussed i n t h e book by Gakhov ( 1 9 6 6 ) . C o n s i d e r a b l e doubt has l o n g e x i s t e d c o n c e r n i n g t h e c o r r e c t n e s s o f t h e a n a l y s i s o f Schmelter and Lewinca@ l CamY ( 1 9 6 4 ) ) and i t i s c l e a r t h a t a d i r e c t v e r i f i c a t i o n o f t h e s o l u t i o n o f ( 3 ) and ( 4 ) by t h i s method would be d e s i r a b l e . I n t h e seque1,a m o d i f i e d f o r m o f t h e a n a l y s i s o f Schmelter and Lewin i s g i v e n and a r e l a t i v e l y s i m p l e f o r m f o r t h e s o l u t i o n i s obtained. F i n a l l y an o u t l i n e o f t h e v e r i f i c a t i o n o f t h i s s o l u t i o n i s d i s c u s s e d . The procedure i s g u i d e d by t h e s p e c i a l case a = 0 , f o r which t h e v e r i f i c a t i o n i s much e a s i e r . SOLCTION OF THE INTEGRAL EQUATION S i n c e f ( x ) i s odd i n by t h e e q u a t i o n
x
, one
5 j-' [
can c o n s i d e r
$(t)
as odd
$ ( t ) K o ( a l x - t l ) d t = -E(x)
,
n
t
XI
> 1
and r e p l a c e ( 5 )
(7)
-m
I f we i n t r o d u c e branch c u t s y = 0 , 1x1 t 1 i n t h e z-plane , then u s i n g p r o p e r t i e s o f t h e Macdonald f u n c t i o n g i v e n i n E r d g l y i e t a l . (1953), we o b t a i n the results If z
+
x i i0
f o r any r e a l
x
Ko(a(z-t))
-f
Ko(alx-tl) T i n H(t-x)Io(a(t-x))
Ko(a(t-z))
+
Ko(alx-t()
*ix
,
H(x-t)Io(alt-x))
, where ,x
H(x) = 1
>
0
= o , x < o and
Io(x)
Define
i s t h e m o d i f i e d Bessel f u n c t i o n .
F(z) =
r1
$ ( t ) Ko(a(z-t))dt +
-m
Then,writing
F,(X)
=
[
$ ( t ) Ko(a(t-z))dt
l i m F(z) , F-(x) = l i m F(z) z+x+i 0 z+x-i0 F,(x)
j-'
= F-(x) =
.
, we
(8)
have
$(t) Ko(a/x-tl)dt
, -1
< x < 1 ,(9)
1
-m
1
.-1
F ( x ) = -nE(x)
+
- in
'
Io(a(t-x))$(t)dt
,x
< -1
(10)
X
F - ( x ) = -nE(x)
Io(a(t-x))$(t)dt
t i a
,x
< -1
X
F+(x) = -ITE(X)
I,,(a(t-xj)$(t)dt
t in
,
x > 1 (11 1
F - ( x ) = -nE(x) -
i a
r"
Io(a(t-x))$(t)dt
,
x > 1
455
An Integral Equation Arising in Diffraction Theory
From ( 9 ) , ( l o ) , ( 1 1 ) we have
,
F + ( x ) + F - ( x ) = -2nE(x)
- F-(x) = 0
F,(x)
,
1x1 > 1
1x1 < 1
These c o n d i t i o n s a r e r e p l a c e d by F i ( X ) + F ' ( X ) = - 2 ~ E ' ( x ) , 1x1 > 1 F k ( x ) - F'(x)
= 0
, 1x1
< 1
Now f r o m ( 8 ) i t can be shown t h a t
F'(z)
= c f osh
(az)
fi
1
, z + =
(13)
and
The d e t e r m i n a t i o n o f p r o b l em. I f we w r i t e
F ' ( z ) = O(l0g (1-z))
,z
F ' ( z ) = O(1og ( l + z ) )
,z
F'(z)
F'(z) =
+
1
-f
-1
(14)
s a t i s f y i n g ( 1 2 ) , ( 1 3 ) and ( 1 4 ) i s a Riemann-Hilbert
e-az +
eaz , t h e n t h e f u n c t i o n s
Z
A(z) , B(z)
s a t i s f y t h e Riemann-Hi1 b e r t problem - n x ~ ' ( x ) e ~ ', 1x1 > 1
A,(x)
+ A-(x)
=
A+(')
- A-(x)
= 0
A(z) = O(&),
z
+
B
<
1
,z
+
z
-f
1 -1
m
A(z) = O(l0g ( 1 - z ) ) = O(l0g ( l + z ) ) and t h e c o n d i t i o n s on
1x1
a r e t h e same e x c e p t
J
a
i s r e p l a c e d by
-a
.
A s o l u t i o n o f t h e homogeneous v e r s i o n o f ( 1 5 ) i s L(z) =
JT-1
.
By t h e method d i s c u s s e d i n Gdkhov (1966), u s i n g t h e P l e m e l j f o r m u l a , i t f o l l o w s that
(16) and
B(z) i s t h e same e x c e p t a i s r e p l a c e d by -a
Finally,
F'(z) =
-fZ l f t 1
From ( 6 ) i t can be proven t h a t
E
2 [coshta!t-z)
-
. t+z cash
dt
(17)
456
R.A. Ross
T a k i n g t h e x - d e r i v a t i v e through ( 1 8 ) and u s i n g ( 1 9 ) t h e n a p p l y i n g t h e F o u r i e r c o s i n e i n v e r s i o n theorem g i v e s
e,,, $ i'
t E i ( t ) [cosh
where
+.!i?$? j' (ux)dx 0 1 & X
71
E'(x) =
-
0
f ( t ) lsgn (x-t)K,(a]x-tl)
a ( t - x ) cosh a ( t + x t-x t+x
- sgn ( x + t ) K 1 ( a l x + t l ) l d t
,
i s a known f u n c t i o n . Now e v a l u a t e t h e c o n t o u r i n t e g r a l
where
C
a
i s t h e s e m i - c i r c u l a r c o n t o u r shown i n F i g u r e 1 . 2 - plane
-----1
-s
F i g u r e 1.
-t
t
s
l
Contour o f I n t e g r a t i o n i n ( 2 1 ) .
.
The branch c u t s a r e as shown and t h e i n t e g r a n d has s i m p l e p o l e s a t z = -I- t I f R '+ m i t can be proved t h a t t h e c o n t r i b u t i o n t o ( 2 1 ) f r o m t h e c u r v e d p a r t o f t h e c o n t o u r tends t o z e r o and we f i n d
457
An Integral Equation Arising in Diffraction Theory
where
G(x,s)
= -
-
cosh a ( s - x )
cashs+x -ax
teat
I1 ( a ( t - s ) ) d t
+
Il(a(t+s))dt
Then ( 2 0 ) becomes
s xeaXIo(a(s-x)) where
F(t,s)
=
!l I f one l e t s
a + 0
4 - 7 (x-t)
dx
.
i n (23) then
and t h i s i s one o f t h e forms o f s o l u t i o n of ( 3 ) and ( 4 ) f o r t h e case i n Sneddon ( 1 966).
a = 0 , given
VERIFICATION OF THE SOLUTION I t can be proved ( E r d e l y i e t a1 . ( 1 9 5 4 ) ) t h a t i f
s i n ux cos u t du =
,t
x > 1 > x ,
t < -x
0 =;,
-x < t < x
So s u b s t i t u t i n g ( 2 3 ) i n t o ( 4 ) and u s i n g ( 2 4 ) we have
-I
-I
The o r d e r o f i n t e g r a t i o n i n ( 2 5 ) may be interchanged, and ( 4 ) i s s a t i s f i e d i f
,/’
f ( s ) G(s)ds = 0
-1 where -I
458
R.A. Ross
C o n d i t i o n ( 2 6 ) i s a necessary c o n d i t i o n on f ( s ) i n o r d e r t h a t ( 2 3 ) be a s o l u t i o n o f e q u a t i o n s ( 3 ) and ( 4 ) . I n t h e s p e c i a l case a = 0 we have
m However
t ( d t = s-t -1
So, i n t h e case
S
1’
-
1
!f$dt
+$!’
-1 a = 0
Qsd- t
= TI
.
-1
t h e c o n d i t i o n ( 2 6 ) becomes
!l$ 1
ds = 0
and ( 2 7 ) i s t h e necessary condi t i o n s a t i s f i e d b y f ( x ) ( G r i f f it h ( 1970) ) i n o r d e r t h a t t h e system ( 3 ) and ( 4 ) , i n t h e case a = 0 , possess a s o l u t i o n .
To v e r i f y ( 3 ) we f i r s t e s t a b l i s h t h e r e s u l t
b where
g ( s ) = sgn s I K l ( a l s l ) + K i s ( j s j ) 1
,
b The i n t e g r a l i n ( 2 8 ) i s i n t e r p r e t e d i n t h e sense o f Abel s u m m a b i l i t y
So, u s i n g ( 2 8 ) we must p r o v e t h a t f(x) =
2 a
j
2n
-1
t
Cg(X+t) + g ( x - t ) l d t
f(S)
Fs(t,S)dS,
0 < x < 1. (29)
-1
g ( s ) = 0(-)1 , so t h e s and t i n t e g r a l s a r e improper as i n t e g r a l s . By t h e P o i n c a r 6 - B e r t r a n d theorem (Gakhov ( 1 9 6 6 ) ) i f t h e o r d e r o f i n t e g r a t i o n i n ( 2 9 ) i s i n t e r c h a n g e d we must v e r i f y t h a t If
s i s small
i‘ -I
f(s)ds
i‘
Cg(x+t) + g ( x - t ) l F s ( t , s ) d t
= 0
,0
< x < 1
.
-I
Now t h e i n n e r i n t e g r a l in t h i s e x p r e s s i o n can be shown t o v a n i s h by a c o n t o u r i n t e g r a l procedure s i m i l a r t o t h a t used i n v e r i f y i n g ( 2 2 ) , so t h e v e r i f i c a t i o n o f ( 4 ) i s accomplished.
459
A n Integral Equation Arising in Diffraction Theory
REFERENCES
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.
[21
Schmeltzer, R.A. and Lewin, M., F u n c t i o n - t h e o r e t i c s o l u t i o n s t o a c l a s s o f dual i n t e g r a l e q u a t i o n s and an a p p l i c a t i o n t o d i f f r a c t i o n theory, Q u a r t e r l y 269 ( 1 9 6 3 ) . o f A p p l i e d Mathematics X X I ,
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C311 Carleman, T.,
Math. Z.
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, New
York,
C71 E r d G l y i , A. e t a l . , Tables o f I n t e g r a l Transforms (McGraw-Hill, New York, 1954). 181 G r i f f i t h , J . L . , On a c l a s s i c a l p a i r o f e q u a t i o n s , Roy. SOC. o f New South Wales, 103, 31 (1970).
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