e,
ressa
•
ng
Iy 980
Giorgio Ferrarese (Ed.)
Wave Propagation Lectures given at a Surruner School of the Centro Intemazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Balzano), Italy, June 8-17,1980
~ Springer
FONDAZIONE
CIME ROBERTO
CONTI
C.LM.E. Foundation c/o Dipattimento di Matematica ''D. Dini" Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected]
ISBN 978-3-642-11064-1 e-ISBN: 978-3-642-Il066-5 DOl: 10. 1007/978-3-642- I 1066-5 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1" Ed. C.LM.E., Ed. Liguori, Napoli & Birkhauser 1982 With kind permission of C.LM.E.
Printed on acid-free paper
Springer.com
CON TEN T S
c
0 U
r
8
e
8
A. JEFFREY
Lectures on nonlinear wave propagation
Psg.
Y. CHOQUET-BRUHAT
Oodes 8symptotiques .•..•••.•••••..•••
"
99
G. BOILLAT
Urti .•••••.•••••.••..•.....•.........
"
167
"
195
continui •••..•....•..•...............
"
215
Onde nei solidi con vincoli intern!
"
231
1
5 e min arB
D. GRAFFI
Sulla teoria dell'ottica non-linear.
G. GRIOLI
Sulla propagazione del calore nei Mezzi
T. MANACORDA T. RUGGERI B. STRAUGHAN
"Entropy principle" and main field for
a non linear covariant Byat••••••.••.
"
Singular surfaces in dipolar materials and possible consequences for continuUN mechanics .....•.........•••••.•••.•.•
"
275
CENTRO INTERNAZlONALE MATEMATlCO ESTlVO (C.l.M.E.)
LECTURES ON NONLINEAR WAVE PROPAGATION A. JEFFREY
CIME Session on wave Propagation Bressanone, June 1980
Department of Engineering Mathematics, The University Newcastle upon Tyne, NEl 7RU, England
9 CONTEIITS
Lecture 1.
Lecture 2.
Fundamental Ideas Concerning Wave Equations 1.
General Ideas
1-1
2.
The Linear Wave Equation
1-2
3.
The Cauchy Problem - Characteristic Surfaces
1-5
4.
Domain of Dependence - Energy Integral
1-9
5.
General Effect of Nonlinearity
1-13
References
1-15
Quasil1near Hyperbolic Systems, Characteristics and Riemann Invariants
2-1
1.
Characteristics
2-1
2.
WSvefronts Bounding a Constant State
2-6
3.
Lecture 3.
Lecture 4.
Riemann invariants
2-8
References
2-12
Simple Waves and the Exceptional Condition
3-1
1.
Simple Waves
3-1
2.
Generalised Simple Waves and Riemann Invariants
3-2
3.
Exceptional Condition and Genuine Nonlinearity
3-6
References
3-9
The Development of Jump Discontinuities in Nonlinear Hyperbolic Systems of Equations
4-1
1.
General Considerations
4-1
2.
The Initial Value Problem
Time and Place of Breakdown of Solution
4-2 4-2
References
4-9
3.
Lecture 5.
1-1
The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying
5-1
Depth and Width
Lecture 6.
1.
Basic Equations
5-1
2.
The Bernoulli Equation for the Acceleration Wave Amplitude
5-2
3.
The Amplitude a(x) and its Implications
5-3
References
5-5
Shocks and Weak Solutions 1. Conservation Systems and Conditions Across a Shock 2. Weak Solutions and Non-Uniquenes&
6-1
6-1
6-4
10
Lecture 7.
3.
Conservation Equations with a Convex Extension
6-11
4.
Interaction of Weak Discontinuities
6-13
References
6-14
The 1l1emann Problem, Glimm.'s Scheme and Unboundedness of Solutions
7-1
1. 2. 3.
The Riemann Problem. for a Scalar Equation
7-1
Riemann Problem for a System
G1im1ll's Ilethod
4.
Non-Global Existence of Solutions
7-3 7-5 7-8 7-10
References Lecture 8.
Far Fields. Solitons and Inverse Scattering
8-1
1.
Far Fields
8-1
2.
Reductive Perturbation Method
3.
Travelling Waves and Solitons Inverse Scattering
8-3 8-6 8-9
References
8-13
4.
11
Lecture 1. 1.
Fundamental Ideas Concerning Wave Equations
General Ideas
The physical concept of a wave is a very general one.
It includes the
cases of a clearly identifiable disturbance, that may either be localised
or non-localised, and which propagates in space with increasing time, a timedependent disturbance throughout space that mayor may not be repetitive in nature and which frequently has no persistent geometrical feature
that can
be said to propagate, and even periodic behaviour in space that is independent of the time.
The most important single feature that characterises a wave
when time is involved, and which separates wave-like behaviour from the mere dependence of a solution on time, 1s that some attribute of it can be shown to propagate in space at a finite speed. In time dependent situattons, the partial differential equations most closely associated with wave propagation are of hyperbolic type, and they may be either linear or nonlinear.
However, when parabolic equations are
considered which have nonlinear terms, then they also can often be regarded as describing wsve propagation in the above-mentioned general sense.
Their
role in the study of nonlinear wave propagation is becoming increasingly important, and knowledge of the properties of their solutions, both qualitative and quantitative, is of considerable value when applications to physical problems are to be made.
These equations frequently arise as a result of the
determination of the asymptotic behaviour of a complicated system. Nonlinearity in waves manifests itself in a variety of ways, and in the case of waves governed by hyperbolic equations, perhaps the most striking is the evolution of discontinuous solutions from arbitrarily well behaved initial data.
In the case of parabolic equations the effect of nonlinearity
is tempered by the effects of dissipation and dispersion that might also be present.
Roughly speaking, when the dispersion effect is weak, long wave
behaviour is possible, whereas when it is strong a highly oscillatory behaviour occurs, though the envelope of the oscillations then exhibits some of the characteristics of long waves.
12 Waves governed by a linear wave equation arise in many familiar physical situations, like electromagnetic theory, vibrations in linear elastic solids, acoustics and in irrotational Inviscid liquids.
However,
these linear equations often arise as a consequence of an approximation involving small amplitude waves, so that when this assumption is violated the equations governing the motion become nonlinear.
Not only does this convert the problem to one involving nonlinear partial differential equations, but it also usually leads to the study of
a system of first order equations, rather than to a nonlinear form of the familiar second order wave equation.
This happens because the wave equation
usually arises as the result of the elimination of certain dependent variables from first order equations (like! or
~
in electromagnetic theory),
and this is often impossible when nonlinearity arises.
Our concern hereafter
will thus be mainly with quasilinear first order systems of equations - that 1s to say with systems that are linear in their first order derivatives, and for the most part we will confine attention to one space dimension and time. 2.
The Linear Wave Equation
Because of the importance of the linear wave equation (1)
let us begin by reviewing some of the basic ideas that are involved, though 1n the more general context of the variable coefficient equation
3
L
~,j·O
3
a
ij
U
i j x x
+
L
i-o
biu i x
+ cu
(2)
f
with aij' b i , c, f functions of the four dimensional vector
~
0 1 (x • x
x
2
Not all linear second-order equations of this form describe wave motion. and 00
account of this it is necessary to produce a method of classification which
readily allows the identification of wave type equations from amongst the other types that are possible (i.e. elliptic and parabolic). The form of classification to be adopted utilizes the coefficients of the highest-order
rl?-r!va~~ve~
and has an algebraic
~asis
but, as will be seen
3
x ).
13 in a subsequent section, this classification in fact effectively distinguishes between equations of wave type and those of other types.
Let us start by
attempting some simplification of the form of equation (2) by changing the independent variables through the linear transformation 3
I ""ijoJ i-<> where the GC
0,1,2,3
i
(3)
are constants.
ij
A transformation of this form gives an affine mapping of the
(xo, x1 , x 2 , x 3 )-space which Is one-one provided det I~j I
~
O.
Employing
the chain rule for differentiation we find that equation (2) may be re-wrltten 3
o•
I
(4)
i,j,k,l"O Hence the coefficients 8
1j
of the derivatives
U i
j' which are functions of
x x
position, transform to the new coefficients
of the derivatives
k l' which are also functions of position.
U
If, now,
f; f;
we confine attention to the set of coefficients a
specific point
~
z
!o
1n
o (x ,
appropriate to some ij 123 x • x • x )-space, we see that this is exactly
the transformation rule which would apply to the coefficients
8
ij
of the
quadra tic form 3
I
i,j-O
a ij nin j •
(5)
when the n i are transformed to Bit by the variable change 3
ni
I k-O
akiak'
Now it is a standard algebraic result that by a suitable transformation a quadratic form with constant coefficients may always be reduced to a sum of squares. though not all of the squared terms need be of the same sign. Furthermore, Sylvester's law of inertia asserts that however this reduction is accomplished, the number of positive terms m and the number of negative
14 terms n will always be the same.
To apply these results to the differential
equation (2) itself with the variable coefficients attention to a fixed poine
the
8
~
the specific values
1j
8
•
!o
ij
8
ij
, let us again confine
o 1 2 3 in (x , x , x , x )-space and attribute to
• alj(~)'
This then implies that some choice of the numbers G
ij
• d
ij
exists for
which 3
I
1,j-O
where m + n < 4.
The number pair (m.D) Is called the signature of the
quadratic form (5) and, being an algebraic invariant, 1s used to classify the quadratic form.
We shall use it to classify the
partial differential equation (2) at each point
~
-
va~lable
coefficient
!o.
1n the transformation The effect on equation (2) of us1ng these numbers Q 1j (3) is co yield at .! • m-1
I 1-0
!!o
a differential equation of the form
1Itkl-1 U
3
I
tit i
U
i-m
1 1 t t
+
I
i-o
biu 1 t
+
f
Equation (6) or, equivalently, (2) is called hyPerbolic
o ( -direction
(6)
0 a~ ~
z
!o
in the
when the signature is (1,3), elliptic when the signature is
(4,0) and parabolic when m + n < 4.
o
If an equation is hyperbolic in the ( -
direction at each point of a region O. then it is said to be hyperbolic in the CO-direction throughout O. Obviously, if an equation has constant coefficients, then one suitable transformation (3) will reduce it to the form of equation (6) throughout all space.
For example, aside from the trivial transformation to remove the
2 constant factor llc , the wave equation (1) is already seen to have the
signature (1,3).
Thus if a transformation is made at one point of space to
convert the factor lIe
2
to unity,then it does so for all points in the space.
The usual effect of variable coefficients and first-order terms in hyperbolic equations of the form (2) is to introduce distortion as the wave profile propagates.
This produces various complications, not the least of
which 1s the fact that the wave velocity becomes ambiguous and requires
15 careful definition.
Only when there is a clearly identifiable feature of
the wave which is preserved throughout propagation 1s it possible to define the propagation speed of this feature unambiguously.
Such is the case with
a wave froDt separating, say, a disturbed and an undisturbed region and across which a derivative of the solution 1s discontinuous.
3.
The Cauchy Problem - Characteristic Surfaces Fundamental, to the study of hyperbolic equations 1s the Cauchy problem,
and the associated notion of a characteristic surface.
In brief, when
working with four independent variables the Cauchy problem amounts to the determination of a unique solution to an initial value problem in which a hypersurface F 1s given, and on it the function u 1s specified together with the derivative of u along some vector directed out of F. directional derivative is
call~d
Such a
an exterior derivative of u with respect
to F, in order to distinguish it from a directional derivative in F which is known as an interior derivative.
In the Cauchy problem it must be
emphasized that the function u and its exterior derivative over the initial hypersurface F are independent, and can be specified arbitrarily. A hypersurface F for which the Cauchy problem is not meaningful because u and its exterior derivative cannot be specified independently is called a characteristic hypersurface.
Let us now see how characteristic hypersurfaces
may be determined. It is convenient to utilize curvi-linear coordinates (0, (1. (2. (3 and to let the hypersurface F on which the initial data is to be specified have the equation ( to (
o
o
~
O.
In terms of the new variables. a derivative with respect
is a directional derivative normal to F so that it is an exterior
1 2 3 derivative. whilst derivatives with respect to ( , ( • ( are interior
derivatives. We now utilize this by rewritine equation (2) in a form in Which the derivative u
~o~o
is separated froa the other second-order derivatives
16
3,
L
i,j,k,.e.""O 3
Uk
i,k-<J
,
Here
,
L
+
r
+
(7)
f •
Cll
signifies that the terms corresponding to k ~ 1 - 0 are omitted from
the summation.
,
Now if we specify u and u 0 independently on F, &s is required in the Cauchy problem, the substitution of their functional forms into equation
,,
(7) enables the determination of u 0 0' provided only that the coefficient of this derivative does not vanish.
Thereafter, the solution may be obtained
in the form of a Taylor series by determining the coefficients of the series by successive differentiation of the initial data and of equation (7) itself.
This is. of course, the idea underlying the classical Cauchy-Kowalewski theorem.
It is, however, very restrictive as an existence theorem since w
it demands that all functions involved are C .
In the event that the coefficient (8)
of
u 0 0
vanishes, neither this nor higher-order derivatives of u with
, ,
0
respect to
t can be found.
,,
Furthermore, the derivative u 0 0 may then be
specified arbitrarily on F, and even differently on opposite sides of F. This is not remarkable, because when the coefficient of
,
u and u 0 cannot be specified independently over
F.
u 0 0
<,
vanishes,
This follows because
they must satisfy the equation which results when the first term is deleted from equation (7), and so we then have As
insuf~icient
initial data.
already mentioned, the hypersurface F with the equation
t o = 0 for
which the coefficient (8) vanishes 1s called a characteristic hypersurface of the differential equation (2). begin by setting Pi
a
a(
olax i
To examine such hypersurfaces further, we
and writing
3
H
L i.j-O
(9)
17 Then the quadratic form H 1s the coefficient of the derivative u 0 0 in t t equation (7), and the characteristic hypersurface F will be given by the eoad! tion H - O.
To interpret the condition H • O. we first recall that 1f , is a differentiable scalar function, then
grad~
Consequently, by analogy, Pi
Y • const.
c
is a vector normal to the surface
atolax i is the itb component of the
four-dimensional gradient of to and so 1s the ith component of a four-dimensional vector £ normal to the hypersurface F.
Hence the equation H - 0 1s a
condition on the orientation of the normal vector
~
to F, and as the
8
1j
are
usually functions of position. it follows that this condition will differ from point to point.
The quadratic form (9) Is, of course, just the same quadratic form we encountered in (5), so that its signature will depend on the type of the equation (7) or, equivalently, (2). ~
• !o
If the equation is hyperbolic at
the signature will be (1,3), and it follows that at the point the
condition H •
° determining the characteristic hypersurface can be reduced to (10)
It is obvious that
DO
real characteristic hypersurface exists for
elliptic equations, since their signature is (4,0) and the components of the vector
~
need to be complex if they are to satisfy the condition 2
2
2
2
B • Po + P1 + P2 + P3
0 .
10 proceed with the hyperbolic case we now simplify matters by setting
xO - t and writing
to
t -
1
Hx • x
so that Po - 1 and Pi •
2 -~
3
x ) xi
(11)
for i • 1,2,3.
Then the quadratic form (10)
becomes ~2 x1
+
~2
x
2
+ ~2 3 x
1
(12)
18
which 1s a differential equation for the function, locally at
~
-
~.
This is, of course, the familiar Eikonal equation from mathematical physics. At any time t 1 4t (x ,
X
2
to a real three-dimensional surface 5 is defined by
~
t
X
3
(13)
)
and this is called a characteristic surface.
If equation (7) Is a constant coefficient equation it can be reduced to the form of equation (6) with m • 1, n - 3 throughout all
8~ace,
80 that
equation (12) then describes the characteristic surface 4' - const for all points In space. In summary, we have established that real characteristic surfaces
oc~ur
In connection with hyperbolic equations, and that across such surfaces a
discontinuity may occur In the second normal derivative of the solution.
This
discontinuity in a derivative of a solution is usually identifiable with
an interesting physical attribute of the solution, since it represents a wavefront bounding two regions. The discontinuity surface, or wavefront. advances with time. as is
shown by the following simple argument. Taking the total differential of
to -
2
o
1
dt - d" • 1
d" • 2
"
0 and using equation (11) we find
"
or. equivalently
1 2 3 where dr is the vector differential with components (dx • dx • dx ).
Hence 1
Igrad, I
(14)
v.n
where
dr dt
!!. •
~ Igrad. I
The vector n 1s the unit normal to the surface • - const. and as the displacement of a position vector with time. v
d~/dt
d~
represents
is the velocity of
19 displacement of a specific point on the surface as the characteristic surface moves from Its position at time t to its position at scalar
~.~
t
+ dt.
The
is the normal velocity of propagation of the characteristic
surface or wavefront and, In general, Is a function of position. By re-writlng equation (7) and differencing it across the characteristic
surface, we shall see that there may also be a discontinuity in the first normal derivative of the solution and this, like the discontinuity in the
second-order derivative, Is propagated with the characteristic surface. The equation governing the development of the discontinuities In first- and second-order derivatives is an ordinary differential equation defined along a curve in space and is called the transport equation. 4.
nomain of Dependence - Energy I~ tegral The dependence of a wave solution on initial data is most easily
illustrated in terms of the one-dimensional wave equation
o ,
(15)
with the initial conditions u(x,O)
and
h(x)
au at
k(x) .
(16)
x+ct k(s) ds
(17)
(x,O)
The explicit d'Alembert solution h(x-
u(x, t)
+.12<
f
x-ct
shows how the solution at (xo,t ) depends only on data in the interval
O
X
o-
cto
.::
x .::
X
o + cto
This is called the domain of dependence of the 901ution at (xo,t )' O
This
same idea generalises to quasilinear hyperbolic systems and we shall employ it later. In conclusion, to illustrate the important notion of an energy integral that arises when working with equations derived from the conservation of physical quantities, let us prove the uniqueness of the solution to the
20
I:
rx.; f-.) \
\
~
'\
~d""":'" \
•t \ \ J./-U"'; '" ~~.1 \ Domain of dependence Cauchy problem for slightly generalised two dimensional wave equation
au q(x,y)u - r -
(18)
at
with u(x,y,O)
a
au (x,y,O) 3t
UO(x,y),
(19)
and where we shall assume p, k, r to be positive constants and q(x,y) > 0. It will be convenient to consider that (18) governs the motion of a membrane with density P, tension k per unit length, distributed springing under the membrane with spring constant q(x,y) per unit area and frictional coefficient
r. Then the potential energy within a fixed region R with boundary B of the (x.y)-plane comprises the energy stored in the springing qu
2
dxdy
and the energy stored in the membrane
~(t)
dxdy
with ~ the outward~wn unit normal to Band ds a length element of B.
The
first integral in ':(t) is the negative of the work done by the tension against the interior of R and the second integral the negative of the work done against the boundary. Green's theorem shows that
so that the total potential energy
21
(20)
The kinetic energy 1s (21)
so the total energy is
or (22)
It then follows after use of Greents theorem that
•
f atau B
au
-k-ds-
an
II
r
R
[:~r
(23)
dx dy ,
which 1s the outward flux of energy across the boundary and the loss due to friction.
7 t, f------,.<--hl--+--\--\-----J o "'------~----------
><:
Now let R vary in such a way that at t = 0 it 1s
Ra
and at
t •
t
1
it 1s
the smaller domain R " The surface between R and R , we write in the form I O I t -
T(x,y).
Integration of the identity
2
o • 1-1.. at {P
-k
[au]2 at + k[[au)2 ax) + [au]2] ay
(a lou au] +- wat"aya: [au au]] at"ax
px
+
+r
qu 2} [au]2 at
22
followed by use of the divergence theorem and some manipulation finally
gives the result
dt dx dy
+
2 rIIk
II
.!
[au + aT auJ2
ax
ax at
(x,y) in R t-T(x,y)
[;~ + ;; ;~J
+ k where c
2
2 +
+- [[;;f c
2
+
[;;n[;~r + 2J dx dy QU
(24)
- kIp and V is the volume of the region concerned.
Now impose on R(t) the condition 1 2
<
(25)
c
Then all terms on the right-hand side of (24) are non-negative, so if U -
ut
at
- 0
t
•
0
in
R O
the right-hand side of (24) must vanish, since
with zero initial conditions the left-hand side vanishes.
In particular this
means that u t must vanish identically on the top and sides of V. the top R corresponds to any 1
follows that u
=0
t
1
80
that u
t
;
0 in V.
However,
Since u : 0 in R it O
in V.
This proves uniqueness, for If
two
different solutions v and w exist
corresponding to the same Cauchy data (19), u = v-w will satisfy the initial data u
~
u
t
- 0 at t - O.
We have seen this implies u ; 0 so that v
= w,
and
the solution is unique at all points that cannot be reached by a disturbance starting 1n RO and travelling with a speed
2 c.
The region
Ra
now plays the
part of the domain of dependence, and the volume V becomes the domain of determinacy. The limiting case 1
- "2 c
may be interpreted in a useful physical manner if we let n be the normal to
23 the ruled surface dn dt 80
T(x,y).
t -
[[;~r + [;~)2r
1
•
TVTT •
•
c ,
We have
that d.!'! dt
showing that c 1s
the
speed of contraction of the region R.
The volume V
in which the solution 1s determined by the Cauchy data on R is thus an O inverted cone with base R ' O 5.
General Effect of Nonlinearity It 1s now necessary to make clear that the effect of nonlinearity in a
wave equation involves more than the 108s of superposib111ty, for it can also change the entire nature of the solution.
This is best shown by a simple
non-physical example.
Consider the single first order partial differential equation
au + feu) au at ax
..
0
(26)
for the scalar u(x,t) that is subject to the initial condition u(x,O)
g(x) .
(27)
Now the total differential du 1s given by
au dt + au dx
du
at
so that if x and
ax
t
are constrained to lie on a curve C, then at any point P on
C we have
au
at
•
au +
at
[dX) dt
where now dx/dt is the
au
(28)
ax gradien~
of curve C at point P.
Comparison of (26) and (27) now shows that we may interpret (26) as the ordinary differential equation du dt
o
(29)
along any member of the family of curves C whi,:h are the solution curves of
24 dx
feu) .
dt
(30)
These curves C are called the characteristic curves of equation (26).
The
solution of the partial differential equation (26) has thus been reduced to
the solution of the pair of simultaneous ordinary differential equations (29) and (30). Equation (29) shows that u • const along each of the characteristic curves C.
The constant value actually associated with any characteristic curve being
equal to the value of u determined by the initial data (27) at the point at which the characteristic curve intersects the initial line U a
const
O.
Setting
1n (30) then shows that the characteristic curves C of (26) form a
family of straight lines. (~tO)
point
t •
50, 1f we consider the characteristic through the
on the initial line, we find after integrating (30) and using
(27) that the family of characteristic curves C have the equation x
!;
+ tf(g«»
(31)
where t now plays the role of a parameter. Expressed slightly differently, we have shown that in terms of the parameter t,u -
get) at every point of the line (31)io the (x,t) plane.
In
physical problems t usually denotes time, so that it is then necessary to confine attention to the upper half plane in which t
~
o.
The solution to (26) and (27) may be found in implicit form if !; is eliminated between u • g«(), which is true along a characteristic, and the equation (31) of the characteristic itself. u
•
g(x - tf(u»
We find the general result
•
(32)
Result (31) shows that 1f the functions f and g are such that two characteris tics intersect for t > 0, then since each one will have associated with it a different constant value of u, it must follow that at such a point the 801ution will not be unique.
This can obviously happen however smooth the
two functions may be, since intersection of two characteristics depends merely on the value of f(g(t» characteristics.
that is associated with each of the straight line
This 1s to say on the two points «(1,0) and (t 2 ,0) of the
25 initial line through which they pass.
We conclude from this that such
behaviour of solutions is not attributable to any irregularity in the coefficient feu), or in the initial data u(x,O) • g(x). Differentiating (32) partially with respect to x gives
au
ax •
g'(x-tf(u») l+tg'(x-tf(u))f' (u)
(33)
showing ou/ax becomes infinite whenever 1 + tgt(x-tf(u»f'(u) • O. what is often called the gradient catastrophe.
This 1s
In order to extend the
solution beyond this point we will need to introduce the concept of a discontinuous solution called a shock.
This will be done later.
General References [1]
Courant. R. t Hilbert, D. Methods of Mathematical Physics, Vol. II. Wiley-Interscience, 1962.
[2]
Garabedian, P. R.
[3]
HellWig, G.
[4]
Roubine, E. (Editor).
[5]
Coulson, C. A., Jeffrey, A.
Partial Differential Equations, Wiley, 1964.
Partial Differential Equations, Blaisdell, 1964. Mathematics Applied to Physics, Springer, 1970. Waves, 2nd Ed. Longman, 1977.
26
Lecture 2.
Qu8silinear Hyperbolic Systems, Characteristics and Riemann
Invariants.
1.
Characteristics The notion of a characteristic curve needs to be introduced in the
context of the quasilinear system + A 3U 3x
+
o
B
(1)
in which U and Bare n element column vectors with elements u ' u 2 ' ••. , 1 un and b I , b2 , b ' respectively, and A is an n x n matrix with o elements a ..• 1J
The system (1) will be quasilinear if, in general, the
elements a ij of A depend nonlinearly on u ' u 2 ' ••• , un' 1
When B
~
0 the
elements hi of B may, or may not, depend linearly on u ' u '· .•• , un0 2 1
It
will be assumed throughout this section that the elements b. and s .. are 1
continuous f1znctions
Although x,
t
1J
of their arguments.
are the natural variables to use when deriving systems
of equations describing motion in space and time, they are not necessarily the most appropriate ones from the mathematical point of view.
So, as we are interested in the way a solution evolves with time,
let us leave the time variable unchanged in system (1), but replace x by some arbitrary curvilinear coordinate , and then try to choose manner which is convenient faT our mathematical arguments.
~
in a
Accordingly,
our starting point will be to change from (x, t) to the arbitrary semicurvilinear coordinates
t(x, t)
t
,
t'
t
.
(2)
If the Jacobian of the transformation (2) i. non-vanishing we may
thus transform (l) by the rule
.L 3t
.L 3x
-
l i .L
-
l i .L
3t
3x
3t aJ;
+
~ 3t
at'
+
~~ 3x
3 l i .L at' .. 3x 3t
3
-
l i .L 3t
at
+
3
at'
27 where, of course, at/at and
3~/3x
are scalar quantities.
This leads
directly to the transformed equation
au
at'
at au
+
at
+
~
at ax
au
A
+
~
B
0
the terms of which may be grouped to yield
au F
(.!i r at
+
+
.!i ax A
au
)
+
~
o•
B
(3)
where I is the u. x n unit matrix. Equation (3) may now be considered to be an algebraic relationship connecting the matrix vector derivatives au/at' and aU/3(.
It is tben
at once apparent that this equation may only be used to determine
au/at
if the inverse of the coefficient matrix of
au/at
to say, if the determinant of the coefficient matrix of
exists.
That is
au/at
000-
is
vanishing.
This condition obviously depends on the nature of the
~urvilinear
coordinate lines (x, t)- const., which so far have been
chosen arbitrarily.
Suppose now that for the particular choice ( ;
~
the determinant does vanish, giving the condition.
at r I !1
o
(4)
Then because of this the derivative family of lines const.,
au/a.
n elements 3u • - const.
~
• const.
au/a~
will be indeterminate on the
Consequently, across such lines
may actually be discontinuous. i
/a+
of
au/a,
~(x, t)~
This means that each of the
may be discontinuous across any of the lines
To find how, when they occur, these discontinuities in
are related one to the other across a curvilinear coordinate line.
auila, =
const.,
it is necessary to reconsider equation (3). We shall now confine attention to solutions U which are everywhere
.
continuous but for which the derivative
.
the particular 110e
t
~
k (say).
au/a.
is discontinuous across
Because of the continuity of U, and the
continuity of the elements s .. of A and b. of B, the matrices A and B will 1J
experience no discontinuity across • • k.
• We
call this a weak discontinuiry.
1
So, in the neighbourhood of a
28
typical point P of this line, A and B may be given their actual values at P. ~
In equation (3) there is no indeterminacy of s
au/at' across the lines
const., and as a/at' denotes differentiation along these lines it
au/at'
must follow that
is everywhere continuous and, in particUlar, that
it is continuous across the line • • k at P.
Taking these facts into account the differencing equation (3) across
=••
the line t
k at p gives
+ , . A) oX
where [ a]
=
Q_
-
Q+
p
[~~] 0, p
Q+
Q_
denoting the value to the immediate left
the value to the immediate right at P.
was any point on this line the suffix P may now be omitted.
a/a+
(5)
O.
signifies the discontinuous jump in the quantity a
across the line • • k, with of the line and
-
As the point P The operator
is differentiation normal to the curves. - const., so that equations
(5) express compatibility conditions to be satisfied by the component of the derivative of U on either side of and normal to these curves in the (x, t)plane. This is a homogeneous system of equations for the n jump quantities
[ 'u./'.] 1.
~
('u./'.) - ('u./'.) and there will only be a non-trivial 1. 1. +
solution if the determinant of the coefficients vanishes.
The condition
for this is
o However, along the lines
••
~
+
••.X
dx dt
••
(6)
const . we have, by differentiation,
0
so that these lines have the gradient dx dt
_!! / .3x. 3t
~
(say).
(7)
Combining (6) and (7) we deduce that A must be", such: thar.
IA
~I
I
o .
(8)
29 Consequently the A in (7) can only be one of the eigenvalues of At
and since (5) can be re-written (A
-
Al)
[:~]
•
(9)
O.
the column vector
[au/a~] must
eigenvector of A.
This, then,
[ au. /a~ ] of ttie vector [ 1
be proportional to the corresponding right ~etermines
au/a. n
the ratios between the n elements
that we were seeking.
As A is an n x n matrix it will have n eigenvalues.
If these are
real and distinct, integration of equations (7) will give rise to n distinct
families of real curves eel) t e(2), ••.• C(n) in the (x, t)-plane:
C(i) • dx = A(i)
i • 1, 2, ••• , n.
• dt
If x denotes a distance and dimensions of a speed.
t
(10)
a time, the eigenvalues will have the
Anyone of these families of curves c(i) may be
'taken for our curvilinear coordinate lines. - const.
The A(i) associated
with each family will then be the speed of propagation of the matrix column vector [ au/a~] along the curves C(i) belonging to that family. When the eigenvalues ~(i) of A are all real and distinct, so that the propagation speeds are also all real and distinct, and there are n distinct linearly independent right eigenvectors rei) of A satisfying the defining relation t
(i)
•
for i • 1, 2, ••• , n.
the system of equations (1) will be said to be totally hyperbolic.
(ll)
We
may, if we desire, replace the words right eigenvector by left eigenvector in this definition, where the left eigenvectors I of A satisfy the defining relation
r(i) A
A(i) r(i).
for i • 1, 2, ••• , n.
(12)
The families of Curves e(i) defined by integration of equations (10) are called the families of characteristic curves of system (1).
30
The relationship between characteristic curves and the solution vector U to system (1) is illustrated in the Figure in the case of a typical element u of U. i
Here it has been assumed that initial conditions
have been specified for system (1) in the form U(x, 0)
•
,(x)
where the ith element u
of U has for its initial condition ui(x, 0)
i
E
Wi(x),
w,,"vell'orot" t""ke
(cJ,."u/;.,,: ,,/;;,)
_ _--'<-_ _
~ -
~
..L_ _ :>r
...I~
-'1.-"'"
-:x,
Since it was not necessary that
aUla.
?t)
should be discontinuous across
the characteristics • • const., it must follow that continuous and differentiable elements of the initial data u.(x, 0) 1
propagate along characteristics.
~.(x) 1
will also
In the case of the element of initial
data at A, this will propagate along the characteristic. - k
1
(say) starting
from the point (xl' 0) which is the projection of A onto the initial line. The characteristic
+•
k
1
is then the projection onto the (x, t)_plane
of the path AB followed by the element of the solution surface S that started at A.
Characteristics corresponding to k • k , k , k , etc., 2 3 4
may be interpreted in similar fashion. To distinguish between initial and boundary value problems "it is necessary to classify arcs r in the (x, t)-plane as being either timelike or spacelike.
This is done by assigning to each characteristic arc
an arrow showing the direction corresponding to increasing t, and then
31 testing to see whether at a point in question all characteristics radiate out to one side of the arc r corresponding to a timelike arc r, or some lie to one side and some the other
arc f.
This is illustrated in the Figure. c!~)
d..-'}
r
timelike arc r at P Suppose that the
~ector r
spacelike arc r at P
(i)
(i)
with elements r 1
(i)
, r2
'
"0'
(i)
rn
is
the i,th right 'eigenvector of A corresponding to the eigenvalue " • A(i) .
Then it followes from (9) that across a wavefront belonging to the C(i) family we may write,
[au/a. ] r
[ aun 13.]
(13)
(I)
1
where the elements of rei) • r(i)(u) have values determined by U on the wavefront. 2.
Wavefronts bounding a constant state In physical situations the solution vector U describes the "state"
of the system described by equations (1).
It is thus convenient to refer
to a region in which U is non-constant as a disturbed state, and a region in which U is constant as a constant state, irrespective of whether or not the system involved described a physical situation.
Our purpose here
will be to examine the simplification that results in equation (13) when a wavefront bounds a constant state.
32
First, as the elements a .. of A are continuous functions of their 1J
arguments, if follows directly that the eigenvalues A(i) of A are continuous functions of a ij , and hence of the elements u ' ll2' l U.
un of
"'J
Since U is itself continuous across a wavefront we conclude that
~(i) • A~i) • const., on a wavefront bounding tbe constant state U - 00' where ~~i) _ ~(i)(Uo).
From equations (10) we thus see that if a charac-
teristic curve from the ith family e(i) bounds a constant state, then it must be a straight line. If such a straight line characteristic
C6i )
belonging to the ith
family e(i) bounds a constant state U • 00 that lies to its right (say), then because (3U/3~)+ - 3u0/3~
= 0,
[~]
3u.
3u •
(at )
(at )+
for j - 1, 2, ...• n. (14)
Now au/at I is continuous across c~i) while aVO/at'
= O.
Thus in
the disturbed region immediately adjacent to c~i) the total differential du. reduces to J
for j • 1, 2, ... , n.
duo J
(15)
By virtue of (13) and (14) this is equivalent to duo J
•
Kr~i) d~
•
(16)
J
where K is some constant of proportionality.
It proves convenient to
choose K so that the first element Kr~i) of Kr(i) becomes unity. j • 1 in (16) then gives dUl •
d~,
Setting
so that all the other differentials
du 2 , du , ••• , dUn become expressible in terms of du , because (16) 3 l becomes r
(i) j
dUl
fo~
j •
I, 2, .•• ,
D
or
dU
-
r
(i)
du
l
. (17)
33 A simple rule that is sometimes useful for deriving results of this
form follows by combining the matrix vector form.of (17) and the defining relationship Ar
l:r
for the right eigenvector corresponding to the eigenvalue
~.
Immediately
adjacent to the constant state 0 • U this gives the result
o
'(A - ~el) dU e
•
0
(18)
where A • A(U ) and ~O • ~(UO). Comparison of this result with system O O (1) from which it was derived now yields the following rule. Rule for compatibility conditoDs for elements of dUo
To find the
relationships that exist between the elements du , du , ••• , dUn of dU 2 l in the disturbed region immediately adjacent to a wavefront that bounds
a constant
state U • U ' the vector B in (1) should be neglected, the O
undifferentiated variables should be replaced by their CODstant state values. and in the differentiated terms the folloWing replacements should be made and 3.
aax ..
d(.).
(19)
Riemann invariants This method applies to any totally hyperbolic system of two
homogeneous first order equations involving two dependent variables u • l u of the general form 2
o,
(20)
which is subject to the initial data and
(21)
The coefficients a ••• a .• (u , u ) will. in general. be assumed to 2 1J 1J l be functions of the two dependent variables u and u • but not to have 2 1
34 any explicit dependence on the independent variables x and t.
The system
(20) will be guasilinear when a •.• a .• (u , u ) and it will be linear in 2 1J 1J r --the special case when the coefficients a .. are all constants. 1J
Defining A and U to be
U
•
[:~J
enables equations (20) to be written .!Q.+A 3U
at
ax
.O,
(22)
when we know that the system will be
totall~
hyperbolic provided the two
eigenvalues ~(i). i • 1, 2 of
IA
-
~I
I •
(23)
0
are real and distinct and A has two linearly independent eigenvectors. In place of the right eigenvectors r that were useful in the previous section, let us now make use of the corresponding left eigenvectors 1 defined
for
i - I . 2.
(24)
If, now, we pre-multiply (22) by l(i) and use (24) we obtain the result •
0 •
for i • 1 ,2.
(25)
In this the bracketed expression will be recognised as the directional derivative of U with respect to time along the family of characteristics e(i).
Denoting differentiation with respect to time along members of
the eel) family of characteristics by dIdo and differentiation with respect to time along members of the C(2) family of characteristics by
d/d8 enables us to replace (25) by the following pair of ordinary differential equations which are defined along the ell) char.cteristic3 by 1 (l) dU
da
-
O.
(26)
35 and along the C(2) characteristics by (27)
Hence e • const., along e(l) characteristics and u • const., along e(2) characteristics as indicated in the Figure.
Setting the left eigenvector t(i) • (t~i), l~i», for i ~ I, 2 then enables (26), (27) to be re-expressed as dU 1 -do
+
o
(l) h .. h e a 1 ong t C e aracter1stLCs
(28)
o
along the e(2) characteristics.
(29)
and
Since, by supposition, A depends only on u
and u ' so also will the 2 coefficients l~i) of the left eigenvectors t(l), t(2). Consequently, l
J
both (28) and a9) will always be integrable along their respective characteristics, though they may first require multiplication by a suitable integrating factor u. (1)
Integrating (28) with respect to a along the C
.
.
character1st1cs,
and (29) with respect to B along the C(2) characteristics gives: along ell) characteristics "I (1) dU
" 2
2
r(8)
(30)
36 and along C
(2)
.
.
character~st~cs
(31)
s (a) ,
where r, s are arbitrary functions of their respective arguments sand a. The two faDdlies of characteristics are themselves given by integration of the equations
for i • I, 2.
(32)
The functions reS) and sea) are called Riemann invariants and, by virtue of their manner of derivation, rand s are constant along their To be more precise, reS) is
respective families of characteristics.
constant along any eel) characteristic, though as it is a function of S, which in turn identifies the characteristics, it will, in general, be
different for different characteristics.
Correspondingly, sea) is
constant along any e(2) characteristic, though here again the constant will be different for different characteristics depending on the value of a associated with each characteristic. Equations (30) and (31) enable u
l
and u
2
to be expressed in terms
of rand s, the values of which are determined at points of the initial line t • 0 by the initial data (21).
Suppose r(B) in (30) is denoted by
R(u , u ) and ala) in (31) i. denoted by S(u , u ). Then along the eli) l 2 2 1 characteristic issuing out from the point (x ' 0) of the initial line in O the sense of increasing time we have from (21) and the property of r(B) that
R(u , u ) 1 2
R(U
(x
)' 1 O
u/xo» .
(33)
Similarly, along the C(2) characteristic issuing out from the point (Xl' 0) of the initial line in the sense of increasing time we have from (21) and the property of s(a) that
(34)
37 Solving these two implicit equations for u
1
and u
2
then determines
the solution at the point P in the Figure which is the point of intersection of the eel) and e(2) characteristics along which the respective constant values of Rand S are transported.
In principle the initial value
problem is now solved, since as the points (x ' 0) and (xl' 0) of the O initial line were arbitrary, in
t~e
upper half plane.
SQ
also is the point P which may be anywhere
However, in any particular case, the task of
solving the two implicit relationships and of finding the characteristic curves in order to determine their point of intersection P is usually difficult.
Nevertheless, this method of solution can often be used to
solve-problems and it is, in any case, of considerable theoretical importance. References [lJ
tlJ (3) [4J
Courant, R., Friedrichs, K. O. Supersonic Flow and Shock Waves, Interscience 1948. Jeffrey, A. Quasilinear Hyperbolic Systems and Waves, Research Note in Mathematics, Pitman Publishing, London. 1976. Coulson. C. A., Jeffrey. A. Waves. 2nd Ed.; Longman, 1977. Garabedian, P. R. Partial Differential Equations. Wiley, 1964.
38 Lecture 3.
Simple Waves and the Exceptional Condition
Simple Waves
1.
When
oDe
of the Riemann invariants r or
9
is identically constant, the
corresponding solutions of equations (20) of Lecture 2 are known as simple wave solutions.
That is, simple wave solutions occur either when r(e)
canst' J or when sea)
= so·
= rO
c
canst., and we now deduce the basic properties
of this fundamental class of solutions directly from this simple definition.
Suppose, for example. that sea)
= so'
then equations (30) and (31) of
Lecture 2 may be written rCB)
along
eel)
characteristics
(1)
along e(2) characteristics •
(2)
and ·0
where (3)
This shows that everywhere along a eel) characteristic specified by S
= 60
• const., say, u
1
and u
2
must also be constant, for they are the
solution of the nonlienar system of simultaneous equations
and
The actual constant values associated with u characteristic are
~
•
uI «(), u 2 • u2 «()
l
and u
determinded
along this
2
by
the values of the
in1tia1 data (21) of Lecture 2 at the point (',0) of the initial line through which this
eel)
characteristic passes.
Any function of u
l
and u
2
will also
be constant along this characteristic as, in particular, will be ,1.(1)(u «(), u «(») • A(l)«(), say. 2 1
Consequently, as the C(l) characteristic
is found from (32) of Lecture 2 by integrating
39
it must be the straight line
~ + tA (1) W
x
As B and hence
O
allowing
~
(4)
t, were arbitrary, this result implies that
by
to move along its permitted interval on the initial line,
(4) will generate a straight line family of C(l) characteristics.
80
Conversely,
had we set reB) ~ r ' it would then have followed that the c(2) family of O characteristics was a family of straight lines along each of which u and 1 U
2
were constant.
Thus simple waves occur adjacent to constant state
regions and one of their main uses 1s to piece together solutions between different constant states. By analogy with the
sit~ation
in gas dynamics, when a straight line
family of characteristics converges, the associated simple wave is often called a compression wave, whereas when it diverges, the associated simple wave i's called 'an expansion wave.
Compression waves generate shocks which
are to be discussed later. The property that in a simple wave u
l
and u
2
are constant along the
straight line characteristics means that simple wave solutions are the simplest type of non-constant solution for the system (1), (2). 2.
Generalised Simple Waves and Riemann Invariants It is reasonable to enquire whether the notion of a simple wave can
be generalised and extended to systems with more than two dependent variables.
Specifically we shall consider homogeneous systems in one space
dimension and time of the type all (U) a
a
21
(U) a
12
22
(U)
aln(U)
ul
(U)
a
u
2n
(U)
2
o•
+
u
n
a t
nn
(U)
u
n"
(5)
~o
which will be said to be reducible in the generalised sense. Since our concern ~l be with generalised. simple waves we seek an extension of a Riemann invariant from amongst the properties of ordinary The property we choose to generalise 1s that in an ordinary
simple waves.
simple wave there is a functional dependence between u u
1
- £(u ). 2
1
and u
2
of the form
Accordingly, we propose to take a generalised simple wave region
to be one in which the solution vector U is a function of only one of its
n-elements, say of u ' so that U· U(u ). 1 1
If 1n (5) we set U· U(u ), 80 that 1 elementary calculation establishes that
U
1
•
ul(~)
for 1 • 2,3, ... ,0, an
·0.
(6)
This system can only have a non-trivial solution 1f
o ,
IA - "II ~
where
- -(3u1/at)!(3u1,ax).
The n solutions
(7) are just the eigenvalues A(i) of A, dU/du
1
80
~
(1)
to the algebraic equation
that when ~ • ~(i) the vector
must be proportional to the right eigenvector rei) of A corresponding
to A(i).
As system (5) is assumed to be hyperollc, there will be n distinct
eigenvectors r(i).
The fact that ~(i) • A(i) then implies that along the
family of characteristic curves C(i) dx
dt
[:~l] / [::]
). (i)
-
(8)
or. equivalently. that au! - - dx
ax
au!
+ -at- dt •
0 along each member of the
e
( ) i
family,
thereby showing that ul(x,t)
•
const. along each member of the e(i) family.
(9)
A corresponding result applies along each of the n different families of characteristics e{i). Consider now the k-th such family and let us determine the nature of the generalised simple wave that is associated with it.
The fact that the proposed
41
generalisation of
8
simple wave region allows a corresponding generalisation
of the notion of Riemann invariants will emerge from the fact that we will
find that we are able to determine the form of these generalised invariants. Setting i • k in (9) shows that u • const along the C(k) family. 1 fact. taken together with U • U(u ), then shows U 1 k-th family of characteristics.
a
This
const along members of the
As A - A(U) and U • const along any C(k)
characteristic we conclude that A(k) • const, thereby proving that the C(k) family comprises a family of straight line characteristics.
Now provided
attention is confined to continuous and differentiable solutions, system (6) may be written in differential form by replacing
dU/d~ by
dU, when the fact
that in the C(k) family of characteristics dU!du
eigenvector r(k) establishes that dU
~
is proportional to the right I r(k) along each member of the family
of straight line characterist1cs c(k). This result gives rise to the set of n differential equations
dU
2 --:--w • r
(10)
2
in which r (k)
1
• r 2 (k) • ...• r n (k) are the elements of the eigenvector r(k).
These n first order ordinary differential equations determine the behaviour of the solution U across what will be called a generalised wave.
~(k) - simple
When integrated. (11) will give rise to n-l linearly independent
relations between the n elements of U. though multiplication of (lQ) by an integrating factor m(u • u , ••. , u) might be necessary because of the fact 2 n 1
that r(k) 1. only determined up to an arbitrary multiplicative factor. These 0-1 invariant relations aloog the k-th family of characteristics
vi11 be denoted by (11)
J (k)(U) - const.for 1.1.2 •••• ,0-1. 1
They will be called generalised A(k)_Riemann invariants to make clear that (k)
they are associated with the k-th family of characteristics C (k)
relations hold throughout the generalised A
•
These
-simple wave region where they
determine the behaviour of a continuous and differentiAble 80lution.
42 On occasions, when deriving the generalised
~(k)_Riemann invariants
from (10), it 18 useful to express them in terms of a parameter
~
by writing
(10) in the form
----m- ----m- . dU
dU l
r
•
r
1
The u
.
du
2
0
r (k)
2
(12)
de •
0
may then be determined in terms of ( by integrating the system
i
•
for
Elimination of
j •
1,2, •.• ,0
(13)
•
t between these n equations gives rise to'the 0-1 generalised
,(k)-Riemaoo iovariaots (11). Definition (Reducible System in Generalised Sense) The system
in which U is an n x 1 column vector and A is an n x n matrix will be said to be reducible in the generalised sense if the elements of A depend explicitly only on U. Definition (Generalised Simple Wave Region) Let the system DC
+
AU
0
x
in which U is an n x 1 column vector and A 1s an n x n matrix be reducible in the generalised sense.
Then any region S in the (x, c)-plane 1n which the
solution vector U is of the form U - U(u ), with u j
U~
j
one particular element of
will be called a generalised simple wave region. The following theorem 1s easily established from the previous results.
Theorem 1 (Generalised Simple Wave Regions) Let the system U
t
+
with U an
AU
0
X
0
x
1 column vector aod A an n x n matrix be reducible in the
generalised sense.
Then if S is a generalised simple wave region:
43
(k)
(a)
there 1s a family of straight line characteristics C
traversing S;
(b)
the solution vector U 1s constant along members of the C(k) family;
(c)
in S there will be 0-1 generalised A(k)-Riemann invariants J (k)(U) i
const for 1 • 1,2'0'0,0-1 which will be determined by integrating equations (10).
3.
Exceptional Condition and Genuine Nonlinearity
It is now appropriate to introduce
two
related concepts in connection
with first order quasilinear hyperbolic systems.
These are the notions of
a solution which is exceptional with respect to a particular characteristic field, and of a system which exhibits genuine nonlinearity with respect to a characteristic field.
Although these ideas may be introduced without
reference to generalised simple waves it will be convenient to use this approach here and to remove thts restriction later. For our starting point we take a generalised .\ (It) -simple wave region and the associated generalised .\(k)-Riemann invariants J (k)(U) • const i for i '. l,2, ••• ,n-l. Each of these invariants defines a manifold in the the i-th of which J (It) must obey the l 2 ... , un)-space, on 1 constraint condition dJ (It) - O. or 1 (U , u '
3J(k)
+ --! du 3u n
o.
(14)
n
Now In a generalised .\(k)_S1mPle wave region we have from (13) that dU
j
•
r (k)dt, so that (14) Is equivalent to the condition j
(V
J
u 1
(k))r(k)
0 •
(15)
These orthogonality conditions for the (V J (k» with u 1 respect to the right eigenvector r(lt) associated with eigenvalue .\(k) of A
with i · 1.2, ...• n-1.
were the ones used by Lax to define generalised .\(k)_Riemann invariants.
He
then used this definition to establish the properties of solutions in a generalised simple wave region that are given in our Theorem 1. Before proceeding to our main objective let us first use condition (15), together with an argument due to
Friedri~hs,
to prove that the solution
44 adjacent to a region of constant state must be a generalised simple wave region.
This result which might have been conjectured from Theorem 1 will then
complement the results of that theorem. First we notice that from Theorem 1 it follows that 1f a region A of constant state exists in the (x,t)-plane, then it will be bounded by a characteristic, say by a member C of the C(k)-family.
Any region adjacent to it will also be
bounded by this same line C. Now
pre-multiplication of system (5) by the left eigenvector t(j) of A
gives the system along the C(j) family • for j - 1.2 •••. ,0. 90
(16)
As the left and right eigenvectors of A are
biorthogona1~
that
o
for j
~
k
it follows directly frem (15) that the vector t(j) must be expressible as a
linear combination of the vectors (VuJ (k». i n-l
l:
sol
b
j
(9 J (k)jfor j
sus
~
Accordingly, we set
k
(17)
Equations (16) then become n-l
l:
.-1
b
js
(9 J (k»
u.
dU dj
-
0
for j
~
(18)
k ,
which by the chain rule reduces to n-l
l:
sol
dJ (k) s
b jS Cij
-
0
for j
~
k.
(19)
This is now a linear hyperbolic system involving (n-l) equations for the
(n-l) generalised l(k)-Riemann invariants J (k) l specified solution vector U the coefficients b
js
j
+k
J (k), .•• , J _ (k)
For any 2 o l will be known. The condition
ensures that the line C common to both the region of constant state and
the generalised simple wave region will not be a characteristic of the new system.
Consequently there exists a unique smooth solution that can be
continued across the line C.
Since the solution on one side of C was the
45 constant state 801utioo, the solution that is continued across it will be one for which all the generalised A(k)-Riemann invariants are constant.
Hence
from the nature of generalised Riemann invariants it may be seen that the solution adjacent to a region of constant state must be a generalised simple wave region.
This result also merits a formal statement.
Theorem 2 (Constant State and Generalised Simple Wave) Let the system
+
U
t
AU
x
•
0
with U an n x 1 column vector and A an n x n matrix be reducible in the generalised sense.
Then 1f A 1s a region of constant state in the (x,t)-plane,
the region S adjacent to it 1s a generalised simple wave region. Let us now examine further the implications of equation (15). the speeial case in which the eigenvalue
~l (k). ~2 (k) •.•.• I _ (k) n-l ,,(k)
(k)
u
"
••
- '{; 1 lJ (k) m
n-l 31. (k)
aJm(k) }
...1
aun
1 aJ (k) m
is expressible as a function of
Then we have
n l
(VI.)
~(k)
Consider
aJm(k)
n-l
~
11I"1
I
(k)
aJ (k)
aJ (k)
aU2
_31.
m
m_
,
...
,
or, equivalently,
n-l
3I.(k)
...1
aJ (k)
I
shOWing that (V A(k» u
generalised
V J (k)
(20)
u m
m
1s a linear combination of the gradients of the
~(k)-R1emann invariants.
After post-multiplication of (20) by
r(k) it then follows directly from (15) that (21)
In general, when a quasi linear hyperbolic system exists for which property (k)
(21) is true with respect to the k-th characteristic field C
associated
with A • A(k), the system will be said to be exceptional with respect to the k-th characteristic field.
This will be true irrespective of whether or not the
46
system permits generalised simple wave solutions.
When (21) is not true, the
system of equations will be said to be genuinely nonlinear with respect to the k-th characteristic field C(k).
Expressed differently, condition (21)
asserts that when a system 1s exceptional with respect to the k-th characteristic field, the directional derivative of
r
(k)
Is zero.
~(k)
in the direction of the eigenvector
We now formulate these ideas generally, without reference to
generalised simple waves or to Riemann invariants. Definition (Exceptional Condition and Genuine Nonlinearity) Consider the quasl1inear hyperbolic system
where U is an n x 1 column vector. A - A(U,x,t) Is an n x n matrix and B - B(U.x,t) is an n x 1 column vector. (a)
Then the system will be said to be:
exceptional with respect to the k-th characteristic field if
o ,. (b)
completely exceptional if it is exceptional with respect to each of the n characteristic fields corresponding to ~~l), A(2), ••. , A(n);
(c)
genuinely nonlinear with respect to the k-th characteristic field if
References [1]
[21 [31 [41
Jeffrey, A. Quasilinear Hyperbolic Systems and Waves, Research Note in Mathematics 5. Pitman Publishing, London, 1976. Coulson. C. A., Jeffrey, A. Waves, 2nd Ed., Longman, 1977. Friedrichs, K. O. Nichtlineare Differenzialgleichungen. Notes of lectures delivered at G8ttingen. 1955. Lax. P. D. Hyperbolic Systems of Conservation Laws II, Comm. Pure App1. Math. 10 (1957), 537-566.
47 Lecture 4. The Development of Jump Discontinuities 1n Nonlinear Hyperbolic Systems of Equations
1.
General Considerations We shall consider initial value problems leading to the propagation of a
wavefront in quasi-linear systems of equations of the form (1)
where U Is a column vector with the D components u 1 , u 2 ' "0' un' A Is an n x n matrix and B Is an n element column vector; A and B are assumed to depend on x, and
80
t
and U.
the system (1) will be considered to be hyperbolic
all the eigenvalues of A are real and A possesses a full set of
linearly independent eigenvector••
The left eigenvectors of A, l(i,k) with k· 1,2, ••• the eigenvalue XCi) with multiplicity ~(1)t(1,k)
,
k
,8
corresponding to
satisfy the equations
8
•
(2)
1.2"'0,8 •
They may be used to display the equations (1) in characteristic form and to introduce the n characteristic curves C(i)
8S
follows.
Pre-multiply
equation (1) by t(i) and, assumdng for the moment that the n eigenvalues of
A are distinct, we obtain n equations written in characteristic form which, by virtue of (2), become
(1 operator
a + at
,(i)
A
a ax
1,2, ••. ,n)
in the ith equation represents
differentiation along the ith characteristic curve C(1) determdned by
• C(1) .• dx dt
~(1)
•
(4)
We shall be concerned later with the propagation of a disturbance or wave
into a state which i8 either known (and non-constant) or is cODstant, when the line bordering these two states, the wavefront, is determined by a relation of the form
+lx,t)
•
O.
(5)
The wavefront • • 0 1s assumed here to be a line across which the solution
48
U 1s continuous but across which the normal derivative of U is discontinuous. The class of 80lut100s U considered 1s thus Lipschitz continuous with exponent unity.
2.
The Initial Value Problem Consider the system U
+
t
AU
x
+
o
B
(6)
subject to the initial condition
U(x,O)
•
t(x).
(7)
where t(x) 18 Lipschitz continuous. Using Haar' B a priori estimate and a special iteration scheme it may
be shown that while the solution on the wavefront remains Lipschitz continuous the solution of (6) and (7) on the wavefront depends boundedly on the initial values and on the inhomogeneous term.
Accordingly, when the advancing wave-
front ceases to be Lipschitz continuous U will attain a bounded value, say V ' behind the wavefront while ahead of the wavefront U will have a value
c
appropriate to the state into which the wave is advancing. critical time t
c
and at some critical distance X
o the
Thus at some
solution ceases to he
Lipschitz continuous on the wavefront and a finite jump or shock like discontinuity appears in U with magnitude U - U. c
We shall now obtain exact
analytical expressions determining the initial time t
c
and the critical
distance xc.
3.
Time and Place of Breakdown of Solution We start with a general quasi-linear hyperbolic system
o
(8)
and assume that the vector B and the eigenvalues and eigenvectors of A are continuously differentiable with respect to their arguments.
We also suppose
that A and B do not depend explicitly on x and t and so there exists a constant solution U satisfying the equation
o
o)
B(U
(9)
0 •
tbt. conataDt state will
oe
denoted by the subscript 0 and we shall consider
49 a wave advancing into this constant state.
The solution is Lipschitz
continuous normal to the wavefront and initial conditions may be prescribed such that at continuous at x - O.
0, U - U for x > 0 and such that U is
t •
o
Denote by
t
c
and Xc the critical time and critical
distance. respectively, at which the solution ceases to be Lipschitz continuous on the wavefront. Let us now assume that there exists at least one positive eigenvalue
of A so that the wave proceeds in the direction of the positive x-axis. We identify the velocity of the wavefront with one of the positive
eigenvalues, say
A~t), and
constant ,
introduce the curvilinear coordinates t'
constant
through the equations t'
(lOa)
t
and.
o.
(lOb)
Ftom equation (lOb) we see that
(11)
but along, • constant we may write
o and so from (11) and (12) we see that the
(12) ~
• constant lines are characteristics
and 80 dx dt
-
~(.) along • • constant.
Thus, since equation (13) is only valid along
~ c
constant and from (lOa) we
have t' • t, equation (13) 1s identical with (13' )
which 1s a result that will be required later. As • is a solution of (lOb) we must specify it by giving initial
conditions.
These should reflect the fact that it is a coordinate variable
50
and so should be assigned monotonically.
We choose
,(x~t)
by imposing the
in1tial condition Hx,O)
x
when the wavefront 1s given by Hx, t)
0
and, in the region of constant state ahead of the wavefront, ~(x,t) >0 •
The transformation introduced through equations (10) is non-singular provided the Jacobian
• ..!... ~x
1s non-zero and finite.
(14)
The initial condition on • ensures that x. 1s initially equal to unity and so we may assume the non-vanishing of the Jacobian for at least a finite time after
t
s
O.
Let us denote by L the open region lying to the left of the
advancing wavefront C(x,t) · 0 and bounded on the left by the characteristic ~(x.t)
- 0 also issuing out of the origin and chosen so that no other
characteristics enter t.
Then. since no characteristics enter L, U will
remain smooth in L for at least a finite time.
All subsequent limiting
operations on the side f < 0 of the wavefront will be assumed to be performed in L.
Let l(j) be the left eigenvector of A corresponding to the eigenvalue ;\ (j) then, frOID equation (3)
o .
I::
I
Employing the identities
<{l=O and
..l.. =.!t..!. + 11'..l.. 3x
- ax 3+
we see that
3x at'
o
$1 Thus. using these results and equation (lOb) together with condition (14) to ensure the non-vanishing of the Jacobian. we obtain l(j)
[x ~ .1...at'
+ (.{J) -
),(~».1...J
U+b·(j)
a~
In particular, if A(j) • A(~), we have
o •
1,2, ••.•
k
o .
x~
(15)
r.
(16)
where r. is the multiplicity of the eigenvalue A(+). By virtue of our choice of coordinates the wavefront • • 0 1s a
characteristic, and since the solution is Lipschitz wavefront, jump
cont~nuous
across the
discontinuities in derivatives with respect to • may take
place across. - O.
Accordingly we define the jumps across
U is continuous :
[U
J
~.o
o or
~.O+
VCO,t')· U o
~
- 0 as follows:
(constant)
o
U ,1s continuous t
.n(t')+o·
U. is discontinuous
and x. is discontinuous
X(t')
+0
We note that since both n and X depend on • they are not independent and we shall later determine their precise relationship [see equation (21)]. From the definition of X we see that X + (x~)~_O+
• x 0 (say) is finite.
(xf)~~O+
•
(x~)~.O_,while
Hence, in a neighbourhood of the wavefront
~
condition (14) is seen to be equivalent to the condition X + x 0 1s finite and non-zero.
(14')
~
The significance of the non-vanishing of the Jacobian may easily be seen by noting that in L and along
• 0 we have
~
whence
So, if
x~
vanishes while
U, remains
finite, U ceases to be Lipschitz continuous
and we have the gradient catastrophe.
52
o•
In the simple case that U
constant the jump conditions on n and X
reduce to
nCt') and
(17)
X(t') .
So, since t(j) is assumed to be continuous across the wavefront and U , e is continuous across the wavefront with Uti
5
0 in the constant region we
have, by using (9) and by considering equation (15) at a point P in Land letting the poioe tend to a point on the wavefront, that
J
r~l'
... ,
(18)
D
where again the subscript 0 signifies the constant stste appropriate to (9). We now differentiate equation (16) with respect to ; at point P in L to obtain
or,
where T denotes the. transpose operation and where Q 1s the gradient operator u with respect to (u ' u ' .•. , uu)-space. 1 2
Again, letting P tend to a point
on the wavefront and using the fact that Uel Is continuous across the wavefront with CUt') ... 0+
"(~,k)n
"'0
t'
'S::
0 we obtain the equation
+ (V b(~,k» u
0
n •
0
•
k
-
1.2 •...• r~
If. now. we differentiate equation (l3') with respect to ~ at a point P in
(19)
L
we obtain
a
~
(;~, )
(9 A<+» u
u~
and so,
a at'
(x~)
(V A<+»U u
•
(20)
53 Thus, again letting P lend to a point on the wavefront, and using (17) we find that (21)
Since the
i~j) are linearly independent vectors we may use equations
(18) to express (n-r.) components of n In terms of a certain r. components of
n.
Introducing these expressions Into equations (19) leads to r. first order
ordinary differential equations with constant coefficients for the r. unknowns, say IT , n , .•. , n . r 2 r~
Introducing n thus determined into equation (21) and
integrating the result we may obtain an expression for X.
However, before
doing this it is necessary to define certain limiting operations that will be necessary in the integration process. If Q is a quantity defined only in L we define the operation Q to be the
Q_ 11m
(Q),-oFor a jump quantity P depending on the state On r'->o adjacent sides of ~ - 0 we define the operation P to be the limit P = lim P t'->o taken along ~ c o. limdt
Thus, integrating equation (21) with respect to t' between 0 and T and noting that X is a jump quantity defined across operation
x -
Xjust
~
c
0 we may use the limiting
defined to obtain the result
x+ ITo (~A(" u 0
ITdt' •
(22)
This equation describes the variation of X along the wavefront
~
• 0 with
advancing time and in writing equation (22) we have tacitly assumed that th~ multiplicity r. of A(t) remains unchanged. and at t' - Tl(T
l
Should this assumption not be true
< T) the multiplicity changes, then n must be re-determined
for the interval t' > T • l
We remark here that although an initial condition
O. this limiting on U may be prescribed arbitrarily by specifying lim (U) x tx x->o operation is not in L and so in general Ux is not equal to this limit and
although on the initial line may not be prescribed arbitrarily. Equation (22) may be displayed in a slightly different form from which the critical time t
c
may be determined as follows.
By definition
54
or
x • when equation (22) becomes
x+
x
l
x +
-
+
JT (V 0
u
~(+»
0
fidt' •
However, from (14') we see that the left-hand side of ,this expression 1s
simply the Jacobian of the transformation and so is required to be finite and non-zero in order that the transformation 1s unique. critical time T • x~
Jacobian
• X
+
t
So, if there is a
at which condition (14') ceases to be valid and the
c
O. this is given by
x 0
+
o.
(23)
Geometrically the vanishing of the Jacobian 1s equivalent to the point at which the
~
• constant lines first intersect the wavefront
family of characteristics,
To determine
1 +
Jto
t
c
z
= O.
(i.e., the
constant intersect at a cusp).
in terms of U we divide equation (23) by x
c (V).(+» 0
u
~
o .
The vector n must be determined from equations (18), (19).
x.
to obtain
(24)
In the particularly
simple case that B • 0 it follows directly from (18) and (19), provided the multiplicity of value
n.
and
A'+>
is constant, that
n is a constant equal to its initial
80
n • and thus
iJx Using the definitions of X and n we see that
55
when
• x.(l + (V A('»
x
,.
u
0
Ux t')
but
and
80
U is given in L by the expression x
UIII x
+ (V A('» u
0
Ux t')
(25)
•
Thus U becomes unbounded if the denominator vanishes for some x
(V A(9»O U
=0
and
is infinite.
80 t
c
t
c
>
O.
If
it follows directly from equation (21) that X is a constant
remains finite for all
The discontinuity in this csse 1s propagated but t~.
Systems for which this property is true are
a special case of those which are exceptional with respect to the A(') characteristic field.
The general case when A, B depend on U and also explicitly on x, has been discussed in detail in [1].
t
A different approach to the problem
that involves three space dimensions and time has been described by Balilat [2,] and Chen [3].
References
{I] [2] [3]
A. Jeffrey, Quasl11near Hyperbolic Systems and Waves. Research Note in Mathematics No.5, Pitman Publishing, London, 1976. G. BoilIat, La Propagation des Ondes. Gauthier-Villars, Paris,1965. P. J. Chen, Selected Topics in Wave Prop~gation. Noordhoff J Leyden, 1914.
50
Lecture
5.
'l'be Gradient Catastrophe and the Breaking of Water Waves in
a Channel of Arbitrarily Varying Depth and Width 1.
Basic Equations To illustrate the gradient catastrophe in a physical context, let us
show how to obtain an explicit form for the amplitude of an acceleration wave that propagates into water at rest which 1s contained in a vertical
walled channel with slowly varying width W(x) and an arbitrarily varying depth hex) below the equilibrium water level.
The method we describe is
taken from the joint paper submitted for publication to ZAMP by the author
and J. Hvungi [1]. As usual, let the x-axis lie in the equilibrium surface of the water
in the direction of propagation, with the y-axis pointing vertically
upwards, and write the equation of the bottom of the channel as
y+
hex) -
o.
Then. if the elevation of the water above the equilibrium level is n(x.t). g is the acceleration due to gravity and the x-component of the water velocity is u(x.t), the equation of motion in the x-direction is as derived by Stoker [2]. namely
o.
(1)
However. the equation corresponding to the conservation of mass will now be different on account of the widtb variation of the channel.
To derive
it. all that 1s necessary 1s to observe that tbe cross-sectional area S(x.t) of the water at any given place and time (x.t) is S(x.t) and that the flow through this area is S(x.t)u(x.t).
~
W(x)(n(x.t) +
hex»~.
Thus, equating the
time rate of change of S to the negative flux through it. we find -(Su)
x
(2)
•
froll which it follows that
"t
+ [u(n+h)] x + u(n+h)(Wx/W)
o
(3)
The governing equations for flow in a variable width channel of arbitrary depth are thus equations (1) and (3).
The assumption of a slow
variation in the width is necessary because tbe transverse movement of the
57 water has been neglected in these one-dimensional long wave equations). and
this will cease to be a good approximation if the width changes too rapidly. 2.
The Bernoulli Equation For The Acceleration Wave Amplitude
Suppose the wave moves 1n the direction of increasing x, starting from x • 0 at
t
-
0, and that it moves into water at rest.
Then, across
the wavefront: (1)
u and n are continuous, with u(x,t) • n(x,t) • 0 ahead of the advancing wave,
(11)
the first aod second derivatives of u and n suffer at most a jump discontinuity, so that the wavefront being propagated on the
surface is an acceleration wave. Usiog a superscript minus sign to denote the value of a function immediately behind the advancing wavefront (i.e. at the edge of the disturbed region) we conclude from (i) that n
U
o
(4)
Taking the total differential of equations (4) gives, just behind the W8?efront, -0
• u;dx + u~dt aoo
0
or, equivalently,
• -eux and nt
ut where c
E
(5)
-en x
dx/dt is the speed of propagation of the wavefront which is, of
course, a characteristic curve for the system (1), (3). Immediately behind the wavefront (1) and (3) become
o
and
n~ + hU; -
(6)
0,
where it 1s understood that h • hex) is the depth at the wavefront. n~ ~
0 equations (5) and (6) imply the standard result c
2
If
• gh.
Now define the amplitude of the acceleration wave to be a
sex)
- nx
(7)
58 when (5) and (6) become and
u z
gale.
(8)
Now notice that the operation of differentiation with respect to x along the characteristic followed by the wavefront, behind which u; and u
t
are defined, takes the form (9)
It then follows immediately from this that
e
2
u
xx
- u
(10)
tt
To obtain the differential equation governing the behaviour of the
amplitude a of the acceleration wave we first differentiate (1) partially with respect to
t
and (3) partially with respect to x.
Then eliminating
n ' and using (7) and (8), we find xt
c 2U - xx
2ih
U
tt
gewz]
3 2
x + - - a +~ a + (- - c W c
2
o.
(11)
Combining (10) and (11) and using (8) brings us to the required Bernoulli type equation for the amplitude a(x),
3h W] 382 da+ [ 4hx + 2~ a + 2h dx
-
o ,
(12)
in which use has been made of the fact that, as c
2
=
gh, we have (dc/dx) •
gh/2e. 3.
The Amplitude sex) And Its Implications The standard substitution a • b-
l
reduces the Bernoulli equation (12)
to a linear first order equation, and a simple calculation then shows that (13)
in which a
l(x)
O
o•
• a(O). W
3h 3/4 w 1/2 o 0 2
W(O) and
(14)
S9 A wave of elevation corresponds to to
8
0
>
8
0
<
0, and a wave of depression
The wave will be said to break if for some x • x
O.
c
the water
surface behind the wavefront becomes vertical, so that the amplitude
sex) c
(a)
Since lex)
.~.
~
0, we conclude from the form of (13) that:
A wave of elevation (sO.
0) 1n a variable width channel always
<
breaks in water of finite depth provided lex) 1s such that 1 + 0, and Xc
>
0 1s finite.
8
0
I(x ) • c
If the depth of the water shelves to zero
at x • i, say, so that h(t) • 0, a wave of elevation propagating towards the shore will break before reaching the shore line if laol (b)
>
1/1(1), and at the shore line if laol ~ 1(1).
A wave of depression (8
0
>
0) in a variable width channel can only
break 1£ the depth of the water shelves to zero, and then only at the shore line provided 1(1) When we set W(x)
= We'
< -.
these general conclusions agree with the
special case of waves climbing a beach that was studied by Greenspan [3]. This is because the
one-dimensi~nal
long wave equations do not distinguish
between flow in a parallel channel and unrestricted one-dimensional flow. Result (14) shows that the integrand of importance in this case combines the depth function hex) and the width function W(x) in the form (h(x»-7/4(W(x»-1/2.
Thus any modification of the depth and width that
leaves this combination invariant will lead to the same conditions for breaking provided a o' he and We are unchanged. As
special cases of results (13) and (14) we observe first that in
a parallel channel of constant depth h we obtain Stoker's result [2], that breaking occurs when 2h
-~o
at a time t
2
c
-~o
Secondly, when hex) • h - mx so that the bottom has a constant slope, we obtain Jeffrey's result [4] that breaking occurs when
(15)
60
(16) Result (16) also shows that when the water deepens at a constant Tace (m
< 0),
then although a wave of elevation will normally break, this will
not occur in the special case that 28
0
- m.
This was the result found in
[4] which used the transport equation approach that has been presented in a
general form in [5].
The equivalence of the method used here and of the
seemingly different one used in [4] and generalised in [5] has been established by Bol11at and Ruggeri [6].
References [1]
A. Jeffrey and J. Mvungi, On the breaking of water waves in a channel of arbitrarily varying depth and width. ZAMP (submitted for
[2] [3]
J. J. Stoker, Water Waves. Wiley-Interscience, New York, 1957. H. P. Greenspan, On the breaking of water waves of finit~ amplitude on a sloping beach. J. Fluid Mech. 4 (1958), 330-334. A. Jeffrey, On a class of noo-breaking finite amplitude water waves. Z. angew. Hath. u. Phys. (ZAHP), 18 (1967), 57-65. See also Addendum~ A. Jeffrey, Z. angew. Math. u. Phys. (ZAMP), 18 (1967), 918. A. Jeffrey~ Quasilinear Hyperbolic Systems and Waves, Research Note in Mathematics 5, Pitman Publishing, London, 1976. G. Boillat and T. Ruggeri, On the evolution law of weak discontinuities for hyperbolic quasilinear systems. Wave MOtion 1 (1979), 149-151.
publication.
[4]
[5] [6]
Lecture 1.
6.
Shocks And Weak Solutions
Conservation Systems and Conditions Across a Shock
In what follows it will be assumed that the system of equations involved is hyperbolic and capable of expression in the generalised conservation form.
That is, when the system involves n dependent
and is formulated in
m3
x
variabJ~s
t, we assume it can be written in the divergence
form
aF at + with U
&
div G
F•
U(~,t),
vectors and G •
UI
B• F(U,~,t)
G~,~,t)
and H
&
H(U,~,t)
an n x 3 matrix.
all n element column matrix
The matrix G in (1) is in
effe~t
to be regarded as a tensor so that div G has the meaning 3
div G
8-1 I
*
a CsI 8
where 8(s) is the s-th column of G. Systems of this type are of considerable importance because of their frequent occurrence in physical problems where they arise from integral formulations of quantities that are conserved.
Indeed, since an integral
formulation is more fundamental than the related differential equation and it permits the integrand to be discontinuous, we shall make use of it to discuss discontinuous solutions for system (1). Discontinuous solutions have considerable physical significance, since they may be interpreted in terms of physical phenomena such as a shock wave in a gas.
If a discontinuous solution exists across a surface, the first
problem to be resolved is how the solutions on adjacent sides of the surface are to be related one to the other and to the speed of propagation of the surface.
In the case of a shock wave in a gas this involves determining the
relationship connecting gas pressures and densities on opposite sides of the shock with the speed of propagation of the shock. Theorem 1 (Integral Rate of Change Theorem) Let F be an n x 1 column matrix with elements· which are continuous scalar
62
functions of position and time defined throughout the volume Vet), which is itself bounded by a surface set) moving with velocity
Then the rate of
~.
change of the volume integral of F is given by
d
dt
of
FdV JV(t)
fV(t)
at
+ J
dV
F .!.d§. S(t)
where d! is the vector element of surface area. Let us now identify the column matrix F 1n Theorem 1 with the n x 1 column matrix F in system (1) and assume that a surface
a(~.t)
• const
exists acrOBS which the matrix vector U, and hence F. G and H are discontinuous. Next we choose the volume Vet) bounded by surface set) moving with velocity 90 that an arbitrary part SO(t) of the discontinuity surface divides it into the two sub-volumes V+(t) and V_(t).
o(~,t)
~
• const
Denote by S+(t) and S_(t)
those parts of Set) that bound V+(t) and V_(t), respectively, excluding the dividing surface So(t) which, we assume, also has velocity Integrating (1) over Vet) - V+(t)
I
v+uv_
~:
dv +
I
v+uv_
d1v G dV
uv_ (t)
fv+uv_
H
~.
gives dV
or, from the matrix form of the Gaussian divergence theorem applied separately to V+ and V_ in which F, G are continuous and differentiable,
of
f V uV at
+ -
dV
+f
G. d§.
(3)
S uS
+ -
where G.dS denotes the scalar product of G now regarded as a tensor and vector dS.
Combining (3) with the result of Theorem 1 applied separately to V+ and
V then gives the next result 1n which. it must be remembered, the dividing surface SO(t) that is part of
a(~,t)
- const also moves with velocity
~
(4 )
If, now, we subtract from (4) the corresponding expressions integrated over the separate .volumes V+(t) and V_(t), and bounded, respectively, by S+(t)USo(t) and S_(t)USo(t) we arrive at the result
63 (5)
where d.4 and dE. _ are the outward directed surface elements with respect to
the volumes V+(t) and V_(t).
This situation is illustrated diagramatical1y
in the figure which shows an arbitrarily thin volume element taken across a(~.t)
• const.
The effect of differencing to obtain (5) Is to make the
volume contribution and the contribution due to the surface element
directed
along~'
d~'
parallel to a(x.t) - const vanish In the limit as the
cylinder collapses onto the area element dS ' O
,,-
r1
Vo1ume element divided by discontinuity. surface d~
Since
a(~,t)
• const.
are both normal to the same discontinuity surface a(x.t)
but are oppositely directed so that n ..
-0
we have d.4 • ...dS
". !!dS
O
c
const.
showing
that (5) may be re-written as
o. The fact that dS
O
(6)
is arbitrary then gives an algebraic jump condition across
o (.!' t) • const of the form (7)
It is useful to re-express this result by observing that the scalar quantities
.!.t-0.!!.
d4 and dS
and .!._ o.!!. are the normal speeds of propagation of the elements
on opposite sides of, and moving with. o(~,t) - const, and as such
must be continuous across SO(t).
So writing ~ .,
.!+'E. - .!._ o!!
enables the
jump condition (7) to be expressed in an alternative form using the speed ~ normal to S
64 (8)
which 1s sometimes written
~[FJI •
[Gl~.'!.
(9)
with [Q]J denoting the jump 1n Q across discontinuity surface SoCt).
The
arbitrary nature of dS O also implies that U+ varies continuously over S. Because of the similarity of (8) to a corresponding condition 1n gas dynamics this result will be called the generalised Rankine-Bugoolat condition for system (1).
In general,
~
1s
~
equal to a characteristic speed A.
Theorem 2 (Generalised Ranklne-Bugonlot Condition) Consider the conservation system
;~ + div with F
a
G
-
F(U,~,t)1
B,
G~
G(U,~,t)
and H •
H(U,~,t).
Then, if this has a
discontinuous solution across a surface 8. on the adjacent sides
± of
S the
solution varies continuously and 1s related by the jump condition
in which n 1s the normal to S and j 1s the normal speed of propagation of S, with C+ regarded as a tensor and unit vector
~
G±.~
denoting the scale of prodyct of G and the
normal to S.
Definition (Shock Solution) A discontinuous solution to a system of equations expressed in conservation form which satisfies the generalised Rankine-Hugoniot condition will be called a shock. 2.
Weak Solutions and Non-Uniqueness In the development of the concept of a solution to a quasilinear hyperbolic
system, care has been taken to distinguish between classical once differentiable so called
c1
solutions, and piecewise differentiable
cl
solutions separated
by shocks across which both U and its derivatives are discontinuous.
It
would be desirable, 1f possible, to unify these two types of solution by generalising the whole concept of a "solution" to system (1) in such a way that strict differentiability and continuity are no longer required.
This is
65 precisely the motivation underlying the notion of a weak solution.
For
simplicity. the argument that follows will be confined to a scalar equation. but the extension to a system may be made without requiring any essentially
new ideas. For our startins point we take the equation
au + feu) au • o.
at
(10)
b.
sub1ect to the 'initial condition u(x,O)
(11)
g(x)
and assume that feu) is a continuous differentiable function of u.
Then the
first point to notice 1s that (10) can be expressed in conservation form by
defining F(u)
(12)
If(U)dU •
to obtain (13)
Let us consider the half-plane
t
> 0 and recall that in general a
unique solution to (10) and (11) will only exist for a finite time.
As we
have seen in Section 1, a conservation equation possesses discontinuous solutioDs or shocks, corresponding to a non-unique solution along an arc.
Accordingly,
and with reference now only to a general function f and initial condition g, let us consider some strip 0 <
t
1
< T in which the classical unique C
solution
exists everywhere except on certain shock lines across which the solution is bounded.
Adapting the notation of Section 1 we denote by u
and u+ the
limiting values of u to the left and right of the shock under consideration, which from Theorem 2 are seen to vary continuously along the shock. Then the bounded function u defined In the half plane t
:>
0 will be
called a weak solution of (10) 1f in this half plane it satisfies the condition
o •
(14)
66
for every twice continuously differentiable function w(x.t) that vanishes outside some finite region in the half plane t > O.
Such functions ware
called test functions and the closure of the region in which they are zero 19 then known as the support of the test functions.
000-
As a general
1 classical C solution to (10) subject to (11) has been found. we already 1
know that if a weak solution satisfying (14) is also piecewise C , then it
must be a classical solution wherever it Is ~. solution coincides with a piecewise
cl
1 Thus a piecewise C weak
classical solution, as would be
expected of any reasonable extension of the concept of a solution. Let us now show that there is a further common property shared between weak and piecewise
c1 classical solutions. This is that a piecewise Cl weak
solution satisfies the generalised Rankine-Hugoniot condition across a shock. Consider the region R bounded by the closed arc oR and traversed by the line L across which a shock occurs.
Denote the two sub-regions so defined
by R_ and R+ and their boundaries by 3R_ and 3R+ 1 and let the directed arcs along adjacent sides of L be oL_ and 3L+. as in the Figure.
t
L
Shock line L dividing R Then R - R_ UR+ and oR • 3R_uoR+.
The test functions w in (14) will be
assumed to have their support in R
80
on oR.
that the test functions w will vanish
Thus (14) may be written
II!:;
u
+ :: F(U») dxdt
o
(15)
67 Now multiply (13) by wand integrate over R_ to obtain
JJ
_ R
(w :~
+
W
:~)
dxdt
-
O.
which may also be written in the form
a(wu) + a(WF») [ at ax
dxdt -
II
dxdt
•
o.
(16)
R
Applying Green's theorem to the first terms in this result then transforms (16) to
I -wFdt 1aa vaL
o.
+ wudx _
(17)
However as the support of the functions w lie in R, w will be zero on itR
80
thOt (17) reduce. to
1 -wF(u_)dt
TaL
+ w u_dx
(18)
A similar result applies with respect to R+ where we find
taL-
F(U+) dt +
W
+
u+dx -
Ill:~
u + :: F] dxdt
-
o.
(19)
+
the integration along
at and 3L+ being oppositely directed, as indicated
in the Figure.
If (18) and (19) are now added. the sign of the line reversed with a corresponding replacement of aL
integral~ln
(18) is
by 3L+ and result (15) 1s
used we find (20)
where as the point (x,t) 1s now constrained to lie on 3L+ the term (dx/dt) represents the speed of propagation
~
of the shock along L.
As
w i8 arbitrary,
(20) can only be true if (21)
which 1s the one dimensional form of the generalised Rankine-Hugooiot condition.
This holds degenerately when u is continuous across L.
If, now, the support of w is allowed to be arbitrary, the same form of
68 argument proves that piecewise
cl
solutions of (13) satIsfying (21) across
a shock will also be a weak solution of (13).
We thus arrIve at 'the following
definition and theorem.
DefInition (Weak Solution) The function u will be called a weak solution of
o if for all twice continuously differentiable test functions w with support
In
t
> 0 the function u Is Buch that
ff"" { Jl at:
I)
aw PCu) ] dxdt + ax
o•
the integration being extended over the upper half plane
t
> O.
Theorem 3 (Properties of Weak Solutions) Let u be a weak solution of au + aF(u)
at
ax
_
o.
The following results are then true: (a)
If u Is piecewise C1 in additIon to being a weak solution it is also a piecewise C1 classical solution.
(b)
1 a piecewise C weak solution satisfies the generalised Rankine-Rugoniot condition
across a discontinuity moving with speed A; (c)
a necessary and sufficient condition for a piecewise
cl
classical
solution to be a weak solution is that across a discontinuity moving with speed l it satisfies the generalised Rankine-Hugoniot condition. The general objective when introducing a weak solution was to lift the requirements of strict continuity and differentiability that Deed to be imposed on classical solutions. solution is successful and,
In this respect the notion of a weak
furtbermore~
because of its method of definition
1 the class of weak solutions is even wider than the class of piecewise C
69 functions
80
that considerable
generality has been achieved.
However~
this generality has been obtained at the cost of the uniqueness of a weak solution.
MOre precisely. unlike a strict classical C1 solution. a weak
solution 1s not determined tmiquely by the initial data.
Th.1e is most easily
deDJostrated by means of a simple example.
Consider a Riemann problem for an equation of the form
[1 3}
au +...!. at a,,"3
•
u
0.
with
80
for x < 0
Ol'
•
ues.O)
{
for x > 0
that in (13) we have F(u) • u 313.
11leu,
the equation
a8
'!,8
ho.:)gea.eous, when it 1s differentiable a
non-constant solution u viII be a function of x/e, and it 1s easily verified that the function
for
0 u(". t)
•
(,,It)1
1 1
1s a
cl
"It for
<
°
° ~ "It ~ 1
for x/t > 1
solution subject to the initial condition.
everywhere for
t
>
This solut10n is continuous
0, and it is differentiable everywhere except along each
of the lines x • 0 and x • t on which, due to the continuity of u, the generalised Rankine-Hugoniot condition holds in a degenerate form.
It is a
simple matter to verify directly that this piecewise Cl classical solution 1s also a weak solution.
The form of this solution is shown in the Figure.
This
1s simply a centred rarefaction wave of the type mentioned in Lecture 3. V.
The
continuous piecewise Cl
a discontinuous initial condition.
70
Another weak. solution follows by observing that
8
discontinuous function
1 that :1s a C solution away from the lines of discontinuity will be a weak
solution provided the discontinuity condition.
Let
U8 8ee~
~atisfles
an even simpler
for x/t
w~ak
the generalised Rankine-Euganiot
solution of the form
< It
u(x,t) for x/t > It , by choosing It to satisfy the generalised Ranltine-Hugonict condition.
Subsdtution into (21) coupled with the fact that the speed of shock propaption A • It then g1ves k(l - 0)
[t - 0]
or k
1/3 .
The second weak solution 1. thus for x/t < 1/3 U(Xt
t)
for x/t > 1/3 , and this weak solution 18 piecewise constant. but is dlscontlnous across the line 3x • t.
In physical problems only one solution 1s permissible. so that if the class of weak solutions is considered some selection principle must be devised to choose a unique weak solution with the appropriate physical properties.
This is usually achieved on the basis of the stability of the
solution and leads to selection methods known as entropy conditions.
This
name derives from the gae dyna-.ic case in which both compression and rarefaction shocks are 'DIlthematically possible, though only the compression ShdCk
is physically real.:1.sable since it is only in that case that the entropy
.toe&" not decrea.se across the shock. -r.eD.U'ed rarefaction- wave nat at:able.
In the example just examined the
is the physical solution since tbe shock wave 1a
71 Some account of entropy conditions and of the associated literature is t.o be found in the work of Lax [1), Jeffrey [2] and in the paper by
Dafermos (3J. 3.
Conservation Equations with A Convex Extension
Yhen the conservation system involved 1s symmetric hyperbolic, the ideas of Section 2 may be pursued in some detail without giving rise to
undue
This we do now, basing our approach on the paper by
dl£flcul~y.
Friedrichs snd Lax (4J. Consider a system of conservation equations
au + 3G at ax
•
0
(22)
•
with U and G • G(U) each n x 1 vectors and integrate it over an arbitrarily large interval [-a,a] of the x-axis.
Integrating the second term by parts
then gives rise to the equation
I'" l!1 J_ at
Ixl.
so that G(± a. t) .. 0 as a + ., we see from the above reault
and the degenerate form of Theorem 1 that
showing that the integral
is a conserved quantity because it is independent of t. the problem we now consider is, when is a new conservation system
o•
(23)
witb V. K functions of U. a direct consequence of the original law (22).
To resolve this we need to make a direct comparison between (22) and (23) .0 that fi.h:t
we perl'or.
bec6iie:. ru~ct1ve1y.;
the indicated differentiations, when these equatl.ontl
72
i!! + (v
at
au _
G)
a;
u
0
(24)
and
au + at
(V V)
u
(V K)
u
au • o.
(25)
ax
Employing the summation convention, the j-th component of (24) may be written
5
+~
at
aUt.
aUt ax
o •
(26)
while equation (25) itself becomes
~~+~~ aU at aUt ax
•
j
(27)
0 .
Consequently, comparing (26) and (27), we conclude that (25) will be a consequence of (26) only if
(28) Let us now assume that this condition is true. Bnd differentiate it with ~.
respect to
when we find
~ [a [av]] + av,u j [~] 3u t a"h ~ a"h ,u t The second term on the left hand side and the right hand side are both symmetric in 1. and h,
BO
that the first term must also be symmetric.
We
have thus shown that if (28) is true, then
~ ,u t [,'"h [,v]] 'U j - ~ '"h [,'ut [av]] aU j If, now, we multiply (26) by a
,2
v 5 at
~Uj~
2
V/aUja~
auja"h aUt aUt ax .
+~ ~
(29)
and sum with respect to j we find
o.
(30)
2 This will be equivalent to (22) if the matrix {a V/auja~} is non-singular. and we here take note of the fact that system (30) is symmetric.
Hence~
whenever (22) 1. hyperbo11c~ and (28) is true~ the equivalent system (30) will be
8yaaetric hyperbolic.
It can be shown that 1nitial value problems
73 for symmetric hyperbolic equations are unique and will exist in some
neighbourhood of the initial data. As the hyperbol1clty of (30) implies that the matrix {a
2
v/aUja~}
is
positive definite we may assert that V 1s a convex function of the elements ~
and so arrive at the following conclusion.
Theorem 4 {Uniqueness Theorem} If the system of conservation equations (22) is such that it implies a new conservation equation (23) with the property that the new conserved
quantity V is a convex function of the original elements u ' u ' 1 2
"0'
un
of U, then the initial value problem for (22) has a unique solution in the neighbourhood of the Initial time. ~.
We~k
lnteraction of
Di,continuities
We conclude this lecture by adding a few remarks about the interaction
ot ~
weak
discontinui~y
propagated along a characteristic C($) and a shock.
Here we use the term weak to refer to a solution which is continuous across
C(~)
though its normal derivative is discontinuous.
This is in contrast to
the strong discontinuity pf the Rankine-Hugoniot type where the solution itself is discontinuous.
This situation is illustrated in the Figure where D is the
shock line corresponding to a conservative system of the form
A(U)U + B(U) x
o
~
p
v.. ~d--....t,=~~~--'----------<~>c
o
In general at
~where c C+) meets D. there will lie r characteristics of
the system to the right of D entering the state U+ and back into
~he
state
)lJ~ •
5
to the left reflected
74 By writing down t.he transport equation for the incident weak discontinuity
along C(+). as in Lecture 4~lt 1s possible to determine its nature as it approaches P from the left. 5
Then, using the fact that the r transmitted and
reflected weak discontinuities must propagate along characteristics, it
is possible to resolve the jumps across all characteristics at P in terms of the original system of equations. the differentiated Rank1ne-Hugonlot
equation across D at P and the initial discontinuities propagating along the
r +
8
characteristics together with C<+). Provided the set of equations that results at P 1s properly determined
the reflected and transmitted weak discontinuities may be determined.
Special cases arise. 11ke the coincidence of D with a characteristic on either side at p. and the fact that the .ystem . .y be except10nal with respect to one or more characteristic fields. A general account of these ideas 18 to be foUlld in Jeffrey [1]. whUe attention was drawn by Bolllat and Ruggeri [5] to tbP necessity to perturb the shock speed in ca.es where the interface D CaD .ave. References [1]
(2] [3]
[4] [5]
Lax. P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock 'Waves. SIAM Regional Conference Series in Applied Mathematics. 11, 1973. Jeffrey. A.. Quasilinear Hyperbolic Systems and Waves. Research Note in Matheaatic8, 5, Pitman Publishing, London, 1976. Dafermos, C. K. Characteristics in Hyperbolic Conservation Laws. A Study of the Structure and the Asymptotic Behaviour of Solutions in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 1. Research Note in Mathematics, 17, Pitman Publishing, London, 1977. Friedrichs, K. 0., Lax, P. D. Systems of conservation equations with a convex extension. Prac. Nat. Acad. SeL, USA 68 (1971), 1686-1688. BoilIat, G., Ruggeri. T. Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks. Proc. Roy. Soc. Edin. 83A (1979), 17-24.
75 The Riemann Problem, Glimm'. Scheme and Unboundedness of
Lecture 7.
Solutions
1.
Riema~~
The
Problem for a Scalar Equation
To illustrate ideas we consider the single equation for the scalar u already encountered in connection with weak 801utions in Lecture 6, namely:
au
+
at
1.-
o
a"
or, equivalently,
~
at
+
u
2
au ax
-
-
(1)
The Riemann problem for this equation is then the resolution of the discontinuous initial data
u(s, where
U
0)
•
and u
o
1
{uo
for
x
u
for
x> 0
1
< 0
are two artitrary constants.
More generally, it may be
extended to include a number of such discontinuities located along the ini~ial
line.
The characteristics of (1) are the curves (2)
along which the equation may be written in the form du
dt
o.
-
(3)
Rence for x < 0 the characteristics are parallel straight lines with slope
~
•
2 uo.
whereas for x > 0 they are parallel straight lines with
2
slope A • u • l If u~ < u~ these two families of characteristics diverge, as in Figure (a), when the wedge shaped region W is not traversed by any of these characteristics.
However, if u~ > u~ the two families of
characteristics intersect from the start, leading to non-uniqueness and shock formation of the type first indicated at the end of Lecture 1.
o.
76
t
lsI-it I i",e..
= ",,4,.:
(a)
I
(b)
Centred simple wave in W
Shock speed
we thus arrive at the result that the condition for a physically admissible ahock solution lor (1) i. (4)
Now (1) is invariant under tbe replacement of x and so that its solution depend. only on tbe ratio t • x/t. all pass through the origin, and along them. u • const..
the wedge shaped region W in (a), and
88
t
by ax and ~tt
The linea t • canst. They thus fill in
tbey are characteristics the wave
solution described by them iD W is called a centred simple wave.
In this
case the centre 1s at 0 which is the location of tbe discontinuity in the initial data.
Taking tbe particular case in (1)
leads
o•
U
to the differentiable solution for region W given in
Lecture 6, and illustrated there by
! 0
u(x, t)
0, u • 1 and setting u(x, t) • u(C) 1
for
(X/t) I
I
for
3
Figure:
x/t < 0
for
0 s xlt S 1
xlt > 1
Notice that the non-physical shock
(5)
(weak solution) liVeD in
Lec~ure
6,
_ly
u(x, t)
. {0 1
for xlt
<
1/3
for x/t
>
1/3
(6)
77
lies in region Wand
is
80
~
produced by the intersection of characteristics.
It is for this reason that it is not physically realisable and
80
must
be rejected.
A physical shock occurs in the situation illustrated in Figure (b) however and emanates from the origin. u(x. 0)
•
{ :
for
z
for
x> 0
Using the initial data
< 0
as a typical example, we find from the Rankine-Hugoniot condition that ~
• 1/3.
Thus in this case the resolution of the initial discontinuity
merely involves its propagation along the shock line
3x.
t •
We conclude from this that for a centred simple wave (rarefaction fan) to occur, the characteristics must diverge from a point, leaving a wedge shaped region to be filled by the centred simple wave.
A shock will only
occur when the characteristics converge and intersect. ~.
Riemann Problem for a System Let us now consider the reducible hyperbolic system
Ut subj~ct
+
Va
0
to the initial data
U(x. 0)
where
A(U) U x
{::
for
x
<
0
for
x
>
0
and Un are constant n element vectors.
(8)
The Riemann problem DOW
becomes the resolution of the initial vector discontinuity at x • 0, though as with the scalar case it may be extended to include a number of such discontinuities along tbe initial line. We look for the solution of this problem in terms of generalized simple waves and shocks, which will be the analogue of the situation just discussed for a single equation.
The generalized Rankine-Hugoniot
condition is of the form [F] •
(9)
78
once (7) has been expressed in the conservation form
~+~(u).o. at _
(10)
This implies n possible types of shock with speeds i(l)~ ••• ~i(n) p~18ical
and we shall need a
we did in the simpler case.
admissibility criterion for them, just as The extension of our earlier result (4) that
provides the criterion we need i. due to Laz who requires that for some integer k with 1 :s; It S D
while
(11) A(k-1) (u ) <
t
l:
< A(k+1) (u)
r
This condition ensures that k characteristics converge ODto the shock line from the left and n - k + 1 frOll. the right.
There is tbu' a
total of n + 1 conditions provided by characteristic. which when taken together with the
D -
1 result. that follow from (9) after
i has been
eliminated enable the determination of the 20 values taken by U on the left (t) and right (r) of the shock.
The sbock that satisfies (11) for SQae
index k is called a k-shock. Now differentiable solutions to system (7) are also expressible in terms of the ratio
~
• x/t, so tbat this system permits a generalisation
of the notion of a centred simple wave.
The general solution to the
Riemann problem (7), (8) tbus consists of n fans of waves, each consisting of shocks and centred simple waves, arranged in order of increasing k from left to right, and separated by sectors in which the solution assumes constant values. As already mentioned, this generalisatioD of the Riemann problem may be extended to the case of an initial vector that is piecewise constant along the line t • O. Gl~'8
It is this very idea that is basic to
method for the numerical solution of conservation laws (1) with
arbitrary initial data in place of (8), and it is this that forms our next topic.
79
3.
Cli.DID.' 5 Method
This is a method for the numerical solution of a system of hyperbolic conservation law,
1!! at
+
1! ax
o
(0)
(12)
with arbitrary initial data
O(x. 0)
•
.(x) •
(13)
It is a method of first order in
a~curacy
and the basic idea is to
replace the arbitrary initial data (13) by a piecewise constant approxtmation in spatial intervals of length h.
Then, until such time 88 the
centred simple waves and shocks that result from the discontinuities interact, an analytical solution to the piecewise constant approximation to the. initial data is Riemann problem.
provi~ed by
the solution to the appropriate
If this solution is used for a suitably small time step
k, a new Riemann problem may be derived from the analytical solution at time
't • k,
and thereafter the process may be repeated to advance the
solution step by step in time. The special feature in
Gl~'s
method lies in the way in which the
new Riemann problem is derived from the analytical solution.
Unlike the
averaging process over the spatial interval h used by Godunov who also employed the Riemann problem approximation, Glimm chose the constant value for each interval of length h by random sampling within that interval. Let us elaborate on this process sufficiently for its basic ideas to become clear.
First, however, we need to recall tbe notion of a
domain of dependence.
In Lecture 1, when discussing a second order wave
equation, the domain of dependence of a point P was defined as tbe interval produced on the initial line by tracing backwards in time the two characteristics passing through P until such time as tbey intersected tbe initial line. P.
Only data on this interval influenced the solution at
Now, in the case of systems (12), the analogue is to trace backwards
80 in time from a point P in the (x, t}-plane tbe D characteristics that are associated with system (12).
The interval on tbe initial line contained
between the extreme characteristics is then the domain of dependence of P, and the Figure shows a typieal example of such a situation.
t·•
o Only iDicial data lying within this interval can influence the solution at P .• 1
Suppose we now think of a domain of dependence of fixed length h, then for the ith interval there will be a time is determined by the data on this interval.
t
i
up to which the solution
If our initial data (13) is
approximated in a piecewise fashion at intervals of length h, then provided our time step k
<
inf {t , l
t
2
, •• ;,
t
n
} there will be no
interactions between the centred simple waves and shocks that will result from the Riemann approximation to our problem.
Expressed differently. if
x. - sup {x(i)} where A(i) are tbe eigenvalues of V F (X* corresponds to U-~
u
the fastest propagation speed). then we need to take
k
<
h/2)'* •
(14)
This is, of course, the familiar Courant-Friedrichs-Lewy stability condition for the numerical solution of hyperbolic equations. Raving thus generated an approximate solution at time t - k, Glimm's method then attributes to each interval of length h at this time a value of the analytical solution at a randomly chosen point in that interval. A new Riemann problem is thus generated and the process is repeated to advance the solution a further time step.
81 Symbolieally, if the subintervals are (sh, (s + 1) h) ,
(IS)
then we set •
for x ~ t
€
I.,
U (8 D
+ a
n
h, t
n+
1)
(16)
where un (x, t) is the exaet solution in tbe nth strip
t
n
~ t
+ , and {m } is a sequence of randDII mmbers bavi!1& • uaifora n n 1
distribution in the interval (0, 1). Let us use an example due to Lax to in the case of
constant atate.
two
shock moving with speed
u(x, t)
•
i,
'1.
and
ill~tr.te
bow the method works
'it ('\ > 'it)
separated by a
which baa the solution
for
x <
it
for
:z >
),t
(17)
For ease of illustration, let U8 take all time Iteps equal and denote them by k.
'l'be first step of eliDa' 8 scheme gives
for (18)
for where now
[~
if
(19) if
The reault of n such stepa with the scheme is to give
[~
for
x < J h
for
x > J h
n
(20)
n
where we have •• t
•
number of
Q
j
<
~k 11
(21)
The law of large numbers tells us, with probability 1,
+
n d
n
(22)
82
where
0(1/ In)
.
(23)
The consequence 1s that (20) differs from the exact solution (17) by the error do 1n the location of the
sbock~
though the discontinuity itself is
represented perfectly sharply as a true shock. By employing stratified sampling, Chorin has obtained more accurate results than by the simple random sequences proposed by Glimm.
4.
Non-Global Existence of Solutioos We conclude this lecture by presenting two simple examples that show
how even when a hyperbolic system 1s 1n the form of a set of conservation
laws. and it has so weak a nonlinearity tbat it 1s completely exceptional (also called linearly degenerate in 80.e papers), tbe 80lut10n itself may stl1l become unbounded within a finite time.
When this happens no extension
of the 901ution 1s possible, so that a global solution no longer exists. Consider the system proposed by Jeffrey
o
and
av + .&i!!L au • o • f(v) ax
at
subject to the initial data u(x,O)
v(x.O)
and
This is easily seen to be hyperbolic, and as the eigenvalues are ±l it follows that it must also be completely exceptional.
The characteristic
curves belong to the two families of parallel straight lines C(!) given by solving dx
dt • Defining
u•
11.
Ig(u)du and
v•
If(v)dv the system reduces to the linear
hyperbolic syste. in conservation fora
au + l! • o ax
at
and
a~
at
+ ail • o ax
The general solution i. simply
83 u(x. t) v(x, t)
F(x
+ t) + G(x - t) •
-F(x
•
+
+
t)
G(x - t) ,
with FtC arbitrary differentiable functions. We now take two special cases to illustrate the unboundedness (blow-up) of tbe 8Olution.
Example 1
2 Take uO(x) • x , vo(x) • u(x,t)
2
x + t
-
2
and
2
~l. f •
ltv
v(x,t)
• _1_.
t
g • 1.
Then it follows that
2xt-1
There 1s thus an escape time t. > 0 for the solution v(x,t) when t.
(x > 0) •
1/2x
This is not due in any way to the intersection of characteristics within a family, for they are parallel straight 11nes. initial data becomes unbounded for large x
80
However, in this case the
that it might be considered
this 1s the cause of the unboundedness of the solution.
To show this 1s
not the caBe cODsider this next example.
_"Ple 2
2 Take uO(x) • a tanh x, vO(x) • 1, f • ltv and g • 1, when we find
ttl
u(x,t)
[tanh(x
+
t)
+
tanh(x - t»)
2 2+a[tanh(x+t)-tanh(x-t)]
vex, t)
Here u(x,t) remains finite for all x,t but v(x,t) becomes unbounded at an excape time t. given by t.
-1
tanh
2 2 4 asech x + a sech x + 4tanh2x 2 tanh 2 x
In this case, by making
a
suitably small, the deviation of the
initial data from constant values may be made as small as desired, but the finite escape time still persists.
84
References
[1] [2J [3] [4J [51 [6] [7J
Quasilinear Hyperbolic Systems and Waves. Research Note 1n Mathematics,S, Pitman Publishing, London, 1976. Lax, P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Applied Mathematic8, II, 1973. Gllmm. J. Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 13 (1965), 697-715. Godunov, S. An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521-523. Godunov, S. Bounds on the discrepancy of approximate solutions constructed for the equations of gas dynamics, J. Comput. Math. and Math. Phys. 1 (1961), 623-637. Charin, A. Random choice solution of hyperbolic systems, J. Compo Phys. 22 (1976), 517-533. Jeffrey, A. The exceptional condition and unboundedness of solutions of hyperbolic systems of conservation type~ Proc. Roy. Soc. Edinb. 77A (1977), 1-8. Jeffrey~
A.
85 Lecture 8. 1.
Far Fields. Solitons and Inverse Scattering
Far Fields
There are many different types of higher order equations and systems of equations that characterise nonlinear wave propagation in RX t, either with or without dispersion.
A simplification frequently takes place in
the representation of the solutions to inltlal value problems to such equations after a suitable lapse of time or. equivalently. suitably far from
the origio, particularly when the initla1 data Is localised and compact support.
80
has
These simplified forms of solution are often asymptotic
solutions, and are appropriately called far fields. The simplest examples of these are the types of far field behaviour exhibited by the ordinary linear wave equation and by a homogeneous quasllinear hyperbolic system with n dependent variables.
Due to its
linearity, the wave equation const. )
(c
(1)
may be written either in the form
+ c..l..] [au _ cau] [..l.. at ax at ax
o
(2)
_c..l..] [au + cau] [..l.. at ax at ax
0
(3)
or as
Then. if u(+) is the solution of
o.
(4)
it follows that u(t) is a degenerate solution of (2) and u(-) is a degenerate solution of (3).
The general 601ution of (4) is then
f(±) (x"+ ct) •
(5)
with f(±) arbitrary C1 functions. These travelling wave solutions are such that u (+) propagates to the right and u(-) to the left with speed c.
We thus have the situation that u(t)
86
are special simple types of solution to the wave equation (1), in the sense that they only satisfy a first order partial differential equation, whereas the wave equation itself Is of second order.
Such special solutions
become of considerable interest when the initial data
f~t)
(+)
with compact support, 60 that £0- (x) initial data lies in
Ixl
f
1
CO'
Is differentiable
Then, if the support of the
d, after an elapsed time dIe the interaction
<
between waves moving to the left and right ceases and only the solutions
u(-) and u(+) are observed to the left and right of the origin, respectively.
These are the far fields of the wave equatton (1).
Since u(+) is transported
along the C(+) characteristics x-ct· ( and u(-) along the C(-) characteristics
x+ct • n, and neither family of characteristics intersects itself, the far fields of the wave equation will propagate indefinitely after the interaction has finished.
The situation 1s different 1n the case of the homogeneous quaal11naar hyperbolic system of hyperbolic type
~ + A(U) 3U
at
o•
ax
(6)
1n which U is an n x 1 vector with elements
UI'
u ' .•• , un and A(U) • 2
[aij(u 1 , u2 ' ••• , un)] i8 an n x n matrix with elements, depending on the elements of
o.
If, now, we seek a special aolution of (6) in which n-1 elements of U are functions of only the one remaining eleeent, aay
Direct substitution lnto 3U
[
_1 I
ot
OU
+ _1
ax
_]
A(U)
J-
~
we may set U - U(~).
ahows that
(6)
..U
UI'
•
o•
in which I is the unit . .trix.
A non-trivial solution of thi.
fo~
o • showing that if
~
1s an eigenvalue of A,
only exists when (8)
87
(9)
Since the system (6) Is hyperbolic there are n real eigenvalues A(l), x(2),
....
A(n) of A, from which it follows that when (6) Is totally hyperbolic there are n different solutions
ui
i
) satisfying
o for 1 • 1,2, ... ,0.
(10)
The solutions (10) are, of course, simple wave solutions,
and for initial data having compact support they represent the far field ~~lutions
after the interaction has finished.
The characteristic curves
C(i) in this csse are given by solving (11)
for l· D 1,2, .•. ,n. The characteristics comprising each family e(i) are again straight lines, but now they are no longer parallel within the family as the gradient of a characteristic depends on the value of the solution that 15 transported along it.
This leads to a breakdown of differentiability when members of a family
of characteristics e(i) intersect. and to the formation of a discontinuous solution at some finite elapsed time t
(i)
c
•
-
Thus the simple waves U - U(u
(i)
l
)
corresponding to the solutions u~1) of (10) can only form far fields 1n the time interval between the end of the interaction period for initial data with compact support and the breakdown time for the system t t
(2)
c
2.
• . ..•
t
(n)
c
c
- min{t(l), c
}.
Reductive Perturbation Method The far field equations discussed so far are very special, since the
equations that gave rise to them involved neither dissipation nor dispersion, and one was. in fact, linear.
In more general situations both dissipation and
dispersion may be present, and typical of the far field equations that then result are the following nonlinear evolution equations:
88 Burgers' Equation (dissipative)
a + v...:!.. a ....:! ax
at
•
v
a2v 2 ax
(\I > 0)
~
(12 )
KdV Equation (weakly dispersive) 3v + at
3 3v + a v v ax 1J ax3
0,
(13)
(\.I > 0)
Nonlinear Schr6dinger Equation (strongly dispersive) 2
i
av + 1l...Y + alvl 2v
at
2 ax2
o.
(14)
An important scalar equation that has either (13) or (14) as a far field equation, depending on the circumstances, Is the BOU8sinesq equation US) in which u(x,t) 1s a one-dimensional field, c is the phase velocity In the long wave limit and
1J
Is the dispersion parameter.
This occurs in the study of
water waves.
A very general quasil1near system that contains as special cases many of the systems that are of physical interest has the form
au + at
A(U)
a~ll
.!!!. + ax
u
= 0 •
(p ~ 2)
(16)
Here U 1. an n x 1 vector with elements u ' u ••••• un' the matrices A, 1 2 HB K6 are all n X.n matrices depending on U and B is an n x 1 vector depending a' a on U.
When wave propagation is involved, it is weakly dispersive when B • 0,
and strongly dispersive when B ~ O. We now outline the so-called reductive perturbation method due to Taniuti and Wei. referring for all the details involved to that paper or to the review by Jeffrey and Kakutani. Considering the weakly dispersive case (B • 0) we apply the GardinerMOrikawa transformation •
.+1 t
•
l/(p-l) for p
to system (16) where ,\ is taken to be a real eigenvalue of A.
~
2
It is not
(17)
89 necessary that all of the eigenvalues of A are real, but when they are.
and the corresponding eigenvectors span the space En aSBociated with A. the first order system comprising the first three terms of (16) will be
hyperbolic.
Set
(18) where V 1s a constant solution of the homogeneous form of (16) (i.e. B • 0).
o
Then, rewriting the system in terms of derivatives with respect to t aod T. and equating like powers of c, we obtain the results
(19)
+
s
l:
P
n
[ -A
S-1 a-I
as +
KS
aO
(20)
U
,P J__ l-
-
O.
aO ,tP
K: O and (VuA)O indicate quantities
u•
~pproprlate
to the solution
U ' while V denotes the gradient operator with respect to the elements O u
of U. Then 1f 1 and r denote the left and right eigenvectors of A corresponding O to the eigenvalue A. 60 that
1(AO - AI)
-
(A - AI). O
0 and
-
(21)
O.
equation (19) may be solved in the form (22)
with 41 one of the elements of 0
and V an arbitrary vector function of T. l
1
The compatibility condition fo. (20) when solving for
aU1
1-'T
+
au1
1[V1 '(V uA)OJ ,t
s
+
1
l:
P
n [_AH S + KS aU
Il-l a-l
Then taking the boundary condition U
~
V
o
as x
aU
~
J ,t..-
,tP
a02/a~
1s
-o.
CD, so that we may set
VI ; 0, we find that • satisfies the nonlinear evolution equation
(23)
o•
(24)
where
and
When p - 2 equation (24) become. Burgers' equation, and When p • 3 the KdV equation.
The scalar equation (24) thus governs the far field behaviour
of the homogeneous form of system (16) that 1s associated with the eigenvalue A.
There will be such a far field for each real eigenvalue A of A. In concluding this section we remark that although in what follows we
shall be referring to properties of exact solutions of some far field equations, it should be remembered that these far field equations are in the main only
asymptotic approximations to the solution that is of interest.
There are,
in fact, a variety of different methods by which the nonlinear evolution equations characterising far fields may be derived, and for an account of three other methods we refer to the paper by Jeffrey and Kawahara, to the Scheveningen Conference paper by Jeffrey and to the AMS paper by Whitham. Before moving on to discuss soliton solutions to equations like (24), we remark that it is a simple matter to show that the coefficient c in fact proportional to (QuA)'r.
in (24) is
This means that when the characteristic
field associated with A 1s exceptional c vanishes.
l
1
- 0 and the nonlinear term in (24)
Further analysis is required to derive the nonlinear evolution
equation that then governs the far field, and it has been shown by Jeffrey and Kakutani that a modified KdV type equation then results. 3.
Travelling Waves and Solitons We have seen that in nonlinear hyperbolic equations, waves propagate
with a shape change so that no travelling waves can exist for such
tio08.
That is to say, there is no reference frame moving with a constant sree1 s
91 in which the wave appears staionary.
However~
when dispersion or dissipation
are present in a nonlinear evolution equation, in the sense that the linearised equation exhibits these effects as described in Lecture 1, travelling waves become possible due to the competition between the effects of nonlinearity and dispersion or dissipation. In :R x t, travelling wave solutions have the form v(x,t)
v(~) •
x - at ,
•
const. ,
and, in addition to the nonlinear evolution equation, they must satisfy Bome appropriate boundary conditions at infinity.
determine the permissible range of values of
8.
In general, these will In the case of Burgers'
equation and the KdV equation which are, respectively, examples of purely
dissipative and purely dispersive nonlinear evolution equations, we find when seeking solutions for which all the derivatives tend to zero as
lxl
~~, the
well known solutions: Burgers' Shock Wave (Purely dissipative)
1-+1-+ + v.) - "l(v. - v.) tanh
"l(v.
-+
[(v. - v.)~/4vJ •
(25)
satisfying
and
s
•
1 + I(v.. + v.>
- - - - -- - - - --- - -- u""
o
(
KdV Solitary Wave (Purely dispersive) (26)
92 satisfying with v
.-
> 0
aDd 8
V
eo + a/3.
f------()..
t
_ o
The Burgers' shock wave, as the solution (25) 1s called, is seen to
+ propagate with a speed s • (v-eo + v.>/2 that 1s uniquely determined by the boundary conditions, but is invariant with respect to an arbitrary fixed
spatial translation.
In general, all solutions v(x,t) of Burgers' equation
are invariant with respect to a Galilean transformation.
This 801ut10n links
two different constant states at plus and minus infinity. The KdV solitary wave, as the solution (26) is called, is different and is a pulse shaped wave that, relative to the same constant value v. at plus and minus infinity, tends to zero together with all its derivatives as Itl~.
Its speed of propagation relative to
Veo
is proportional to the
amplitude a, and its width is inversely proportional to the square root of the amplitude.
In this travelling wave solution the speed is not determined
by the boundary conditions, but by the amplitude a > O.
Like Burgers' equation,
the KdV solitary wave 1s also invariant with respect to a Galilean transformation. zabusky and Kruskal found numerically that a KdV solitary wavj:!. behaves like a particle.
Specifically they
fo~nd
that when two different amplitude
waves of this type are such that the one with the greater amplitude starts to the left of the one with thelesset amplitude, then the larger one overtakes the smaller one and, after interacting with it, the waves have merely inter-
93 changed positions.
This is a nonlinear interaction yet the pulse shapes
are preserved exactly after the interaction, though the phases of the
pulses (the location of their peaks) is affected by this process.
On
account of this Zabusky and Kruskal invented the word "soliton" for a wave
that preserves its identity exactly in this sense after a nonlinear
interaction.
Thus KdV solitary waves are solitons.
The recent interest in solitons derives from the fact that the KdV equation is often found to arise as a far field equation and, furthermore, arbitrary initial data for the KdV equation evolves into a train of solitons together with, possibly, an oscillatory tail.
This means that solitons are,
in a sense, fundamental solutions of the KdV equation.
An
extensive
literature now exists on this topic, and we refer to the articles and to the references contained therein, in Jeffrey and Kakutani and in the various articles by Kruskal, Lax, Ablowitz, Newell and Segur 1n the AHS publication Nonlinear Wave Motion listed at the end of this lecture.
Many different
typea of nonlinear evolution equation have been found to possess soliton solutions and for more information on this topic we refer to the review paper by Scott,
Ch~_
and McLaughlin for both a good account of some of them
and also for the basic references, and also to the edited collection of papers by Bullough and Caudrey. 4.
Inverse Scattering The behaviour of soliton solutions to the KdV equation is suggestive
of linear behaviour and this motivated Gardner, Greene, Kruekal and Miura to try to find a linearising transformation of the type used by Hopf and Cole to transform Burgers' equation to the heat equation (see Scott, Chu, Mclaughlin).
No such transformation was found, but during their search
they discovered an important connection between the KdV equation and an eigenvalue problem for the scattering method used in
Schr~dinger q~ntum
equation in terms of the inverse
mechanics.
It 1s this result that has
94 come to be known as the inverse scattering method in the context of solitons. We can do no more bere than outline the ideas that are involved. The b.sic problem to be considered 1s how a general solution of the KdV equation
au
o
3<-
(27)
subject to arbitrary initial data u(x,O) • uO(x) may be obtained.
The factor
-6 1s included here fot' convenience, but it . .y easUy be removed by a
trivial transformation 1f required.
In e••enee, the approach to this
question by Gardner et ale proceeded .s follows.
When v satisfies the
modified KdV equation
v
t
2 - 6v v
x
+ v
o,
xxx
(28)
they noticed that the quantity u which 1s given by u
2
v + Vx •
(29)
satisfies the KdV equation u-6uu+u t x xxx
o
(30)
Equation (29) Is a Riccatl equation for v 1f we consider u to be given. Therefore, we can use the well known transformation which linearizes the Riccati equation, (31)
This gives
o,
(32)
where u Is a solution of the KdV equation (30).
This ia a natural extension
of the Hopf-Cole transformation since the KdV equation has a third order space derivative and it 18 one order higher than that of the Burgers' equation. However. if we merely use (32) in the KdV equation we obtain a complicated result that is not useful.
Now the KdV equation is Galilean invariant. and
so allows the replacements u -+ generalised to
U -
A, :x:
+
x + 6At. so that (32) can be
95 (33) fo~
This Is simply the eigenvalue problem
the Schr6dinger equation
for ljI with the "potential" u, where u is the solution we are seeking.
Equation (33) differs essentially from the eigenvalue problem of the Schr6d1nger equation in quantum mechanics because, as u must be a solution to the KdV equation, it is time-dependent.
considered as a parameter in (33).
That Is, the time should be
In other words, it Is required that
(33) must hold at every instant with uU t t) at that same instant.
Thus,
generally speaking, the eigenvalues A would be expected to be time-dependent. Rather surprisingly, after some calculation, it caD be shown that they are time independent (and constant), provided u decreases sufficiently rapidly at infinity. Let us deduce the relationship between the KdV equation (30) and the Schrodinger equation (33).
If we let u
03) gives the dispersion relation
2 becomes _k .
(j)
~
0 at infinity,
the KdV equation
+ k 3 • 0, and the phase velocity ).p
For .sufficiently small lui, therefore, we have a plane wave
propagating in the negative direction.
For large values of lui, the
nonlinearity dominates to give rise to solitons.
The linear approximation
is also valid at infinity, since lui becomes 0 at infinity. In the case of solitons, the wave decreases exponentially at infinity and k becomes purely imaginary with k • propagates in the positive direction. the
Schr~dinger
h: ,
p
).
=
•
k
2 p
:>
O.
Thus a soliton
On the other hand, in the case of
equation (33), it follows that
small luI, and we obtain.
p
e±ikx, l • k 2 •
-~
xx
~ l~
for sufficiently
This approximation is still
valid for an arbitrary value of u, provided k is thought to be sufficiently large.
For a bound state, the eigenvalue becomes ). •
and. decreases exponentially at infinity. one bound state corresponds to one soliton. soliton solution
-K
2
p
<
0, i.e. k - iK p '
Therefore we can deduce that In fact, if we substitute the
96
(34)
at
t
-
0 into (33) and solve the eigenvalue problem we can get only the
bound state
~
2
/4. P Thus we can see that the eigenvalue 1 corresponds to the speed of a
•
soliton p
2
•
-K
If the "potential" u(x,O) (our initial data) provides N-bound
states, the solution u(x.t)
88 t
1s given by N-solitons propagating with
~ •
speeds four times as large as the value of each eigenvalue and by wavetra1ns decreasing algebraically with respect to time.
The wavetrains can be
determined in relation to the .cact«red ICace for the potential u(x.O).
This
1s one of the most important r ••ulc. and, in lca.ral, the far field of purely dispersive systems for which the lt4V 69 t ...
00
equacioD u the far field equation may
be approximated by If.'lIo11toM.
For a given arbitrary VAlul of
~(x,O),
u(x,t) can b. obt&1u.d exactly
by the following procedure:
(1)
Direct problem Find the discrete eigenvalue 1 for a liVeD pot.ntial u(x,O) of the Schr6dinger equation (33) aftd all. find chi I ..CClcina dlca at
Ixl • -
for the wave function, (i •••• thl r.lliction or tranlm1•• ion coefficients for ulx,O». (2)
Time evolution of 8catterlna data
Find the time evolution of the scattering data and the asymptotic form of (3)
+ at Ixl • -.
Inverse problem Find the potential u(x, t) in terms of the scattering data at time t. This potential 18 then the exact solution to the KdV equation (30) subject to the arbitrary initial data u(x.O) • uO(x).
97 Inverse Scattering Method
+ 1u(x,O) 1---------------------------u(x, u t -6uu x
u xxx • 0
~
t)
(1) direct problem
eigenvalue ). scattering data at t • 0, Ixl • •
(2)
time evolution of
scattering data at t • t,
Ixl ...
scattering data
The properties of the evolution of initial data comprising only solitons follows directly from this approach and gives rise to nonlinear superposition
laws for solitons.
There are, however, easier ways of obtaining these than
by means of the inverse scattering method which takes account of arbitrary initial data, and not just data comprising a train of so11tons.
For the
details of the various steps involved we refer, for example. to the paper by
Scott, Chu, McLaughlin.
The basic paper by Ablowitz, Kamp, Newell and Segur
presents an alternative treatment of this same problem. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Taniuti, T., Wei, C. C. Reductive perturbation method in nonlinear wave propagation I, J. Phys. Soc. Japan 24 (1968), 941-946. Jeffrey, A., Kakutani. T. Weak nonlinear dispersive waves: a discussion centred on the Korteweg-de Vries equation, SlAM Appl. Math. Rev. 14 (1972), 582-643. Jeffrey, A., Kawahara, T. Multiple scale Fourier transform: an application to nonlinear dispersive waves, Wave Motion, 1 (1979), 249-258. Jeffrey, A. Far fields. Nonlinear evolution equations, the B§ck1und transformation and inverse scattering, Scheveningen Differential Equations Conference, August 1979, Springer Lecture Notes (in press). Whitham, G. B. Two-timing, variational principles and waves, in Nonlinear Wave Motion, Ed. A. C. Newell, Lectures in Applied Maths, Vol. 15, Am. Math. Soc. (1974), 97-123. Zabusky. N. J., Kruskal. M. D. Interaction of solitons in a colilsionless plasma and recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240-243. Scott. A. C., Chu, F. Y. F., McLaughlin, D. W. The soliton. A new concept in applied science, Proc. IEEE, 61 (1973), 1443-1483. Bullough, R. K., Caudrey, P. J. (Editors). Solitons, Lecture Notes in Physics, Springer 1979. Gardner, C. S., Greene, J. M., Kruskal, M. D•• Miura, R. M. Method for solving the KdV equation, Phys. Rev. Lett. 19 (1976), 1095-1097. Ablowitz, M. J., Kamp, D. J., Newell, A. C., Segur, H. The il1verse scattering transform - Fourier analysis for nonlinear problems, Studies io App1. Math. 53 (1974), 249-315.
CF2iTRO INTERNAZIONALE MATEMATICO ESTIVO
(e.l.M.B.)
ONDES ASYMPTOTIQUES
YVONNE CHOQUET-BRUHAT
101
ONDES AS'lMPTOT1QUES.
Yvonne Choquet-Bruhat.
I.M.T.A. Universit€ Paris VI
Dedie a
la memoire de Carlo Cattaneo.
Introduction.
Neus allons. dans ces
le~ons.
exposer les grandes lignes de la methode
ge-
nerate de construction des andes asymptotiques et approchees pour les systemes d'equations aux derivees partielles. Le procede utilise a son origine
dans la methode W.K.B (Wentzel-Kramers-Briliouin) consistant
a
une equation differentielle une solution de 18 forme Ae i $
A est une ampli-
au
chercher pour
tude lentement variable et $ une phase rapidement oscillatoire. Dans la premiere
le~on
nous exposerons la construction de solutions des
equations de Maxwell qui donueut, en premiere approximation, les lois de l'optique geometriques. Dans les
le~ons
II et III nous exposerons, dans le
102
cas dlun systeme du I~ ordre, la theorie generate de J. Leray des developpements
as)~ptotiques
pour les equations lineaires, dans la
le~on
IV nous mon-
trerons comment les resultats sont modifies quand les equations ne sont pas
lineaires, et nous donnerons dans 1&
le~on
V une application aux andes dans
les fluides parfaits relativistes. Dans la
le~on
VI
DOUS
formulerons en termes de geometrie symplectique les
constructions effectuees; dans les
le~ons
VII et VIII nous etudierons les pa-
rametrisations des varietes lagrangiennes et
DOUS
ment asymptotique au voisinage d'une caustique.
construirons un developpe-
103
I LES EQUATIONS DE MAXl/ELL E1 L'OPTIQUE GECtlETRIQUE.
Les solutions asymptotiques dont
DOUS
allons parler dans ces conferences
ont leur origine dans les travaux de Debye. Sommerfeld et Runge pour retrouver les lois de l'optique geometrique a partir de 18 theorie electromagnetique. On a montre depuis que cette methode de solution s'appliquait 1 beaucoup d'autres problemes et permettait d'obtenir des renseignements. tant qualitatifs que quantitatifs sur des phenomenes physiques varies rapidement oscillants. Pour justifier l'introduction des developpement8 asymptotiques generaux que nous etudierons par la suite nous allons considerer d'abord Ie cas original des equations de Maxwell.
I - Milieux isotropes non conducteurs.
Les equations de Maxwell verifiees par Ie champ electrique E et Ie champ magnetique H dans un dielectrique parfait homogene .'ecrivent :
aE
(I-I)
rot H - <
at
(1-2)
rot E + lJ.
at
aH
. .
j 0
104
~
au j est Ie vecteur courant Electriquei E et
sont des constantes.
Prenons pour j Ie courant produit par un dipole sieue a I'origiae, de frequence w. Alors j s'ecrit, en representation complexe (Ie courant reel est
;R e
j)
j • 6 M ei t
au 6(x) est la mesure de Dirac A l'origine (de l'espace
~
3
) et M un vecteur
donne de ,3(8i H est reel, Ie dipole est dit polarise rectilignement). Si on cherche un regime "statiannaire". solution des equations de Maxwell,
clest-A-dire de 18 forme, en representation complexe H - vex) e
wt
3 xe.lR.
t,,1R
on trouve que lea vecteurs complexes u et v doivent verifier lea equations (1-3)
rot v - i E W u • M 6
(1-4)
rot u -+ i lJ W v • 0
on introduit Ie potentiel vecteur A en posant
v· rot A
d'apres (1-4)
, div v - 0), alors u est de 18 fO~t d'.pres (1-4)
u - - i IJ
W
A +
srad.
V
Vest appele potentiel scalaire. On Ie choisit en lui imposant 13 relation dite "condition de jauge de Lorentz"
V
(1-5)
alors ( 1-5 (1-0)
1
div A
- iwe:
) devient M.+w
2
£IJA-- 15M
105
avec u • -
i(lJ w A
+...!.. WE
grad div A)
2 Une solution elementaire de l'operateur A + W ment
est 18 fonction locale-
£ 1.1
int~grable
1
Ixl
exp(-
G(x) • - 4tt
Ixl • (
3 1: i •
on eo deduit une solution A de (1-6) A
- - Gt
15M
_ exp(-
iwl£ii Ixl )
4nlxl
H
Pour cette solution A on a v • rot A • exp (- i
w ~
...
avec v
o
- .. .,.. x , 2 'E~
•
i
~3
Ixl
...... x
- i wH
4n
on trouve aUBei
u· exp(- i w IE~ Ixl)(uo + ~ u + ~ w
v
o
u
Ixl
I
"i
u2
)
• i rot v I£"o
-,IF.£
Lea champs E et B correspondanta, solutions exactes des equations de Max-
well
a source
dipolaire Boot :
E
-
exp
[i w (-
I£~ Ixl
+
t)]
R
-
exp
[iw (-
IE~ Ixl
+
t)] (vo +
(uo + ~ u. +:2 u2 I
t
VI')
00 coostate que ceB solutions sont produit d'un terme oscillatoire. de
106
frequence w comme la source, par une somme de termes qui ne dependent de w
-k
que par un facteur w
0
• 5i west grand Ies termes en w • c'est:-a-dire
U
V ' o o
sont preponderants. Le terme oscillatoire est de 18 forme
'f(x,t) - -
l&it IXI
exp i
OJ ~x.
t) avec
+ t. La fanction ~(x.t) s8tisfait a l'equation dice ei-
konale de l'optique
geo~trique
:
=
'f.
Les surfaces rayon R • t/~
cte aout pour chaque
t
donne des spheres de centre
0,
de
• elles sont identiques sux fronts d'onde (normales aux
rayons lumineux) correspondant
a une
source ponctuelle eo a, pour une vitesse
de propagation de 18 lumiere II ~' On cons tate de plus que aiasi aussi de
at
V
o
et U sont ortbogonaux au rayon x (il en est o
mals pas de E). done tangents au front d'onde et orthogo-
naux entre eux. On volt sussi qu'lls verifient lea equations : d dr
oil d dr
(r u )
0
o
.
xi a ax'
~
d
dr"
et
r
(r vol
z•
I i
0
(:1h Z
done
qui exprime 1a conservation aur Ie front d'onde Ie long du rayon x de 1a densite associee au scalaire luol2 + Ivol2 • Ces differentes proprietes sont carasteristiques du champ de l'optique geometrique.
107
2 - Milieux non homogenes, et conducteurs.
Les equations de Maxwell sont maintenant (2-1)
rot B - e:
'E • at
j + 4 TT
(2-2)
rot E + ~
.U • at
0
£ ,
~
lJ
ainsi que Is conductivite electrique
E
a sont des fauctions continues du
point x. On cherche encore une solution de representation complexe E .. e
(2-3)
iwt u(x)
H .. e
correspondant i la source dipolaire j - e
iwt
wt H
v(x)
o. On
peut encore introduire Ie
potentiel vecteur A, tel que v • rot A et fixer Ie potentiel 8calaire V par Ie chaix de
fa~on
1-5
, mais l'equation satisfaite par A ne peut plus etre resolue
exacte explicite.
On cherche done directement u et v sous 18 forme (2-4)
(2-5)
u - e
v - e
u (x)
i OJ ~(x) U
E_o_ _
U •
o (iw)o i OJ ~(x) V
v (x) E_ o__
V •
n (iW)D
E et B satisfont aux equations de Maxwell si U et V satisfont en dehors de l'origine x • 0 aux equations suivantes : (2-6)
rot V .... i UI V IP
V - i Ul
£
(2-7)
rot U .... iwVIP
U + i UI
II
au
00
U- 4 V
iT (]
U
o
o
a pose 'VIP • gr ad tP •
En reportant dans ces
~quations
lea series 2-4 et 2-5 et annulant Ie terme
108
en WO on trouve (2-8)
VljI A
v
(2-9)
VljI A
U
0
0
- <
U
-
v
~
0
0
0
0
les equations ont une solution non nulle (uo ' vol 81 et seulement 81
c'est-i-dire si 'f(x,t) - llI(x) + t
a.tisfait l'equation eikonale i coeffi-
cient eu dependant de x 3 l: i •
Les vecteurs
U
o
o
et V sont encore perpendiculaires en chaque point x 1 la dio
rection (spatiale) du rayon lumineux,
V~
, perpendiculaire au front d'ODde
q,(x) • constante, et perpendiculaires entre eux : la solution geMrale de 2-8,9 pour IVlill
2
-
£~
est telle que
el1e est d'terminee, par exempte, par Ie choix d'un vecteur
U
o
orthogonal A
VljI •
Un moyen remarquable, qui s'averera tout equations de propagation de
et
U
o
V
o
a
consiste
fait general, pour trouver les
a
ecrire que lea equations doi-
vent etre verifiees a l'ordre sULvant, c'est-a-dire a annuler les termes en -J W dans 2-6, 2-7. on trouve: VljI A v
VljI A u
J
- E u
l
J + Jl VI
- rot Vo + 411" a - rot u
U
o
0
on sait que ai IVljI!2 _ <~ (impose par les equations d lordre zero) les premiers
109
membres de ces equations ne sont pas lineairement iode-pendants - il en est done de meme des premiers membres : un caieul purement algebrique fournit les relations satisfaites par ces seconds membres, qui seront done des equations aux derivees partiel1es du premier ordre pour
U
o
et v ' On trouve que ces o
e-
quations differentielles Ie long des rayons, trajectoires du vecteur VW, qui s'ecrivent
.•
~)v·".O " dv o I + - (6 d't' 2 E
avec
d
dT _3_
a~i Lea equations differentielles que l'on vient d'obtenir soot appelees "equations de transport" - la methode qui les iournit s'etend aux systemes quel-
conquest comme nOUB Ie verrons dans la prochaine
le~on.
References.
M. Kline, loW. Kay "Electromagnetic theory and geometrical optics" Interscien-
ce. 1965.
110
II ONDES ASYKPTOTIQUES POUR LES SYSTEIlES
D'EQUATIONS AUX DERlVEES PARTlELLES LINEAIRES.
La methode indiquee dans la
le~on
I pour la recherche d'ondes de frequence
elevee solution des equations de Maxwell~ diveloppee par Luneburg Kline
121
a ete generalisee par P. Lax
131
111
puis par
qui construit, pour un 8yst~me
d'equations sux derivees partiel1es lineaires du premier ordre 1 caracteristiques simples un developpement asymptotique
u
'"
e
iw
P. Lax utilise ce developpement pour la resolution approchee de Cauchy i donnees initiales oscillatoires, et pour la demonstration de certains theo-
remes generaux concernant les equations hyperboliques. D. Ludwig
151
generalise encore la methode en
par une famille de fauctions f
j
'"
~
j • 0
i la fanction e W
, et construit des developpements asymptoti-
ques de 18 forme :
u
rempla~ant
avec
111
pour les equations aux derivees partielles lineaires hyperboliques dans Ie cas d'un systeme du premier ordre ou d'une equation d'ordre quelconque. J. Leray 141 puis L. Garding, T. Rotake. J. Leray 161construisent pour des systemes d'equations aux derivees partielles lineaires d10rdre quelconque, des
developpements asymptotiques tres gene raux, du type
u
W- P u
E
'"
p - 0
(x point d'une vari€te differentiable, reel) dans Ie cas ou la phase
\f
~(x)
P
(x'Wf)
fonction scalaire, w parametre
correspond A une caracteristique simple du
systeme.
C'est 18 methode de J. Leray qui s'etend Ie mieux aux systemes d'equations
aus derivees partielles non lineaires. Naus allons }'exposer, dans un but de simplicite, dans Ie cas d'un systeme du premier ordre.On sait d'ailleurs que tout systeme d'iquations aux dirivies partie lIes peut s'icrire sous forme d'un systeme du premier orrlre.
I - Definitions
Nous designons par X une variite differentiable de classe C~ de dimension ,
h,
" de X," x xO, , I '" par x un pOlnt
l i es d ex. x1-] sont 1 es coord onnees oca
Un operateur differentiel [syste-me d'equations aux derivees partielles] lineaire du premier ordre sur X est une application L de l'espace des sections
C~ d'un fibre vectoriel E sur X dans l'espace des sections C~ d 'un autre fibre F sur X. Nous supposons iei dim E
G
dim F • 1 + N, aiors L s'eerit en coor-
donnees locales (1-1 )
_•.i A L
I, ••• N ; A • 0, ... 1-1
112
au les a~AJ b. i sont des fenctions donn~e8, c~ 1
1
Designons par
, des coordonnees locales.
u~(x,~), q - 0, I, ••• , des fanctions a valeurs reelles eu com-
plexes, derivables, de x E X et d'un parametre numerique reel
~
, par w un pa-
rametre reel (nomme frequence), par If(x) une fonction reelle derivable sur X (nOIllDi!e phase). On defiait la fouction u
i
q. w \0 I sur X par :
(U~
(1-2)
•
I4 I
Nous dirons avec J. Leray
OJ
'f )(x)
U~(x.
OJ
,\,(x))
que la serie formelle
i
( 1-3)
•
U
E
'"
q •
a
w-
q
u~
0
W
'f
pour Ie systeme d1equations aux derivees partielles
est une code asrmptoti9ue
9i en reportane formellement 1-)
dans
I-I
J
compte tenu de 1-2
on trouve un
dEveloppement en serie de puissance de w qui a'eerit, formellement : ~
t
( 1-4)
w-qFj.W'f
q • 0
q
dont chaque coefficient F i est nul quels que scient x et q
~
•
Les developpements de Lax correspondent au cas particulier :
2 - Determination de 1a phase
~
•
Derivons forme11ement J i1 vient en posant i
(2-1)
-i
U
q
aUg
(x,.)
a.
113
(2-2)
(u
i
.w'fl '"
r q -
Dans Ie cas des developpements de Lax les formules 2-) et 2-2 se reduisent
1 :
c' est-a.-dire w
.w
w- q+ 1
'f'):: ( r
q - 0
On trouve dans Ie cas general en reportant dans )-)
forme
J -4
aU. lea coefficients
(2-3)
yi
- a~A ,
(2-4)
yi
-
-I
0
F~
;,i
o
-A ("i u 1
,
a~
une expression de la
sont :
aA- 'P aA 'f
+ "A
u
i)
0
+
b~, u 0i
(2-5)
i Les equations 2-3 ont une solution non nulle u en x € X 5i et seulement 8i o -A la matrice 8 J aX fest singuliere, c I est-ii-dire 5i Ie symbole principal de i
l'operateur L est non injectif pour Ie vecteur covariant p
-
ce symbo-
Ie etant suppose non identiquement nul (systeme regulier au sens de Cauchy Kovalevski) : Ies equations 2-3 auront une solution· non nulle sur un ouvert
n de
X 8i et seulement si, dans cet ouvert 18 phase
derivees partielles du
pre~er
If
verifie l'equation aux
ordre de degre N - dite equation eikonale du
systeme : (2-6)
A(x •
'f x)
• 0
qui s'ecrit en coordonnees locales
ou
'f x
•
(V
'f) (x)
114
"
det (a/ ~
Uoe phase le~on
a, If) •
0
verifiant l'equation eikonale 2-6 itant supposee choisie (cf.
la resolution du syste-me lineaire 2-3 - puis les equations 8ui-
VI),
vautes - depend du rang de ce systeme. Naus etudieroos Ie cas Ie plus simple au ce rang est N-).
3 - Ca. au
~
correspond
a une
caracteristique simple.
Designons par A(x,p) Ie polynome caracteristique du systime I-I
I
c'est-a-.
dire Ie polyname des composautes P du vecteur covariant p defini par : A "
det ( a~
A(x,p)
1
l' equation 2-6 exprime que Ie gradient
'f x
de la phase test une racine du poly-
nome caracteristique; suppos6ns que cette racine soit simple, (A'(X,p)} p •
au
OQ
a pose
,
A
(x, p)
'f x
,
a
~O
c'est-a~ire
pour tout x E.. X
A(x,p)
a - ap,
Le determinant A est alors necessairement de rang N-) pour p •
'fx
i Determination de uo~) .
'f etant choisi (3-1 )
solution de 2-6 on deduit de 2-3
oio
que
(x,C)
ou bi(x) est une solution du systeme d'equations lineaires homogenes
' "Ix
~ X
115
o
(3-2)
et
Vo(x,~) une fanction arbitraire. Puisque A(x, If'x> est de rang N-I, hi .est
determine a un facteur On de-dult de 3-1
pr~s.
:
(3-3)
(3-4)
i et wo(x)
des fauctions arhitraires de x.
Les equations
pi
les N inconnues
u..i
0
• 0
sont, si
'f
a determinant
verifie 2- 6 nul. Les
u,.i
•
N equations lineaires pour
ne pourront done exister,
re-
guliers, que 81
o
(3-5) oil on a pose (3-6)
et ou on a designe par h. une solution du systeme transpose de 3-2, c'est-aJ
dire telle que (3-7)
)-5 s t eerit
h.J
hi
at
.~
U a~ hi 3AVa + h. a~~ 0 J 1
+ bi h
h.
On a. puisque hi (respectivement
J
.~
hh u + aJ 0
1
i
a~ w0 +
b~1 h.
J
i
w
0
h.) est proportionnel a Ai pour chaque i J
J
116
fixe (respectivement chaque i fixe) (3-8)
i
au Aj(x,P) disigne Ie mineur de
8
jA P dans Ie determinant A(x,p) et k une i A
fonction de x qui depend du chalx de hi et
a avoir
h.
3
(on peut lea choisir de
k - I). Or, d'apres la 10i de derivation
(3-9)
•
A~ 3
d~un
maniE~re
determinant
(x,p)
D'oil
(3-10)
On diduit de 3-10 que l'iquation 3-5 a'eerit •
(3-11)
0
est la derivation Ie long du rayon correspondant
'f' (x)
a
• ete • On a pose :
(3-12)
6
o
est Dul 8i wi • 0 0
(3-13)
s(x)
ne depend que des equations donnees et du cholx de
'f .
ealclll de 8 Une expression interessante de Best obtenue aisement
lenme
(3-14)
on a. 5i k • 1
A
S(x) • .!. aA 2 ax +
1
A (x,
-
"2/ hj
a a.iA 1 hi h ( -, + x) j _2 axA
.3·A ___ ah i
i
ai'
hi .i ~ , ax
I
+
h. i )
,
18 surface d'onde
117
un simple calcul, puisque slors
Preuve
Corollaire
81 Itoperateur differentiel L : L
est autoadjoint, on peut choisir hi de sorte que $ se reduise A (3-15)
6(x) •
'21
L'equation de propagation 3-)!, avec 00 • 0, se de conservation pour
~eduit
slors
a une
equation
u~ Ie long des rayons.
Preuve: Lest autoadjoint ai, pour u et v
a
support compact on 8 l'egalite
' ' sea l8treS L2 d es pr 00 Ults (u,Lv)
cleat-A-dire si lea matrices a 1
A
(v,tu)
'A (ail) sont symetriques et 8i
a
'2;;;
o
On a slors h et h proportionnels. On peut choisir h tel que h • h (avec tou-
jours k
c
I).
L'equation )-11 peut alors s'ecrire (3-16)
Remarque
o D'apres les regles classiques de derivation d'un determinant on a:
(3-17)
5i x • x(t,y) verifie Ie systeme differentiel des rayons aSBocies
If':
a
la phase
118
et on a designe par
d d't- AA __a_
a/'
la derivation Ie long de ces rayons.
L'equation 3-16 exprime la conservation Ie long des rayons de la densite de
dt d
[u2 D(x)] o D(y)]
0
119
III ONDES ASYMPTOTIQUES D'ORDRE q. ONDES APPROC1IEES. EXEIlPLE.
i
1 - Determination des termes 8uccessifs uq
Uo(x,t) etant determine verifiant 3-5 de la
le~on II on pourra trouver u~
verifiant ,i • 0, sa forme generale sera o
(1-1 )
eu V1 (x,;) est une feaction arbitraire et
i
VI(Xt~)
est une solution des equa-
tions linea ires (1-2)
i On deduit de I-I en designant par U(x,() et U (x,() des primitives par rapport
at
de V et Vi :
( 1-3)
Supposons alors construites des fenctions u tions ( 1-4)
F~
d
O. P < q-) et soit
u~
de 18 forme
i ,p < q • verifiant lea equap
120
au U
q
(x,~)
est arbitraire et
Uq1 est
une solution quelconque. mais fixee. des
equations lineaires "). i _ a~J ~,u
(1-5)
...
Pour ces fonctious
u~
"q-l
.
+ b. J u 1
i
q-l
u i on a aussi q
Fi q-I
(1-6)
o
Les equations Fi - 0 s'ecrivent q
.i
(1-7)
u q+ l
on pourra trouver
u~+1
+ gj • 0 q
9i et seulement 8i
o
(1-8)
Cette equation est une equation differentielle du premier ordre lineaire pour U • analogue a II. 3-5, qui s'ecrit : q
o
(1-9)
eu Best 18 fauction de x
est conou quand,
U~_l
II - 3-13 et :
etant determine.
2 - Ondes approchees.
NOlls diroos que Ie developpement fioi
u~ a ete choisi.
121
(2-1)
u
r
i
",-p
E p • 0
u
i p
. '"
est oode approche.e d'ordre r-I si il existe line constante M celIe que (2-2)
au
< M '"
I ILj(u)! Ix
-r+J
pour tout w
designe une norme convenable, par exemple {I)
IILi(U)ll x
{2-3)
On voit que la condition 2-2 sera
Sup
ILi(u)1
x€X realisee si lea
u
i verifient F ip p
0, p < r
et sont bornes, ainsi que leurs derivees partielles par rapport aux variables x. .i i D'apres lea equations )-7 lea u + dependent lineairement des llq ; on pourq 1 L ra trouver une oode approchee d'ordre r-l 81 u admet r primitives par rapport o
a E:
uDifonnement bornees, pour E; €:. R, sinsi que leurs derivees partielles par
rapport aux variables x. Cette condition pourra toujours etre verifiee, par un chaLx convenable dlune donnee initiale. au moins dans un domaine de regu-
larite des rayons. Cette construction d'ondes approchees d'ordre quelconque ne sera en general pas possible pour les equations non lineaires.
3 - Exemple
ondes dans les fluides compressibles I-dimensionnels linearises.
Les equations classiques non lineaires sont
(1) Pour les equations hyperboliques on est amene normes.
a
considerer d'autres
122
au
t
est Ie temps, x la variable spatiale, p • w et
respec:tivement la dens i-
u
teo 18 pression et la vitesse. Les equations lineairsees au voisinage d'un etat de repos sont
(3-1)
• o au r et a sont des fenctions donnees de x. Cherchons une solution asymptotique approchee d 'ordre I, avec
'f. 'of (x, t):
(3-2)
on trouve pour I'annulation des termes en W,avee 'fx· l r -I
- Po 1ft
(3-3)
2 r -) On pourra verifier
F~I
•
+ a
I
o 'fx
voIf t + Po 'fx
F~I
- 0 avec Po et
'rt • Clf/at
0
r V
r
alp/ax.
0
va
pour tous deux nuls si
I'e-
quation eikonale suivante est verifiee : (3-4)
A(x,t,
'f t '
2 2 2 'f'x) " 'ft - a 'f'x· 0
II Y a iei deux familIes de phases possibles qui sont 5i a est une constante 'r(x,t) • W(x - at)
ot
r(x, t) • W(x + at)
Lea rayons correspondants etant alors lea droites x - at - ct~ et x + at - ct~ 5i a est une fauction donnee de x les deux familIes d'ondes sont obtenues par
123
resolution des equations lineaires du premier ordre
'f't •
(3-4 a)
'f't - a 't'x • 0
(3-4 b)
et
a'fx - 0
Les rayons spatio-temporels correspondant a la premiere f&mille sont les cour-
bes solutions de I'equation differentiel1e dx dt
a(x)
soit x • £(t,y)
Ie rayon issu d'un point (o,y). jours Ie cas pour
Y
£(o,y) •
t
SUppOSODS
I'equation inversible (c'eat tou-
assez petit puisque f(o,y) • y) on en deduit que si (x,t)
est un point du rayon issu du point y on a : y
g(t,x)
g(o,y)
y
La lolution de (3-4 a) telle que
If (0.")
(3-5)
au
~
-
lj/(g)
est une {oDction donnee, est
(3-6 a)
puisque
'f(t,,,) ~
est constant sur Ie
r~yon
lj/(g(t,,,»
spatio-temporel (bicaracteristique de
l'eikonale) issu de y. De meme la solution de (3-4 b) telles que (3-5) soit verifiee est
'f (t,x) au g-Ct,x) est obtenu par inversion de 1& solution de dx dt • -
a (x, t)
La solution genera Ie des equations pour un
'P de
Ia premiere famille est
124
(3-7)
(on a integre en
~
~
et supprime Ie terme independant de
L'annulation des termes en
).
WO est equivalente a :
(3-8)
Pour
22 If 2 t • a f x
ees equations impliquent
(3-9)
rempla~ant V
et Po par )-7 on trouve l'equation de propagation pour U ' de o
o
1a premiere famille (\oft" - a
fx)
(3-10)
equation de conservation Ie long du rayon du facteur a
r~
, independaIlllleot
de i;. On resoud ensuite les equations pi - O. i • 1,2, I v
au PI' VI
I
SOUS
18 forme
•
est une solution particuliere. par exemple, U satisfaisant o
a
)-10
dlau
oil
U
0
designe une primitive de Uo(t,x,F;) par rapport iI i;
5i on calcule Lu
.
I
2
(L (p,v), L (p, v» pour 1. somme )-2
sont les faDctions qu'on vient de determiner on trouve :
oil Po' PI' v ' VI o
125
L I (p,v) _
.!. pi w I
.!. (a t w
•
p
+.2 r
I
a:x:
vI
)
2
L (p,v) _
done ",-I H
8i lea derivees partielles en ceci sera realise 8i et
~,
t
UO(tIX.~)
et ceci pour tout
~
et x de et
Pl(x.~.~)
Uo(t,x.~)
et
vl(x,t,~)
sont bornees
soot uniformemeot bornes en t, x
E R.
U satisfaisant A l'equation de transport 1-10 (condition de conservation) o sera uniformement borne en
~,
dans tout domaine du plan (x.t)
au
lea rayons
ne representent pas de caustiques (equations x • £(t,y) donnant les rayons
inversibles) slil en est sinsi
a
l'instant initial
est : W
o
(x) sin (
t
-
O. Un choix possible
126
IV DEVELOPPEMENTS ASYMPTOTIQUES POUR DES
E~ATIONS
QUASI LlNEAlRES A CARACTERISTIQUES SDIPLES.
:- Definitions
m
NOllS designons par X une variete differentiable de classe C de dimension o
t. par x un point- de X; x,x
1
.•• , x
1-1
sont des coordonnees locales de x.
Nous considerons un systeme de N equations sux derivees partielles du premier
ordre sur X, quasi lineaires, aux N inconnues uK(x). fanctions(l) sur X
a
valeurs complexes (1-1 )
en posant u -
L u 1
u • • ••• u
N
o
et en designant par u .... L u une application (non
lineaire) de l'espace des suites {uk(x)} de N fanctions differentiables contenues dans un polycylindre P :
(1) les inconnues uk(x) peuvent Bussi etre des tenseurs sur X. I'application u .... L u s'ikrit encore 1-1 en coordonnees locales. les u i etant les com-
posantes des tenseurs envisages.
127
k
(1-2)
uo(x) fonctions donnees.
dans l'espace des suites de N feuctions, ayant dans chaque systeme de coord onnees locales une expression de la forme
0, ,
_ a~ (x, u)
(1-3)
0,
i,j • I, .•• N; A -0,.,,1-1
°
a1 (x,u) et bJ(x,u) sont des feuctions de classe
P et egales
a
em
en x, analytiques en u sur
la somme de leur developpement en serie entiere
(1-4)
oil on a pose
a~' '0
j,
0
&i (x,u )
3aJA ,
·0
---(it,u)
h 3u
i
Designons par uq(x,() • q. 0,1, ••.• des feuctions i valeurs complexes, derivables, de x ex et d'un parametre numerique reel
(nomme frequence), par
~x)
~
par w un parametre reel
une fauction derivable sur X (nommee phase).
POSODS
(1-5)
u
i q
Nells dirons comme dans Ie cas lineaire (cf. II) que la serie formel1e (1"-6)
u
i
m
E
w-q u~
0
w'f
q - 0
est une cude 8symptotique pour Ie systeme d'equatioDs aux derivees partielles
8i en reportant formellement 1-6 dans 1-3. compte-tenu de 1-4 on trouve un developpement en serie de puissance de w qui s'ecrit. formellement :
128
",-qp~.",'f-
E q • 0
(1-7)
dont chaque coefficient
piq
0
est nul quels que soient x et
2 - Determination de 14 phase
t.
'f .
On pose. comme dans II i au (x.!;) q
.i
(2-1)
u
q
on a (2-2)
a ax
-).
•
't')
(u i .oW
• "''f')C).'f+ t
q
q-O
",-q(c).u~.",'f)
On trouve alers en reportant dans 1-3 une expression de 1. forme 1-7 aU les
coefficients
piq
sont
(2-3)
P~I "ni~ u~
(2-5)
P qi - ni.).(u·i 10 q+1
c). 'f
a).'f
+
a).
u i ) + i). ( h(.i q nih. u 1 u q
a).'f
,i + a). u q _ 1 )
129 i NOlls etudierons Ie cas au U est one solution donnie, independante de S, des o
equations 1-3. (2-6)
Les equations
piq •
0 s'ecrivent alars
-, - 0
(2-7)
Fi
(2-8)
Fi
(2-9)
Fi
0
- .!l. 10 al. If
,
-
.!l. al.'f' 10 +
u,.i .i
U
2
• 0 +
8
il. a)., U i io J
'l. i (a~h al. u0
+
il. a).'f U h 1
a ih
h + b i ) u, h
u,.i
0
(2-10)
au
fiq
ne depend que des u
i p
ili
p'
aA up1 oil
L'equation 2-8 entraine, si l'on veut que (2-11 )
A(x,
'I' x)
p < q.
u~ t
0
o
2-11 est une equation aux derivees partielles du premier ordre qui exprime que la phase
\f
est solution de l'equation caracteristique approchee. obtenue
en donnant aux coefficients principaux de 1-3 18 valeur qu'ils prennent pour la solution donnie uiex). o
On designe par A(x,p) Ie polynome caracteristique du systeme )-3 correspondant i la solution
u~
, c'est
covariant p defini par : A(x,p)
a dire
Ie polynome des composantes p). du vecteur
130
l'equation 2-11 exprime que Ie gradient
'f x
de la phase
'f
est une racine de
ce polynome et on suppose que cette racine est simple, c'est (A'<X,p)}
+0 p -
pour tout
a dire
que
x E. X
'f'x
eu on a pose
/'-
aX A(x,p)
Le determinant A est alors necessairement de rang N-I pour p •
~
'f x'
I/x Eo
x.
etant choisi solution de 2-11 on deduit de 2-8
(3-1)
ou hi (x) est une solution du systeme d'equations lineaires homogenes (3-2)
o
et VI (x,() une fanction arbitraire. Puisque A(x,
'f' x)
est de rang N-), hi est
determine a un facteur pres.
On choisit : (3-3)
ou U (x,;) est une primitive de V (x.t) 1 l (3-4)
Les equations Fi - 0 sont, si 1 If veri fie 2-11, N equations lineaires pour . i .i les N inconnues u , a determinant nul. Les "2 ne pourront done exister • 2
131
reguliers, que 8i h. J
(3-5)
gi •
0
I
eu. on a pos6 (3-6)
et au on a designe par h. une solution du systeme transpose de 3-2, c'est J
a
dire tel Ie que
h.a~~3,'f'. 0 J 10 1\
(3-7) On a (d. II)
(3-8)
.
i~
eu Aj(x,P) designe Ie mineur de a io P dans Ie determinant A(x,p), et k une A fauction de x qui depend du choix de hi et
a avoir
k •
1)
h.
J
(on peut les choisir de
mani~re
et
(3-9)
On deduit de (3-9) que l'equation 3-5 a'eerit (3-10)
•
A~ _3_
3x~
'f (x)
est la derivation Ie long du rayon correspondant
~
la surface d'onde
• cte
On a pose
(3-11)
(3-12) a et
ane
i~
a(x)
-
g(x)
" h.J
ail
d~peDdent que des
a~ 'f .~
hi h. h
t
J
.~
{aio 3), hi + (ait
a~
u
i 0
+ bi ) h
t
0
t
} i
equations donnees et de la solution u • 0
132
Le coefficient 8 a une expression analogue aire, en 11.3.14, une fois la solution u j j). 11.3.14 par b i + ali
a
celIe obtenue dans Ie cas line-
i choisie. et en o
rempla~ant b~1 dans
t
cA Uo
Le terme aU) til n'existe pas dans le cas lineaire. La non nullite du coef-
ficient a va provoquer un effet, lie a la non linearite. de dis torsion des signaux.
Pour calculer Ie coefficient a on remarque que (3-13)
or, d'apres 3-8 on a (3-14)
h
i
j).
h j a if.
en considerant que
(3-15)
A(x.u.p)
on demontre (cf Boillat IV-~) que la quantite 3-14 est proportionnelle A la derivee
a~
au
de 18 vitesse de propagation correspond ant A l'onde 'r(x) • cte.
On aura :
a
0
si
o
(3-16)
les varietes caracteristiques possedant 1& propriete 3-16. renconttee par
P. Lax (mecanique classique) et G. Boillat (mecanique relativiste) dans l'itude des discontinuites ont
et~
appelees par eux exceptionnelles.
133
4 - Integration •
L'equation 3-11
est une equation aux derivees partielles du jer ordre qua-
81 lineaire, son integration se ramene done
a
celIe d'un systeme differentiel
ordinaire. soo systeme bicaracteristique. Celui-ci a'eerit : d!; a(x) U (x,!;) 1
dt
(4-1)
Is solution (4-2)
x • x(t,y)
x(o,y) - y
a
des t+1 premiers rapports est Ie rayon correspondant ~x)
Is surface d'onde
• cte passant par Ie point y.
La solution
UI(x.~)
(4-3)
( 1: )
passant par 18 variete initiale sly) · 0 ,
(U,(x,!;)} x _ y - W,(y,n) !; -
n
(8 et WI fanctions donnees) est engendree par les courbes bicaracteristiques
(solutions de 4-1) s'appuyant sur E • cfest
a dire
est obtenue par €limina-
tion de t, y, n entre les solutions de 4-) (4-4 a)
x • x(t,y)
(4-4 b)
(4-4 c)
!; • f;(t,y,n)
verifiant x(o,y) • y • UI(o,y,n) - W1 (y,n),
~(o,Y,n)
- n avec s(y) - O.
Remarque : Si la sous-variete S de Vn , s(y) - 0, est differentiable et transversale aux rayons (c'est
a
dire que son plan tangent ne contient pas le vecteur tangent
134
au rayon en ce point} les equations 4-4 a sont inversibles pour t assez petit; en dehors d'un voisinage de s(y) - 0 cette inversion peut ne pas etre possible (cas au les rayons admettent des caustiques). UI(t~y.n)
Les rayons x - x(t,y) etant supposes cannus lea fonctions ~(t,Y,n)
et
sont donnees par les quadratures t
(4-5 a)
(4-5 b)
c'eat
Uj
-
t
-
exp( -
n +
I:
S(X(T,y» dT) WI (y,n)
o
"(X(T,y» UI (T,y,n) dT
a dire
(4-6 a)
UI - WI (y,n)
(4-6 b)
!;
Remarque
~
(t,y)
n + WI (y,n) $(t,y)
ou
~
ou
$'
I:
- exp (-
I>
~
S dT
)
dT
on a
~ an
(4-7)
d'ou
I
(t,Ytn)
($ (o,y) - 0) :
~ an
(4-8)
(a,Y,n) •
~ etant une fanction continue de
t, d'apres lea hypotheses faites au f I on
deduit de 4-8 l'existence pour t petit d'une fauction differentiable
n
.
•
n(t,y,!;)
.
Plus preclsement on pourra tlrer pour
It I ~
t
o
• On
n(t,y,()
de
4-6 b pour
It I ~
. li an ..r
to 51
a , pour t petit (done ~ petit) n '"
t - WI (y,!;) $ (t,y)
qui traduit, reporte dans 4-6 b de petites perturbations de 1a forme du
0
135 signal. dependant de cette forme. On remarque d'autre part que. puisque
$ > 0 8i a est de signe constant
west une fanction monotone de
(independamment du phenomene des caustiques) ilWI
valeurs de t (dependant de y) si
La plus petite valeur de
t
an--
t
et en general
s'aonulera pour certaines
est de signe oppose a
Q.
pour laquelle l'equation 4-6 b cease d'etre in-
versible est appelee Ie temps critique.
Le temps critique effectif a
ete
d€-
termine dans un certain nambre de situations physiques realistes de la mecanique des fluides compressibles et de la magn€tohydrodynamique par A. Greco •
M. Anile et leurs collaborateurs.
5 - Conclusions.
Le premier terme u
u~
• '"
i 1
0
w'f
de l'onde asymptotique 1-6 est done
'f - WI(y(x).n(t(x),
y(x), "''f(x»)
c'est-a-dire que ce terme est, comme dans le cas vecteur propre
a droite
hi de la matrice A(x,
~
(t(x), y(x»
lineaire~
hi(x)
proportionnel au
~x) (valeur propre zero). Le
facteur de proportionnalit€ WI 41 est, cOllIDe dans Ie cas lineaire, produit d'une fonction $ qui ne depend que de l'equation et de la surface d'onde donnee (et ici de la solution
u~) et que lIon calcule par integration Ie long
des rayons correspondants par une fonction WI' dite facteur de forme, qui
de-
pend d'un choix initial. Cette fonction est, dans Ie cas lineaire, constante Ie long des rayons; elle ne l'est plus dans Ie cas non lineaire, si a n'est
136
pas nul : nous dirons que la non linearite provoque en general une distorsion des signaux, et nous enoncerons Ie theoreme : I. Le long de varietes caracteristiques exceptionnelles lea signaux se propagent sans dis torsion. 2. 5i Ie facteur de dis torsion a est de aigne constant il existe un temps critique au deli duquel 18 distorsion du signal conduit
a
la disparition de
l'onde asymptotique (formation d'un choc).
[11
Leray J.
t
Particules et singularites des andes ... Cahiers de Physique,
t. 15, 1961, P 373-381.
[2J
Garding L., Kotake T•• , Leray J.
t
Uniformisation et solution du probleme
de Cauchy lineaire •.. Bull. Soc. Math. 92, 1964, P 263-361.
[31
Lax P., Asymptotic solutions of oscillatory initial value problem, Duke mat. J., 24, 1957, p. 627-646.
[4]
Ludwig D., Exact and asymptotic solutions of the Cauchy problem, Comm. on pure and appl. Math. 13, 1960, P 473-508.
[5)
Choquet-Bruhat Y., Ondes asymptotiques pour un systeme d'equations aux derivees partielles non lineaires, J. Maths pures et appliquees, 48.
1969, p. 117-158. [6]
Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics. Ann. I.R.P. vol XXIX.no3, p. 257,197&
[7]
Boillat G., Ondes asymptotiques non lioeaires, Ann. Mat. pura et Appl. ~,
1976, p. 31-44.
137
V APPLICATION AUX EQUATIONS DES PLUlDES PARFAITS RELATIV1STES.
1 - Equations des fluides parfaits relativistes.
En
l'~bsence
de courant de chaleur, les equations s'ecrivent a
(1-1 )
(p + p) u
a a u d p+ (p + p) Va u - 0 a
(1-2)
a u d S a
(1-3)
P
t
V uB _yaB daP - 0 a (E)
0
P et S sont respectivement la densite d'energie, la pression et l'entro-
pie spikifique. Ce sont des "grandeurs d'etat" fanctions donnees de deux d'entre elles, par exemple p et S pour l'equation d'etat
( 1-4)
On sait que
p
5i
p(p • S)
1 1 00 pose
p
r(l + e)
ou rest la densite materielle propre et e l'energie interne specifique.
138
1-) est, compte-tenu de l'equation thermodynamique
de. T d S + P r- 2 dr equivalente
a
l'equation de conservation de la matiere introduite par Taub [2]
On a pose
a,B • 0,1,2,3
(1-5)
ou gQB est une metrique hyperbolique donnee; Va designe la derivee covariante correspondante et dQ
a/axQ
•
•
On peut ecrire les equations 1-2, 3, 4, sous la forme (1-6)
n
"I
V~ u
1
I,J - 0,1, •.•• 5
- 0
en posant • ua u
pour I - 0, .•• 3
4 • p
2 - Determination de 13 phase'f •
Cherchons une solution asymptotique des ces equations, au voisinage d'une solution donnee u ' Po' So' cleat o
a dire
perturbe d'un mouvement donne. Soit
un mouvement vibratoire du fluide,
139
~
Po +
P
E
w-q p q
q •
ow'f
~
S
S
0
+
E q •
S
ow
q
If'
en supposant que I'equation dletat
p • p(p • S)
est une fonetion analytique de p et 8. et on eerit (2-1 )
p(P.S)
s
Po + P~
(p - po) + P
o
(8 - So) + •••
0
en posaor.
Les termes independants de w donnent.
as \D
Y
o
,a
(p I Po
.a-O •.. .,3
o
(2-2)
On a pose
Ces equations lineaires en ~~ Ie
dete~inant
PI'
5,
n'ont une solution non nulle que 5i
A est nul. On trouve pour A, comme prevu. Ie determinant carac-
te.ristique du systeme (E) A
p'
Po au on a pose
2 + a )
140
a -
u
~
o 'P I~
La condition A - 0 exprime que les variites t.p(x) - c te sont des varietes
caracteristiques, c'est
a dire
des surfaces d'onde, des equations (E) pour la
I 2 solution u ' cndes acoustiques (a +b p' -0) ou cndes materielles (a - 0). o Po
3 - cas ou 18 phase verifie l'equation des andes acoustiques.
Supposons que
(3-1 )
A
A est, pour A
-
+ a
bp'
2
~a
g
" (p'
Po
- u
Po-
0
u (pt
o
Po
-
I )
'fa 'fa'
0
a
I
0, de rang N-) - 5, lUI }
u 1 ' PI' 5, ) est donne par
(3-2)
OU
UI(~Jx)
I est une fanction scalaire et lea h (x) sont une solution de 2-2,
cleat a dire proportionnels aux mineurs d'une meme ligne de A. Calculons, par I
exemple lea mineurs de la quatrieme ligne A , on trouve : 4 A4 - a
4
QA Y
If ~
4 4 5 A4 - (p + p) 0 a
Naus prendrons
(3-3)
A ~A h - p' Yo 'fa h4 _ Po (p + p) 0 a hS 0
3
p' (p + p) 0
Po
141
Appliquons la methode generale pour trouver l'equation differentielle verifiee par U
en egalant
1
:
a
-I
zero lea termes en w
des equations on trouve
(3-4)
o~
(l'indice I indique que l'on prend dans la quantite ecrite Ie coefficient -)
de w
)
(3-5)
.
solut~on
Les equations 3-4 ont une
u·r 8i 2
(3-6)
au lea
pIe i
bJ
AI4 .
meme
sont proportionnels aux mineurs d 'une
colonne de A, par exem-
On •
4 4
A4A
(p + p)o
A4
Yo
5
aa
\fa
•
p'
S
(p + 0
On prendra
hA -
A4 4
'fA
• 'fA
4
p)o 'fa·
3
-
4 5
(p + p)o
•
142
(3-7)
a
J
Formans hJ gl en
2
rempla~ant
u
I
I
par U h . On trelive que U do it sstisfaire I 1
1
l'equation differentielle du premier ordre suivante (3-8)
avec
(3-9)
(ia
2 k - - 2 a (p + p)o
- y~a 'f' a
est bien Ie rayon acoustique)
et? compte-tenu de 3-) : 2
(3-10)
a =
• 2"
(p + p)o )
(2(J - p' )
Po
alars que -I
2
a-
+.!.... 'iJ 2
u
a
+ p"
(p+p)o
a
-
o
2
(P+P) 0
2
p'
P Po -I (p+p) o ( , Ps
-~
Po
au on a designe par Cl l'operateur des andes acoustiques
O'f (on a
:; (pI
Po
2 0 - 2"1 (op a !OP).} a
5i la solution de base est
a).
Yo
PI.
+ ua u).) V 0 0
a
- 'fA
V a
a densite
'fA
°A ) et entropie constantes, et
~
lignes de
143
courant
a divergence
nulle
o
(3-12)
8
8e
reduit a
B-tO,!, on remarque que lea conditions 3-12 impliquent, si u~, Po' Po' So est une so-
lution de (E) que uno V(l uS. 0 • d'ou on deduit que 0
Remarque a est nul si et
c'eat
a dire
viste cf
en particulier si Ie fluide est incompressible (au sens relati-
(IJ>,
e'est
1
dire admet I'equation
d'etat
p - p + cte
Dans ce cas les andes acoustiques sont exceptionnelles au sens defini precedeument.
Pour les fluides reels on a pI! p
~
0
done
a
5i
P~ <
I, ou
P~ >
~
0
0 on a raidissement des signaux.
On peut aussi trouver un deve10ppement asymptotique correspond ant aux andes
materielles; celles ci apparaissent comme multiples (en accord avec la theode generate de Bailiat, cf [4]). Le probleme de Cauchy oscillatoire est
144
resoluble~
au premier ordre d'approximation. pour les equations des fluides
parfaits.
II olen est plus de meme en presence de phenomenes dissipatifs, par exemple
pour un fluide charge
a
conductivite non nulle (c£ (3J) .
References.
[I] [2]
Lichnerowicz A., Hydrodynamique et magnetohydrodynamique. Benjamin 1967. Tauh A.H., High frequency gravitational waves and average lagrangian, General Relativity and Gravitation, Einstein Centenary volume, A. Held ed, Plenum.
[3J
Choquet-Bruhat Y., Ondes asymptotiques pour un syst~me d'equations aux
derivees pareielles non lineaires, J. Maths pures et appliquees, 48, 1969, p. 117-158. Coupling of high frequency gravitational and electromagnetic waves, Actes du Congres Marcel Grossmann, Trieste, Juillet 1975).
r4}
Boillat G., Ondes asymptotiques non lineaires, Ann. Mat. Pura et Appl. ~,
[5}
1976, 31-44.
Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. I.R.P., vol.XXIX. n03, 1978.p.257.
[6]
Breuer R.A. and Ehlers J •• Propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time. to appear in Proc. Roy. Soc. A.
145
VI DETERMINATION DE LA PHASE. BICARACTERISTIQUES. VARIETES LAGRANGIENNES.
I. D€finition
des variEtes lagrangiennes.
L'equation eikonale i. laquelle doit satisfaire 18 phase (1-1)
'f:
A(x. '!'x) • 0
est une equation aux derivees partielles du premier ordre non lineaire, A est un polynome de degre N, homage-ne, en
1 x."
NOllS
allons rappeler
CODIDent
on in-
tegre une tel Ie equation. en utilisant Ie langage de la geometrie symplectique. On
*
designe par T X l'espace fibre cotangent
a
Is variete x de dimension 1.
Un point de T*I est un couple (x,p) ou pest une I-forme sur llespace tangent
TxX a x en x, clest a dire un vecteur covariant. Une solution de l' equation I-I dans 8u-desSU8 de
Q
c. X
est une section du fibre T:l.X x lR,
n par
telle que
(d
'f) x
• < P. dx >
). - 0, .•. t-I
146
A(x,p)
et
o
sur une telle section on a
Soit n : T%X ~ X la projection canonique (x,p) l~ x de r*x sur X. On defiDit une J-forme sur TXx. appelee I-forme fondamentale ou forme de soudure par
u~T
6(x,p) (u) - p(fl' (u))
x,p
T"x
son expression en coordonnees locales est ;
6 La 2-forme
o -
d6
est fermee et de rang 2t, une telle 2-forme est dite symplectigue. La 2-forme cr munit
T*x
de sa structure symplectique canonique. Une sous-variete de T*x
qui annule a et qui est de la dimension maximum possible, c'est
a dire:
t •
est dite lagrangienne.
5i Vest une variete. de dimension t. immergee dans
T*x
par une application
f, on dit que (V. f) est une sous-variete lagrangieone [ilDl1ergeeJ de T*x si
sur V
au
£
2 Recherche dlune sous-vari€te lagrangienne (V.f) de TXx telle que A(x.p) ~ O.
Le probleme est celui de la recherche des varietes integrales (immergees
147 dans T~)de dimension t du systeme differentiel exterieur
o
(2-1)
o
(2-2)
A(x,p) '" 0
La fermeture de ce s~steme contient, outre les equations precedentes. l'equa-
T*x :
tion exterieure sur
o
dA •
(2-3)
Le systeme caracteristique de 2-1. 2-2 est le systeme associe de 2-1, 2-3. II est constitue par les champs de vecteurs v sur TXx tels que :
(2-4)
ia-kdA
avec
v
c'est
a dire
k€lR
en coordonnees canoniques (xA,p.>..) de TXX
ou
v-
A - O. • ..• 1-1 VA+!
(2-5)
- aA/a/'
Un champ de vecteurs VA possedant la propriete 2-4 est dit hamiltonien pour la structure symplectique a et l'hamiltonien A. On remarque que VA est tangent
a
Jt (sous-variete
A(x,p) • 0 de T*x)
Une trajectoire du champ de vecteurs hamiltonien VA est appelee une (courbe)
bicaracteristique de Itequation aux derivees partielles A(x. On
• O.
suppose que I'hamiltonien A nta pas de point critique sur cf};(c'est
dire que dA sur
~x}
cfC.
Theoreme
~
a
0 quand A • O}. Ie champ hamiItonien VA n'a alors pas de zero
et on demontre Ie theoreme fondamental suivant (cf par exemple
Ill):
Soit Y une sous-variete compacte de dimension t-I de T*x verifiant
o • 0 et A • O. transversale en chaque point au champ hamiltonien VA" Soit Ie flat du champ de vecteurs VA' alors (y x ~.f)
au
f : Y x:R ~T*x
par (y.t) .... ft(y)
f
t
148
est une sous-variete lagrangienne immergee de T~.
3. Determination de 18 phase.
Etant donnie une sous-variete. lagrangienne (V,f) immergee dans X, il existe dans chaque ouvert U C V simplement connexe une phase $ • de.terminee a une
constante additive pres, satisfaisant a l 1 equation d ~ •
(3-1 )
f'to
de
puisque l'on a. sur U. ft:
f"
e
_ o. v
Remarque : II existe toujours, globalement. sur Ie recouvrement universel V v
de V. projete sur V par
n.
v
une fanction v d ~
On de-duit de
IT
0
f
:
W•
COnDU
(f
$ telIe que 0
v IT)"
dans U C. V. une phase
f
e dans
nC
X si l' application
U + nest inversible.
L1 application f etant une immersion il existe toujours un sous-ouvert, encore note U. de U. tel que f soit un diffeomorphisme de U sur feU). L'applica-
tion
n0
fest alers inversible sur U si 1a projection IT : T*X ~ X restreinte
a £(U) est inversib1e : i1 en sera ainsi au voisinage de tout point OU f(U) n1est pas tangent
a
1a fibre de T*X, c 1 est
a dire
n 1 a pas un plan tangent
"vertical". Soit x Eo
n c.
X. tel que
n-I (x)
ne soit pas tangent :it f(U). Soit Y1 "··· Yk
les points de feU) tels que n(Yi) ex:
149 Le point x admettra un voisinage dans X, encore note Q tel que IT
-)
(0) soit
l l union disjointe d'ouverts de feU) : f(U )
U
(3-2)
i
i • I ••• _,k
A chaque U, pour une meme phase $ sur V. correspond une phase
f
i sur
n donnie
par :
If i
• ~
0
(IT i
0
f)
-)
•
Dans les applications la donnee physique est souvent la variete lagrangienne
V, provenant de la geometrie du probleme et de sa dynamique : les bicaracteristiques. c'est
a dire
les trajectoires du vecteur hamiltonien vA sont les
rayons lumineux (dans l'espace des phases) dans les problemes d'optique. les trajectoires des particules materielles dans d'autres problemes. Nous allons considerer Ie cas OU la p~ojection de V, supposee sous-variete de T*x pour simplifier, sur
n c:
T~X n'est pas bijective, mais
ble I: de V (son "contour apparent") tel que (3-3)
-I
n
au
(0). V
U
i • 1, ••• k
au
chaque restriction
J':i de n a
il existe un sous-ensem-
,1:
soit de 1a forme
U.
•
U i
u..... n
•
est un diffeomorphisme.
4. Solutions asymptotiques.
II est naturel de chercher une solution asymptotique du systeme differentiel
150
d'equation eikonale A(x.
~x)
a
- O. correspondant
une variete lagrangienne du
type 3-) sous la forme : k u(x)
(4-1 )
r
i ou les
~i
sont des phases. sur nc x. correspondant a la variete V. On sera
aide dans ce caleuI par la methode de la phase stationnaire qui montre (ef VII)
comment 4-1 liee a l'evaluation asymptotique d'une integrate. Des developpements du type 4-1, et les equations de transport correspondantes. sont utili-
sees pour determiner l'intensite lumineuse en presence de caustiques (enveloppe des rayons lumineux en projection sur l'espace-temps X). Remarque : Chaque phase
"Pi
n'est connue quia une constanCe additive pres.
puisque la variete lagrangienne V
De
determine
~
qu'a l'addition pres d'UDe
constante. depourvue de signification physique. La theorie des integrales asymptotiques et de ltincide de Maslov permet de determiner ces
constantes~
tification de
et
[51
relations entre
puis des conditions sur la variete V. dites conditions de quan-
Maslov~
pour qu'il corresponde
globale. avec une phase
[31
d~s
de VIII).
If
a v une
solution asymptotique
determinee globalement sur n • fi(V) (cf references
151
VII PHASE STATIONNAlRE. PARAMETRlSATION
D' UNE VARlETE LAGRANGlENNE.
1. Methode de 18 phase stationnaire (one variable).
On
se propose d'evaluer. pour w grand, une integrale de la forme l(w) m
au a et f sont des fonctions C , et a est )0)
On suppose que ~~
a
support compact.
ne s'annule pas sur Ie support de a; on deduit alers
de
a ( iwf)
aa
e
af
-iwaae
iwf
que l(w) -
1 iw
-
iw
l'integrale etant bornee par un nombre M independant de w on a
Mw
-1
152
Par iteration du
prod~de
on trouve, pour tout n €.
~
2°) On suppose que f s'annule en un point et un seul 0
if
a dire
que ---2 ' 0 pour a • a ' c'est o
aa
0
du support de a et
que f a un point critique, et un seu!,
non degenere sur Ie support de a. On montre que f peut alers s'ecrire dans un voisinage de ment de variable a --
t (0)
0
0
,
par un change-
tel que t (0 ) - 0, sous la forme 0
g( t) _ f(a(t» • g(o) +
f
t
2
e:
K
sign
done
I(w)
I
+
~
<Xl
elwE/2
t
2 b(t) dt
_00
_ iwg(o) ou b(t) - e .(a(t»
da
Cit
l'integrale existe, absolument convergente, puisque b a un support compact. On peut en particulier la calculer par passage
A
elwE/2
f
l(w)
t
a
la limite
2 b (t) d t
-A
On pose alars b(t) - b(o) " t e(t)
On sait que +A
lim A=oo
I
b (0) e iWE/2 t
2 dt-b(o)
-A
On va. estimer
I, (w)"
"A
lim A-+OD
I
e
iwe:/2 t
2 c(t) t dt
-A
Puisque b a un support compact on a c(A) - c(- A)
done
(31!.) 1/2
w
e
iITE/4
153
+A
( ) __I_
I,
lim
iwe:
W
A-
J-A
~
2
e iw£/2 t
c'(t) dt
2 l'integrale est bornee parce que c' est une fauction de classe C bornee ain2 iwE 2 / t c' (t) dt tend si que ses derivees 2 (on montre que e
J:
d'ordre~
vers zero quand A et B tendenr vers l'infini en faisant des integrations par
a nouveau
parties). Le raisonnement peut s'appliquer
pour l'estimarion de
I'integrate figurant dans 1 (00). 1
On a montre que :
l(w) On. b(o) -
On.
is.. at dt
+~ 30
do 30 dt
au point t - 0, a
=
a
o
on a done
2
do
E •
dt ) taO
done do
dt
I t-O
_
32~
1- 1/ 2
aa
n-o
o
D'oll finalement l'evaluation asymptotique
l(w)
ou 3°) Supposons que f a un ensemble fioi
OJ'
j Go J, de points critiques non
degeneres sor Ie support de a. En utilisant une partition de l'unite sur ce
154
support, isolant lea points critiques, on demontre que
ill£./4
•
E
I(w)
j
E
J
iwf(a.) J a(a.) J
•
J
ou
2. Methode de 1& phase stationnaire (plusieurs variables).
Considerons l'integrale I(w) •
JY
a(y) .iwf(y)
d~(y) m
oil a et f sont des fanctions a valeurs reelles. C
sur une variete riemanien-
ne C~J Y, d'element de volume d~(Y). de dimension n. to) Supposons que a est a support compact et que f o'a pas de point critique sur Ie support de a, on a alars, pour tout.N E. E
:
-N I(w)-O(w) En effet on a :
g
iwl
ij
•
d'oil
•Lwl
I
-
iw
iwl
•
en posane (si f n'a pas de point critique on a
g
dX v
i
af
ij H i
dX
j
i
0)
155 c'est
a
dire Vf v -
•
IVfl 2
i<.Jf
I
- iw
done
Jy .i<.Jf(y) div(av) duly) -I
lew) - O(w ). Le resultat
l'integrale etant bornee on a
oew-N )
s'obtient
par iteration.
2°) Supposons que f a un point critique et un seuI, Yo' sur Ie support de
a. et que ce point critique est non degenere c'est a2 f en ce point est une forme quadratique non
a dire
que Ie hessien
degeneree, a
q carres po-
sitifs et p - n - q cards negatifs. n'apres Ie lelIlDe de Morse i1 existe des
coordonnees locales
t
J
, ••• t
i(t) • f(y(t»
n
au voisinage de y
o
telies que:
- i(o) + Q(t)
avec I
Q(t) '"2
p+q
q
E i -
E i - q+1
Dans ces coordonnees :
I(w) •
J
bet) .iwQ(t) dt
I
avec
bet) - ",wi(o) a(y(t»
Idet
1/2 g.. 1 1J
au gij sont les composantes de la metrique dans les coordonnees t. La quanti-
te det Hess
£1
det g est un scalaire; on a au point Yo' en coordonnees
I• done, en ce point
t
156
c1est
a dire b(o) - e
iwf(y ) 0
a(y) Idet o
g .• 1 ~J
1/2 /
On mantre, par un raisonnement analogue
ou a - sign Hess
£1 Y·Yo -
Idet Hess
Y-Yo
a
celui fait pour une variable que
a(y ) Idet
gl 1/2
o Idet Hess
12 fl yayo
P
0
q-p
3°) 8i f a un nombre fioi de points critiques Yj non degeneres sur Ie support de a, 1(00) est estime par une somme de termes de la forme de l'expression precedente.
3. Integrate dependant dlun point x£.
x.
Dans Ie cas OU Ies fonctions f et a sont definies sur Ie produit de deux
varietes X x Y et ou on considere l'integrale (3-1 )
l(w)'
fy
a(x,y) eiwf(x,y)
on obtient I'estimation, pour w grand: (3-2)
OU
I(w)
E j€.J
OJ. sign Hess fly_yj(x) , au Yj(x), j £ J designe l'ensemble suppose £i-
ni et non vide des points critiques, supposes non degeneres, de f ree cOIlllile fanction de y pour x fixe et au on a pose
'fj (x)
consid~-
- f(x, Yj (x».
157
a. (x) '" a(x,y. (X'). J J
Remarque
L'expression 3-2 est a la base de la methode de quantification de
Maslov-Leray. Pour que 18 phase
~
soit determinee globalement sur n il faut
que modulo 2 II
ce qu'on ecrira, et d ($
0
~-
8i
0 ~
y est un I-cycle sur V, puisque 'fi(x) - ($
-)
)(Yi(x»
e
J) =
mod I
131. \41. 151.
oil [nJy est la variation sur y de "l'indice de Maslov" de V (cf
161
de VIII). Dans les applications
a
la mecanique quantique on prend w -
I/~.
4. Parametrisation d'une variete lagrangienne V.
Soit V une sous-variete lagrangienne de T~. Si la restriction projection II :
r*x ~
av
de la
X est un diffeomorphisme V + n.des coordonnees locales
x A sur X fournissent aussi des coordonnees locales sur V. Vue phase ~ sur V faurnit une phase
'f
sur
n
et I' application
n -+ T~
par x
l-+
(x?
'f)
est un
diffeomorphisme de n sur V? qu'on appelle une parametrisation de V. On generalise la definition de parametrisation. dans Ie cas ou TI pas un diffeomorphisme de dimensions.
NOllS
en plongeant X dans un espace
a
un
1v
n'est
plus grand nombre
exposerons iei Ie cas Ie plus simple Oll l'on introduit un
seul parametre supplementaire. Designons par f(x.a) une fonction sur X )( :R et par C (x.a) ItC- X x E.
af tels que 'aa (x. a) - O.
f
1 'ensemble des points
158
Definition
La faoction f
x x E
-+
IR est dite une parametrisation de V si
I' application
est un diffeomorphisme de C£ sur V. 5i fest une parametrisation de V la faoction VI c'est a dire verifie sur cette variete d$
~
- f
0 ~
-I
est une phase sur
ite ou i est I'inclusion de V
dans T~. Une condition equivalente, puisque ~ est un diffeomorphisme de C
f
sur V, est en eifet "vC.T X,a Cf
Cette relation est verifiee d'apres la definition de 8 et Ie fait que, sur
Une faoction f parametrisant V fournit done aussi une fODction phase
n de
un ouvert
X diffeomorphe par la projection
n a un
ouvert
~i
sur
U. de V. On 1
pose : -1
'fi - f c'est
"
0
-I
0
IT i
ou
IT i - ITlu
i
a dire
If i (x) au l'application ~
-I 0
• f(x, ai(x)) TI
-I
i
: x'" (x.ai(x»
est determine par resolution en a.
de l'equation de Cfl ~~ (x.a) • O. Cette resolution est possible (au moins 2
loca1ement) si
a f (x,a) , ---2 aa
o.
L'ensemble singulier I de 1a variete lagran-
gienne V (cycle de Mas1ov) est l' image par lJ des points de X x lR . au
if
et - - •
aa 2
~~ .. 0
o.
La definition d'une parametrisation d'une variet~lagrangienne s'etend en remplac;ant
X'x
lR par un espace fibre de dimension R. + m au dessus de X (par
159
exemple
r*x
dans lequel Vest
deja
plange). L'obtention d'une parametrisation
donne la possibilite de fixer dans I' ouvert n C
dant
X une phase
'Pi
c.orrespon-
a chaque Vi diffeomorphe par ITlu. a n et de remplacer 18 valeur asymp1
tatique d'une integrate du type 3-1 en utilisant )-2 par une samme de termes
faisant intervenir ces phases. Exemple : supposons que nous ayons une seule variable d'espace x et une varie-
te
lagrangienne dans Ie plan (x,p) qui passe par Ie point (0,0), et est tan-
gente en ce point i l 1 axe des p. p
v
11 est slors naturel pour Ilde-plier 18 singula-
rite", clest a dire iei pour trouver la varie,te
C et 18 faDction f(x,a), de chercher cet enf
x
semble C dans X x Eo saus la forme: f x
af aa·
ce sera l'ensemble
a
2
0 si f est de la forme f(x,a) • o(x) + xa -
f sera une parametrisation de V si
at
ax
0' (x) +
a
est tel que l'on a v _
c'est
a dire
{x~
0' (x) + n}
x • a2
si Vest compose des deux ouverts u
1
..
°
{x~ o'(x) + Ix}
x >
lx, a'(x) - Ix )
x > 0
et U ' 2
et de l'ensemble singulier :
r •
(0,0)
160
VIII DEVELOPPEHENT ASlMPTOTIQUE AD VOISINAGE D'1JNE CAUSTlQUE.
1. Caustiques du premier type.
La forme dlune parametrisation d'une variite lagrangienne est liee a la nature des singularites de l'inverse de la projection
n : T*x
+
X restreinte
a
V. Lea singularites des applications differentiables ont ite classifiees par Thorn. On montre que dans Ie cas Ie plus simple ( pliage, suquel correspond l'exemple
a 18
fin de VII) on peut parametriser V par une fanction de la
forme :
f(x,a) - a(x) + p(x)a ou a et p sont des fanctions regulieres du point x E X.
L'ensemble C est slars f
af - p(x) aa
- a
2
0
. a2 f L'ensemble singulier E de Vest l'image de l' ensemble --2 := - 2a - 0 de Cf" aa
II se projette sur X en p(x) - 0 et on a
U E
±
+ EP
1/2
V p)
x p(x) > 0
161
La,
figure represente la projection sur
X : l'ensemble singulier
~
se projette
sur l' ensemle C appete "caustique".
L'ensemble p(x) < 0 n'est la projection
p(x) < 0
d'aucun point de V, I'ensemble p(x) > 0 est recouvert deux lois par la projec-
p(x) > 0
tion de V : deux "rayons" psssent par
chacun de sea points.
2. Integrate asymptotique, fonction d'Airy.
Pour construire des solutions asymptotiques d'un systeme differentiel correspondant i une variete lagrangienne du type precedent on va etudier Is valeur asymptotique de 11 integrale
u(x) -
3
I
- e iwo(x) e iwf(x,a) a ( x,a ) da =
On sait que
(theor~me
I
iw(p(x)a ~ a ) 3
e
.(x,a) da
de preparation de Malgrange) il existe des Jonctions
regulieres 8 (X), al(x) et h(x,a) telles que: 0 2 + b(x,a) (p(x) - a )
. done puuque
Ie iwf
(x, a)
p(x) _ a 2 •
af an Jeiwf(X,Q)da +
a(x,a)da • •0 (x)
+
Le dernier terme peut s'ecrire
I
a)
(x)
Jeiwf(X,a) ada
e iwf(x,<» b( x,a ) at aa d a
162
ao.a
J
iw
il est done d'ordre
1 . w
h(x,a) do
En repetant Ie meme raisonnement on trouve un develop-
pement de 13 forme feiWf{X,Q> do +
u(x)
t
w-q c (x) q
q
Je
iwf(x,a)
a da
Dans Ie cas considere f(x.a)
_
- a 3 /3
a(x) + p(x)
on introduit la fonction d l Airy
..t (t)
J "
2IT
on a alars
et. en derivant sous Ie signe d'integration par rapport i
Je iwf (x,a) ada. -+n
e iwo(x)
.;t, (w2/3
p:
p(x»
w
D'ou
r
q
w-q b (x) + ;fl:.'(w2/3 p(x» q
Iw-q - 2/3 ·c (x») q q
On a les estimations suivantes pour 18 fanction d'Airy Je(t)
~
.It. (t)
:II.
J
~4 _ t l/4
AI
2
3/2
2
312
cos() t
.
81.0(3 t
-
.!!4 )
pour t > 0
.!! )
pourt>O
4
gTand
•
.gTand
(peut etre obtenu par 18 mithode de 18 phase stacionnaire).
On volt que pour p > 0 et w2/3 grand on a pour Ie premier terme du developpement :
163
eil.l.lO(X) u (x)
2
2/3
cos(~
~ -tiI7"'W~IF{.1l1·(W""'2"J'l'3p·)"T07'T.4
p
J /2
. (2wp2/3
s~n
3
IT ]
-"4
cette expression coincide avec celIe obtenue dans Ies etudes d'optique geometrique en presence de caustique, on cons tate en particulier un changement de
phase de n/2 entre les deux termes. Remarque
pour t <
o.
on.
grand
1 tl
2..n (_ t)1I4
it- (t) '"
Un
I
J
.'
_ 2/3 (_ t)3/2
ill: (t) '"
(_t)I/4 a - 2/3 (- t) 3/2
Si l' on admet que l'integrale donne aussi une estimation de u pour. p <
On trouve que
U
devient rapidement petit avec
a dire
lei une estimation pour p • 0 (c'eat fait que pour
t
voisin de zEro on :!t(t)
~
8
I/lpl ,
P
<
o.
O. On trouve aussi
sur la caustique). en utilisant Ie
:
0.355) 0.259
t
3. Onde a haute frequence, solution approchee d'ordre I d'un systeme differentiel lineaire.
Considerons Ie systeme differentiel line-sire (3-1)
o
j • I, ... N
de polynome caracteristique A(x,p). On cherche une solution approchee, pour
w grand, sous la forme :
164
(3-2)
ou b
i
et e
i
soot des feuctions de z de la facme
(3-3)
b
tandia que f(x,a)
~
CJ
i _ bi + 0
et P sont
1.
bi
"'
I
li~s
c
i
iI 1. geomitrie des rayons du
probl~me
pos~
:
3
,,(x) + p(x)a - a /3 est une parametrisation de 18 variete lagran-
gienne engendree par les rayons dans l'espace des phases. On voit alae-ment sur
80n expression que la fauction d'Airy verifie l'equa-
tian differentielle : (3-4)
o
'*"(t) + t Jl:(t)
En reportant )-2 dans )-1 et annulant Ie terme en w on trouve, compte tenu de 3-) et 3-4, et du fait que
systeme lineaire en b
i
eft et JJ;I
sont des fauctions independantes, Ie
i
o· c 0
o qui est equiv.lent pour p> 0 nus en faisant
£ •
+ 1 ou E
a l'ensemble
des deux systemes lineaires, obte-
I dans
00 en conelut, en accord avec les resultats precedents, que les fenctions
If••
£ •
doivent etre solutions de I'equation eikonale
+ I
ou
165
Un calcul analogue
a celui
fait en l'absence de caustique (cf II) peut alors
etre fait pour determiner les equations de transport de b
'[lJ
o
et co'
Choquet-Bruhat Y., De Witt-Marette C•• "Analysis manifolds and physics"
nd 2 edition. North Bolland 1980. [2]
LUdwig D. Uniform asymptotic expansion at a caustic. Comm. pure and app.
Math•• Vol XIX p. [3J
Leray J'
t
215-2~O.
1966.
Solutions 8symptotiques des equations aux derivees partielles
(une adaptation du traite de V.P .• Maslov). CODvegno International. Metodi valutativa della fisLes matematic8, Acad. Naz. dei Lineei 1972. [4J
Leray J •• Seuunaires du College de France 1976-1977 et livre a parattre
aux H.I.T University press. [5]
Gui11emin V. et Sternberg S. "Geometric Asymptotics" A.M.S., Providence
1977 (Math. Surveys nO 14).
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
U R T I
GUY BOILLAT
URTI
Guy BOILLAT
Introduciamo un vettore che dipendenti
x•
e
un inBieme di N funzlonl delle variabili In-
(CI. 1, 2, ••• In)
u'
•
u(x )
u' c
La varlabl1e xO • t rappresenta usualmente 11 tempo mentre xi (1 n) 80no variabili di spazio.
1,2 •.•. ,
Scrlvl~o
A"'Cu, xlt)u. • fCu.
x~
(1.01)
dove
e Ie AII.
80no matrici NX N generalmente dipendenti dal campo u e delle varia-
bili x~ • La somma ~ sottointesa sugl! indict rlpetutL .• Un tale sistema 61 chiama quasi lineare. 5e Ie matrici
60no
indipendenti da
semi linearei se poi anche la fUnzlone sorgente
U
61 he un sistema
r non dlpende da u il siste-
ma s1 dice lineare. Dimentichiamo ora la dipendenza esplicita de ~ .11 sistema pub easere ri.&cri tto coal
170
o A (U)U
t
1 + A (ulu 1
= £(u)
(1.02)
Definizione di IperbolicitA. Gli autovalori di A
n
Ain. 1
(ir veraore
delle
spazio) rispetto a AO 80no tutti reali per egoi ~ ed estate una base di autovetteri delle spazio di u. Questa implies Ie regolarita di AO e pertanto 11 sistema (1.02) s1 puo mettere Botto Ie forma 1
+ A (U)U
1
All'autovalore
= £(U)
(1.03)
(i)
di molteplicitA m
(1 )
devono corrispondere m
Butovetto-
ri (destri e sinistri) linearmente indlpendenti cosl definiti = 0, (1.04)
J '"' 1,2, ••. ,m
che denoteremo
anche
se~pllcemente
IJ
(j)
dIe
In particolare se tutte Ie metric! di (1.01) sono simmetriche,
ed inoltre AO
e
cio~
definite positive, (1.01) viene chiamato sistema di Friedrichs.
E chiarO che tali sistemi 80no Iperbollci. In generale un sistema qualunque non s1 puc mettere nella forma 61mmetrica; perc 1 ~tica
616~eml
della Fisica mate-
si possono rieondurre. come vedremo • ad una tale forma.
2~
Sistemconservativl
Questi hanno una (orma speciale nel senso ehe si Bcrivono come divergenza nello spazio-tempo di certi vettor! t~(u)
1 £"(u) •
£(u)
(2.01)
oppure. 'con l'introduzione del aradiente rispetto a1 campo u.
(2.02) che corrisponde 8 (l.Ol) con
(2.03)
171
Poiche AO non ~ 8~ngolare s1 pub quindi scegliere f
O
come campo u ed allore
(2.04)
Per un fluida, ad esempio.
('U u*
1 k u
rik + p, ~u
i
(f+ p)u
dove,f
=eu2 /2
+
e
e
(2.05) i
l'energia totale, e l'energia interns, 1
• +
pit
l'entalpia libera legata al1'entropia da di • TdS + dr/e
(2.06)
Come conseguenza segue 18 legge di conservazione dell'entropia
(2.07)
Abbiamo qui un sistema con un equazione in piu; l'ultima e pera una conseguenZB delle altre. In generale data un sistema conservativo
(2.08)
e un'equazione scalare canseguenzB (2.09)
se s1 fa 18 derivata rispetto a t e 81 sostituisce (2.08) s1 he l'identlta
che deve essere vera per ogni U da cui 1 i
1,2 •••• to
(2.10)
Friedrichs e Lax hanna fatto vedere [1] che. definita 18 matrice hessians ~
Hor;:YVh, la matrice HAL ottenere
Wl
• slmmetrica. Basta allors moltLpllcare Is (2.08) per H per
s1stema del tipo
172
(2.11)
HC
che e un sistema simmetrico nel sensa di Friedrichs purche Is funzione h(u) sia una funzione convessa di u, cioe H
definite positiva.
Quando s1 moltiPlic8 (2.08) per H s1 perde 18 forma conservativa, pero con
&.
un cambia di variabili s1 pub ritrovare e si perviene ad un sistema conservativo e simmetricQ. S1 introduce 11 campo nuovo data da
u'
e
quat~ro
C2.J
.-
=Vh
(2.12)
funzioni scalar! (2.13)
In particola~ per ~ : 0 h' = u·.u _ h.
e una
h'
O.
h'.
(2.13' )
trasformata di Legendre.
Deriviamo la (2.13) rispetto 8 u'
e da (2.10) risulta
'f-. V'h.tl
(2.14)
~ • "'h'
(2.15)
In particolare
Ne segue
"/'(UI)U~
(2,]6)
=C,
cioe Ie nuove metric! non solo Bono simmetriche rna anche hessiane. La forma (2.16) estate introdotta de Godunov
Notare 18 differenza
in (2.11) • (2.16).
131
con tre esempi.
Ie matrici simmetriche sono
ri8pettivam~nte
173 3. Eguazionl di Eulero Applicando un principia variazionale alIa lagrangiBna L = L(q: ' qB)
(3.01)
dove Ie q B (x" ) Bono funzioni delle spazia-tempo e \ B
:z::
" -UllI(q.
31 arriv8 aIle
equazioni di Eulero (3.02)
che s1 possono mettere nella forma conservativB (2.04) con 'ClL;)qB
,r!
B qi
u
q
•
~L~·
')L/)q;
0
=
-qB ~ j o i
, f =
0
(3.03)
0 q
• 0
Se L non dipende esplicitamente de qS basta eliminare 18 terza riga. C'altra parte s1
pu~
definire una quantitA con due indici (3.04)
tale che
,..
')T"-O -
(3.05)
se Ie equazioni di campo (3.02) sono soddisfatte. Le (3.05) rappresentano quattro equazioni supplementeri. La conservazione dell'energia corrisponde a ~
= 0 che seeglieremo
come equazlone (2.09) con (3.06)
51 deve ricavare u'. 11 che signifies che dobbiamo valutare Ie derivate par-
z1a11 di h rispetto aIle component! della u. Indichiamo le componenti di u nella maniera seguente U
o B
u.
U
i B
u·
174
s1 ricava u· in
questo modo
• q
r
o
e similmente per Ie a!tre componenti. Infine s1 ha
u' u'
u'
S
S
0
• qo
1 • _ ')L!)qs 1
(3.07)
S
_ ')L/~s
u'
S
(51 taIga la terza riga se L 01 conseguenza - u
,j
- u'
•
S
o
f
u'S
o cio~ ~,
f
(3.08)
o
sono funzloni linear! di
U'j
Ie equazionl di Eulero prendono la
forma (4J • (5J, 1 H'(u')u' + AI u' "" B'u' t 1
(3.09)
L'uniea nonlinearita e dovuta a1 coefficiente della derivata temporale. Le 1 matric! A' • 5' 50no costanti, Ie prime simmetriche, l'ultima emisimmetrica
.-
B'
:c _
S'
Se L non dipende esplicitamente da qS, f e il secondo membra della (3.09) sono nul11; un caso gill considerate da Godunov [3j .
Studiamo ades60 la convessitA di h.
Con (3.03.07) s1 ottiene tornando aIle variabili 101z1ati
175
'1>u.h' (3.10)
somma di tre forme quadratiche ciascuna delle qual! deve essere definita po-
sitive. Se cerchiamo Ie velocitA caratteristiche basta far corrispondere
e 81 ottiene
Quando la velocita
e
nulla segue Bubito de (3.08)
n'lu,j J r
'" O.
Se invece supponiamo ~ ~ 0
'i u tr
:: O.
(3.11)
0
de (3.02) 81 he
o. Deve risultare nullo 11 determinante delle quantitA tra parentesl. Le sue radiei
~ (1)
usualmente sono diversi de zero. Ad ognt autovalore
~ I:
0 e asso-
ciato un certo 8utovettore. Supponiamo che per un certo ~alore di ~ Is veloetta diventa nulla, signifies che la molteplicita dell'autovalore ~= 0
menta, rna 11 numero di 8utovettori associati
e
8U-
sempre 10 stesso. date de (11).
51 perde cod l' iperbolici tAo 51 deve dunque evl tare che
c\ (;r)
= 0; Is matri-
ce
deve essere regolare. Una condizione ovviamente soddisfatte se vale la con-
vessita (vedi (3.10).
176
4. Urti Supponiamo che attraverso una superficie ~(x~)
o che s1 muove col tempo,
di normale,
e veloci til
il campo u
~
Itt /19'(\
~ -
6
discontinuo
[uJ • u -
U
~
o
O.
(4.01)
Al sistema (2.01) 51 applies i1 teorema di fluBso-divergenza e 81 arriva alle equazioni di Rankine-Hugoniot che s1 scrlvono (6)
[r16[U)= n
r
0,
ri
n
(u)n.
•
i.e .. f (u) n
B U
r
=
n
(u ) 0
IS U
0
e
trovare i1 campo dopa l'urto in termini del campo U prima o l'urto e di s (velocitA dell'urto) Il problema
U
:::: U
(4.03)
o
Derlviamo (4.02) rispetto 8 8 e u o (A
(A
n
- 01)& - h
- 61)(1 +
(4.04)
• 0
V h)
• A
noon
- 61
Supponiamo che l'urto sia debale. Facciamo tendere b a zero; s1 he (A
on
II che signifies che s
6
- 61)&
O.
0
e
autovalore di A oo
& =rJ.d(u~). o
0
(4.05)
177
Os (4.03) segue
h(u •
',n) ~ 0
(4.
I
0
0
(4.06)
e, facendo 18 derivata rispetto a u
o
(4.07) Prendendo 11 dterminante della
(4.05) s1 ottiene
Voh)
dot (1 +
• dot (A
- sI)/ dot(A
on
n
- .1).
L'equazione precedente d~ Bubito 11 valore del primo membra
dot(I + V h) • 1 - V ~ .Ii • 1 - ~. o
0
11 secondo membra s1 presents -! II (~ o
-
1 .)/II(~
-
'0 una
0
0
0
forma indeterminate
s)
11 cui lim! te - 1/(~' - 1)
o
derive facilmente della regale di L'Hospital. Pertanto
1-/\ So ~:
o
o
=l/(l-~·). 0
V>. o
o signifies che
vo ~.d 0
0
•
0
.Ii
0
• 0
li(~ 0
0
0
-2).0.
(4.08)
qulndi
(4.09)
O.
Torneremo eu questa condizione di "eccezionalita. ". cosi chiamata da Lax e supporremo per 11 momento
i.. o
= 2,
•
(~+
che non
verificata (urto normale). Allere
~ = ~ + ( (. - ~ ) + •••• o "'0 0
l- o )/2
+ 0(. _). )2
cioe 18 velocita di un urto debale ristica.
e
(4.10)
0
e
11 vel ore medic della velocita. caratte-
178
Questa mostra che s u
e
compresa fra
~
o
~. 5i pub quindi sempre chiamare
e
10 stato tale che
o
(4.11)
escludendo 11 caso ~eccezionale" (4.09).
Unatale disuguaglianza insieme aIle equazioni (4.5) conduce aIle condizioni di Lax [71
: s non raggtunge rna! una velociU. cratteristica rna
e
compreso
fra due autovalori consegutlvi sia per 11 campo u che per 11 campo u
o
(4.12)
(4.13) per ogni
1'.
in modo tale che (11) viene verificata qua Ie che sia k. Cosl. gl1
urti vengano classificati secondo 11 valore dl k.
5. Entropla. Funzione generatrice
Supponiamo adesso l'esistenz8 di una legge di conservazione supplementare (2.09) e, in analogta con Ie equazioni di Rankine-Hugoniot consideriamo la funzione dell'urto (5.01)
In condizione di differenziabllitA (2.09)
e
ebbe pensare che ~ o
"J.
e
conseguenza d1 (2.08) e 81 potr-
nulla se Yalgono (4.02). Inyece, di solito, non e Ye-
Se s1 inserisce (4.03) in (1) 51 ottiene (5.02)
Deriviamo rispetto a s
i
=
• Vh)h -
0,]
:;. (A
• n
- 61)h - [h]=
W
(5.03)
(5.04)
tenendo conto di (2.10) e (4.04). Essendo h una funzione convessa di u.
179
h(u ) - h(u) -
'lh(u
o
=~~
- u)
2.
0
u
u C
+ ~(U
o
-
U
~
~
- u)H(u ) (u C
0
),
0
0
~
- u)
"'? 0,
o Cl: (1
e pertanto w .., 0,
~
V h
O.
(5.05)
Oa (3) e dalle condizioni di Lax segue
i=j"WdS )..
(5.06)
purch~ I' urto sia nullo quando s = ).. • 01 conseguenza [
o
81 (5.07)
che traduce come vedremo la crescenza dell'entropia.
Facciamo la derivata di ( v Vj=U'(A
o
1
n
= (~I - ~')(A o
ispetto a
U
o
v -sI)(I+Vh)-u'(A 0
on
0
on
- 51)
-sI)
...
in virtu di (4.05). Se conosciamo la funzione ~ (uo,s,o) possiamo rieavare l'urto in termini delle variabili u'
r vu')
= V"'(A
o'
on
- sI)
-1
(5.08)
L'entropia genera l'urta. Accanto a questa proprieta la formula mette in rtsaIto il ruolo importante di u' chiamato da T. Ruggeri
pale
[91 ."
campo prine i-
11
Per 11 fluido (§ 2) la legge di conservazione supplementare (2.09) che sceglieremo
e
e
quells dell'entropia (2.07) e vedremo che
funzione convessa di u
(2.05). Infatti [10)
......,
2
Tdh • u.d(eu) + (G - u (2)d~ - d
dove G
- TS
e
l'entalpia libera e
t.
(6.02)
180
....u u'
-1 T
2 G - u /2
(6.03)
- 1 che coincidono con Ie variabili introdotte direttamente da Godunov [3J .
Per la convessi ta basta considerare [10
'1
rf./T)( U 2/2_rGJ+S [T]-V [p1)
w =
o
0
0
Mo
e pertanto deve essere
Ne risulta che - G deve essere una funzione convessa di peT. una condizione verificata per l'equilibrio termodinamico [11)-
D'altra parte
1
(6.04)
ma dall'equazione di conserv8zione della massa segue la continuita di
e dunque
i
=
"
\0
(0 - u
on
l(S) .
(6.05)
De (5.07) deriva allora la crescenza dell1entropia
-'>
'ir(u ' s. n) s1 esprime nel modo seguente o
Infine la funzione generatrice
(7/C vooe c (' M )
exp
dove c
e la
2 • M 'ljO 0
velocita del 9uono, M
o
r
2
•
«(-
"
= (uon
1)/(1+ 1).
_,1
l"'
+~M2)Y~O +f).l" _t'2~
- s)/c Cp /C v
0
0
0
11 numero di Mach,
181
C
p
e C sana i calori specific! costanti per un fluido politropico. v II grafieo della ~ si compone di due rami corrispondenti all'urto lenta e
all'urto veloce, con due punti di flesso per S
s
=
u
+ on -
c
0
V <'1-
= u on
e un punto isolato
1)/2Y
~
asintot~
due
Co
~= 0 per 5= u
on
per
corrispon-
dente all'urto caratteristico. Per i material! iperelastici della meccanica de! continui (6.06)
dove T • (T .. ) 1J
e
il tensore di Piola-Kirchhoff, F . (F.
1
gradiente di spostamento. Se poi v. =')u./~
e
j
.';lu.ldX.) J
1
tensore
la velocita di spostamento, 0_
1 1 ' +
la densita costante nello state di referimento, hi Ie forze esterne, il siste-
ma (2.04)
e definite
cosl (su una base ortonormale di vettor!~) 1
Tji~ F
u·
• £
i
~
•
-
V
!!fl6
-0
CI e
£
i
0 -+-0
e
ej>b.v
v/ ji
con 18 conservazione dell'entropia
~S/~ = 0 . Da h
~
- S e (6.06) ai deduce ~
v
u· •
T
-1
(6.07)
- T - 1
51 dimostra come nel caso del fluido che -5
che questa sia vera
~e(s.
18 velocita dell'urto
e
e
una funzione ccnveSS8 di u pur-
F ). Ne segue la crescenza dell'entropia quando ij
positiv8
£12 J .
11 quadrivettore (2.13) he Ie componenti
182
7. Urti caratterlsticl Floora a1
e
conslderata una soIuzlone (4.03) delle equazioni di Rankine-
Hugoniot (4.02) dipendente da un solo parametro s. Adesso ci chiediamo se e possibile di trovare una Boluzione = u
u
o
+ h(u
• u
I
~
(7.01)
,n).
0
ehe dipende da piu parametri u
I
(1::l:" 1. 2 •..•• p). 5e p : 1 posslamo sce-
gliere generalmente s come parametro. Deriviamo (4.02) rispetto a u (A
poi rispetto a u
I'
n
-"r)')
I
h -
1 (7.02)
')1" h • 0
e eliminiamo h
':lI" ~,h) = O. Se
p.., 1 esiste
almena uno de! vettori che non
sarlamente un autovalore
e nullo.
Ne segue che B
e neces-
di A
n
Studiamo, in generale, questa possibilita. Supponiamo per primo che 6
...
= stuo ,n),
autovalore di A on
Sostituendo nella (4.02) e facendo la derivata rispetto a
e
u
I
viene
pertanto s deve essere soche autovalore di A . Scriviamo dunque Ie equazioni n
(4.02) con
(7.03) per ottenere
(7.04) e d'altra parte. facendo Is derivate rispetto a u
o
(7.05) Mol~lichiamo
queste equazioni per un autovettore corrispondente all'autova-
lore ~ di mol teplici tA
(il
m
183
(7.06) (I ,·h)Vc\(I + 'loh) I
-
I'=1.2 •.•• ,m Se l ,.h I
l ,(A -~I), on I
(7.07)
(i)
~I ~
"0, dalla prima segue
O. Se invece vale l'uguaglianza.
18 seconds equBzione d~ 1
cioe
r.
I'
(A
on
- ~I) = 0,
e autovalore anche di A
• dipende soltanto da U e dunque abbiamo'
o
on
encore
(7.08) e d.
(7.04) (7.09)
i. e. h puc dipendere dB tanti parametri quanti pOBsono essere i vettori (1)
f)Ih indipendenti. vale a dire m
(1)
leI. 2,
.'Of
(7.10)
m
La (8) s1 scrive
oppure tenendo conto di (9) (7.11)
Questa uguaglianza, ehe deve vel ere qual~he sia u
erie
la condizione di ec-
cezionalita di Lax.
La (8) fa vedere ehe ~ zione nulla u I
=u
o
e
indipendente di u
I
e siccome ammeUlamo la 601u-
(corrispondente per esempio alIa nullita di tutti i para-
metri u ), segue
...
...
~(u,n) = ).(u ,n).
o
(7.12)
La condizione (11) 51 incontra speaso in Fisica matematiCBj b3sta citare
184
Ie oode di materia, di Alfven, gravitationali, di Born-Infeld, della corda relativistica, ecc. [13]
• E'sempre verificata per Ie oode moltiple di un sis-
tema iperbollco conservativo (14) • Deriviamo (A-~I)d_O n 1
nella direzione dell'autovettore d
I'
Scrivlamo la steese cosa cambiando gIL indict e facciamo 18 differenz8. Perch~ 11 sistema ~ conservativo An ..
Vfn e
VVtndI,d
J
"VVCndrdII .Resta
Ne rlsulta necessariamente (11). Nel caso di motepllcltA esistera dunQue sempre un urto che 81 propSSB con una velocitA caratteriatlca e che chiameremo
urto caratterlstico. Nel caso di un autovalore aemplice la condizione (11) di ortogonali tA del gradients dl ~ edell' autovalors corrispondente pu~ anche
essere noddisfatta. p. e. per ali urti di Alfv6n. 5ia ~ (~)
=0
l'equazione del frente d'urto che verifiea l'equBzione carat-
teristica
If' (u,'fill ) - \flu0 •'I) ill
(7.13)
0,
'f t +\''fhl. (u, 1)
\f'(u,'t.t) -
(7.14)
51 ha (7.15) e derivando rispetto a
1/'
+'OO(ll"'fj\-:>':1 +'#''$\
MoitipUch1amo Ie equaz10nl del campo per 1 1 AJ. 1
Me lIA~).
e
u
fI
(7.16)
1 (7.17)
- lIt
un operatore di derivata tangenziale aIle superCicle d'urto come
a1 vede da (l3,lS). 01 conaeguenza a1
pu~
aostituire nella (17) il velore
185
(1) di U Bulla 8uperCicie
~(u0 + h( u0' UI',n») "" II' f ; I, I 1 I A" ~ I~un
I
(I ) "" 1,2, ...• m
I
sistema di equazioni differenziali ordinarie per ali u . Infatti, easen-
dO')rh un autovettore destro di An
(vedi 9) Ie derivate de! coefficienti ap-
tramite termini del tlpo
patono
(7.18)
che tenendo conto di (16) 81 mettono Botto la forma (7.19) dove
dId . . .
~"'f '"
A la derivate lunge i raga! dell'urto (7.20) Po~eh~
11 sistema
~
iperbolico la matrice dl coefficient! II.dr' i invertibile
e (14) pub essere rfsolto rispetto aIle derivate. Le quantitA ~
mente
~i~ Bono Ie component! della velocltA radiale. La ~ chiars-
una funzione
omogent~del
primo grade rispetto aIle
f•.
Pertanto
del tearema di Eulero
oppure
E'interessante di notare che mentre Is velocita normale ~
e vero, in generale,per 1a ve1oc1tA rlspetto a 'f.c. tenendo U costante s1 he o
questo non (13)
~~ r =
e continua (12), radiale [141. Derivando
'4"rr +V'f'')~h
pl'/'1. -lV'fIVc\.')loh e i1 secondo membro non
e.
di solito, nullo
Per esempio, 1n un fluido esia-
te un urto. cosl chiamato di contatto, che 61 propaga con la veloc1tA carat-
186
teristica
continua s "" -:.n
"t.n. Invece it bene conosciuto che, per questa o urto, la velocita radiale ~ e discontinua [~1 ~ o.
8. Soluzione esplicita Abbiamo
gta
debole, il saIto di u appartiene al-
~ • Quando l'urto non
10 sottospazio degli autovettori di pu~
e
vista che, quando l'urto o
e piu
debale 51
encore dare una forma espllcita del saIto. non di u rna di u'.
1 . Derivando
alIa funzione aeneratrice
r.>"'.';;'(A r t n
Ie (5.01) con s
grazi~
=~ .
.0,
-l-I}?h
I
in virtu di (7.09). Siccome ~ ~ nulla per l'urto nullo ne risulta '" (u
I
0
,~ ,~) !! 0
(8.01)
0
Pertanto, la sua derivate rispetto a u
o
e
anche nulla e tenendo conto dalla
(5.03) viene
"'1 '0
.-wV~ 0 0
che inserita nella (5.08) foroisee
= u'
h'
(8.02)
- u·
o
Adesso introduciamo un vettore a(u.~) cosl definito I • 1,2 ••.. ,m
(i)
(8.03)
La soluzione estate proprio perche vale la condizione di ecceziona1ita (7.11),
e
unica perche
e
ortogona1e ag11 autovettori d
puo mettersi sot to la forma
r .
11 saIto del campo princi~~
[151 (8.04)
Nel caso lineare g sarebbe nullo. 11 secondo termine rappresenta dunque 1a parte non lineare dell'urte caratteristico. I due vettori dipendona sol tanto I
delle state prima dell'urto e 90no cono9ciuti. Resta da determinare ..... (u ,u ). o I Deriviamo (5.04) rispetto a u =
';l
I
'"h'.h
187
cio~
• w• 1 I
10
.h/(l - g .h)
(8.05)
0
purch~
(8.06)
Per la derivata seconda
v +') h'H'~ hi I I'
•
(8.07)
9. Stabll! ti. dell' urto caratteristlco Sia (9.01)
01 = cost .•
una soluzione ovvia di (7.13). Scegliamo come parametri u sono del tutto arbltrari ) delle funzioni lineari di
Se supponiamo che
If.
1
(chef a priori,
Segue da (B.05. 07)
quat!he sia l'urta, (9.02)
l-gh>O o
come 10
e
quando l'urto
e
debale, ailars
la derivata prima cresce e siccome
e
nulla quando l'urto
Ne risulta che w tende all'infinito con h.h' ~ w
't .
e
nullo,
e
positiva.
Ma s1 vede facilmente che
188
e
e dunQue anche l'urto non rappresentata ad
e~empio
f e Questa
e
- '(
limitato appena s1 sposta 18 superficie d'urto
da un'equazione del tipo (1) i
'no.
)I·0 t
= x ",, -
i l caso dell'urto di contatto gil citato. In un urto stabile Is quan-
tita (B.D6) cambia di segno [161 . Vedremo come 51 traduce Questa condizione per Ie equazioni di Eulero.
10. Evoluzione dell' urto earatteristico di Eulero
DaIle equazioni di Eulero scritte nella forma
U
t
(vedi §3)
+
AIlu'
= Btu·
i
(10.1)
derivano Ie condizioni di Rankine-Hugoniot A'h' -I-h
(10.2)
= O.
" Da cui segue, tenendo conto del legame (§ 2)
e di (8.04). h = H'h' -
o
wV')./ 0
0
At
(10.3)
Definiamo gIl Butovettori 1'(A' - ~H')= O. I "
tale chef I' =- 1 I
.-
d'
I
I
(A' - ~H' )d' n I
H'd' I
(10,4)
0,
~II'
.
Allers da (3) viene semplicemente,
mentre g .h =
o
dove 0(.0 =o«u ' o
"1) e
0( w ,
(10.5)
0
una ftJn:z1one conosciuta del campo prima dell'urto
189
(10.6)
La (8.05) diventa u 1(1 -0( w)
'I
(10.7)
0
che s1 integra subito -' 2 (1 -",w) = - 0(
o
lui 2
0
+
a(u • ~). lui o
La costante di integrazione s1 determine sapendo che l'urto I
que anche g11 u • vedi 6.04) quando w
~
nullo
(5.05) e
~
nullo (e dun-
r17] (10.8)
Vediamo dB (5) che 1-ah-1-o(W o 0 L I instabi!i tlt corrisponde
guenzB l'urto
Non
e
e
8
-<.o "
O. Invece se 0(
limitato; 18 quantita
1 - goh
0
"> o. Iu I
e di conse-
~
continua. Derivando (4)
HI )V'd' - H'd' V'~ -~V'H'd' = O. I I I
Moltiplichiamo a slnistra per
vata del autovettore
W
cambia allore di segno .
difficile di vedere che Ie velocita radi31e (A' n
t
Ii
e a destra per ')i
di
che rappresenta Is deri-
rispetto a ~i
V't, ~id'
• -l
I
Se adeaso s1 moltiplica
8
destra per
di
(10.9)
Qottlene
~
Derivando rispetto a ~ e ricordando che H' = V'V' h I s1 trovs che Ie quanti tA al secondo membro di (9)
e
nulla. A causa della condizione di eccezionalita
(7.11) risulta che
'i ,')i't'di • -V 'cf)~i - 0 ehe equivale aneora a ') ')iu. "i '''] I ' ; : 0 e pertanto[ II T. '"' O.
190
Una conseguenza immediata
e
che, per quanta abbiamo detto alIa fine del §7,
non esiste per i l fluido un principie variazionale con una densitA h (3.06) convessa. Dopo passagi lunghi assai [17J la Legge di evoluzione (7.17) dell 1 urto s1 · d·ff . 1·1 esprime in un modo molto sernp 1 ice; un sistema di m(1) equaz 1 on~ 1 erenZ18
per quantita VI legate in modo non lineare agl1 u
r
. Quando ~
0
e
negativo
l'urto puo effettivamente tendere all'infinito dovuto in particolare alIa formazione della caustica.
11. Limltatezza della velocltA dell'urto Per un urto norma Ie , cioe non caratteristico. l'evoluzione non s1 fa lungo Ie caratteristiche. Da ciaseun lata della superficie i campi u e u
soddiso fano i1 slstema a derivate parziale. Le due soluzioni si devono poi raccordare sulla superficie tramite Ie equazione dt Rankine-Hugoniot. II salta del campo dipende come abbiamo vista della velocita dell'urto che, a priori, se non si assumono Ie condizioni di Lax, puO anche andare all'infinito.lnvece can una densita di energia convessa la velocita Abbiamo il seguente teorema (19)
e
limitata [IS)
(dove fu pubblicato per la prima volta ?).
Sia f una applicazione continuamente differenziabile di un aperto convesso di R in R . Se la parte simmetrica della matrice jacobiana
N
cazione
N
e
e
definita, l'appli-
glabaimente invertibile.
La matrice jacobiana dt (4.02) rispetto aIle variabili u '
e
simmetrica
A' - s H' • n
Basta allara che i3
)
sup u'60 '
max
inf
min
u'fiO'
i
1
oppure
s
<.
(0' aperto canvesso) purche
~siste
,( 1)
r
una soluzione unica che ovviamente e
191
u' = u'
o
i. e. 8SSenZa d'urto. D1
con~nza
i
valori estremi delle velocita caratte_
ristiche costituiscono limiti per Ie velocita dell'urto in accordo con Ie
condizloni di Lax. In particolare in una teoria relativistica tutte Ie veloci-
ta
caratteristiche sono minori,in velere assolutto,della velocita della luce.
Ne segue, a causa della convessita, anche 18 limitatezza della velocita
dell'urto
lsi, c.
Ringrazio la Dott.ssa Franca Franchi per gli appunti prest de lei durante 11 corso e che sana serviti come base per la stesura del testa.
192
8ibl1ografia
[1]
K. O. FRIEDRICHS & P. D. LAX. Proe. Natl. Acad. Sci. U. S. A., 68 (1971) 1686.
[2J
G. SOILLAT, Comptes rendus, 278 A (1974) 909.
[3J
S. K. GODUNOV, SOY. Math., 2 (l9611 947.
[4J
G. BOILLAT, Ann. Mat. pura ed appl., 111 (1976) 31.
[5}
100' Comptes rendus, 283 A (1976) 539.
[6]
A. JEFFREY, Corso C.r. M. E.• in questa libra.
[7j
P. D. LAX, Camm. Pure Appl. Math., 10 (1957) 537.
[aJ
G. BOILLAT, Comptes rendus, 283 A (1976) 409.
[9}
T. RUGGERI, Corso C. 1. M. E. • inquesto libra; T. RUGGERI &- A. STRUMIA. Ann. lost. H. Poincare, in corso di stamps.
[IOJ
D. FUSCO, Rend. Sem. Mat. di Modena, in corso di stampa.
[11J
D. ter HAAR & H. WERGELAND, Elements of Thermodynamics, Addison-Wesley
Publ. Co., Reading, Mass., 1966. [12.1
G. BOILLAT & T. RUGGERI, Acta Mech., 35 (1980) 27l.
r13J
10 •• Boll. Un. Mat. Ital., 15 A (1978) 197.
[14j
G. BOILLAT. Comptes renduB, 274 A (1972) 1018.
1)5]
10., Compte. rendu., 280 A (1975) 1325.
(lQl
ro., Ibid., 284 A (1977) 1481.
[17]
ID.. J. Math. pure. et appl., 56 (1977) 137.
[IS]
G.BOILLAT & T. RUGGERI, Comptes rendus. 289 A (1979) 257.
[l~J
M. BERGER & M. BERGER. P~rspectives in nonlinearity, W.A. Benjamin, Inc. New York (1968), pag.137.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
8ULLA TEORIA DELL'OTTICA NON-LINEARE
DARIO GRAFF!
SUlla
t~oria d~ll'ottica non-lin~~
Dario Graffi Uaiv~rsita
di Bologaa
Com~ ~ noto. un campo ~l~ttromagn~tico in un dominio ~ del-
1.
10 spazio
. . . _--+~lettromagnetici) E, H. D. B. J detti.
tori po
~ rapp~s~ntato da cinqu~ v~ttori (ch~ chiam~remo
~l~ttrico.
zione,
campo magnetico.
d~nsitA
rispettivament~.cam-
v~tto~ spostam~nto, v~tto~
indu-
di corrente. Questi vettor! sono, in generate,
funzioni d~l punto)(~ P
~ del t~mpo t quindi. a rigo~, si do-
~bb~ scrivere, in luogo di altri
vet-
E: E(X, t)
~ analogam~nt~ p~r gli
v~ttori.~l~ttromagn~tici.P~r~, p~r
semplicitA,
quest'ultima scrittura solo nei casi in cui
~
us~remo
necessaria per
ewit~rf ~~uivoci.
Come
e
noto, i vettori elettromagnetici sonO legati Bra loro
dall~ ~quazioni
di
Ma",,~ll:
, (1.2) ch~
esprimono
l~ggi
sempr~ ~ ovunqu~.
Ma
sono
valid~
anch~ corr~dat~
da op-
fisiche fondamentali l~
(1.1)
~
(1.2),
~ perci~
portune condizioni iniziali e alIa frontiera, non sono su££ic~nti p~r d~t~rminar~
un campo
~l~ttromagnetico ~ p~rci~
biso-
198
gma aggiUAgere opportune equazioni costitutive che nell'elettr<JllagnetiSJllo ordinario (suppoaendo, come faremo sempre i. seguito, esterne a ~ Ie sorgenti del campo elettrOlllagmetico e,j>e"' .....
- --
isotropo il mezzo ia 'f' ) SODO: (1.3)
t> ~
Eo Eo
- -
, (1.4) ~ = .. '" ./
1'"
dove E. '.Y' SOIlO rispettivamellte la cost ante dielettrica, la peraeabilitA magnetica, la conduttivitA del mezzo ael punta
X in
cui si cOllsiderano i vettori che cOlllpaiollo rispet-
tivamente in (1.3),(1.4),(1.5). Be poi il mezzo e anisotropo Ie £ ,
r-
Poiche E ,
' f vanno sostituite COil tensori -doppi.
r:J'"
dipendollo solo dal mezzo e nOll dal campo e-
lettromagnetico, le (1.1),(1.a),(1.3),(1.4) e (1.5) costituisCODO un sistema 1iaeare, perci6 l'elettromagnetiSJllo ordilla-._ rio si puO chiamare anche e1ettromagnetismo liaeare. Per6 in quei dielettrici (ai quali ci riferiremo sempre ill seguito) dove si manifestano i fenOllleni dell' ottica noa lineare J mentre restano val ide Ie (1.4) e (1.5). 1a (1.3) va sostituita
COD
UJla relazione non lineare ira 0 ~
E
che scriveremo:
(1.3') sicche i mezzi in cui vale (1.3') si possono chiamare dielettrici non 1ineari. Ia questa lezione cercher6 di.stabilire alcune proprietA
del~e
onde elettromagnetiche che si propa-
,,\\--..t.o
gano ne1 die1ettricorcosi da interpret are qualche fenomeno del1'ottica nOll lineare.
2.
Riferiamo i punti della spazio a un sistema di coordinate
cartesiane ortogonali (O,x,y.z) e supponiamo il dominio coincidente con una 1amiaa di spes sore
5
riempita da Ull
~
199
dielettrico non lineare omogeneo. Porremo ~
l'origin~
0 e l'asse
del sistema di assi in modo che le facce della lamina abbiano
equazione z=O, z=s. All'esterno della lamina supporremo il vuo-
to che, dal punto di vista elettromagnetico, si pu6 identifica_ re con I'aria. Indicheremo con £.la costante dielettrica del
vuoto, mentre ammetteremo la
r
che compare nella (1.4) ugua-
le a quella del vuoto (ipotesi non restrittiva dal punta di vista fisico)
ci~ ammetteremo~ identica
i. tutto 10 spa~io.
Nel semispazio z < 0 sia posta una sorgente che generi un' onda
elettromagnetica piana con all'asse
~.
dire~ione
di
propaga~ione
parallela
Supponiamo la lamina tagliata e disposta in modo
che il campo elettromagnetico dipenda solo da
e t; anzi, con
~
un'opportuna disposizione degli assi x e y si possa scrivere, per ogni punta dello spazio, : (2.1)
"E.
E. (l,lo)L"
laoltre supporrenioIi parallelo ad
E ci~; avremo:
(2.3) Allora la (1.3') diventa (sottintendendo le variabili z e t)· l'equazione scalare:
(2.4) e la fUnzitne D(E) verrA supposta di classe
c~
in qualunque
intervallo limitato dell'asse reale. Le equadoni di Maxwell nella lamina si riducono a:
200
La. (2.5) e (2.6) valgol1o aache all'esterno della lamil1a purchi!
r
si pOl1ga .0, e. ill luogo di ~ ~ • Ammetteremo inoltre, conPorme llesperienz~: (2.4')
cioi! D Pul1zione crescente di E e D(O)_O. Stabiliamo ora alcune condiziOl1i sui pian! che limitaao la lamina,pi~
precisamente sui piani z=+O, z=s-O ; si i! scritto +0 e
s-O per identi£icare Ie Pacce dei piani rivolte verso l'interno della lamina
o,pi~
brevemente,£acce interne.
Ora, nel semispazio z < 0 s1 avranno due onde, una che diremo
diretta, emessa dalla sorgente e che si propaga nel verso positivo dell'asse z, l'altra ri£lessa dal1a lamiaa e che si pro-
paga nel verso negativo dell'asse z. Detti E~(z,t), ~(z,t)
g~(z,t),
HdJz,t),
rispettivamente il campo dell'onda diretta e
il campo dell'onda rifles sa, si hal (2.7 )
Ora,
} c~e
e
noto, su un piano che separa due mezzi divers! so-
no continue Ie cOlllponenti taagenziali al piano del campo elettromagnetico (ovviamente z=-O, z=s+O sono Ie facce della lamina rivolte verso l'esterno
0
facee esterne). 5i ha cos!:
(~.:l)
£.olJ_O,!;)+fo",(-o,~). E (d,t)
(2.9)
Hd.(-o,~) .. I-1,,{_O,~). 1-1 (+o,~).
Ora, per note proprietA delle onde elettromagnetiche piane si ha:
(2.10) l4.l(-O,!;).~ Eol(-O,~) ./
Sostituendo (2.10) e (2.11) in (2.9) e sommando con (2.8) moltiplicata per \;~
.r
si eliminano E,.,. e HJ(... • Allora, riservando
201
il simbolo E{z,t), H{z,t) al campo entro la lamina ed ometteado, per semplicita di scrittura e perche ora non vi
e
luogo ad
equivoco, i segni + e - davanti allo 0, si hal
Hel semispazio z;> s si ha solo un'onda che diremo trasmessa e che si propaga nel verso positivo dell'asse z (non si possono avere riflessioni perche per z >s il mezzo e omogeneo) i cui campi indicheremo con
E~(z,t), H~(z,t).
Per la continuita del-
le componenti del campo elettromagnetico sul piano z=s (ora si possono evitare i simbcli +0 e _0) si hal
e poiche "~vale
Viii- If~ si ha subito:
(2.14) Le (2,12), (2.14) in cui
E~(O,t)
si suppone assegnato, cost i-
tuiscono condizioni alla frontiera per (2.§) e (2.6). Ad esse si possono eventualmente associare opportune condizioni iniziaIi, sicche i1 campo entro la lamina resta determinato.
Le (2;12) e (2.14) si devono al Frof.Cesari (11 (2) [3] [4] il quale ha dimostrato importanti teoremi di esistenza, di unieita, di dipendenza continua dai dati per Ie soluzioni delle
equazioni
(2.~)
e (2.6) corredate da (2.13) e (2.14), qual ora
sia noto Eo\,{O, t) per ogni t (positivo " negativo). Nel caso, importantissimo per le applicazioni, in cui co rispetto al tempo e con periodo T,
~nche
E~{O,t)
e periodi-
i1 campo entro la
lamina risulta periodico con 10 stesso periodo. Noto i1 campo
entro la lamina, mediante (2.8), (2.9) e (2.13)
e
colare i1 campo ri£lesso e trasmesso dal1a lamina.
facile cal-
.202
I teoremi di Cesari sono stati dimostrati per valori dello spessore s della lamina non trappo elevati. Torner6 in seguito su
[5]
questi risultati, per ora noterO che il Prof. Bassanini
ha dimostrato che i valori di s per cui sono validi i teoremi ora citati risultano superiori allo spessore delle lamine usate in pratica_
3.
Passiamo ora a
ricerca~
una soluzione di notevole interes-
se di (2.5) e (2.6) supponendo (come faremo sempre in seguito)
fhO. A questa scopo poniamo, ricordando (2.4'), (3.1) ~
Nel caso lineare (si ricordi (1.3»
= E.
(cost ante die-
lettrica) ed esiste una soluzione delle (2.5) e (2.6) per cui i1 campo elettrico ha l'espressione:
E(Z,t) dove G(u)
~
= G(u)
,
una fUnzione di classe C. della u per u variabile
in qualunque intervallo limitato dell'asse reale; G(u) se u=t vale il campo elettrico suI piano z=O e all'istante t,
sicch~
Ie proprietA della ru,zione di t,E(O,t), sono Ie stesse di G(t) o G(u).
Ora. nel caso lineare p(E)=
J8y' ;
viene perciO naturale con-
getturare valide Ie (3.2) anche nel caso generale sostituendo perO nell'espressione di u a V~, peE) come definita do. (3.1) e con seguo positivo. Si ha cosi:
(3.3)
E .. G(u)
(3.3')
u .. t - p(E)z.
203
~rimo m~mbro
di (3.3) si ha l'~quazion~ implicitam~nt~ E in £unzion~ di t ~ z :
Ora, portando G(u) al ch~ d~fiftisc~
E _ G(
(3.4)
t -
p(E)z ) • 0 z=o.
Qu~sta ~quazioft~ ~ ovviament~ risolubil~ p~r
risolubil~ p~r z~O ~ sufPic~ftt~, p~r ch~
implicite,
il
£Unzioni
)
condizione
c~rtam~nt~
discuter~
Cal n.5) di ~
t~or~ma d~ll~
sia
sin I
(3.5)
sto
AffiRCh~
intuitivo, che
z e (O,h), t
E:
soddisfatta
z=O. Ora,
p~r
amm~ttiamo, com~ d~l r~
meglio la (3.5), ~sista
(-'J;,T) ( T,
~
Uft
h> 0
ftum~ro
~
T positivi avv~rt~nza
sia valida (3.5). Fino ad
ris~rvandoci
tal~
del
che
r~sto
p~r
ogni
arbitrari)
ift contrario,
amm~tt~r~
i1
campo magne-
mo z e t Dei limiti ora indicati.
Cic
prem~sso, v~diamo
di
d~t~rminar~
tico H che, associato al campo
~lettrico ~spr~sso Maxv~ll
(2.5)
A questo scopo ricordiamo che i l Prof.
J~ffr~y
ha dimostrato,
n~lla
da
prima
soddisPa
d~ll~ Su~ l~ziofti, ch~
all'~quazion~
fi
(3.6)
soddisfar~ l~ ~quazioni
da (3.3),
di
sia
ta1~
valor~ d~l
~ rL~)
a
d~rivat~
%f "
E,
com~ ~spr~sso
~
(2.6).
da (3.3),
parziali:
0
che ora veri£icheremo direttameate.
A
qu~sto
a z
~
(3.7)
scopo
poi ';IE
III
oss~rviamo
risp~tto
che, derivando (3.3) prima
a t, si hal
(-L~ c;.'(I.<) ~11) BE
+ (,,'(...
JrCE)
= 0
risp~tto
204
p~r
Sommando (3.7) con (3.8) moltiplicata s~nt~
In
(3.5),
bas~ all~
segu~
p(E),
t~n~ndo p~
subito (3.6). l~
(3.1), (3.6)
(2.5)
~
(2.6) si possono
scriv~r~:
; Poniamo ora:
f
Ii
(3.10)
H. L
pCB) dB
.Yo In
bas~
alIa (3.10) H
Dalla (3.10),
~
una
Punzion~
d~rivando risp~tto
~H.~)9E !If / ' £it-
(3.11)
che coincidono con
l~
a z
di E ~
~ tramit~
E di t
~
z.
a t si hal
Gi!l&-LrCe).ge ~~
<;It-./''"
(3.9), quindi (3.3)
tanG una coppia di valori di E e H
ch~
~
rapp~s~n
(3.10)
soddisfano
all~ ~quazio
ni di Maxwell.
4.
di
Passiamo ora al10 studio delle sOluzioni delle equazioni Maxv~ll trovat~ n~l
numero
pr~c~d~nt~.
Supponiamo anzitutto 1a lamina di spessore in£inito, ossia s: -=>
,
sicch~
....
1a lamina occupa i1 semispazio z
~O •
Supponiamo che per t=t;" z=oYE. =G(t o ) ~ proponiamoci di d~t~rminare i valori di z e t per cui E riman@ uguale a Eo, valori ch~ rappres~nt~ranno una caratt~ristica d~ll'equazione
come ha osservato i1 Prof. Jeffrey. Si ha perci6 l'equazione:
(4.1) c~rtam~nt~ soddisrat~a
(4.2)
t - p(E.)z
se:
= to
•
(3.6),
205
Dif'f'ereuiando si ha subitOI (4.3)
Ora dz
~
10 spostamento del campo elettrico di valore
REo)
E~
nel
teJllPO dt, quiadi ~ la veloci tA con cui si propaga i l campo elettrico che all'istaate t. aveva il valore Eo. Hotiamo che la (4.2) si pu6 ricavare in altro modo di validitA
pi~
ampia. 5i osservi infatti che se G(t - p(E.)z)
~
costaa-
te e uguale a Eo, per la (3.6) si hal
da eui integrando e teaendo
CODto
che per t-to • z.O , s1 r1-
trova (4.2). E' bene notare che, essendo ~~!~t positiva, il campo si propaga nel verso positivo dell'asse z, quindi la soluzione del numero precedente rappresenta un'onda che si propaga nel verso positivo dell'asse z.
In particolare. se il campo
~
nullo per zsO
in un certo istaa-
te t •• esso si propaga COn velocitA l/p(O). Hel caso per noi
pi~
interessante in cui G(t)'CO per t .. 0, si
cOOlprende c£e per t:> 0 si avrA un f'ronte d' onda, n0.lche si sposta col
temp~ di
ci~
Ull pia-
ascissa z. s z.(t) tale che per
z >z., E(z.t)=O. per z 0 e del resto qualsiasi. Poich~ E ~ uguale a zero per ogni t suI f'ronte d'onda. la sua velocitA sarA la velocitA del campo nullo,
ci~
il fronte d'onda si sposta con velocitA l/p(O).
5. Passiamo ora a discutere la (3.5). Anzitutto se G' (u) e 'df>P/; haJUlo (se diversi da zero) per ogni u e per ogni E 10 stesso seguo (per esempio G(u) e peE) sono
206
Punzioni crescenti, la prima rispetto a u, l'altra rispettG a E), la (3,S)
e
sempre soddisfatta e It
zOO
per ogni t, ~O
IB questa caso, se Ie condiziOBi iBiziali sonG Bulle per z • suI piaao z=O
e
assegnato per ogni t positive il campo elet-
trico, per UB teorema di unicitA del campo elettromagnetico, (3.3) e (3.10) (purche si asswu G(U)=O per u ... 0) rappreseJltaJlO 11 campo elettromagnetico caapatibile con Ie condiziOBi iJliziali e alIa frontiera e che si propaga nel verso positivo del-
l'asse
2..
Tornando al caso geJlerale, cerehiamo di dimostrare l'esisteJl_ za del Dumero h;> 0 di cui si
e
accennato al B.3.
A questo scopo, aggiungeremo un'ipotesi
pi~
che plausibile dal
punta di vista fisico. Ciee la funzione G(t) (0 che e 10 stesso G(u»
che rappresenta il campo E(O,t) sia limitata assieme
alla sua derivata G' (u) per t" (-- ,T); ia altre parole esistaao dUe lNIIleri positivi: H e H' tali che per ogni u .. (-..., ,'1') sia:
(61(....) I ~ M
(S,l)
Inoltre per Ie nostre
I g I'(E¥c>E \
Ci~
I ~ '(.. . ) I E; M'. ipotesi -fi.. o a c-k...
liJlitata da
U1l
I B \ .. H
sarA
numero II.
premesso, fissato un istante t, esisterA un numero positi-
vo h(t) tale che per z £ [O,h(t») , (3.S) e verificata e quiadi (3,3) risolubile. Allora per questi valori di z, t, che
I~
1< N,
I E(z,t)l::: IG(U)\ ~ H,
inoltre IG'(u)l~ H'.
Dimostriamo ora che esiste un numero h o tale che h(t);;. h o t E. (to che
<>0
e
sic-
'
,T). Infatti sostituendo h(t) in (3.S) e tenendo COnsoddisfatta se G' (u)
?lff)
si hal
207
Quindi: (5.3)
h(t)
> ~,= ~~
cCllle si era a1'i'e1'lllato. Assumeremo h~h. l'estremo ini'eriore delJli h(t) per te(-oc.T). T puc) essere anche ini'inito Perc) nel caso G(t)-O per soddisi'atte) e t non
~
purch~
t~
0
sia soddisi'atta (5.1).
(sicch~
(5.1) sono certamente
molto elevato, segue h-
sia p_ ~ 0 i l minimo valore di peE) per gono
le relazioai:
(5.4)
t .. T
t < P.
.-..
ho =
00.
I III ~ M,
lni'atti
aHora se val-
n"",..
N M'
i l valore di u che CClllpare- nella (5.2) ~:
-
u = t - p(ll)h(t) (; t - " h < P.
(5.5)
....
Ma allora il G'(U) della (5.2)
~
-
h - p h - O.
nullo e questa equazioRe noa
puc) essere soddisfatta per h(t) finito. Deve essere h= 00,0, che
~ 10
stesso, la soluzione (3.3)
~
valida,per valori di t
soddisfacenti (5.4), per ogni z, ed essa rappresenta il campo elettromagnetico in tutto il semispazio. Si noti che. come ved~~o
nel
num~ro segu~ntet
tempo in cui la (3.3)
~
N e malta piccolo; l'intervallo di
valida puc) essere sui'ficentemente gran-
de per le applicazioni pratiche. Hel caso in cui non siano soddisfatte Ie ipotesi ora esposte,
fissato t puc) esistere un valore z di z per cui la (3.5)
~
nul-
la, e se G(t - p(E)z) risulta diverso da zero. da (3.6) e (3.7) segue che
IH.'
te di E per z
:;l~J:E -to
z
1-
-1" DO
•
n.' BE 1~IM;lt' - ~~ --
,cioe Ie deriva-
tendono a di vent are infinite. 51 ha cioe,
conforme a una locuzione del Prof.Jeffrey, una catastroi'e. Si puc) cosi interpret are l'accennato risultato di Cesari per qui i suoi teoremi sono validi solo mina
~
sufficentemente piccolo.
~e
10 spes sore della la-
208
In seguito comvaque ammetteremo che (3.3) e (3.10) rappresentiao il campo elettromagnetico, almeno per valori di t e z suI£icentemente grandi per Ie question! pratiche.
6.
Nel casO s _... aotiamo che, mentre (2.12) rimae valida,
(2.14) non ha
pi~
signiEicato e si
pu~
sostituirla con la con-
dizione che il campo sia nullo all'inEinito, 0 meglio che il canapo rappresenti un'onda che si propaga nel verso positivo del-
l'asse z, condizione questa, come si
~
osservato al ••4, sod-
dis£atta dalle (3.3) e (3.10). Supponiamo ora I' onda Eot incidente suI piano z=O, col campo elettrico' (e quindi allChe i l campo magnetico) per t
a.. seali>t
Ie a
(<1. e
~
0 ugua-
w costanti) •
Converra introdurre la Eunzione di Heaviside l(t), (l(t)-l per l~l1.IL)
t ;1-0, let) -0 per t< 0) sicchl! si avra (sostitueJ:ldo.' per brevita di scrittura, E.t(t) a &..l.(O,t), e analoga selllPlilicazilJlle£aremo in segui to per i termini che campaiono in questa equazione): (6.1) Ci~
premesso, per ottenere £ormule semplici, supporremo, come
avviene spesso in pratica, debole la non linearita ciol! che sia leeito scrivere: (6.2)
I>(E)=
dove F(E) guit ..,
'I..
~
e. E
+
1. Eo
F(E)
una Eunzione di E che speciEicheremo meglio in se-
ua parametro adimensionale molto piccolo in modo da
poter trascurare in seguito termini in
'1.'- •
SUpporremo inoltre,
il che non l! afEatto restrittivo, F(O)=O. Quindi si avra. da (3.1) :
209
(6.3)
e da (3.10) : (6.4)
A110ra sostituendo in (2.12) si ha l'equazione per E(t), campo e1ettrico su1 piano z=O e su11a faccia rivo1ta verso l'internO della lamina:
Non sarebbe di£fici1e dimostrare che 1a soluzione E(t) della (6.5)
~
1imitata; ometteremo per brevit! questa dimostrazione'
Per riso1vere esp1icitamente (6.5) useremo un ben noto procedimento di approssimazione ponendo: (6.6)
Ora, come
~
noto, si pu6 scrivere, applicando poi i1 teorema
del valor medio e indicando con
~
nn numero compreso fra 0
e,1
(6.7)
=
1. F ( ~ (t) + "t ~.(tl)'1 f(E"I~))" "l (F(~(t) .. 1. £,I~V - F (eo (~»)
1 F(fol~l)+"l"F'LE',.(t)''O''lE,ltl)E.l~)
tenendo conto che l'u1timo termine l' ordine di
~
~ "l. F(£.(t-l)
trascurabi1e
perch~
del-
1.'1..
Con questo ragionamento si giunge a11a formula piu genera1e (A(t) e B(t) due funzioni limitate di t ):
Si ha cosi, sostituendo (6.6) in (6.5)
=
210
Quindi: (6.9) E !!:-) = 11!r. o l VF;;• J£
dove a=
vto
e
4 ......
•
..,t ..Jet):
_t_ It
.... ""
0"'" wI:- -i{C)
l'indice di rifrazione del dielettrico in assen-
za di non 1inearitA. Si ha poi:
Specia1izziamo ora 1a F(E) considerando 1a re1azione non-1ineare pill semplice Era DeE
Quindi essendo (l(t» (6.12) E,(t). _ ~
'L
",(4_)
ci~
= l(t)
« _/.,_ (4....)..
(c( costante) :
a: ~",t
Posta: (6.13) si ha quindi:
e i1 val ore di
E~(t), ci~
de~
campo e1ettrico de11'onda ri-
f1essa suI piano z=o, ricordando (2.8) e (6.12) :
Per avere il campo ri£lesso nel punta di coordinata z, basta
porre nella (6.15) al posta di t, t • ~ z. percM, ora 1'on./
da si propaga nel verso ~egativo dell'asse z, con velocitA ~ • olE;; ./
211
Nell'onda riElessa vi sona tre termini, uno di
£requ~nza ~
che si ha nel caso lineare, mentre la non linearitA porta per
'';U-
~
t -
z> 0 a un termine cost ante (rispetto al tempo) di scar-
so interesse pratico e a un termine
sinusoidal~
di frequenza
2 cJ , cioe nell'onda riflessa si ha un'onda di frequenza doppia dell'onda incidente. In altre parole, dalla riflessione su un dielettrico non lineare si pu6 avere i1 cosiddetto fenomeno della
duplicazione di frequenza. Per esempio, inviando 1a lu-
ce emessa da un laser, di lunghezza d'onda
6940 AO (cioe luce
rossa), si pu3 avere, nella luce riflessa, anche luce di lun-
ghezza d'onda 3470 AO, cioe luce violetta.
7.
Passiamo ora allo studio della propagazione dell'onda tra-
smessa dal dielettrico. Per la (3.3),
poich~
G(u)
~
espressa
da (6.14), si hal (7.1)
dr, t) <1
1) _ U)
- 't [ b -
(t -Hel", ~\)~) .{ (t-- r C€(',~I) 1,) l,
l
..,,2.w t-
-
r(etf,~)ll)1i( t-- r( E(~,H)I-)
Ora, per (6.3),(6.11), trascurando sempre i termini in ~'l. ha: (7.2)
,
si
rc~ l!,~)l • V€j (.l..~ '101 E(t,l))
Sostituendo (7.1) in (7.2) e trascurando i termini in
..
t,
te-
nendo presente anche (6.7'), si hal
Ora, ponendo in (7.3) (7.2) e sempre trascurando i termini in
...
, si hal (7.4)
)
r(E(~.~)).JF ({. "1O(Q../'-4t-~ ~l-i(t-"Y l);
212
Ora poniamo per brevit.!:
(7.5)
Ir.
~-~E)
Sostituendo (7.4) e (7.5) in (7.1) cazioni suggerite (7.6)
d~
si trova, con Ie semplifi-
(6.7')
e(~,l). 0.)<- (~_"1. ~ u'lIo(o.,...-,.&- A(t_~t)).(t-~l)- "L (b - beo-.l9-){ (r- "V'- I) .
Ora, con UJlO sv11uppo di Maclauri.relativo alIa variabile accorciato fillO al termine ill
t... ,
t.
e
si ha, per il primo termiDO!
a secondo membro di (7.6) (a meno s'intende del termine in'!,'-):
()..,I'",.+-- "tJVo. u>1-
Quindi in un punta " e ill tutti gli istanti t > che il campo dell'onda trasmessa di frequenza
u)
,
~
J"5-- 1
si ha
la somma di tre campi, uno
cioo il campo qualora si trascura la non li-
nea:rita, uno CQstallte e infine uno di frequenza 2 w
,cioe, co-
me nelltonda riPlessa, si ha una duplicazione di frequenza. Si noti che per z _"", il campo di frequenza doppia tendereb-
be all'infinito, perO per z grande nOD valgono Ie precedenti approssimazioni, anzi non sarebbe neppure valida'la (3.5) e quindi la soluzione (3.3) e (3.10) delle equazioni di Maxwell.
E' bene anzi notare ehe nelle ricercpe sperimentali dell·attica non lineare si considerano ovviamente lamine di spessore finite s. Se s
~
interiore a1 valore di h calcolato a1 n.5. i1
213
che avviene in pratica. le (7.3) e (3.10) sana soluzioni (senza catastroti) delle equazioni di Maxwell. Esse pera rappresentano l'onda entro la lamina e quel1a che ne esce
do~averla
.....
~
attraversata solo se si trascuraol'influenza dell~tacce interne della lamina stessa. Comunque non mi sembra inutile la seguente osservazione.
Eseguiamo il rapporto r fra i due termini di frequenza 2
~
•
dhe compaiono in (7.7). Ricordando il valore (6.12') di b e che
w~.
"'''''Jf.J'- • ,z,1f'_/ ~
dove'). <1! la lunghezza d'onda.
nel vuoto. corrispondente alla trequenza oJ • si ha:
Ora s vale 10·1 mm •• poich<1! n+l/2 <1! dell'ordine di unitA, r all'uscita dalla lamina (cio<1! per z=s) <1! dell'ordine del miglia io. Percia nella (7.7). nei termini in
2~
, prevale il primo.
almena all'uscita dalla lamina. Poich<1! b <1! l'ampiezza dell'onda riflessa. si pua atfermare che <1! molto piu facile osservare la duplicazione di frequenza nell'onda trasmessa che nella ri£lessa.
Notiamo che la velocitA di propagazione delle onde che compaiono in (7.7) vale 1/.JEj-
• cio<1!
tale velocitA non <1! altera-
ta dalla non linearita. 11 risultato non <1! pera generale. ad esempio se peE)
e
proporzionale al cubo di E si ha alterazio-
ne di velocita, rna su cia non insisto.
214
Bibliografia
1
L.C~sari-R~ftd.S~m.Mat.Fis.Univ.Hilano
2
L.C~sari-Ann.Scuola Normal~
Sup.Pisa 4. (311) (1974).
3
L.C~sari-Riv.Mat.Univ.Parma
13, 107 (1974).
4
L.C~sari-Rend.Accad.Naz.Linc~i
5
P.Bassanini-ZAHP 27, 409 (1976).
6
D.Graf£i-"Non linear partial differential equations in
45, 139 (1974) •
56-1 (1974); 57,303 (1974).
physical problems· Pitman London (1980).
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(C.I.M.E.)
SULLA PROPAGAZIONE DEL CALORE NEI MEZZI CONTINUI
GIUSEPPE GRIOLI
BULLA PROPAGAZIONE DEL CALORE NEI
~~ZZI
CONTmUI
Giuseppe Griol1 Un1versi~a di Padova In~roduzione
Il problema della propagszione del calore nei mezzi con~inui ha richiama~o in questi ultim1 tempi l'attenzione di mol~i studiosi i quali hanno prospe~ta~o ~eorie di vario ~ipo con 10 scopo di arrece~e un con~ributo alIa formulazione delle equazioni cos~itutive dei continui e nel contempo superare il cosidetto paradosso della propagazione del calore con velocita infinita di cui e affetta la teoria classica. Le varie teorie propos~e poggiano 0 su generalizzszioni di talune funzioni termodinamiche di state (a parer mio, discu~ibili del punta di vista fisico mstematico) 0 au una quslcbe modifica della claasica legge di Fourier cbe lega 11 vettore flus so termico al gradiente delle temperatura 0 su smbedue Ie case.
Ben nota e vsstissima e la ~ett~ratura relativa alla termodinamica dei continui e aIle equazioni costitutive.Mi limitere a citare taluni lavori attinenti aIle questioni considerate. Una soddisfacente formulazione della teoria della propagazione del calore per conduzione non pue prescindere dall'influenza dei renameni meccani91 concomitant! ad eccezione,se mai, de! caai ideali di continui a temeratura ignorabile e dei corpi rigidi. Gia il tener conto della completa interazione tra fenomeni termici e fenomeni meccanici da luogo a equazioni non pin affet~e dall'accennato paradosso me rende la velocita di propagazione del eelore uguale a quella del suono. Tale risultato PUQ essere accettabile me non e certo un privilegio della teoria classica il fatto cbe la velocita di propagezione delle onde meccanicbe non sie sostanzialmente influenzate dells propsgszione
218
termica enche in casi non isotermi ns adiabatici. Cia e dovu~ a1 fatto che nella trattazione abitua1e 1e derivate prime della temperatura sono ritenute continue anehe attraverso 11 fronte d'onda. Recentemente e stata proposta una teoria in cui si continua ad ammettere 1a va1idita della 1egse di Fourier ma si abbandona l'ipotesi di der~ate prime della temperatura continue attraverso i1 fronte d'onda.Ne nasce un problema ana1itico del tutto nuovo i1 cui studio ri@hiede l'uso delle discontinuita iterate secondo Thomas[121. Un'ipotesi costitutiva ben diversa da11a 1egge di Fourier e stata gia considerata da I-laxwel1(l]. In essa si ammette un 1egame lineare tra vettore flusso di cl1ore, la sua derivata temporale e i1 gradiente della temperatura.L'ipotesi,abbandonata dallo stesso ~-;axwell che ha ritenuto di dovere sopprimere il termine con la derivate prima te~porale ricadendo nella legge di Fourier, s stata ripresa da vari Autori ([2], ••• ,[6] ) e anche genera1izzata(lO]. La re1azione,associata a11a nota uguaglianza dell'entropia (che in generale e in realta una disuguag1ianza) da luogo a un problema non piu parabolico e implica ve10cita di propagazione finita. Tuttavia, i1 necessario procedimento di eliminaztpne t ove 81 tenga conto. com'e desiderabile~del1'inte razione @cccanica, porta la presenza di derivate terze delle componenti di sI.ostamento che insieme aIle seconde sana discontinue
attraverso i1 fronte d'otda, modificando tota1mente il c1sssico problema delle onde di discontinuita.L'inconveniente non si presenta soltanto ne1 caso dei corpi rigidi. Una teoria della propagazione termiea fondata su un'ipotesi 00stitutiva analo~a a quella proposta da :-:axwell che sembra molta interessante ma che tenga conto 1n modo completo dell'interaziozione tra tenemeni ter~ici e fenemeni meccanici a1 ott1ene dando
alla relazione di I-,axwell un significato un po piu generale, adatta aIle esigenze della meccanica dei continui, come mi propongo di mostrare. Le ipotesi aD1Clesse sono Ie seguenti: continuita attreverso il fronte d'onda della temperatura, del vettore f1usso di calore, dello $posta~ento e delle sue derivate prime; possibilita di disccntinuita di prima specie per 1e derivate prime della teffiperatura,de1 vettore f1us~o di calore e delle derivate
219
seconde della spostamento.A titolo applicativo e di indagine ho considerato il caso dei fluidi non viscosi comprimibili e inc omprimibi1i e que110 dei corpi e1astici isotropi poco de!ormabi1i. In ognuno dei casi considerati. si giunge. per la determinazione delle possibi1i velocita di propagazione, a un'equazione risolvente di quarto grado. In assenza di vinco1i interni, tale equazione ammette in generale due radiei reali positive, una malGiore e una minore di que11a ben nota che da la ve10cita di propagazione delle onde acustiche 10ngitudina1i. Mentre ne1 caso e1astico Ie onde possibi1i sono ne trasversa1i ne 10ngitudina1i, nel caso di un !luido non viscoso comprimibi1e esse sono,invece, solo 10ngitudina1i ma, come ne1 caso elastico, sono possibi1i due distinte ve10cita di propagazione, una maggiore e una minore di que11a solitamente ammessa a11a qua1e esse si riducono se si !a tendere a zero i1 coe!!iciente di ri1assamento. Si presenta,cioe, un secondo suono 1a cii possibi1ita e gia stata segnalata (vedi, ad es., [111). Di!!erentemente vanno 1e cose nel caso dei !luidi non viscosi incomprimibili in quanto, pur dipendendo ancora il problema da un'equazione di quarto grado" puo accadere che sia possibile una sola ve10cita di propagazione, come capita certamente ne1 caso che la pe~urbazione si propaghi in un mezzo in quiete a temperatura uniforme. EI interessante not are che ove Itincomprimibilita sia totale (cioe, la densita non dipende neppure dal1a temperatura) i1 comportamento del !luido per quanto concerne 1a propagazione di onde termomeccaniche e ana10go a quello dei corpi rigidi. Ne1 seguito ri~eriro in modo esp1icito solo su1 caso dei !luidi non viscosi incomprimibili, rinviando per a1tri casi a una nota lincea in corso di stampa [141. 1.- Qualche osservazione sull'eguazione dt Fourier.
Siano C e C' due contigurazioni del continuo in evo1uzione termomeccanica, delle qua1i 1a prima,!issa, e 1a contigurazione di ri!erimento, i'a1tra que11a attuale (all'istante t). Denotero con x r ' Yr Ie coordinate rispetto a una cedesima terna di riferimento trirettango1a 1evogira di punti corrispondenti P, P' di
220
C e C'.La corrispondenza tra C e C' glianze
e
caratterizzata delle ugua-
eontinue,invertibili e a jacobiano,D, positivo. Denotero,inoltre, eon T e S la temperatura assoluta e il vettore che caratterizza il flus so di calore in P'. Denotando con E l'entropia per unita di massa, dalla disuguaglianza di Clausius-Duhem,in assenza di un'eventuale sorgente di ealore,inescenziale per quanto seguira, segue (1.2)
ove il punto denota derivazione materiale rispetto al tempo. Tradizionalmente, alla (1.2) si associa la relazione eositutiva g = -L gradp,T , ove L rappresenta un operatore matriciale soddisfacente alla relazione
Supporro che l'entropia, i l'enercia interna, 11 cui val ore per unita di massa indichero eon J, dipendano dalla temperatura e dal complesso di altre variabili al,a~, ••• ,an caratterizzanti ~ la deformazione e il moto del continuo, mentre eventuali vincoli interni siano esprimibili nella forma (1,5)
Sussiste d1
r • 1,2, ... conse~uenza
la nota relazione dell'entropia
221
(1.6) ove i coefficienti Pi rappresentano delle incognite reazioni vincoleri da considerare tutti nulli in assenza di vincoli interni. Da (1.2),(1.6) segue (1.7)
)2 • cT - T 7.naJ s
9:s-
2; (Piti+ •
Piti) + divp.S
~
0,
ove (1.8)
c
-T
)2 J
17
denota il calore specifico sotto configurazione costante. E' ben noto che nel caso di sistemi a trasformazioni reversibiIi, come accade,ad es., nel caso di corpi iperelastici e di fluidi non viscosi, nella (1.7) vale il segno di uguaglianza, rna snche in Caso di irreversibilitA si suole ritenere cbe la (1.7~ considerata come uguaglisnza e associata alIa (1.3),rap~resenti l'equazione che regola la propacazione del calora. Spesso si ritiene trascurabile l'interazione tra fenomeno termico e fenomeno meccsnico,sopprimendo nella (1.7) il secondo termine e provocando cosi l'insorgere del paradosso della velocita infihita di propagazione del calore. Si supponga, per semplicita, cbe non vi siano vincoli interni e si interpreti l'uguaglianza associata alIa (1.7), associata alIa (1.3), come equazione del calore. Si denoti con la sbarretta la derivazione rispetto aIle x r e con Cw] la discontinuita (di prima specie) attraverso il fronte d'onda di una qualunque funzione w. Supponendo cbe L come J possa dipendere dalle as e da T e t c.om1e abituale,continua la temperatura attraverso 11
fronte d'onda, da (1.3),(1.7) segue - T
ft (a 1 J
•
s
a
+
222
L'ipotesi di eontinuita delle derivate prine della temperatura fa si ehe il problema della determinazione della veloeita delle onde aeustiehe non risentevdella propaJazione termiea.La (1.9) serve Buccessivamente,per la determinazione delle discontinuita
delle derivate seconde di T Ie quali si propagano, pertanto, con la medesima veloeita delle onde puramente acustiehe. E' possibile riconoscere che nel caso dei corpi elastici poco deformabili solo Ie on~e lon~itudinali trasportano una discontinuita termica. A parte ogni considerazione di tipo fisico, una trattazione di quella appena richiamate presenta il grosso inconveniente di non coosentire di asseeoare a piacere i valori iniziali delle disc ontint~ita
termiche e cl0 fa ritenere che i1 problema sia cal posta.
Osservando che Ie (1.2),(1.6),(1.7),(1.9) discendono dai principi generali della meccanica e della termodinalica, si conclude che unico modo possibile di modificare Ie cose e quello di riconsiderare l'equazione costitutiva (1.3) e,al fine di avere una effettiva interazione tra problema meccanico e problema termico,
di ritenere ehe Ie derivate prime della temperatura possano non easere continue attravereo 11 fronte dtonda ma possano 1vi pre-
sentare delle diseontinuita di prima specie. 2.-~
nossibile teoria della
propa~azione
La seneralizzezione della (1.3) e cui si colta da veri Autori
(2.1)
e espressa
termomeccanica
e gia
aeeennato e ac-
dall'equazione
o ,
ove h e L sono dei coefficienti non neBativi.;:D.X\<1ell gia. considero un'ipotesl del Eenere ma poi riten~e di sop~rimere i1 termine in g ricedendo nella (1.3). Facendo sistema tre la (1.7) Iriva del terwine nelle as e la (2.1) 81 ottiene un sistema che ~ediante IfeliDin~zione dei vettori S e porta a un'oquazione piu ~enerale di quella di Fourier e non pili parabolica e che d~ 1uozo a velocite di propa8azione
g
223
J•
finita[2 La relazione (~.l) e stata giustificata con considerazioni di meccanica statisticar21 e anche generalizzata[lOl • L4eliminazione dei vettori S e tra le (1.7),(2.1) non e agevole (e spesso impossibile) nel caso di deformazioni finite, facile nel caso di corpi rigidi. Comunque, essa da luogo a un gros80 inconveniente qualora nella (1.7) s1 tengano 1 termini nelle ;8' corote carretta fare, per 11 fatto aha compaiono Ie derivate delle as e cic crea delle compltcazioni non lievi e in un certo senso stravolge il classico problema delle onde di accelerazione.Ad es., nel caso dei corpi elastici il sistema differenziale risolvente dipende dalle derivate terze delle componenti di sposts8ento, dando luoGo a un problema del tutto nuovo e alIa necessita dell'applicazione della teoria delle discontinuita iterate. I~oltre, l'ipotesi di co~tinuita delle derivate prime della temperatura toglie l'influenza della propagazione termica BU quells meccanica. ~ostrero come eia possibile formulare una teoria priva dei Veri inconvenient! segnalati sempliceoente usanda una relazione differenziale piu generale della (2.1) nella quale si tenga conto in modo completo della necessaria in~erazione tra renemen! terQlci e fenemeni meccanici.A tal fine e fondamentale l'osservazione che la (2.1) si puc derivere dalla relazione eostitu-
S
tiva (~.2)
s
=
-hL
..S
e-hSg(t_e)de ,
o
ove g denota il gradiente spaziale di T, nell'ipotesi di h e L indipendenti sia dalla deformazione sia dalla temperatura e da t. Volendo tener conto della naturale interazione tra fenomeni termici e fenOiileni meecanici, si ammetta,invece, aneors valida la relazione costitutivB (2.2) senza escludere pero che L posea d~ pendere, in un modo che per ora non occorre precisare, dalla temperatura e dalla deformazione del continuo. In tale ipotesi e facile verificare che S soddisfa non alla (2.1) ma all'equazione differenziale
224
ove z L
e un
e un
coefficiente costante di rilassamento (L=l/h) mentre
operatore matriciale dipendente da T e dalle as. Si denoti con V la velocita d1 propagazione, con a e B 10 sca-
lere e il vettore caratteristici delle discontinuita delle darivate prime d1 T e g, con n 11 versore della normaIe al fronte d'onda supposto dotato di normale orientata (nel verso dell'avanzamento). Supposto che, come generalmente aeoade, i coef-
•
ficien~
trs,t
r
cheo compaiono nelle equazioni dei
vi~coli
si-
ano funzioni note delle as e di ~, l'applicaz1one delle formuIe d1 HUGoniot-Hadamard alIa (1.7) (pensata come uguaglianza) e alIa (2.~), de luogo aIle relazioni
(2i:Pil;f - cV)a (2.4-)
{ -zVl2
+
a(Lg
+
+
l2·n
zV ~~ L-l~)
- z
mentre dalle (1.5) 5i deduce
o. Da (2.4-,2) si trae
(2.6)
o.
La (2.6), confrOnteta con la (2.4-,1), de
o.
225
La (2.7) va associata alle equazioni sulle discontinuita che provengono dalle equazioni dinamiche, all'equazione di continuita e alle (~.5). Osservazione I~ II caso £.B = 0 va considerato a parte; e da preBurnersi che soltanto in casi eccezionali esso non dia luogo a iocompatibilita. Osservazione II~ Se 11 continuo e assimilabile a un corpo r~gl do le equazioni della dinamica non hanno influenza sulla velocita di propagazione di onde termiche.ln tal caso il problema e descritto dalla sola equazione (2.7), ridotta al solo termine in a e supposto che L sia un coefficiente di proporzionalita dipendente al piu dalla temperatura.Si ha, pertanto,
(2.8)
zcV
2
- z
,L
~ S.~
V- L
0,
la quale ammette due radici reali di cui una sola positiva, per ogni val ore positivo di z. Osservazione IlIa. II valcre z=o,posto nelle relazioni precedenti, porta a incompatibilita, avendo supposto S continuo anche attraverso 11 fronte d'onda e,invece, discontinue Ie derivate prime di T.Cie non puc sorprendere: infetti, per z = 0 Ie (2.3) si riduee alIa leGee di Fourier, incompatibile con tale supposizione. r~eglio si pub dire che l'ipotesi Z = 0 ha come conseguenza la continuita delle derivate prime di T e riporta alla teoria tra-dizionale con gli inconvenienti in essa contenuti. D'altronde, se si riflette che S in realta esprime una velocita di flusso di calore Be~bra naturale attribuire ad esso una -densita e a z i1 significato di coefficiente d'inerzia termica.ln tal mpdo il calore viene assimilato a un fluido dotato di massa e l'annullarsi di z non appere plausibile.
3.- Fluidi n2B viscoei
inco~primibi1i
Supporro che 11 continuo sia un fluido non viscoso BOr.Setto a1 vinco10 interne di incompri~libilita.Intenderotale vincolo in senso Beneralizzato rispetto all a sua co~sueta accezione; riterro,cioe, che oeni sua porzione possa variare di vo~ume se e s01-
226
tanto se varia Is tenperatura.Basta richiamare l'equ8zione di continuita per riconoscere che un tale tipo dt vincolo e esprimibi-
Ie nella forma 0.1)
ove
~
denota la densita all'istante t e F(T)
e
una funzione del-
la temperatura. La (~.l) Bostituisce il gruppo di equazioni (1.5).Denotando con
pIa pressione e con Y lao velocita in P', il gruppo delle equazioni dinamiche e di continuita e
•
0.2)
~+
~ divp'Y = o.
Datta pi l'incoGnita pressione di reazione vincolare e osservato che l'energia libera deve ritenersi funzione di T e SI Ie
equazioni costitutive, tenuto conto del vincolo di incomprimibilita, 51 scrivono
*
+ E - p'F
= 0,
+ p'
=
o.
Tenuto conto di (3.3), si ha
avando indicato con l'apice la derivazione rispetto aT. Si denoti con ~ 11 vettore che caratterizza La discontinuita delle derivate pri~e della velocite. ~ata la continuita attraverso 11 fronte d'onda di T,S,y e p, l'applic2zione dell~ note formule di Hugoniot-Hadamard porta aIle relazioni
[T J
227
In base alle equazion1 (3.1),(3.2), a1 ha
In definitive, tenuto conto di (3.5),(3.6),da (3.4) ai deduce
- (F
V3 ---~- +
2
mV)a,
pur d'intendere (3.8)
Tenuto conto di (3.5),(3.7) e ritenendo L funzione di T e di il sisteoa delle equezioni (1.2),(2.3) da luo~o elle seGuenti relazioni sulle discontinuita
~,
2
(F 2 _ V
+
~
m )Va -
{ z]2V -
[( ; ;
F +
1
~ £.~
0,
1~ + )z
q + Lq] a
=
o.
Escludendo l'ipotesi £~u = 0 che in generale porta a discontinuita teroica nulla (cioe,ad a = 0 ), da (3.9) segue l'equazione risolvente (3.10)
A
v4 z,~2 __
+
o.
~ Risulta, O.ll)
COrnie
A( 0)
c
facile riconoscere, -L
< 0,
Cia assicura che la (5.10)
A(tO")
>
a",~ette
o.
in ogni casm due radici
228
reali, una posit iva l'altra nesative. Se em>
0
Ie (3.10) non
ha altre radici realijinfatti,in tal caso, la funzione A(V) ha derivate seconda seillpre positiva e i1 corrispondente diagramma
la concavita rivolta senpre verso Ie A positive. Se s1 riflette che V indica una velocita di yropasazione rispetto 81 mezzo continuo ed e, pertanto, espressa de V = VI_y.~, se con V' s1 denota Ie velocita di avanzamento nello spaziox della superficie d'onda e si 8uppone che 10 stato di riferimento coincida con quello attuale, s1 comprende corne possano esaere accet-
tabili anche valori
neg~tivi
di V. La scelta del segno di V pre-
euppone una discussione che ~inuncio a fare. Se Ie perturbazione a1 propaga in un mezzo inizialmente in quiete e a temperatura uniforme, nella regione imperturbata e per 000tinuit3 suI fronte d'onds risulta ~= cost., 1).1 = cost., ! = g =
°
e la (5.10), tenuto conto di (3.8), diviene zv~
0.12)
+
mzV 2 -
L
~
= 0,
la quaIe ammette due radici reali in V2 di cui Q3a sola positiva. Pert ant in un mezzo non perturbato e possibile solo una velocita di propa~azione.
°,
Os~ervezione.
Be i1 vincolo di ihcomprimibilita
e
assoluto,
cioe se nella (,;1) si suppone F = 0 e,pertanto, risulta ~ = cost., l'eq~ione (3.10) si identifiea con la (2.8), valida nel caso dei corpi rigidi.Pertanto, sotto questo sspetto il fluido non viscoso assolutamente incmpricibile si com?orta come un corpo
rigido. BIBLIOGRAFIA [lJ [2]
"~~.ell,J.C.Phil.Trans.aoy.30e.157A 49 (1867) Caotaneo,C.Atti del Se~inario natecatieo e fisico dell'universita di ';odena.~.(194E)
[3]
Vernotte,P.Compt.Rend.Acad.Sci.~'6(1958)
[41 [5] [6]
Cattaneo,C. Compt.Rend.Aead.3ci.247 (1958 Rettleton,R.F.Phys.Fluids 3 (1960) Chester,M Phys.Rev.13l (1963) Gurtin,r. and Pipkin,A.Arch.Rat.liech.and Anal.3l (1968)
[7J
229
La]
I;eixner Arch.Rat.~ech.andAnal.39 (1970) ~ Carressi,M e florro,A. Nuovo Cimento 9B (1972) ~~ Carrassi,M. Nuovo Cimento 4b B (1~78) r~ LindsaY,K.A. and Straughan,B. Arch.2at.Mech. and Anal. Ge (1978) Bressan,A. In corso di stampa nelle Eemprie dell'Accademia Nazionale dei Lincei Grioli,G. Nota r a , in corso di stampa nei Rend. dell'Accademia Nazionale dei Lincei Grioli,G. ibidem
CENTRO INTERNAZIONAlE MATEMATICO ESTIVO
(C.I.M.E.)
ONDE DEI SOLIDI CON VINeOlI INTERNI
TRISTANO MANACORDA
ONDE NEI SOLIDI COif VI1fCOLt IB1rERNI
Tristano Manacorda Universit~
di Pisa
I-Vincoli interni La nozione di vincolo di
e ben
incomprimibilit~ in
nota fin dai primordi;la stessa nozione
ta nella teoria dei solidi,ben piu
~luldo
un
e stata
ideale
introdot=
recentement~.Per quanto ~
mia conoscenza la si trova in Poincar~ [lll) nel ~ondamentele articolo di Hellinger
119]
a
e soprattutto
[i2J .Solo assai piu re=
centemente,sono state sViluppate considerazioni generali suI
v~
colo di incomprimibilitll. "uggerite inizialmente dallo studio del comportamento della gomma la quale
ea
dilatazione cubica nulla
in ogni sua tras~ormazione isoterma [5] colo cinematico studieto di·recente
~
[24] .Altro tipo diva:
quello della inestendibili
til. in una direzione introdotto da Rivlin [20] te studiato da Adkins
[2]
e poi ampiamen=
ad altri,anch'esso suggerito dal
comportamento della gamma rinforzata da una
~itta
trama di fili
di nylon.Vincoli piu complessi sono atati considerati da Wozniak [26] .Una teoria generale dei vinvoli cinematici non pub non far riferimento all'articolo di rruesdell e Noll
[25]
dello
Handbuch der Physik. E' quasi spontaneo, a questo punto,la introduzione di vincoli in= tarni non puramente cinematici me dipendenti anche dalla tempe= ratura assoluta (11 (1) Nella teoria termodinamica di MUller,sviluppata ampiamente da Alts 3] per solidi vincolati,l'esistenze della temperatu= ra assoluta ~ provata invece che ammeasa.Qui per semplicitll. ai accetterll. la temperatura assoluta come nozione primitiva.
r
Una prima est ens ions si ha quando ai smmetta cbe la dilatazio=
234
ne oubics s1s una funzione della temperatura aBsoluta la quale
ai riduce a zero nelle trasformazioni isoterme ( solidi incomp primibili secondo Signorini
[11], cfr. anche J.!anacorda [15].
Fiu in generale si pub ammettere come vincolo interno una
fra deformazione e temperatura,cfr.Amendola [4]
lazione finita e J.!anacorda
r~
(15] .Green,Haghdi,Trapp (91
hanno introdotto
vincoli espressi da forme differenziali non com)letamente
int~
grabill nelle.quali perb non compare la derivata temporale de! la temperatura. La teoria piu generale di vincoli termomeccanici
In1
.Easi hanno provato la
che il vincolo riguardi tutta la storia della de=
formazione e della variazione di temperatura Be a1 vuole
Boddi~
fatta la disuguaglianza di Clausius-Duhem da ogni proceeso pos= sibile.I lore risultati
sono stati ritrovati in modo piu ele=
mentare,ma con condizioni piu restrittive da Manacorda
[15] .
Reetrizioni suI gradiente di temperatura sono state consider:! te da Trapp [23] Benersl.
.Per 11 seguito,
di un solido vincolato nell'ambito della moderna mec=
canioa dei continuL
Di un continuo @.l
,fonnato da elementi
canto ad una configurazione di riferimento
X ,a1 considers ae=
Bo,la configurazio=
X indica la terna di coordinate di X in Bo rispetto ad un sistema cartesiano fiseo, mentre: ~ la terns
ne ietantanea
B.
corrispondente in II
B
B.
Data la corrispondenzs biunivoca
traB o
.!
( 1.1 )
=
b (~ ,..t)
mentre il gradiente di deformazione
, !=
X' 1 (~, 1)
- -X· 1'=
G~d
(2)ha Ie coml!
(2) le lettere maiuscole indicano che Ie derivate sono fatte r! spetto alle !. ponenti
cartesiane
235
t
1.2 )
r.
~
=~
PH· x 'H
Naturalmente ( 1.3 )
J pi l;l
J = det
e ai ammette
o,
J)
oX
0
durante tutto il moto del corpo.
Altri tensori di interesse cinematico nella meccanica
dei
continui Bono : i1 tens ore di Cauchy-Green l"~ £ = pT P
(
)
(!'f' a 11 trasposto
di
! ) e il tens ore di
( 1.5 ) E 1 e il tensore unitario. Lo stress di Cauohy solido
=1 ( C - 1 ).
'2 -
-)
rappresenta gli s~orzi interni nel
T
2:.!! , e ad esso corrispondono,11 tensore degli
j;~.
sforzi di
de~ormazione
Piola-Kirchhof~
T R
l
( 1.6 ) 3R = J 2: ( e il tensore lagrangiano degli sforzi
)-1 (
0
secondo tensore di
Piola-Kirchoff
Zo
( 1.7 ) Men~e
!R
non
a
[ 1 !R = J !-l
a simmetrico,l'equazione
! (
l
)-1
di bilancio
del mo=
mento della quantita di mota e l'assenza di coppie interne im= plica la simmetria di T e di !o' Le equazioni fondamentali di bilancio per un continuo sono: a)
l'eguazione di conaervazione della masea. Se ~
ta materiale di ll?>
in
per ogni sottodominio ( 1.8 ) dove
bo
e
11 , e
b
di
11.
la densita
di iB
dens!
in 110
Hla
S~~ dv"
l'insieme di
S. a
e la
)..S'. dV corrispondente a
b • Sotto
OV=
vie di regolarita ( 1.8 ) equivale a ( in forma rispettivamen= te lagrangiana ed euleriana ) ( 1. 8'
)
'i
J = ~o
236
b)
l'e uazione di bilancio della
~ r~
( 1.9 )
dt
:f, dv =
b
\.
Sotto opportune condizioni • ( 1.10 ) ~
~
in questa
.!
uantita di moto. Si Bcrive
~f.o...r ~
(i", cW
+
1,,;
di regolarita , ( 1.9 l equLvale a
=~! + div
!
-),V -;. ==X=;·; ..
la densita di forze di massa e
-
1a de=
v = X( X , t
rivata molecolere della velocita,
~
~
~
~
( 1.10' l + Div l R c) bilancio del momento della guantita di moto.In assenza
di
Alla
puo dare • ~! = ~.!.
(1.10) si
forma lagrangiana. Si ottiene
coppie distribuite, si limit a ad imporre la simmetria
-
:!;
= T
TT
( 1.11 )
T
=
pT
!o
condizions per J ' R
e in conseguenza (cfr. ( 1.6 -R -
=
di
d) bilanoio della enarKie. In 8Bsenza di Borgent1 interne,st sorive per ogoi
(0 f -d J.)
( 1.12 )
ove estema a
e la b
b E:: B ,
dt
dv
= ( q. n d6" + ( tr( T grad X ldv ~L '" "V )1 "'bdensita di flusso ten:ico, n 1a normale
u b.
nei punti di
.
Questa equivale ,nelle consuete ipotesi di regolarita a ( 1.13 )
5 f.
= div q
S
+ tr (
! !! )
.£ = 1 (
grad v
+11,,.J·f)
+ ~( !grady ) = div
tenuto conto della simme-;ria
di
!
,con
Alle equazioni di bilancio va aggiunta
la
2
disuguaglian=
za dell'entropia.Qui ,per semplicita,in assenza di sorgenti,e a.sunt .. nella forma ( 1.14)
~ (~iJ..r - h:~~.I6"
:?-O
equivalente,nelle conauete ipotesi di regolarita,a ( 1.15 )
~6i
- ..d.:v~ + .~ .,...J.9 0 Q
237
e
In questa, d1v S
tna (
e
la temperatura assoluta.L'eliminaz1one d1 e
1.13·
) consente d1 scr1vere la
(1.15
) nella forma della d1suguagl1anza d1 Claus1us-Duhem.
(1.15
Le ( 1.13)
e
(1.15) s1 possono scr1vere in forma
grangiana,rispettivamente, . ( 1.13:) ~o £ = D1v ~R
e'1 -
+
*
T tr (~R
di scrivere
r
0
&-
"I ()
,conaente
e ( 1.15' ) nella forma
(1.15)
1 q. grad e ~ 0 T·"5"~ (:!'R!) - 1 ;J.R.Grad ~ 0 Le equaz10ni fondamental1 ( 1.10 e ( 1.13 ) vanno comple=
( 1.15' )
~(r' "I 8) -
•
!),
e;,
$. D1v ,gR + 9 R ' Grad L'introduz1one dell' energia libera =
( 1.15')
18=
tr (
"J ~o(r .. ,8 - tr
X .!! ) -
e
Y
tate da equazion1 cost1tut1ve. Un solido e detto termoelast1co
'f = If' ( !. e,!! )
se ( 1.16 )
-
ove g = grad
e
!
=
! ( !. e, ~ ),
~ = 9 ( !' e, § ) e sottintesa I'eventuale
ed
1=1( !' e, ~ ) ~.
d1pendenza da
La incond1z1onata valid1ta d1 ( 1.15 ) per oen1 processo am= m1se1b11e, implica che
!§
, e che s1 abbia
( 1.17 )
TR
'f' : 1
'.e
!
non
po;e~o
d1pendere
1 e ) - tr (.J R l ) = 0 = - vef • ! = J-~f!'T
~"(f +
=~o}!'"
i
da
c10e
ine1eme a
,s . .g;ro
( 1.18 )
( $R •
Q~ 0 )
3i richiamano infine le condiz1oni di d1scontinuita dovute alle equazion1 di b11ancio quando 11 solido sia attravereato da un'onda di diecontinuita del primo ordine; esse eono
[~u] = ( 1.19 )
[ ') ;!
[~"1 ( 3)
uJ +[!-e]=0 uJ + [:!! . ;3] = 0
0
[')'£
u] + ['!
.eJ =0
( 3 )
La validita di ( 1.19 )4 richiede che le eorgent1
entropia,eaterne e
di
intrinseche,siano limitate a1 tendere di
238
volume di b a zero. Green e Naghdi non ammettono tale condi zione t16] .S1 puc invece ammettere 11 esistenza di Wl flusso
=
intrinseco di entropia.ln tal caso esso dave essere sempre non negativo. h
e
i1 fluseo di entropia. E ' facile acrivare la versione la=
grangiana di ( 1.19 ) la quale fa intervenire la velocita
Bo
dell'immagine in Osservazione
1
UN
dell'onda. 8i assuma
-
h = q/9 ; se
~
G8
continua at=
traverso l'onda, la ( 1.19 )4 ' tenuto conto di ( 1.19 )3' di=
[S
viene
9, U.] -
[3£l.l] = 0
Ci08 [cfr. ( 1.19 )]
[£-'1 8]
=
['1"1
= 0
31 osserva che l'introduzione di un vincolo ha dei riflesai
anche meccanici , in quanto il tens ore degli sforzi non
e piu
completamente determinato da una equazione costitutiva
come
( 1.17 )1 ' ma contiene una parte che rappresenta gli
sfo~ziodi
reazione dovuti a1 vincolo.Per un vincolo termomeccanico,anche l'entropia e
S
non sono completamente determinati,mentre B1
puo assumere completamente nota la forma di Osservazione
2 -
In un solido
'Y
nelle ( 1.16 )
termoelastico non Boggetto
a vincali, Itessere 11 calare specifieD a configurazione stante
e
1 '
cv
positiVQ impliea corrispondenza biunivoca tra
co=
e
per cui s1 pub assumere come variabile termodinamica Ia
entropia al posto della temperatura. Tale corrispondenza viene a cadere per solidi vincolati,onde si potrebbero istituire due teorie paraIIele,assumendo in una ,come variabile termodinami= ca,percio soggetta al vincolo,la temperatura; nell'altra,invece l' entropia.
239
2 - Vincoli interni nei solidi
Esempi di vincoli a)
Vincoli ci..n:ematici
Vincolo di incomprimibilita ( 2.1 )
= det
J
F
~
= V det
C
= 1
~
Vincolo di inestendibilita 3i emmette che in
Bo
esista un campo vettoriale
~
linee vettoria11 conservano lunghezza inalterata in
(X) le cui
B. 3e
~
ha modulo unitar10,1l vincolo e tradotto in
Fe. _"..., F e
( 2.2 )
J'VIY
=1
c10e = 1
Vincolo superfic1ale : 31 emmette l'esistenza d1 una famig11a
1 di
Buperfici I. I dens a in
Eo' la cui dilatazione superfi=
B.
ciale e nulla nella trasformazione
--t BJ •
Poiche la dilatazione superficiale e ellS""
con
~
norn.ale nei punti
= J ()2'C-
di.2:.
l
:Q BoJla condizione di inell
in
-.oJ ) -L :r
stendibilita Buperficiale a1 acrive J ('V'C ~
per ogni punto d1 L ed ogn1
= 1
€.
b) Vincoli termomeccanic1 Sene 8stensioni di vinceli precedenti.
Vincolo di incomprimibilita ( 3ignorini ) J = f
( 2.4 ) ~
(e}",~)
f
(l:',7:j2S) = 1
e la temperatura ( supposta uniforme ) in
Bo '
Vincolo di inestendibilita (2.5)
F,3'
Vincolo anolonomo A
f(9;t',X); f(c,'l) = 1
(Green, Naghdi,Trapp ) tr ( /
2.6 )
ove
1:2
e
i
) +
1( !
,e) ..§
sono un tensore ad un vet tore funzioni di..!
=0
e
240
e
e di Si assumera ,d'OTa innanzi,l'esistenza di un vincolo nella f'onna
e ,X ) :
r ( -F,
( 2.7 )
(1) 11 principio di obiettivita
1,
de
solo per 11 trami te di
imp1i~a
.£ =! !
(1)
0
~
r
che t
possa dipendere
ma 91 conserve Is
forma ( 2.7 ) per comodita.
-----------------------------------Lungo un processo F: f ( t ), e: e(t) ,in condizioni • di sufficiente regolarita di r e identicamente r = 0, ~
~
~
qUindi
-
( 2.8
'I' •
tr ( r F ) + ~
ove
r
2.9
~
Osservazione
1
-
: -;)1"
r
fe
0
:
()e r r:l'(g ,f),xl; in una
~
51 assume
:
trasformazione linearizzata a part ire da Bo ~ LX) tr (k'l' E l") 0 ['I f ~ (") ill loT ove ~ e 1a 1inearizzazione di J, 2 E - : Grad~ +( Grad-II: ) ~~Ispostamento linearizzato, e e~) la linearizzazione della
+f, ell':
dO e
variazione di temperatura;
la determinazione in
~, quella di ~ 1'. In :r:srticolare, se
di ~ Te
B:o
#- : i:!. , 1a (~)
diviene
div con
a
costante.Cio accade
-u.'"1 : se r
a el!}
1.
dipende da
tramite dei suoi tre invarianti
per 11
IUE •
In un solido termoelastico soggetto ad in vinco10 del 2.7 ),10 stress
tipo
e l'entropia risultano non completamente
determinati da equazioni costitutive : precisamente si ha ( 2. 10 )
ove
p
T
e un
~R
:
-
p l' + -
5'..? 'f,;: i"
r
1"~
f- r <>8
parametro 1agrangiano atto ad individuare 1a rea:
zione vincolare interne.
241
Qsservazione 2_
La potenza delle reazioni vincolari
da
e
data
-••
-tr(prl!')
e quindl
ed
~
nulla,in eorrlspondenza al vineolo,in ognl trasfoE
maziane isoterma .Piu in generale,essendo la
prod~ione
flea interna dl entropla data , da
5• ..y .. ~.1J
speci=
•
e
t)
la produzlone dl entropla dovuta alle reaziani interne - p
r
~
p
ed
e
ns assumendo
3-
- ..... 'F"
+
tr (r l!' »
qUindl zero in virtu del vine 010
Osserva.ione ta
(pe
( efr. ( 2.8 »
Una teoria perfettamente analoga si ottie=
1 come variabile
termodinamlea
e
da una equazione costitutiva. Naturalmente,
ta con
C.
e determina =
'r
va Bostitui
242
3 -
Onde di accelerazione
II quadro delle equazioni fondamentali in preeenza di un vin:
colo come(2.7),e in assenza di forze di massa e di sorgenti termiche e di entropia
~ei
= Div
( cfr. ( 1.10' ), ( 1.15' ), ( 2.10 »
.,
~o ~
~
,
:£R
= Div
of
~=J'!g
( 3.1 )
SR ='sR (2 ,§,G) G = Grad II 81 comincia ,qui, ad esaminare Ie condizioni per la propaga=
zione di secondo ordine ( onde di accelerazione ). A queeto scopo a1 pone
-a=[X] -
3.2)
Una notevole semplificazione e dovuta al fatto che,in un
co~
duttore definito,tale cioe che la parte simmetrica del tensore
~ = ~~R
e definita positiva, possono snche in presenza di
vinco11 (1)
(l),propagarsi solo onde omoterme ,per le quali,cioe
Per un solido non vincolato
[6 J
=
[ ~]
V.ad es.
o
=
'f
5i osservi ,infatti,che pur essendo ~
minata da un'equazione costitutiv8,
E
=
[B,71
completamente non 10
e
,mB a1 ha
'Y + 1 e = '1 - e~ + f" f e
tuttavia f. e da ritenersi continua attraverso l'onda (2) (2) La
(1.19)2
implica
[1,e]
= 0, cioe
Dalla equazione della energia si ha allora
[5'. 11k G]
= [SR
1
N
e perci/>
( 3.3 )
.!!
=
0
[p]
dete~
=
0
243
.l! e
ove
8ia
la normals all' immagine dell' onda in discontinuQ, perdo
G
~ = Grade
continua e
~;
=
[
~]
B. •
=o(!! '
e
dato che
a
; si ha aHora
,e )..
Sa ( ~+
Sa (~i""
+O(~, B )
qUindl
If'fx)=r.~a]·l!
= (Sa
(G-
+ ... ~,e)
-Sa(2- ,9) )~!=
0
Derlvando questa ldentlta rlspetto ado(
-- -
= 0
K+ N • N
che a impossibl1e per l'lpoteel che 11 conduttore eia definito.
31
puo
dunque assumere che Itonda sia omoterma.
Dopo dl cio,dalla ( 3.1)6 [ t;:
od anche
{ [
el ha
T'''p]
= 0
iJ= l.... ~ If i, ~
e
tr
l' T ( ~ lID
1i
UN
-- -
=l'N.a
-flN.a=O: - .--
( 3.4 )
le onde ( omoterme ) dl accelerazlone dl un eolldo termoelaet! co eono traevereall. 81 prende in eeeme,ora: la ( 3.1 )1 (
Tal
[dlV
=
-!... U II
L~a] = -[ :r] ove
g:
e 11
+
[Tal! )
Hi] _ei[e]
teneore quadruplo ()" ~ • 81 ha pol
([i]
= -
f (1"
-
ove
!:-
zlone dl
~ ~!! )
UN
=
-!... ~ ~ UN
indica l1n conveniente tensore del secondo ordine,fun = N,.
r p'T'r
=
[;JJ> p[d!'l'(i )J
e cloa,nelle condlzlonl presentl
( 3.5 )
ha
el
- [P!]!!. = -[pll'!
+ L
\r:' If
244
avendo poato
(
d P.2'( if 18l!J )
)N =
J~
Si ottiene infine la c-ondizione di c~patibilita dinamica
UN[pJ2'!+(Q-t~l)~ =
(3.6)
0
ove !!=~-p~
(3.7)
e un
tens ore doppio simmetrico.
Le implicazioni della equazione dell'energia si ottengono ra=
[2}
pidamente. Per essere
N
= 0 con
l.... [sR1 .Jl
[DiV SR1 = -
e
1lj.,
:j
=
a.9. 1. R . Essendo
L =
~
ottiene
-- aeN ) • JJi
Poiche
un tens ore
-
e una funzione doppio~tale che N L ( a 8 N )
-
e si ottiene,alla fine
~oe[iJ = A!J!.! -
( 3.8 )
lineare di L ( a
__
l....~!J
=f [pJ
con I
~
3.10
+
" - 5. d~! 't'
-
esiste
=.11. N.
a
"-"",,o.J
•~
UN
D'altra parte ,se B1 ricorda la ( 3.1)4
3.9 )
liP'; )
~
, s i ha
(~- p.f) .!!-.~
1 = - 1:! T'
Si ottiene quindi la condizione di compatibilita
3.11 )
"'!J!.J! -l....!!-.!!. ~= feEpJ
+fJ(e'- Pl) N.a ~
~
UN
~oiche
[p J
si
a1 limits ad esprimere
e di
p •
puo
ricavare dalla ( 3.6 ), la (3.11 )
A in funzione
di grandezze continue
Moltiplicando, infatti, la ( 3.6 ) per
T'
N
-~
si
245
0]
ottiene
G1
( 3.12 )
=
J.JL
~ =
50stituendo questa determinazione di
IT!I
nella ( 3.6 )
a1 perviene a ( 1 - \I ClII ) ( Q _~ U..
3.13 )
...........
5i rieordi ora che pendicolari a
a
( 3.15 )
-
Q a l
~
II problema
=
0
posto
'"
( 1 _::!Ilf~ .8.1 = la condizione ( 3.13 ) a1 scrive ( 3.14 )
a:
deve essere Bcelto tra 1 vettori per=
~
\I
1 )
"\J)ON---"",
~
e quindi
) ,g ,
= ~ U'l a "
N
-
-
a • \l =
~
0
ridotto a bidimensionale,ed esistono
pereio due autovettori almeno diatinti e,ae i eorrispondenti autovalori sono poaitivi, easi individuano Ie direzioni di di' ,= aeontinuita e Ie rispettive velocita di propagazions. In particolare,sia per
--
e
A
u
A
A =
v'l.
u
un Rut ovett ore comune, ( se esiste )
11 ~
!
u
=
:;2.
U
si ottiene
~l ~ =l]oU; ~= (v.. _pV2.)u
(3.16)
la quale mostra che,anche Be
ehe
U 'N
<.
0
v 1) 0 ,
V 'I.;)
0
,puo darai
e quindi che non ai abbia propagazione.
246
4 - Onde d'urto Lo studio delle onde d'urto
e ricondotto
allo studio delle
condizioni dinamiche di compatibilita (1.19) cui va aggiunta la condizione che si ottiene dall'equazione del vincolo. Conviene scrivere la (1.19) in rorma lagrengiena
t'lo uNl = 0 + [!R!!l=o
( 4.1)
=
alle quali va aggiunto
0
~O
N,
( 1)
[\1 ~ 0 • una relazio=
(1) Questa equivale ad imporre ne dal significato fisico evidente •
• la condizione derivante dall'equ8zione del vincolo
( 4.2 )
e evidentamente
Lo studio di tale tipo di onde meno di non riiursi
complesso, a
a casi unidimensionali.Qui ci si occupa
solo di onde di paccola intensita,per Ie quali a1 vuole dire,
sono valide le versioni linearizzate delle ( 4.1 ) e ( 4.2 ), in solidi
iso~pi.ln
tali condizioni,si introduca la differen=
za di temperatura tra la temperatur~ attuale e quella (che a1
suppone uniforme ) di riferimento, e si indichi
e
con
la
configuzazi~
sua linearizzazione.lnfine,si Bupponga naturale la
ne di riferimento.In tali condizioni,per 11 tensore
linearizz~
to degli sforzi si ha Tl') =
( 4.3 ) ove
)(.
e
T('I =
~
-R
-
1 +
-)"(.
la linearizzB.zione di
=7 e
nearizzato di deformazione,
e
rizzazione della spostamento, ?I ,
fA-'
quali?,
e certamente
positiva.
~
P
?• e
e
~
div
,
1 + 2}1-~
-r(91
11 tens ore 11= ~
,
u
line!!;
sana costanti delle
247
Allo steeso tempo, per la linearizzazione di
"ll'J=
4.4)
!!....~ + l
-
~. -e
con
~ 'l:
e
~ e
+
f.-e
la temperatura dt riferimento,mentre la condizione
di vincolo dlviene
4.5 )
div
u
=
a
e.
I campl linearizzati devono 8odd18fare alle linearlzzaeioni delle
(4.1) e ( 4.2)
: prec18amente,po8to m•. =
-m.[il = ~J, m.C'Z~'I+CS~).NJ =
r.{'
(4.6)
~ UN
m.[~"·l=[~~J.!l 0
1;-
(div ul
= are).
~
Conviene d18tinguere due ca81 :
(eJ
a)
A.
s.
(ande meccaniche)
= 0
e il parametro caratteri8tlco delle d18continul=
ta delle derivate prime di
~,11
vincolo impone
( 4.7 ) ~ • • .!! = 0 l'anda e tra8ver8ale (1). Dalla prima delle
(4.6) 8i ha
---------------------------~----------
(1)
Propriamente perc.!/) non e pill un' onda d' urto [~l = -
all ora
UN 1l. )
- - 4.3-J
r T(')
(4' .7 )
lfi
N
•
e cioe, tenuto conto dl (
=
0
),della continuita dl
e
e di
(div ~l = 0 [x) ilia
e
e = 1
[~_ ~
N AJ
=
2'
1= 2'_0 1 :>-
Segue dunque
2fl[~
.!! 1.
N
T
l)
e infine (e N 1 .N
=
i grad ~ +
( grad
,... -
l::. )
-
L~ J= ~~ ~411 ~ 1 ;>. • N = 0
2" .... 0
,...,
+
!!;illI~,}
248
( 4.8 )
frtl =
0
C"L(" 1 =
0
(4.4 )
e,da~la
( 4.9 ) Aneora dalla
4.6 )1 ,molt1pl1cando scalarmente per
iI"
a1 ottiene
e percH. ( 4.9 ) che aasegna la 'i,(')
=
la ( 4.6 )2
d1 propagaz1one dell'onda.Infine,po1che +/f1l.'i) [t e.)] = 0
veloc1t~
(f)~)
"'It: •
e
~~ •
e
e aodd1afatta da
'
[~'.!! 1
sodd1afatta anche La ( 4 .6 )3 (2)
=
0, che rende
aubordinatamente
1
["It GlJ.1 • N = 0'11 gra= temperatura normale deve es~ere continuo .•
Se vale 18 lagge d1 l'OURIER , 4iente
di
alla
( 4.9 ):11 fluaao normale d1 calore
e continuo.
249
[e 1 f
b)
0
(onde term1che ) ,
a
~ 0
In questa caso,11 vincolo essendo rappresentato da (4.5 ),s1
ha
-
( 4.10 )
".'
~
( 4.11 )
= a@
N
~
da'ora,molt1pl1cando scalarmente per
La(4.6)1
u;
= [- 'It +
a @
A
[~1.1I
Se
~.
non
e
=,.u.~.
parallelo ad
~
UN la determinaz10ne
la ( 4.12 )
e
~
-(~.'.!!)!i.)
=~ •• ! = a e .
t
.. " ~UN~.·~
mentre,molt1pl1cando per (4.12)
+ 2)LA Gl
= t(~.
!!.].
[!.
eJ
~
e -
N,
~
t
,s1 ott1ene ancora per
(4.9). Se invece l'onda
U:
una 1dent1til. e
Per onde non longitudinal1
(e
e
e longitudinale
indeterminato.
neanche trasversal1, v. (4.10)
s1 ott1ene dalla ( 4.11 ) per la ( 4.9 )
[}GJ
( 4.13
a
=
e (;\
+)J. -
["l"11 =
ofr.n.l Oss .1 ) Segue.po~endos1
4 .15 )
-a(~+).l»
( .!L. ( 2'{
r.'t-
D' altro canto,la ( 4.1 )4
)
~
e ( 4.14 )
~
+!!...
implica
che
1f
sia continua
e quindi ahe sia continua anche assumere
(€("] =
)@.
~.'t;
"!'('J .
=0
~
L'P '!] l
e quind1, da ( 4.6 )2 ( 4.16 ) che coincide con
(q~11.
N,
=
m. 'l:
['1("].
(4.6)3' Le cond1zion1 di comp~tibilita
termomeccaniche Bono qUindi atte a determinare
UN ed insie=
250
-
me le diseontinuita di
e
e
di '7.. (Ii in funzione di ® .
3i ha snehe ( efr. ( 1.19 )1
c~ con
uJ
=
U =
0
u
n
- v
n
u;..c veloeita dl avsnzamento ( locale dl propagazlone )
dell' onda. Linearlzzsndo
quindl
[~l"
UN' =
-t, (#lJ
=
~o[l\J
= -
go af> UN
e quindi
[ ~ ('ll =
4.17 ) l'onda
e
compresaiva
se
-\,oa6:
>
a @
O. espansiva nel
caso
opplilsto (3) (3) 31 osserva ehe ( 4.17 ) vale snehe se l' onda e longltudl= ( ® = 0 ) impliea lOa nale, mentre,per onde meccaniche cont inulta dl ~(". Per onde longitudinall,
UN
non pub eSsere determinato dal=
le sole eondlzlonl di diseontinuita,mentre per le dlseontinul= ta di )C , di
"'L 1') [}Cl
( 4.18 )
sl ha
=
a@(?I+21J-1l'
r
-
a
-oU") I~ N
251
mentre
e
anche
BDlLIOGRAFIA A
Vincoli
1 - J.E.
Adkins
: A three-dimensional problem for
highly
elastic materials subject to constraints,Quart.J.Mech.Appl. Math,11,88-97 ( 1958 ) 2 - J.E. Adkins - R.S.Rivlin : Large elastic deformation of isotropic materials : X • Reinforcement by inextensible cor: ds, Phil.Trans.Roy.Soc.London A, 248 201-223 ( 1955 ) 3
T.Alts -
Termodynamics of thermoelastic bodies With
kinematic constraints : fiber reinforced materials.
Arch. Rat.Mech An. 61 (1976) 253-289. 4 - G.B. Amendola
A genaral theory of thermoelastic solids
restrained by an internal thermomechanical constraints,
Atti Sem. Mat. Fis. Univ. Mod. 27 (1978) 1-14. 5 -
J. Bell
: The experimental foundations of solid mecha=
nics.Encycl.of Phys.Vol
VI a/I ,Springer Verlag ,Berlin
1973. 6 - G.Capriz - P.Podio- Guidug1i
Formal structure and
classification of theories of oriented materials,Ann.mat.
pura appl.(4) 115 (1977)17-39. 7 -
H.H. Erbe
W~rmeleiter
mit thermomechanischer in;
neres Zwangsbedingung,ZAM,Sonderheft
76-79 (1975)
252
8 - J.L. EriksGn- R.S.Rivlin
: Large elastic deformations of
homogeneous anisotropic materials, J.Rat.Mech.Anal.3,281361, '1954) 9 - A.E.Green-J.E.Adkins : Large
elastic deformations. and
non -linear continuum mechanics,. Oxford,Clarendon l'ress 1960 ~O-
A.E. Green- P.M.Naghdi-J.A.Trapp : Thermodynamics of a con= bnuum with internal constraints,lnt.J.Enw1g.
.Sei.8 (1970)
891-908. 11- M.F.Gurtin-P.Podio-Guidugli : The thermodynamics of constrai ned materials, Arch.Rat.Mech An.51 (1973) 192-208. 12 -E. Hellinger , Die allgemeinen Ansatze der Mechsnik der Kon= 4 tinua,Enz.math.Wiss. 4 , 602-694 (1914). 13 -S.Levoni:Sulla integrazione delle equazioni della termoela= sticita di solidi incomprimibili,Ann.Sc.Bbrm.Sup. Pisa (3), 22,(1968),515-525.. 14 -T.Manacorda:Una osservazione sui vincoli interni in un soli= do,Riv.Mat.Univ.Parma (3),3,(1974),169-174. 15 -T.Manacorda:Introduzione alla termomeccanica dei continui, QUaderni UMI,n.14,Pitagora Ed. ,Bologna 1979. 16 -S.L.Passman:Constraints in Cosserat surface theory,lst.Lomb. Sc.Lett. Rend. A,106,(1972),781-815. 17 -A.C.Pipkin:Constraints in linearly elastic materials,J.of El. 6,(1976),179-193. 18 - H.Poincare:Le90ns sur la theorie mathematique de la lumie= re,Paris, 1889. 19 -H.Poincare:Le90ns sur la theorie de l'elastcite,Paris,1892. 20 -S.Rivlin:Plane strain of a net formed by inextensible cords, J •Rat.Mech.Anal. 4, (1955) ,951-974. 21 -A.Signorini:Trasformazioni termoelastiche finite,Mem.III, Ann.Mat.pure app1. (4),39,(1955),147-201.
253
22 - A.J.M.Spencer:Finite deformations of an almost incompres= sible elastic solid,Second-order Effect in Elasjicity, Haifa 1962,Oxford,Pergamon Press,1964. 23 - J. A. Trapp: Reinforced materials with thermo-mechanical
COIl=
straints,Int.J.Engng.Sci. 9,(1971),757-778. 24 - L.R.G.Treloar:The physics of rubber elasticitY,Clarendon Press,Ox~ord,1949.
25 - C.Truesdell - W.Noll:The non-linear field theories of Me= chanics,Encycl.of Phys. Vol.III/3,Springer Verlag,Berlin, 1965.
Bl
) Omie.
1 - A.Agostinelli:Sulla propagazione di onde termoelastiche in un mezzo amogeneo e isotropo,Lincei Rend. (8),50 (1971) 163-171,304-312. 2 -G.B.Amendcla : On the propagation of first order waves in incompressible thermoelastic solidS, B.U.M.I. (4) 1 267-284. 3 -
G.B. Amendola rials. Atti
Acceleration waves in incompressible mate.
Sem.Mat.Fis.Modena
24
(1975) 381-395.
4 - C.E.B.evers : Evolutionary dilational shock waves generalized
(1~73)
in
a
theory of thermoelasticity,Acta Mech. 20(1974)
67 -'19.
5 -
B.R.Bland: On shock waveS in hyperelastic media,IUTAM, Second order effects in Elasticity,Plasticity and Fluid Dynamics,Pergamon Press,1964.
6 - P.Chadwick - P. Powdrill:3ingulllr surfaces ilL linear thermo=
254
elasticity,Int.J.Engng Sci. 1 (1965) 561-596. 7 -
P.Chadwick-P~.Currie
, The propagation and growth of acce=
leration waves in heat-conducting elastic materials,Arch.
Rat.Mech.Ah 8 - P.J.Chen
49
(1972) 137-158.
Growth and decay of waves in solids,Enc.of Phy=
sics, VI a/ 3 Springer Verlag,Berlin, 1973. 9 - P.J .Chen - M.E.Burtin
: On wave propagation in inextensi=
ble elastic bodies, Int.J.Solids Struct.1Q
(1974) 275-281.
10 - P.Chen-J.W.Nunziato :On wave propagation in perfectly heat conducting inexteneible elastic bodies,J.of El.5(1975) 155-160. 11 - B.D.Coleman, M.E.Gurtin,I.Herrera,C.Truesdell : Wave prop! gation in dissipative materials,A reprinj of five memoirs, Springer Verlag, New York 1965. 12 - WEiD.Collins : One-dimensional non linear wave propagation in incompressible elastic materials,Q.J.Mech.appl.Math.19 (1966) 259-328. 13 - J •Dunwoody
: One dimensional shock waVeS in heat conduct!
ng materials with memory : 1- Thermodynamics-Arch.Rat.Mech. An.~7
(1972) 117-148; 2-Shock analysis, ibidem, 192-204;
3- Evolutionary behaviour, ibidem 50;278-289. 14 - J.Dunwoody:On weak shock waves in thermoelastic solids,Q. J.Mech.appl.Math.30 (1977) 203-208. 15 - J.L.Eriksen:On the propagation of waves in isotropic incom= pressible perfectly elastic materials,J.Rat.Mech.An. 2 , (1953) ,329-338. 16 - A.E.Green-P.M.Naghdi,A derivation of jump condition for entropy in thermomechanics,J.of Elast. 8 (1978) 179-182. 17 - E.Inan:Decay of weak shock waves in hyperelastic solids, Acta Mech. 23 (1975) 103-112.
255
18 - T.Manacorda:On the propagation of discontinuity wave~ in ther-moelastic incompressible solids,Arch.Mech.Stos.(2),
24 (1971) 277-285. 19 - T.Manacorda: Zagadnienia Elastodynamiki,Ossolineum,Varsa=
via,1978; Cfr.anche A - 15. 20 -
R.W.Ogden:~rowth
and decay of acceleration waves in incom=
pressible elastic solids,Q.J.Mech.appl.Math. 27 (1974) 451-464. 21 - N.Scott:Acceleration waves in constrained elastic materials
Arch.Rat.Mech.An. 58 (1975) 57-75. 22 -
N~Scott:Acceleration
waves in incompressible elastic so=
1ids,Q.J.Mech.appl.Math. 29 (1976) 295-310. 23 - N.Scott-M.HaYes:Small vibrations of a fiber-reinforced composite,Q.J.Mech.app1.Math. 29 (1976) 467-486. 24 - C.Trimarco:Onde di accelerazione in materiali termoelast±:
ci con vincolo di inestendibi1ita,in pubbl. su Atti Ace. Sci.Modena. 25 -
C.Truesdell:~enera1
and exact theory of waves in finite
elastic strain,Arch.Rat.Mech.An. 8 (1961),263-296.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(C.I.M.E.)
"ENTROPY PRINCIPLE" AND MAIN FIELD FOR A NON
LINEAR COVARIANT SYSTEM
TOMMASO RUGGERI
INTERNATIONAL MATHEMATICAL SUMMER CENTER (C.I.M.E.) 10 1980 C. LM.E. Session: "Wave propagation".
Bressanone 8-17 giugno 1980. "ENTROPY PRINCIPLE" AND MAIN FIELD
FOR A NON LINEAR COVARIANT SYSTEM by
TOMMASO RUGGERI Istituto di Matematica Applicata -
Universita di Bologna
Via Vallescura 2 - 40136 Bologna (Italy).
1.Introduction My lecture is complementary to the lectures given by G.Boillat in the first
part of this course. In
part~cular
I am shall deal with some problems concer-
niog quasi-linear hyperbolic system compatible with a supplementary conservation
law; relativistic theories will be considered with special emphasis. I start with a brief bibliographical introduction to the subject I shall be
concerned with. In 1970-71 1. MUller, in Bome works [1] on "rational" thermomechanics of continuous media, proposed the "entropy principle" as a criterion for selecting the constitutive equations. This author considers the equation governing the evolution of a thermomechanic system: a) balance of momentum, b) balance of mass and c) balance of energy eqlJations. Adding the constitutive equations to the previous system one gets a system of 5 equations in 5 unknowns. Each solution of this system is called a IIthermodynamic process". Then MUller postu- lates the existence of an additive function cr (entropy) such that: +
(PSv
i
+
~
i
) = cr
>
o
¥ thermodynamic process.
(1)
Furthermore he supposes that both the entropy density S and the flux .1 are ~
constitutive functions (p and v are respectively the mass density and the velocity). Hence, from (1) further constrains arise for the constitutive rela-
260 tions,
besides
the
usual
ones
which
can
be
imposed
according
to
the
principle of material objectivity. In particular
which
he
the
author shows
with
identifies
the
the existence of' a universal function,
absolute
temperature;
hence,
he
deduces
the first principle of thermodynamics. In a different conte>d: in 1971 K.O.Friedrichs and P.D.Lax the
:former
in
1974
[3] Friedrichs,
[3J
examined
a
similar
in a covariant formalism,
problem.
In
[2] and later particular
in
considers a conservative quasi(0)
linear hyperbolic system of r first order equations of the type ,
a
3
a
(2)
(U) = .2(U)
In (2), N eqs. may be identified with the field equations, while the remaining r - N are supplementary conservation laws. Then
comp~
tibility conditions are required in order that the system has a solution. In particular, when r=N+l (one supplementary law), as the system is quasi-linear, compatibility is ensured by the existence of an r-vector
z(U), such that: a l"a!!
Introducing the operator V
• U,
= l".2
= a/au
• , U•
a
we have = 0
(3)
(condi tion I)
Moreover Friedrichs supposes another condition holds: it exist at least a time-like covector
{~
a
l, independent of the field, such that the quadratic
form (condition II)
is positive definite.
Here 6U is an arbitrary variation of the field and
a
=6U,VV!!
6U.
Using condition I and II Friedrichs shows that the system of the field equations is a hyperbolic symmetric system.
(*) To avoid misunderstanding the vectorErn
r
are underlined.
261 Later several authors [4J,
[5].
[6J provided further contributions on this
subject, especially concerning shock waves in non-covariant formalism.
Now we shall obtain the above mentioned results in an explicitly covariant formalism, dealing with
the physically relevant case of one supplementary
law. The covariant formulation allows to apply the results to explicitly covariant
theories and, moreover, to emphasize some conceptual aspects that, in our opi-
nion, have not yet been pointed out. 2. Main field and Covariant convex density.
Let V~ be a C·, 4-dimensional manifold and x a point of V~. x a being local
coordinates of x. The manifold is supposed to endowed with a pseudo-Riemannian a
metric. In the local coordinates x ,g
aB
represents the components of the
metric tensor of signature (+ - - -). On V4 we consider a quasi-linear conservative system of N first order partial differential equations for the unknown N-vector U(x
Q )
ERN (5)
a
the components of F
and U are contravariant tensors and
~Q
is intended as a
covariant derivative operator. We suppose that the system (5) is hyperbolic. i.e.: ~ a time-like covector (tal, such that the following two statements hold: det(Aat ) ~ 0
i)
•
a
(A.a •• Fa: • • a/aU)
ii) V covector{t a } of space-like the eigenvalue problem (t-."t)Aad.O a a'
(7)
has only real proper solutions p(k) and a set of linearly independent eigenvectors d
(k)
(k=1.2 ••••• N).
The covectors{C
- pta} • where p is solution of (7) are called "characteristic", a while the {tel} fulfilling 1). 11) are said "subcharacteristic ll •
262
When a differentiability conditions holds, let us suppose that (5) is compatible with a supplementary conservation law
a. h
G
(8)
h ·(U) = g(U).
being a contravariant vector and g a covariant scalar.
In this case we may write the conditions I and II of Friedrichs in a more convenient form. We have:
Since by (3) 1 is defined up to a scalar factor, we may write
l
-
then Friedrichs conditions lock like U'·VFO= Vho,
(9)
U'·C
(10)
g.
(11)
We remark that (9)
I
multiplied by 15 U, can be written
equivalently~
(12)
"6 U.
The identity (12) show the first important result:
u'is invariant mth respect to field transformatiors:in fact d!'! fa and
{t and then
U' depends only
it does not depend on the choice of the field variables.
By applying the operator li
to (12) and replacing into (11) we get
Q "" OUI • oyQ
~ >0
•
(13)
Hence (13) too is independent of the choice of the field; we may then choose the field in the convenient form U"" yQf;
•
(14)
263 We put also (15)
and contracting (12) with
to we
U' ·6 U
have
= 6h
+4
U' = Vh
(16)
For the particular choice (14) of the field variables, U' is espressed by the
gradient of a covariant scalar function h only. In the case of continuum mechanics, the expression (16) is equivalent to
the first principle of thermodynamics. We point out that the components of U' play the same role of the Lagrange
mul tipliers introduced by I-Shih Liu [7]
in the context of entropy principle of MUller. Condition (13) is now equivalent to (17 )
i.e. to the convexity of the covariant scalar density h=ha~ with respect to a the field U ::: F
a
to'
If for a system (5) there exists a vector U' and at least a caveator (£';a}such that
(12)
hold~ we say
and (17)
that the system is a convex covariant density
system. Conditions (16) and (17) ensure also that the mapping U' +-+U is globally invertible, becauseVU' =VVh and this gradient matrix is symmetric and positive definite; then for a theorem about globally univalence ([8J) ur +-+ U is gloN bally univalent in every convex open domain D ~ R Therefore it is possible to choose the vector U' itself as field variables and prove that in this case system (5) has the form L (U·)
where the operator
~
(t)
(18)
flU' )
is given by A' ().
-
a
a
(19)
(t) For the proof of the statements proposed in this lecture one may see [9]
.
and (20)
System (18) is symmetric hyperbolic: in fact a system A,a a U' :z: f is sima a aT metric hyperbolic if ~' is positive definite, and in our ~' a case we have ='Q' V'h', but from (20) h,Q t = hi = •A' I;a a = U'· U - h is the Legendre conjugate function of h and then it is a convex function of U' •
We remark also that the differential operator in (19) depends only on oneiQ
four-vector h
and this justifies our definition of "four vector generating func-
tion II for the symmetric system. We have seen that any convex covariant density system is endowed with a vector U' that may be expressed as a function of the field variable and is invariant with respect to transformations of field. In fact it is determined completely law
(8).
the
system
is
only
by
Moreover
well
assumes
posed.
the we a
conservative
pointed
out
system
that.
symmetric form,
Such
remarkable
so
(5)
and
the
supplementary
when
V'
is
that
the
local Cauchy
properties
suggest
chosen
us
to
as
field, problem
call
VI
the
"main field" of the system. We remark possess a the
that
not only on
the
mathematical
special role with respect
physical
point of view,
point of view U'
to other quantities,
and h I a
but also from
they are privileged, since they are related
to the Itobservables" of the physical system, as we shall see later. System a
(5),
sui table
easy
;-.0
compatible
choice of
prove
that
with
the
the
field
system
(8),
is
riducible
variables (18)
to
the
and vicevers8;
form
(18)
in fact
it is
provides always a supplementary law
(8):
let h
Q
= U'
• V'h lQ
_
h l o, we have a
U'·!' = g.
Finally we have shown that
for
a
h
Q
aa u'
U"f{U'}
265
A necessary and sufficient conditions for the system (5) to be compatible IJ'ith a supplementary conservation Z<w peat to
a
(8J
with
choice of the fieLd
convex function with res-
sY8tems)~
(convex covariant density
U = F (a
h(U)
is that there exists a
U' (invariant respect field transformations and indepen-
dent of the congruence defined by the time-like caveator {, )), so that the a
system (5) assumes the syrrrnetric form (18) with h'=h,QE; This
proposition
is
a
first
contribution
of
convex function of U
a questlon
the
proposed
I
by
I.MUller (liS challenge to mathematicians" [1]). At least we point out that if in (8) we impose the condi ticn g
>
O.
then, by (10) all solutions of (18) satisfy U' • ~ {U·}~O.
3. Shock Waves Theory for Convex Covariant Density Systems.
i) Entropy growth across a shock wave. Let
a
[1
connected
open
into two open subset Gh of r :
we shall
set
G2 _ Let".
indentify r
\1+
of
a
r
and
a hypersurface
) = 0, ~ E em
cutting II:
(m> 2). be the equation
with a shock hypersurface for the field U.
It is known that the Rankine-Hugoniot conditions must hold
[Fa]~
=O,onf a where brackets denote the jump across rand q. a = 0a q. • Formally
the
Rankine-Hugoniot
equations
are
(21)
obtained
from
the
field
eqs. (5) through the correspondence rule
aa However
this
rule
does
+
~
[
(22)
]
a not work
when
applied
to
the
supplementary
equation (8); in fact (23)
does
not.
in general,
n is
non negative.
vanish.
This
Furthermore
result was proven
1. t
is
possible
to show
that
in
a non covariant formalism
by P.D.Lax [4J introducing an artifical viscosity in the field equations; a different proof was given in [51. It is know that the positive signature of
n for the non relativistic perfect fluid implies the growth of thermo-
dynamic
entropy across
n
is
> 0
often
the
called
in
shock. the
That
is
the
literature
reason why the condition
"entropy growth condition"
•
266
aoo
is
assumed as
a criterion to pich ur> the physical shocKs among the
solutions of the Rankine-Hugoniot equations. In this
section we
suggest the
proof of the fact that Let It
I
main steps of an explicitly covariant
r.
is non negative on
Tl
1 and a a
be a 6ubcharacteristic covector such that
" covariant scalar
defined 8S:
r. a +a
a = -
(24)
th8'l there exists a space-like covector {t )
such that
"
(25)
Let
cp(x
Cl
)
= 0 be the equation of a characteristic hypersurface which
locally has the same Itdirection of propagation" Le.
det(A" -
(jl )
det{A"(~
•
a
-
a
t )}
-"
a
to of the shock surface. (26)
0,
(27) where
Ik)
are
(k-=l,2 •...• N)
~
the
solution of
(26);
these
eigenvalues
are real by the hyperboliclty condition. Now
we
consider
a
solution
U U.
r
U* (in
field
being the
the
U(U',9),
perturbed
following
computed
~
for
*
"
and
will
U ::
of
the
denote
U*).
Here
the
U
Rankine-Hugoniot ~
U'
we
values take
(21) (28)
unperturbed the
equations
fields of
any
respectively
on
function of the
only k-shocks
according to
the following
Definition of k-shock
We
shall
say
that
a
shock
is
a
k-shock
if
there exists a number k (=1.2 •...• N) such that lim C"'Il (k)
•
U
:: U·
(29)
267
Roughly speaking a k-shock
approaches
is a
shock that
to a characteristic velocity
weak Sh DC k 5 Wh en a
We suppose
is near to
to know
the
(k) )
~*
solution
vanishes
(of course,
(28)
for
a
shock
speed
these shocks become
k-shock
and
replace
it
a
a
differentiating
the
.
into (23): then we get n as function of U* and ¢I
By
when
a
"(U*'.a) • h (U(U*'.e».a - h (U*).a
(30)
to
ifa
(9).
a
J -
(30)
respect
and
taking
into account
after some calculations we obtain [h
Vh· [F
a
)
Thus (31)
Since h is a convex function of U, defined in a convex domain D, we have: w(U,U-) = -h(U) ... h(U·) + Vh·CU - Ufo) > 0,
lJ. U
f. U* oS D
So the r.h.s. in (31) is equal to -w, restricted to r. Hence 3 n Ja¢la
(a
Furthermore.
in
the
<
(32)
0
frame S • in which .; 0 =1.
~
i
=0,
locally,
condi tl-on
(32) can be written
ani CIa:>
(33)
0
As an/CIa is a scalar quantity, inequality (33) is independent of the
frame; So n is a strictly increasing function of
in any frame.
0
Since our shock is supposed to be a k-shock we have lim
n
= 0
0"" (k)
•
hence we get
For a convex covariant density system and a k-shock one Iuw when
0
;:.
(k)
~*
(on rJ.
268
1i) Jl
If
as generating f'unction of' the shock.
is a know function of u· and
1)
+" a
it is easy to prove that the
to llo»iYl{/ re lations ho lde on r
v· '1
A~ • a
J
[U'
..
(34)
whsre
EQ. a
(34)
that if ve kno... only the scalar function
D'le8nS
function oC U·
Jump
of
U'
and
and
• a ' w1 th
(which 1s
non characteristic) we may find the
•Q
therefore of U;
Tl
"behaves
like a
"potential"
for
the
shock. Of course,
in practice,
ll(U·, +0)
is computed when the shock is known
8S a solution of the Rankine-Hugoniot equations. However it is interesting the
fact
that.
were
it
possible
to
determine
n
through
experimental
tests, we should be able to have all information of the shock. iii) Relativistic bound of the shock speed.
The Rankine-Hugoniot equations Fa (U)+
(35)
a
provide N equations for the perturbed field U if U· and fa are knaYn. Eqs. (35) are equations of the kind
which always possess
feu. +) = f(U·.+ ) a a the trivial solution U
have also non trivial solutions U .solution)
which
in turn are
~
(36) :=
U"
for any
~a
They may
U* (branching solutions of the trivial
physically
acceptable only
aB
if g
tate
~
o.
so that the speed of' the shocks does not exceed that of light, according to relativity theory. We put no'"
the followin&; question: function
f
is
does it exist a set of values of
+0 such
that
the
globally
invertible
U for a
fixed
to? If the answer is affermative,
with
respect
to
then only the trivial
269 solution U
= U·
is allowed.
The problem has been examined by G.
[6]
SoHIat and T. Ruggeri
I
who proved
that non vanishing schoks take place only if their speed is greater
than
the
smallest
characteristic
speed
and
smaller
than
the
greatest
one. It is possible
to provide an explicitly covariant formulation of the
[6J
proof given in
and show that:
Fop hyperbolic convez covariant density systems the speed of the non vanishing shoek fuLfa the eondition: (37)
where m
mini \l
in!
UfD
As a consequence the
(k) }
I
M
sup
Max {\l
UfD
k
8h~ck
(k)
k
manifoLds are time-like or light-like if so are
the characteristic manifotds. In fact
if
(37)
holds,
and the characteristic manifolds are time-like
or light-like we have: .B (k) (k)} ( gcp~ Ma x <
sup
<
UED
I.
Relativistic
Hydrodynamics
-
k
Existence
•
B
of a
o .
Convex
Covariant Density.
The equations of relativistic hydrodynamics are
(38) (39)
CI
•
denoting the covariant derivative operator and r
tensor T
as
aB
the energy-momentum
a 8" as =rfuu -pg
(40)
•
Here r is the matter density. f the index of the fluid. u (u • u
one.
•
1)
and p
the 4-veloci ty
the pressure. The speed of light is assumed equal to
270
By (38) and (39). taking into account the thermodynamic relations, redS
(41)
rdf - dp
(42)
rf "'" p+ p
we can prove the existence of the supplementary conservation law (43)
e
where S is the specific entropy (entropy of the unit mass)
t
dynamic
dens!ty
absolute
temperature
and
p
the
proper
energy
the thermoof
the
fluid.
The system (38), (39) may be put in the compact form (5) on choosing:
f
_ 0; <6",,0,1,2,3).
The supplementary law (43) coincides with (8) when
o.
g
It is possible to show that the system of relativistic fluid dynamics
possesses a convex covariant density. In fact: i) there exist the main field satisfying (12)
u· _ 1 e where G ::;
f
-
eS
-
(,~', )
1 is the free enthalpy (it is remarkable that the
components of the main field,
me velocity.
(44)
all
independet,
essentially coincide ....ith
a (because u u
=1) and the free a enthalpy, Le. "observables" of the system) ; ii) it is possible to prove
the absolute temperature
that the covariant density - rSu is a convex function of the field
a
E;.a
271
u
F
a
(45)
for any unity time-like covector{(a}
oriented towards the future, provided
the usual thermodynamic conditions _ {G
0;
are
satiened
and
the
velocity
Sound
ep
)2 > 0
1s
(G
e
=
aetae.
G
p
=,G/'p)
smaller than the velocity of
light in vacuum. Hence
the
system
of
hydrodynamics
equations 1s a
particular case of
the general theory. As a consequence we have:
The system of 1'e'Lativistic hydPodynamiCB equations is syrrunetrie hyperbo'Lie
1)
in the fieLd U' (44.) with the four-vector generatirI{J fwwtions simp'ly given
by
h
[s]
2)
This is a
r#
on
0
>
,Q
when
conseq~ence
of i) since it is possible to show that
n:: _r·ua~ • a where
p*
is a
Q
""pu /8
[s J
r'(a_~')ua•
=
characteristic
(46)
speed of the corresponding material wave:
a
IJ* = (u.
6 r; )/(u. t ) .
6
a
3) The know7.edge of [s ] as a function of u' and ~
deternnnes the shock.
a
This is a consequence of ii) and (46).
4) The vetoC!ity of propagation of the relativistic hydrodynamics shocks never exceedS the speed og Light.
We have seen above that the main field U' of a convex covariant density system is may = h
be a
invariant
expressed
component
under
transformations
the
the gradient of the convex a , wi th respect to the field U :: F
the
as
of
proper
density
of
the
field
U,
and
that
covariant densi ty h
conserved
it ::
here h represents the quantity
h
a
in the
272
{~
congruence defined by the time-like covector It
is
another
{u} a
remarkable convex
that,
density
in the
h.
fluid
relative
to
1.
a case, i t is the
possible
field-dependent
to define congruence
with the same properties of h
h
= hau
rS • a whose gradient with respect to the field
Fa u
a
is still equal to the same main field U' (44) U'
Le.
Vh. (V -d(rS)
=
waU)
= -(ue/e)d(pu e)
+ (f/6-S)dr
as it immediately verified taking into account (41) and (42).
Moreover it is possible to prove that wi th respect to
U
h = -rS
is still a convex function,
and the prove does not require the auxiliary condition
(ap/ap)s~l. Hence the mapping U'
-U
is globally univalent.
REF ERE NeE S [IJ I.MOLLER. Habilitationsschrift an der RWTH Aachen (1970). Arch.Rat.Mecb. Anal. ~. 1-36 (1971). (See also "Entropy in non-equilibrium - a challenge to mathematicians" in Trend in Apj)lication of Pure Mat.ieiilatics to :«echanics. ·"01. II; Ed. H.Zorski I Pitman London, 281-295 (1979). [2J
K.O.FRIEDRICHS
and
P.D.LAX,
Proc.Nat.Acad.ScLU.S.A.
68
1686-1688
(1971). [3J
K.O.FRIEDRICHS. Comm. Pure Appl. Math. ,gz, also Comm. Pure Appl. Math. 31, 123-131 (1978)~
[4J
P.D.LAX. "'3hock waves and entropy" in Contribution to non linear functional analysis Ed. E.H. Zarantonello , 603-634 New York; Academic Press (1971).
749-808
(1974).
[5J G.BOILLAT, C.R. Acad.Sc. Paris 283A, 409-412 (1976). [~
G.BOILLAT and T.RUGGERI, C.R. Acad.Sc. Paris ~, 257-258 (1979).
[7J I-SHIH LlU. Arch. Rat. Mech. Anal., 46, 131-148 (1972).
(See
273
[8J
M.BERGER
and
M.BERGER,
Perspectives
in
nonlinearity.
>
III.A.Benjamin;,
Inc. New York, p.137 (1968). [9]
T . RUGGERI and A.STRUMIA, "Main Held and convex for quasi-linear hyperbolic systems. Relativistic (to appear).
density dynamic:.;".
covar~ant
fluid
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
SINGULAR SURFACES IN DIPOLAR MATERIALS AND POSSIBLE CONSEQUENCES FOR CONTINUUM MECHANICS
B. STRAUGHAN
CIME Session on Wave Propagation Bressanone, June 1980
Department of Mathematics, University of Glasgow University Gardens, Glasgow G12 8QW
Singular svrfQces·in dipolar materials and possible consequences for continuum mechanics. B. Straughan, University of Glasgow.
1,
Introduction In this paper we study the evolutionary behaviour of a
propagating singular surface in two types of nonlinear dipolar materials;
a compressible inviscid dipolar fluid and an elastic
dipolar solid. The basic theory we use was introduced by Green and Rivlin
[1]
and from the constitutive theory viewpoint essentially extends
classical continuum mechanics by including gradients of the independent variables present in non-polar theories.
Gradient type
theories were suggested earlier by, for example, !·1axwell and by Korteweg, see Truesdell and Noll [2], §125; ~rteweg
in particular,
developed an interesting theory of surface tension by
allowing for the possibility of rupidly changing density gradients in an interface.
Since in a singular surface quantities such as
density and its gradients of various orders may change very rapidly a study of wave motion in mUltipolar materials may prove of value. for an elastic dipOlar material the theory we employ was derived by Green and Rivlin
[1],
whereas the constitutiv~ develop-
ment for dipolar fluid theory is due to Bleustein and Green [3]
278
(later modified by Green and Naghdi [4J).
This theory allows
for an additional dipolar stress as well as the normal one. In contrast with Newtonian theory the constitutive variables include temperature, velocity and density gradients. This is
in one sense a generalization of the Maxwellian fluid of Truesdell (see [2], §125 and the references therein) in that a dipolar stress is included from the outset, although Truesdell's
Maxwellian fluid involves a constitutive theory which includes density, temperature and velocity gradients of arbitrary orders. We pay particular attention to the
compr~ssible
dipolar fluid
since as Truesdell and floll [2J point out, the Maxwellian theory,
••• "is Bet up in such a way as to emphasize effects of
cocrpressibi lity". The dipolar fluid equations given by Bleustein and Green [3] are now reviewed.
The equations of momer.tum and continuity
are
a.. .
(1.1)
Jl,~J
't
pf1,.•
(1.2)
where standard notation (see ego
[3])
is employed throughout.
However. the energy equation takes form (1.3)
pr -
where Akji
= V klji
p(A
't
TS + TS) - qi~i
't
tjidij
't
r(ij)kA~Ji ~ O.
and t ji and rijk are a symmetric tensor and
the dipolar stress tensor. respectively, related by the t ij
= °ij
eq\lad~Jr.
+ r.kijlk + p(Fi j - rij )·
Here Fi j are components of a dipolar hody force and r i j is the dipolar inertia which has form (see Green and Naghdi [4]) (1.5)
rJ1.. .. = a2 (V 't.J • • - v.'LII'::,y,.telJ. - v.1.... kV'JI k + Vk IJ.V,.1".:~1..•),
d2 being the inertia coefficient. inequality takes the fonn (1.6)
- peA
q.T •
+
sT) _....::....a..::+ T
Furthermore, the entropy
279
Bleustein and Green [3] develop the equations for a
compressible, viscous dipolar fluid.
However, as we are
primarily interested in wave motion we shall derive the
constitutive equations for a compressible inviscid dipolar fluid, following the procedure given by Bleustein and Green. Suppose then, that A, S, qi' variables (1.7)
P. P
0'
,,1.
ij and L(ij}k depend on the
T
P • 0' T, T 0' T ••• "l,J .It. ,,1,J
(3), the entropy inequality
Using (3.4) of Bleusteir. and Green (1.6) may be written as
'A -'.It.
(1.8)
."...- T d"J"
.A
rp-:-:-
(0
.'J
,A
-'-
o~T ..
-
,ij
.'
(0
+ 0
0
•
.1,J
.d ,. + odkk + 0 k"k 0) k~ "t.",t. 0
"t.
.d + odkk + pod, k . + 0 .dkk kk , '1. .~ JJ ,J . • 1. • '.I 'LJ
.'J 0
o.
0
+ vk.jP"ki + vk"iP.y...j + Ako.o k)
J' •
and the constitutive relations may then be reduced with the standard Coleman-Noll procedure whereby the terms appearing
linearly in (1.8) may be chosen arbitrariLy, the momentum and energy equations being balanced by suitable Ii or r; coefficients of these terms are zero.
Hence,
(1, 9)
A
= A(p,p
thus the
.,T).
.'
Since, as in [3], A must be a hemitropic function of its arguments, (1.9)2 reduces to
where (l.ll)
\I
=n
.p
.•
,,'l. .. 1.
280
What remains of the entropy inequality may be written as (ct. [3]. (3.14» (1.12)
[Ti,j + p2
:~ ~ij
... 2"
~~
P.,iP.,j)]dij
(vo ij ...
3A
]
+ [ t(ij)k'" p2 iV ( P i 6jk ~ P.,jOik) Akji, 1
Next, since f
¥q ,T
,
~
O.
i and r are at our disposal in (1.1) and (1.3) we
may choose a motion such that at a particular tenr. in (1.12) is non-zero.
~.t
only the second
Moreover, A. ok appears only linearly ~.1
and so we may select the motion in such a way as to invalidate the Thus, the term multiplying A
inequality.
ijk must be zero.
wise, we may argue that the term multiplying d .. is zero. ~J
LikeThis
leads to the constitutive relations 0.13)
(1.14)
T ..
= _ 1'2
3A
ap °ij -
J~
r.(ij'k
=
_ 02
3A
TV
2P aA (void ... p .p .). ., 1. IJ
av
+ 0 .,J'O'k)' 1..
(0 .,1.'O'k J
For a dipolar elastic material I find it easier with wave motion problems to use the equations as referred to a reference configuration, here taken to be a homogeneous one.
In this form
the equations are ([1J, p.342) for the isothermal case, (1.15)
'lI'Ai
+
• POFAi
+ "&4.i.. B
3U :: Po ~
t
<.A
3U "(BA)4'.. :: Po dX i .. AB
(1.16 )
0.17)
-Ai.. A +
poFi ::
POb it
' 1 energyt %i.. A -- 3x t.,f 'XA' h U i s t h e l.nterna were <,J
stress tensor,
I
EMi is the dipolar stress and P Ai is given by
(see Green and Naghdi [4.5]) (1.18)
A1: represents a
'It
281
with F
Ai components of dipolar body force and f Ai the dipolar
intertia defined by ([4J
t
eg. (11»
(1.19)
In these equations the internal
NAB being a constant tensor.
energy is a function of %i~A and %i"AB' i. e.
(1.20)
V = VIz.1-.. A' z.t... ABI.
We should remark at this stage that the constitutive functions
in (1.14) and (1.16) ar, given only for the
symm~tric
part with
respect to the first two indices of '8.4' or r. '1." This does not 'L 'LJY.. make any difference to the equations of motion, as the skew symmetric part is left unspecified.
However, it is important in
considering boundary conditions as is discussed in [1,3,5].
The
skew-symmetric part also plays an important role in three-dimensional wave motion studies (see [6]), but in this article we shall an~
not consider these matters
only investigate the novel effects
of the theory present in a one-dimensional analysis. It is pertinent to draw attention to a paper of Mindlin [7] who discusses other ways of describing elastic materials allowing for micro-structure effects and also includes a detailed account of linear wave motion and a critical comparison with results from discrete-type lattice theories.
2.
Dipolar stress waves Restrict attention now to the one-dimensional form of equations
(1.1), (1.2), (1.4), (1.5), (1.9), (1.13) and (1.14) with
f i = Fij
=0
and suppose p,
only on x and t.
v, the density and velocity, now depend
The one-dimensional form of the stress tensor,
dipolar stress tensor and dipolar inertia are denoted by and satisfy the relation (2.1)
(1
=
t
-
Ex ... pr,
0,
E, r
282
where the notation ~% signifies utive equations become
~:t ~
:
~~
+
V~=.
The consti t-
P",
(2.2)
T :: -
(2.3)
r :: - q.
(2.4 )
r = d 2 (~
p -
'"
(2.5)
~
_
q,
v2 ) ",'
A :: A(p,pz)'
where the pressure p and dipolar (2.6 )
P
= P 2 apt aA
(2.7)
q
= p2 ~
ap",
pressu~
q are defined by
•
The continuity and momentum equations are (2.8)
(, + av.:::; :: O.
(2.9)
Suppose (2.8) and (2.9) hold on R2 and p and v are continuously differentiable functions of :z: and t on ~2.
moreover, that there is a surface r x ft} for each t gating in the
v,
~%1
on
(R,
V:r'
The
t)
~direction
with speed
un{~
0).
Suppose. ~
0 propa-
The quantities
~, ~x' Pxx are assumed continuous functions of %, t
x
R and
ju~p
have at most jump discontinuities across r.
in a quantity P is defined by
(2.10)
where a superscript +
si~n
denotes the region ahead of the wave,
a negative sign denoting the region behind the wave. A surface as defined above is said to be a one-dimensional dipolar stress wave and the corresponding wave aDplitudes are defined by (2.il )
B
=
(0).
(2.12)
C
=
[v,,).
283
To obtain the first discontinuity equation, (2.8) is differentiated with respect to X and limits either side of r are taken to find
(2.13)
r•px ]
+
p
[v:x:r:] =
O.
To further analyse this equation we observe that for a function W it can be shown (see Truesdell and Toupin [8J) (2.14)
m = ~t
[.] -
U[.,,].
where 6 jat is the displu,'ement derivative defined by
(2.15 )
and the relative wavespeed is (2.16)
With the aid of (2.14), (2.13) may thus be rearranged as (2.17)
- UB + pC
= o.
Until this point our analysis has proceeded as for a third order wave for a classical perfect fluid;
it is now things change.
For, if we apply the jump conditions to (2.9) and uSe (2.14) we find the resulting equation already involves the displacement derivative of C.
Thus, it is necessary to adopt another approach
to find a suitable equation to employ in conjunction with (2.17). We instead adopt the approach which is used with shock waves in a classical perfect fluid and use the discontinuous equation of balance form of (2.9) (see Green and Xaghdi [9]).
For this
reason we refer to the third order waves studied here as dipolar stress waves. The appropriate equation of balance is (2.18)
-U[pv] = [oj.
Of course, for the waves considered here, the left hand side of this equation is zero, and so (2.19)
p[r] = [rxl - [T].
284
By the assumed regularity conditions at the wavefront, (2.2) allows us to deduce that [~J
=0
and similarly, with the aid
of (2.4),
(2.20)
orr] :
Also, employing (2.3) and
(2.7~
(2.21)
[IZ] :
pd2 C.
Inserting these expressions into (2.19) leads to pd 2 UC _
(2.22)
3 2A
0 2 - 8 : O.
dP;
rinally we solve for U between (2.17) and (2.22) to obtain U2 :
(2.23)
L!L tfl
30
t
z
(cf. the corresponding expression for sound waves in a classical perfect fluid, ego Truesdell [10].)
To determine the behaviour of the amplitude as time evolves we differentiate (2.8) first materially with respect to t, and then with respect to
%.
and then take jumps across the wave;
also, we take jumps of (2.9) to find a second equation.
The
details of these calculations are contained in [6J and so we simply describe the results here. 68 6t
The relevant equations are
2 pV + Pz - U ~) + 2V,t: B + C ( 'U x
-
UE + pG ~ 0,
and t
Although the principal part of (2.8) and (2.9) suggests these equations aay be written as a hyperbolic system this is not evidently so.
While we would have two real wavp.speeds as
in (2.23), the remaining two would be complex.
I am indebted
to Professor T. Ruggeri for pointing this out to me.
285
(2.25)
1 a~ U (lp.:c
+B ( 2p -_
U
2
_~+-q
pU
+ a'n
1
an
U3
ao 6t
6U
an)
2
..;....;I..+_~_-_v.:.:l.-
=
ap'
'"
where
'"
U'
'"
ao
= o.
'"
(2.26 )
Finally. equations
(2.2~)
and (2.25) may be combined
together to obtain
2! + _1_ ~
(2.27)
6t
2U'a'
where X is given by
(2.28)
5 K=-v
2
B' + lII1
= O.
ao'
'"
,
+,
Up", aq aq 6U --+---+-----. '" 2p ua' aO ao Ua' ap' 2U 6t 0",
0=
'"
1
'"
The amplitude equation (2.2?) is of Bernoulli type and can be solved explicitly, see Chen [11] who also includes several theorems describing the behaviour for various initial data ([11). pp. 387-395).
Of course, once BCt) is known
Cet) may be found from (2.17).
Rather than describe in
detail the solution to (2.27) we suppose the reginn ahead of the wave is one of
constan~
density for which K
= O.
For
this case (2.27) has solution (2.29)
B(t)
=
B(O) /
(1
If the initial wave amplitude is such that sgn B(O)
a'n = -sgn~, dp2
then the amplitude becomes infinite as t
~
-2u'a' B(O) ~
ao'
'"
and the
'"
286
dipolar stress wave in some sense breaks down. (While the treatment of the continuity and momentum
equations is different to the usual procedure for acceleration waves in a perfect fluid, we have essentially used the compatibili ty relations in the manner employed by Chen and his assoc-
iates, see ego [11J.
This approach would initially appear
-different to that of Jeffrey, see ego [12];
but, the two
methods are in fact equivalent as was shown by Boillat and Ruggeri [13
3.
J•)
Elastic dipolar stress waves. While it would be possible to develop a theory of three
dimensional elastic dipolar waves, in the spirit of the work described by Chen [11]. it Seems more revealing at this stage to concentrate on the novel effects caused by the presence of strain gradients.
As these effects are present in one-dimen-
sional wave motion we restrict attention to this case.
To
this end, therefore, let us rewrite equations (1.15) - (1.20) in their one-dimensional form. Let V and F
= aX/ax
be functions of
X and t and supposing
the body forces are zero the equation of motion is then
(3.1 ) where
(3.2)
all& au • = Pl)i!wl;x + Po aF - Po
where in this and the next time derivative, F
§
au ;w.:.
x
a superposed dot denotes the partial
= ax/ 3X , M is
the dipolar inertia coefficient
and
(3.3) In the form (3.1) it is clear that the dipolar theory leads to a differential equation of fourth order.
287
An elastic dipolar stress wave is defined to be 8 surface
across which the third derivatives of
%
are the
finite discontinuities, i.e. x € C2 OR2).
firs~ ~o
L
have
Because of the
order in (3.1) it is clear that to obtain the wavespeed of such a wave we must consider the discontinuity relation given
by
[9]
Green and Naghdi
in Chen [11]).
(ct. the shock wave equation (5~lfi).
If we den'Jte the wavespeed by V the ar,proprii'te
balance relation is
Subsequent analysis makes use of the relation (equivalent to (2.14»)
~t
<3.5 )
[r] : [i]
+
v[~]
In view of the assumed regularity of x, (3.4) reduces to (3.6)
[PXx]
32U -3p2
: U
[r X]·
X
From use of (3.5) this may be changed to 2 3 y : I' V 2
(3.7)
3p2 X
[Fxx ] t
When
0, which we are implicitly assuming, (3.7) leads
immediately to an expression for the wavespeed, V, <3.8)
V2 :
l
2 3 y
M 3pi
It is worth comparing (3.8) with the analogous expression for an
acceleration wave in a nonlinear (ordinary) elastic material I see ego Chen [11]. equation (6.12).
To derive an expression for the wave amplitude we now return to equation (3.1) and take the jump of this. (3.9)
o:
[LCY)~ 3X• 31'
[~(EL)] 3X2 3F .
X
The result is
+
If ["
] }'}f
288
Expanding the first two terms on the right gives
C3.10}
~hJ
(3.11)
These expressions are inserted into (3.9) and together with (3.5) and the relation
(3.12) we may derive the following ordinary differential equation for the amplitude, a( t) ; [EoX];
(3.13)
•
6a . a {/
a3u
y2
ap'X
• --- a
2
= O.
Again, this equation may be solved as in Chen [11].
ahead of the wave is in constant strain.
If the region
F; = F;x = 0, Vis
constant and then
(3.14)
aCt}
= a(O}!
(1 •
t a(O)
"U).
2HV2 ,F'
X
a'u Obviously, under the right conditions on a(O), H, V and - there
,p'X
will be breakdown of aCt). In both this and the preceding
§
it is evident that higher
order waves, i.e. of one order higher than what is usually defined as an acceleration wave, suffer a type of breakdown analogous to that observed in acceleration waves in classical theories. usually ~hought
to be associated with formation of a wave of lower order.
If the behaviour encountered here is not to be accepted. then one
289
may argue:
(a)
the dipolar theories should not be used as exact theories for wave propagation problems.
(b)
the classical case is the first order approx-
imation and dipolar terms should be included in an expansion procedure. If (a) is thought to be the case then perhaps one has to look at a viscoelastic type theory in which gradients of all orders
are present.
Certainly, it would appear
nec~ssary
to invest-
igate higher order theories. If we try to study shock waves or acceleration waves in the theories considered here, we find that the basic governing equations are not in a suital-Ie form.
For example, equation
(3.1) contains terms of second, third and fourth orders and it
is not obvious how to proceed in that case.
It is perhaps
likely that the higher order nature of the equation prevents the formation of waves of lower order.
Certainly the phen-
omenon of focussing (see eg. Knops and Wilkes [l~]) is not likely to occur.
To see this and complete this work we consider
a linear dipolar elastic material and show that, unlike classical linear elasticity. stability in the CO norm is possihle.
(This
possibility was pointed out by Knops and Wilkes [14J and Knops
and Straughan [15]).
4.
Uniform stability of a linear dipolar elastic material. The appropriate equations for an anisotropic linear dipolar
elastic material. which may be derived from the cited papers of Green and Naghdi and Green and Rivlin, are (see [15]) oU. - pr,. , 1. J"£.,J
To. , • r(k'.JoJ 1.,......, = - of., J'L,J 1.J,J 0
,."
0
•
pf"
where U are the components of displacement about a reference
i
configuration and the other terms are named in §l. inertia coefficient satisfies
Here, the
t.
290
rji
::
Mjk
(~) Ui~kt
where mo. is a symmetric tensor which defines a non-negative .J bilinear form. The constitutive equations for tij and I(kj)i
are
[(.")., "k" p,q + ....,. = bpq1-J
"k" p,qm • pqm1-J
C
where the "elasticities" satisfy the following symmetries ('.5) C ijkhmn
:: c1urm.ijk
We suppose (4.1) are defined on a bounded region with u
i
and 3u i prescribed on the boundary of 0,
au
"i = qi 3" • ....-:. :: 30
hi
(~.t)
on
r
x
r.
Q
of R'
Say
[0.-).
(~.t)
(Green and Rivlin [1] suggest that (4.6)2 should be replaced by conditions on the dipolar traction.
However, we have chosen
these conditions so that we may apply the appropriate Soholev inequality for illustration of the new stability effect.) Let 'Vi and
v~
• be
two solutions to (4.1) in
body forces and same boundary data (u.6).
n with the same :: V": - v. i ,
Define u.
•
and then a routine calculation enables us to show the total
energy E(t) (".7)
. • )d.r + 2 broctij " 1',,8 tU 't"J
=
E(O) •
V t > 0.
If now, the potential energy is positive definite in the following sense
(' ., OJ,. we deduce from (4.7) that (4.9)
E(O)
~
•
J
u. jk U
n However, for n c JP.
w%,2
Ul)
1- a
,
'k
t..J
az.
C COCO), see Gilbar-g and i'rudinger
[16], corollary 7.11, and so there is a constan~ ~ depending on
n.
A such that
(4.10)
[E(O)]!
~
" sup
n
1~(~.t)l.
Thus, by making F(O) arbitrarily small
luI
is likewise a~bi~ra~lly
small and so the solution is stable in the CO norm~
Acknowl edgments
I should like to express my sincere thanks to Professor D. Graffi for his kind invitation to study in Italy in June - July 1990 and to both Professor Graff! and Professor G. Ferrarese for the opportunity to hold a seminar in Bressanone.
finally, I
wish to express my appreciation to Professors D. Graffi and M.
Fabrizio and to Dr. F. Franchi for many stimulating discussions.
292
REFERENCES 1.
Arch. Rational Hech. Anal.
A.E. GREEN and R.S. RIVLIN.
16 (1964). 325-353. The nonlinemo field theories of mechanic8. Randbuch der physik, Vol. 111/3. Springer-Verlag, Berlin-Heidelberg-New York, 1965.
2.
C. TRUESDELL and W. HaLL.
3.'
J.L. BLEUSTEIN snd A.E. GREEN.
Int. J. P.ngng. Sci. 5
(1967). 323-340. 4.
~t.
A.E. GREEN and P.H. NAGHDI.
App.
~bth.
28 (1970).
458-460. S.
J. Hecanique. 7 (1968),
A.t. GREEN and P.M. NAGfIDI.
465-474. B. STRAUGHAN.
Ann. f..fat. Pura Appl.
7.
R.D. HI1IDLIN.
J. El.o.sticity
8.
C. TRUESDELL and R. TOUPIN.
6.
Handbuch dero physik,
To be published.
2 (1972), 217-282.
The classical field theories.
Vol. II 1/1.
Springer-Verlag,
Berlin-~ttingen-Heidelberg,1960.
9.
A.E. GREEN and P.H. NAGHDI.
Int. J. Engng. Sci. 2 (1965),
621-624. 10.
C. TRUESDELL.
11.
P.J. CHEN. de~
J. Rational Mech. Anal. 2 (1953). 643-741.
Growth and deaay of t:la'V6S in Bolids. physik, Vol. Vla/3. Springer-Verlag,
Bandbuch
Berlin-Heidelberg-New York, 1973. 12.
13. 14.
Lectures on nonlinear lJave propagation. CIHE session on wave propagation, Bressanone, June 1980. G. BOILLAT and T. RUGGERI. Wave ~otion 1 (1979), 149-152. R.J. IQIOPS and E.W. WILKES. Theory of elastic stabiHty. 8andbuch dar physik, Vol. VIaI3. Springer-Verlag, A. JEFFREY.
Berlin-Heidelberg-llew York, 1973. 15.
R.J. IQIOPS and B. STRAUGHAN.
Aroch. Ilech. StDs.
28 (1976).
431-441-
16.
D. GILBARG and N.S. TRUDINGER. Elliptic parotial differential
equations of second order.
Springer-Verlag, Berlin-Heidelberg-New York, 1977.
