Theory of Elasti Waves Gerhard Müller
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Theory of Elasti Waves Gerhard Müller
Editors Mi hael Weber - GFZ & Universität Potsdam Georg Rümpker - Universität Frankfurt Dirk Gajewski - Universität Hamburg Germany tew_2007.ps + tew_2007.pdf
2
Prefa e When Gerhard Müller hose to leave us on 9 July 2002 be ause of his illness, we lost a tea her and olleague. Part of his lega y is several le ture notes whi h he had worked on for more then 20 years. These notes have be ome the ba kbone of tea hing seismi s and seismology at basi ally all German universities. When asked some years before his death if he had onsidered to translate "Theorie Elastis her Wellen" into English and publish it as a book, his answer was "I plan to do it when I am retired". We hope that our eort would have found his approval. We would like to thank R. and I. Coman (Universität Hamburg) for preparing a rst, German draft in LATEX of this s ript, A. Siebert (GFZ Potsdam) for her help in preparing the gures and our students for pointing out errors and asking questions. We would like to thank A. Priestley for proof-reading the s ript and turning Deuts hlish into English and K. Priestley for his many omments. We thank the GFZ Potsdam and the Dublin Institute for Advan ed Studies for their support during a sabbati al of MW in Dublin, where most of this book was prepared. We would also like to thank the GFZ for ontinuing support in the preparation of this book.
M. Weber
G. Rümpker
D. Gajewski
Potsdam, Frankfurt, Hamburg January 2007
This le an be downloaded from
http://gfz-potsdam.de/mhw/tew/
tew _2007.ps(64MB) + tew _2007.pdf (3.5MB) 3
4
Prefa e of the German Le ture Notes This s ript is the revised and extended version of a manus ript whi h was used for several years in a 1- to 2-semester le ture on the theory of elasti waves at the universities of Karlsruhe and Frankfurt. The aim of this manus ript is to give students with some ba kground in mathemati s and theoreti al physi s the basi knowledge of the theory of elasti waves, whi h is ne essary for the study of spe ial literature in monographs and s ienti journals. Sin e this is an introdu tory text, theory and methods are explained with simple models to keep the omputational omplexity and the formulae as simple as possible. This is why often liquid media instead of solid media are onsidered, and only horizontally polarised waves (SH-waves) are dis ussed, when shear waves in layered, solid media are onsidered. A third example is that the normal mode theory for point sour es is derived for an ideal wave guide with free or rigid boundaries.
These simpli ations o
asionally hide the dire t onne tion to
seismology. In my opinion, there is no other approa h if one aims at presenting theory and methods in detail and introdu ing at least some aspe t from the wide eld of seismology. After working through this s ript students should, I hope, be better prepared to read the advan ed text books of Pilant (1979), Aki and Ri hards (1980, 2000), Ben-Menahem and Sing (1981), Dahlen and Tromp (1998), Kennett (2002) and Chapman (2004), whi h treat models as realisti ally as possible. This manus ript has its emphasis in the wave seismi treatment of elasti body and surfa e waves in layered media. The understanding of the dynami properties of these two wave types, i.e., their amplitudes, frequen ies and impulse forms, are a basi prerequisite-requisite for the study of the stru ture of the Earth, may it be in the rust, the mantle or the ore, and for the study of pro esses in the earthquake sour e. Ray seismi s in inhomogeneous media and their relation with wave seismi s are dis ussed in more detail than in earlier versions of the s ript, but seismologi ally interesting topi s like eigen-modes of the Earth and extended sour es of elasti waves are still not treated, sin e they would ex eed the s ope of an introdu tory le ture. At several pla es of the manus ript, exer ises are in luded, the solution of these is an important part in understanding the material. One of the appendi es tries to over in ompa t form the basi s of the Lapla e and Fourier transform and of the delta fun tion, so that these topi s an be used in the main part of the s ript. I would like to thank Ingrid Hörn hen for the often tedious writing and orre ting of this manus ript.
Gerhard Müller
Contents 1 Literature
9
2 Foundations of elasti ity theory
11
2.1
Analysis of strain
. . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2
Analysis of stress . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3
Equilibrium onditions . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4
Stress-strain relations
. . . . . . . . . . . . . . . . . . . . . . . .
26
2.5
Equation of motion, boundary and initial ... . . . . . . . . . . . .
31
2.6
Displa ement potentials and wave types
34
. . . . . . . . . . . . . .
3 Body waves
39
3.1
Plane body waves . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2
The initial value problem for plane waves
40
3.3
Simple boundary value problems for plane waves
3.4
Spheri al waves from explosion point sour es
3.5
Spheri al waves from single for e and dipole ...
3.6
. . . . . . . . . . . . . . . . . . . . . .
43
. . . . . . . . . . .
45
. . . . . . . . . .
49
3.5.1
Single for e point sour e . . . . . . . . . . . . . . . . . . .
49
3.5.2
Dipole point sour es . . . . . . . . . . . . . . . . . . . . .
56
Re e tion and refra tion of plane waves ... . . . . . . . . . . . . .
62
3.6.1
Plane waves with arbitrary propagation dire tion . . . . .
62
3.6.2
Basi equations . . . . . . . . . . . . . . . . . . . . . . . .
63
3.6.3
Re e tion and refra tion of SH-waves
. . . . . . . . . . .
66
3.6.4
Re e tion of P-waves at a free surfa e . . . . . . . . . . .
76
5
6
CONTENTS
3.6.5 3.7
3.8
3.9
3.10
Re e tion and refra tion oe ients for layered media . .
Re e tivity method: Re e tion of ...
84
. . . . . . . . . . . . . . . .
92
3.7.1
Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.7.2
Re e tion and head waves . . . . . . . . . . . . . . . . . .
95
3.7.3
Complete seismograms . . . . . . . . . . . . . . . . . . . .
97
Exa t or generalised ray theory - GRT . . . . . . . . . . . . . . .
98
3.8.1
In ident ylindri al wave . . . . . . . . . . . . . . . . . . .
99
3.8.2
Wavefront approximation for
3.8.3
Re e tion and refra tion of the ylindri al wave . . . . . . 103
3.8.4
Dis ussion of re e ted wave types
UR
. . . . . . . . . . . . . . 102
. . . . . . . . . . . . . 110
Ray seismi s in ontinuous inhomogeneous media . . . . . . . . . 113 3.9.1
Fermat's prin iple and the ray equation
3.9.2
High frequen y approximation of the equation of motion . 119
3.9.3
Eikonal equation and seismi rays
3.9.4
Amplitudes in ray seismi approximation
. . . . . . . . . . 114
. . . . . . . . . . . . . 121 . . . . . . . . . 123
WKBJ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.10.1 Harmoni ex itation and re e tion oe ient . . . . . . . 126 3.10.2 Impulsive ex itation and WKBJ-seismograms . . . . . . . 131
4 Surfa e waves 4.1
135
Free surfa e waves in layered media . . . . . . . . . . . . . . . . . 135 4.1.1
Basi equations . . . . . . . . . . . . . . . . . . . . . . . . 135
4.1.2
Rayleigh waves at the surfa e of an homogeneous half-spa e138
4.1.3
Love waves at the surfa e of a layered half-spa e
4.1.4
Determination of the phase velo ity of surfa e waves from
. . . . . 142
observations . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.1.5
The group velo ity . . . . . . . . . . . . . . . . . . . . . . 153
4.1.6
Des ription of surfa e waves by onstru tive interferen e of body waves . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.2
Surfa e waves from point sour es . . . . . . . . . . . . . . . . . . 160 4.2.1
Ideal wave guide for harmoni ex itation . . . . . . . . . . 160
4.2.2
The modal seismogram of the ideal wave guide
4.2.3
Computation of modal seismograms with the method of stationary phase
4.2.4
. . . . . . 167
. . . . . . . . . . . . . . . . . . . . . . . 170
Ray representation of the eld in an ideal wave guide
. . 173
7
CONTENTS
A Lapla e transform and delta fun tion A.1
A.2
179
Introdu tion to the Lapla e transform
. . . . . . . . . . . . . . . 179
A.1.1
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.1.2
De nition of the Lapla e transform . . . . . . . . . . . . . 179
A.1.3
Assumptions on F(t) . . . . . . . . . . . . . . . . . . . . . 180
A.1.4
Examples
A.1.5
Properties of the Lapla e transform
A.1.6
Ba k-transform . . . . . . . . . . . . . . . . . . . . . . . . 183
A.1.7
Relation with the Fourier transform
. . . . . . . . . . . . . . . . . . . . . . . . . . . 180 . . . . . . . . . . . . 181
. . . . . . . . . . . . 184
Appli ation of the Lapla e transform . . . . . . . . . . . . . . . . 184 A.2.1
Linear ordinary dierential equations with onstant oef ients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.2.2 A.3
Partial dierential equations
The delta fun tion
δ(t)
. . . . . . . . . . . . . . . . 192
. . . . . . . . . . . . . . . . . . . . . . . . 194
δ(t)
. . . . . . . . . . . . . . . . . . . . . 194
A.3.1
Introdu tion of
A.3.2
Properties of
A.3.3
Appli ation of
A.3.4
Duhamel's law and linear systems
A.3.5
Pra ti al approa h for the onsideration of non-zero initial
δ(t)
. . . . . . . . . . . . . . . . . . . . . . . 197
δ(t)
. . . . . . . . . . . . . . . . . . . . . . 200 . . . . . . . . . . . . . 202
values of the perturbation fun tion F(t) of a linear problem205
B Hilbert transform
209
B.1
The Hilbert transform pair
. . . . . . . . . . . . . . . . . . . . . 209
B.2
The Hilbert transform as a lter
. . . . . . . . . . . . . . . . . . 210
C Bessel fun tions
215
D The Sommerfeld integral
219
E The omputation of modal seismograms
221
E.1
Numeri al al ulations . . . . . . . . . . . . . . . . . . . . . . . . 221
E.2
Method of stationary phase
E.3
Airy phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Index
226
. . . . . . . . . . . . . . . . . . . . . 222
8
CONTENTS
Chapter 1
Literature The following list ontains only books, that treat the propagation of elasti waves and a few text books on ontinuum me hani s. Arti les in journals are mentioned if ne essary. Their number is kept to a minimum.
A henba h, J.D.: Wave propagation in elasti solid, North-Holland Publ. Comp., Amsterdam, 1973 Aki, K. and P.G. Ri hards: Quantitative seismology - theory and methods (2 volumes), Freeman and Co., San Fran is o, 1980 and 2002 Ben-Menahem, A. and S.J. Singh: Seismi waves and sour es, Springer, Heidelberg, 1981 Bleistein, N.:
Mathemati al methods for wave phenomena, A ademi Press,
New York, 1984 Brekhovskikh, L.M.: Waves in layered media, A ademi Press, New York, 1960 Brekhovskikh, L., and Gon harov, V.: Me hani s of ontinua and wave dynami s, Springer-Verlag, Berlin, 1985 Budden, K.G.: The wave-guide mode theory of wave propagation, Logos Press, London, 1961 Bullen, K.E., and Bolt, B.A.:
An introdu tion to the theory of seismology,
Cambridge University Press, Cambridge, 1985 Cagniard, L.: Re e tion and refra tion of progressive waves, M Graw-Hill Book Comp., New York, 1962
ervéný, V., I.A. Molotov and I. P² en ík:
Ray method in seismology, Univerzita
Karlova, Prague, 1977 Chapman, Ch.: Fundamentals of seismi wave propagation, Cambridge University Press, 2004 9
10
CHAPTER 1.
LITERATURE
Dahlen, F. A. and J. Tromp: Theoreti al global seismology, Prin eton University, Prin eton, New Jersey, 1998 Ewing, M., W.S. Jardetzky and F. Press:
Elasti waves in layered media,
M Graw-Hill Book Comp., New York, 1957 Fung, Y.C.: Foundations of solid me hani s, Prenti e-Hall, Englewood Clis, N.Y., 1965 Grant, F.S. and G.F. West: Interpretation theory in applied geophysi s, M GrawHill Book Comp., New York, 1965 Hudson, J.A.: The ex itation and propagation of elasti waves, Cambridge University Press, Cambridge, 1980 Kennett, B.L.N.:
The seismi wave eld (2 volumes), Cambridge University
Press, Cambridge, 2002 Landau, L.D. and E.M. Lifs hitz: Elastizitätstheorie, Akademie Verlag, Berlin, 1977 Love, A.E.H.: A treatise on the mathemati al theory of elasti ity, 4th edition, Dover Publi ations, New York, 1944 Pilant, W.L.: Elasti waves in the Earth, Elsevier, Amsterdam, 1979 Riley, K.F., M.P. Hobson and S.J. Ben e: Mathemati al methods for physi s and engineering, A omprehensive guide, Cambridge University Press, Cambridge, 2nd edition, 2002 Sommerfeld, A.: Me hanik der deformierbaren Medien, Akad. Verlagsgesells haft, Leipzig, 1964 Tolstoy, I.: Wave propagation, M Graw-Hill Book Comp., New York, 1973 Tolstoy, I. and C.S. Clay: O ean a ousti s-theory and experiment in underwater sound, M Graw-Hill Book Comp., New York, 1966 White, J.E.: Seismi waves-radiation, transmission and attenuation, M GrawHill Book Comp., New York, 1965
Chapter 2
Foundations of elasti ity theory Comments In this hapter symboli and index notation is used, i.e., a ve tor (symboli
− → f ) is also written as fi ( omponents f1 , f2 , . . . , fn ), the lo ation ve tor → (symboli − x ) as xi ( omponents x1, x2, x3 ), and a matrix (symboli a) as aij
notation (i
= line index = 1, 2, . . . , m, j = row index= 1, 2, . . . , n). aij with the ve tor fj is the ve tor gi =
n X
aij fj
The produ t of matrix
(i = 1, 2, . . . , m).
j=1
summation onvention = SC )
A short notation for this is (
gi = aij fj . In the following text, if a produ t on the right o
urs in whi h there is a repeated index, this index takes all values from 1, 2, ..., n (usually n = 3) and all produ ts are summed. If the symboli notation is simpler, e.g., for the ross produ t of two ve tors or for divergen e or rotation, the symboli notation is used.
11
12
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
2.1 Analysis of strain 2.1.1 Components of the displa ement ve tor Consider a body that is deformed by an external for e. Before deformation, the
xi and the in nitesimal lose point Q has the xi + yi . The omponents of yi are assumed to be independent variables; this is why dxi was not used. After deformation, P has been displa ed by the displa ement ve tor ui to P', and Q has been displa ed to Q' by the point
P
has the lo ation ve tor
lo ation ve tor
ve tor (expansion up to linear terms)
zi = ui + dui = ui +
∂ui yj ∂xj
(SC).
Fig. 2.1: Neighbourhood of P and Q before and after deformation.
Ve tor
P
zi
des ribes (for variable
Q
in the neighbourhood of
P ) the hanges near
due to the deformation. In general, these hanges onsist of: a translation,
a rotation of the
whole
region around an axis through
P
and the a tual de-
formation, whi h hanges the length of lines (rotation and deformation will be dis ussed later in more detail)
zi = ui + dui =
ui translation
ǫij
=
ǫij ξij
= =
+
ǫij yj deformation
+
ξij yj rotation
1 ∂ui 1 ∂ui ∂uj ∂uj , ξij := + − 2 ∂xj ∂xi 2 ∂xj ∂xj ǫji −ξji (⇒ ξ11 = ξ22 = ξ33 = 0) .
(2.1) (2.2) (2.3)
2.1.
The matri es and
13
ANALYSIS OF STRAIN
ξij
ǫij
and
ξij
are
tensors of 2nd degree. ǫij is symmetri due to (2.2) ǫij is alled deformation tensor and ξij
is anti-symmetri due to (2.3).
is alled
rotation tensor.
2.1.2 Tensors of 2nd degree A tensor of 2nd degree, transforms ve tor
yi
tij ,
transforms a ve tor into another ve tor (e.g.,
into the deformation part of
dui ;
ǫij
another example of this is
the inertial tensor transforms the ve tor of the angular velo ity into the rotation impulse ve tor (rotation of a rigid body)). If the oordinate system is rotated, the tensor omponents have to be transformed as follows to yield the original ve tor
t,kl amn
= =
aik ajl tij cos γmn
(SC twi e) (see sket h).
(2.4)
t,kl = Tensor omponent in the rotated oordinate system (dashed line in sket h).
Fig. 2.2: Coordinate system of tensors of 2nd degree.
For a ertain orientation of the rotated system, the non-diagonal elements
t′12 , t′13 , t′21 ,
... vanish. These oordinate axis are alled
main axes
of the tensor,
and the tensor is in diagonal form. In the diagonal form, many physi al relations be ome simpler. Certain ombinations of tensor omponents are independent of the oordinate system of the tensor. These are the three
T3
invariants (T1 , T2 ,
are the diagonal elements of the tensor in diagonal form). More on tensors
an be found in, e.g., Riley, Hobson and Ben e.
14
CHAPTER 2.
I1 = I2 = I3 =
t11 t12 t13 t21 t22 t23 t31 t32 t33 t11 + t22 + t33
FOUNDATIONS OF ELASTICITY THEORY
= T1 T2 T3
(determinant)
= T1 + T2 + T3
(tra e)
t11 t22 + t22 t33 + t33 t11 − t12 t21 − t23 t32 − t31 t13
= T1 T2 + T2 T3 + T3 T1 .
2.1.3 Rotation omponent of displa ement The rotation omponent of displa ement follows from
0 −ξ12 −ξ13
ξ12 y2 + ξ13 y3 ξ13 y1 − → → ξ23 y2 = −ξ12 y1 + ξ23 y3 = ξ × − y −ξ13 y1 − ξ23 y2 0 y3
ξ12 0 −ξ23
with
1 − → → u. ξ = (−ξ23 , ξ13 , −ξ12 ) = ∇ × − 2
− → − ξ ×→ y des ribes an in nitesimal rotation of the region of P around − → an axis through P with the dire tion of ξ . The rotation angle has the abso− → → − → lute value ξ and is independent of − y (show). A prerequisite is that ξ is Ve tor
in nitesimal. A su ient ondition for this is that
∂ui ∂xj ≪ 1
for all
i
and
j.
(2.5)
2.1.4 Deformation omponent of displa ement After separating out the rotation term, only the deformation term is of interest sin e it des ribes the for es whi h a t in a body. The deformation is des ribed
ompletely by the six omponents
ǫij
whi h are, in general, dierent.
These
dimensionless omponents will now be interpreted physi ally. The starting point is
dui = ǫij yj ,
i.e., we assume no rotation.
a) During this transformation, a line remains a line, a plane remains a plane, a sphere be omes an ellipsoid and parallel lines remain parallel.
Deformation omponents ǫ11 , ǫ22 , ǫ33 Coordinate origin at P and spe ial sele tion of Q : y1 6= 0, y2 = y3 = 0.
b)
2.1.
15
ANALYSIS OF STRAIN
2 Q’
du 2
y1
P
Q
du 1
1
Fig. 2.3: Sket h for deformation omponents.
du1
=
ǫ11 y1
du2 du3
= =
ǫ21 y1 ǫ31 y1 = 0
(assumption :
ǫ
31
not the relative hange
du1 y1 is the relative hange in length in dire tion 1 ( in length of , see also d). Stret hing o
urs if ǫ11 >
ǫ11 =
ǫ11 < 0.
PQ
Similarly,
ǫ22
and
ǫ33
0
and shortening if
are the relative length hanges in dire tion 2 and
3.
)
= 0).
Shear omponents ǫ12 , ǫ13 , ǫ23
Fig. 2.4: Sket h for shear omponents.
Q1 → Q′1 : du2 Q2 → Q′2 : du1
= =
ǫ21 y1 = ǫ12 y1 ǫ12 y2
16
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
du2 y1 du1 tan β ≃ β ≃ y2
tan α ≃ α ≃
This means, right angle at
ǫ12
P
=
ǫ12
=
ǫ12 .
is the angle around whi h the 1- or 2-axis is rotated. is redu ed by
after deformation (sin e
2ǫ12 .
ǫ13 , ǫ23
or
The
If the parallelogram is not in the 1-2 plane
ǫ33
is non-zero), these statements hold for
the verti al proje tion in this plane. Similar results hold for d)
ǫ13
ǫ23 .
and
Length hanges of distan e P Q
P Q.
Fig. 2.5: Sket h for length hanges of distan e
PQ =
P ′ Q′
=
=
l0 =
l= (
(
(
3 X
3 X i=1
3 X
yi2
)1/2 2
(yi + ǫij yj )
i=1
yi2
+ 2ǫij yi yj +
)1/2
3 X i=1
i=1
= 2
(ǫij yj )
)1/2
.
The 1st, 2nd and 3rd term require SC on e, twi e and three times, respe tively. The 3rd term ontains only squares of
in nitesimal strain theory
ǫij
and an, within the framework of
treated here, be negle ted relative to the 2nd term
(the prerequisite for this is (2-5))
2.1.
17
ANALYSIS OF STRAIN
12 2 1 l = l0 1 + 2 ǫij yi yj = l0 + ǫij yi yj . l0 l0 Relative length hanges
yi yj l − l0 = ǫij 2 = ǫij ni nj (SC l0 l0 ni = Approa hes for
yi = unit l0
twi e; quadrati form in
ve tor in dire tion of
nite strain theory
nk )
yi .
exist (see, e.g., Bullen and Bolt). Su h a
theory has to be developed from the very start. Then, for example, the simple separation of the rotation term in the displa ement ve tor, whi h is possible for in nitesimal deformations, is no longer possible. The deformation tensor
ǫij
also be omes more ompli ated. e)
Volume hanges ( ubi dilatation)
V ontaining point P surfa e S. After deformation, for whi h we assume without loss of generality that P remains in its position, volume V is hanged by ∆V .
We onsider a nite (not in nitesimal) volume with
Fig. 2.6: Sket h for volume hanges.
∆V =
Z
un df.
S
Transformation of this surfa e integral with Gauss' law gives
∆V =
Z
V
→ ∇·− u dV,
18
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
and the relative volume hange an be written as
∆V 1 = V V Into the limit
V →0
V
(shrinking to point
lim
V →0 This limit is alled
Z
→ ∇·− u dV.
(2.6)
P ), this be omes
∆V = Θ. V
ubi dilatation.
From (2.6) with (2.1), it follows that
∂u1 ∂u2 ∂u3 → + + = ǫ11 +ǫ22 +ǫ33 Θ = ∇·− u := ∂x1 ∂x2 ∂x3 For
Θ>0
the volume in reases, for
Θ<0
(tra e
of the deformation tensor).
the volume de reases.
2.1.5 Components of the deformation tensor in ylindri al and spheri al oordinates P z
ϕ
Fig. 2.7: Cylindri al oordinates
− → u = ǫrr
=
ǫϕϕ
=
ǫzz
=
2ǫrϕ
=
r
r, ϕ, z .
(ur , uϕ , uz ) ∂ur ∂r 1 ∂uϕ ur + r ∂ϕ r ∂uz ∂z ∂uϕ uϕ 1 ∂ur + − r ∂ϕ ∂r r
2.1.
19
ANALYSIS OF STRAIN
2ǫrz
=
2ǫϕz
=
∂ur ∂uz + ∂z ∂r 1 ∂uz ∂uϕ + . ∂z r ∂ϕ
The omponents refer to the lo al Cartesian oordinate system in P.
P υ r λ
Fig. 2.8: Spheri al oordinates
r, ϑ, λ.
− → u = (ur , uϑ , uλ) ∂ur ǫrr = ∂r 1 ∂uϑ ur + ǫϑϑ = r ∂ϑ r 1 ∂uλ ur cot ϑ ǫλλ = + + uϑ r sin ϑ ∂λ r r ∂uϑ uϑ 1 ∂ur + − 2ǫrϑ = r ∂ϑ ∂r r ∂uλ uλ 1 ∂ur + − 2ǫrλ = r sin ϑ ∂λ ∂r r cot ϑ 1 ∂uϑ 1 ∂uλ + − uλ . 2ǫϑλ = r sin ϑ ∂λ r ∂ϑ r
Exer ise 2.1 How does a re tangular ube with edges parallel to the main axis system of the deformation tensor deform (length of edges a, b, )? Con rm the equation
Θ = ǫ11 + ǫ22 + ǫ33
.
20
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
Exer ise 2.2 Split the deformation tensor into one part that is pure shear (no volume hange) and another part that is pure volume hange (no shear).
Exer ise 2.3: Derive the omponents of the deformation tensor in ylindri al oordinates. Hint:
y’i =yi +du i
Q’ u+du i i
P’ ui
yi Q(r+dr, ϕ+dϕ ,z+dz)
P(r, ϕ ,z) Fig. 2.9: Displa ement ve tors to be used.
With respe t to the lo al Cartesian oordinate system in P, it holds that
y1 y2 y3
= = =
dr rdϕ dz
Determine rst the ylindri al oordinates of
u1 u2 u3
= = =
P'
and
ur uϕ uz .
Q'
under the ondition of
in nitesimal displa ement and deformation. Then give the omponents of the ′ , similar to ve tor y = yi + dui in the lo al Cartesian oordinate system of i
the de nition of
yi ,
in the system of
allows the derivation of ve tor
dui
P.
P'
This requires linearisation. This then
in the form
dui = vij yj and the determination of tensor part of
vij .
The deformation tensor is the symmetri
vij
ǫij =
1 (vij + vji ). 2
2.2.
21
ANALYSIS OF STRESS
2.2 Analysis of stress 2.2.1 Stress body for es
In a deformed body, a volume element is subje t to
to volume, e.g., gravity, entrifugal for e, inertial for e) and to
(proportional
surfa e for es,
whi h originate from neighbouring volume elements (proportional to surfa e). The later is the topi here. We onsider a body
K2
a deformed body
(see Fig.
2.10).
K2
If
assume a new equilibrium on guration. for es through
S
K1 .
on
− → P ∆f (∆f =surfa e
To bring
tra tion.
K1
This indi ates that
− → K2 . P
S
within
will, in general,
K2
S.
has exerted
Ersatz
for es
with the dimension for e/surfa e is
Its dire tion and size depend on:
1. The lo ation of the surfa e element 2. Its normal dire tion
Fig. 2.10: Body
K1
The omponent of
− → n
The omponent of
∆f
(de ned as the dire tion pointing
within a deformed body
− → P
parallel to
tra tion).
− → P
out
− → n
of
K1 ).
K2 .
− → n is alled normal tra tion (= pull
perpendi ular to
tra tion or thrust tra tion. If
with the surfa e
ba k to its original form,
element) have to be applied on
The same for es were exerted by
alled
K1
K1
is removed,
or pressure
is alled tangential tra tion, shear
− → → P is known everywhere in the body and for all dire tions − n , the stress within
the body is known. For this, six fun tions must be known.
2.2.2 Stress tensor pij We onsider a body in an in nitesimal tetrahedron
ABCD
the tra tion of the three sides
are known.
ABD, ABC, and ACD
and assume, that
22
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
Fig. 2.11: In nitesimal tetrahedron
ABCD.
From this, we will ompute the tra tion tensor
− → P
of
BCD. Be ause the tetrahe-
dron is small, all tra tions are onstant over their orresponding surfa es. The normal dire tions and surfa es are
ABD : negative 2-dire tion, ∆f2 ABC : negative 3-dire tion, ∆f3 ACD : negative 1-dire tion, ∆f1 → BCD: − n = (n1 , n2 , n3 ), ∆f ∆fj = ∆f nj . We assume that the for es and tra tion ve tors on known for the
positive
(2.7)
ABD, ABC
and
ACD
are
2-, 3- and 1-dire tion, respe tively
→ ABD: −P→2 ∆f2 , − P2 = (p21 , p22 , p23 ) → ABC: −P→3 ∆f3 , − P3 = (p31 , p32 , p33 ) → P1 = (p11 , p12 , p13 ). ACD: −P→1 ∆f1 , − This means that nine fun tions
pij
are known. After negle ting the body for es
(whi h de rease faster then the surfa e for es for a shrinking tetrahedron), the for e balan e at the tetrahedron an be written as
− → − → −Pj ∆fj + P ∆f = 0. With (2.7), it follows (SC)
− → − → P = Pj nj .
(2.8)
2.3.
23
EQUILIBRIUM CONDITIONS
Therefore, it is su ient to know the tra tion ve tors of three perpendi ular surfa e elements to determine the tra tion ve tor for an arbitrarily oriented surfa e element.
omponent of
In index notation, (2.8) an be written as (note:
− → − → P , Pj
Pj
is a
is a ve tor)
P1 = p11 n1 + p21 n2 + p31 n3 . In general, it holds that The nine fun tions
pij
Pj = pij ni .
form the
(SC)
stress tensor.
oordinate system. The omponents
It is valid for a ertain right-angle
pi1 , pi2 , pi3
give the tra tion ve tor for a
surfa e element, the normal of whi h is in the dire tion of the positive i-axis.
pii (e.i., p11 , p22 ,
or p33 ) is the normal stress, the two other omponents are the
tangential stresses, respe tively. As will be shown in the next se tion (see also exer ise 2.6), the stress tensor is symmetri , i.e.,
pij = pji. Therefore,
Pj = pji ni
or in the usual notation
Pi = pij nj .
(2.9)
In general, the stress tensor has six independent omponents.
Exer ise 2.4 a) Give the stress tensor for hydrostati pressure
p.
b) Give the stress tensor for the interior of an in nite plate whi h is xed at one side (bottom), whereas at the other side (top) the shear tra tion
τ
a ts
everywhere in the same dire tion.
Exer ise 2.5 − → P is the tra tion − → →→ − → ′ − n. P n = P ′−
Show that if holds that
for dire tion
− → n,
and
− → P′
for the dire tion
− →′ n,
it
2.3 Equilibrium onditions The equilibrium onditions for a nite volume
V
in a deformable body require
that the resulting for e and the resulting angular moment vanish
24
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
Z − → − → P df = 0 F dV + S V Z Z − → − → → → x × P )df = 0 (− x × F )dV + (− Z
(resulting (resulting
for e)
(2.10)
moment)
(2.11)
S
V
S V
P
dV O
− → F=
n
F
Fig. 2.12: Finite volume
where
df
x
V
in a deformable body.
body for es in luding inertial for e (dimension:
for e/ volume =
for e density) and
− → P=
tra tion ve tor on
S
(normal
− → n
towards the outside).
Equation (2.10) gives the equation of motion of the elasti ontinuum. For ea h
omponent (only Cartesian omponents an be used)
Z
V
Fi dV +
Z
Pi df =
S
Z
Fi dV +
V
Z
pij nj df = 0.
S
− → pij nj an be understood as the normal omponent Pin of the tra tion Pi = (pi1 , pi2 , pi3 ) relative to the ith-dire tion. Appli ation of Gauss' theorem gives Z
Z
Pin df =
Z
− → (Fi + ∇ · Pi )dV = 0.
V
S
− → ∇ · Pi dV,
therefore,
V
This holds for every arbitrary volume
Fi +
V. Therefore, the integrand has to vanish
∂pi1 ∂pi2 ∂pi3 + + =0 ∂x1 ∂x2 ∂x3
2.3.
25
EQUILIBRIUM CONDITIONS
or
Fi +
with the omponents of
∂pij =0 ∂xj
(SC)
2
F : Fi = −ρ ddtu2i + fi
The rst term is the inertial for e;
fi
(ρ = density).
ontains all other body for es. Within the
framework of the theory of in nitesimal deformation, the impli it dierentiation d ∂ dt an be repla ed by the lo al dierentiation, i.e., partial dierentiation ∂t
dA ∂A ∂A ∂xi ∂A = + ≈ dt ∂t ∂xi ∂t ∂t Then, this gives the
equation of motion ρ
At rest, normally
(A = in nitesimal
pij 6= 0
parameter, e.g.
ui ).
(SC)
∂ 2 ui ∂pij + fi . = 2 ∂t ∂xj
(2.12)
and the remaining stress is alled the initial stress.
It exists, when obje ts omposed of materials with dierent thermal expansions
oe ients are ooled, or by the self- ompression of obje ts in their own gravity eld (in this ase the initial stress is the hydrostati pressure). Assume that for (0) (0) a body at rest pij = pij and fi = fi . Then (2.12) holds and
(0)
∂pij (0) + fi = 0. ∂xj
(2.13)
The pre-stressed body will be deformed by time-dependent body for es (e.g., an earthquake in the Earth's rust). In the ase of a su iently small additional stress (and only then), the following separation is valid
(0)
(0)
(1)
fi = fi
pij = pij + pij
(1)
+ fi .
With (2.13), it follows from (2.12), that
(1)
∂pij ∂ 2 ui (1) + fi . ρ 2 = ∂t ∂xj This means that the displa ement
ui
from the pre-stressed state depends only
pij pij = 0 and
on the additional stress and the additional body for es. In the following, and
fi
in (2.12) will always be understood in that sense, i.e., at rest
26
CHAPTER 2.
fi = 0 .
FOUNDATIONS OF ELASTICITY THEORY
This assumption is su ient for the study of elasti body and surfa e
waves in the Earth. In the ase of normal modes and tides, where large depth
pij
ranges and even the whole Earth moves,
in (2.12) is the omplete stress
tensor, in luding the hydrostati ontribution.
In this ase,
fi
represents all
external for es, in luding the gravitational for e of the Earth itself. The reason for this is that, in this ase, be ause of the large size of the hydrostati pressure during deformation, the hange of this pressure annot be negle ted. A simple example is seen in the the radial modes of a sphere whi h have
larger
periods if
hydrostati pressure and gravitational for e are in luded.
Exer ise 2.6 Derive the equation of motion without assuming the symmetry of the stress tensor
pij ;
then derive this symmetry from the moment equation (2.11). Hint:
In the rst part, use the stress ve tor in the form
Pi = pji nj
instead of (2.9). In
the se ond part, write (2.11) by omponents and use the result of the rst part.
2.4 Stress-strain relations 2.4.1 Generalised Hooke's Law If a body in an unperturbed on guration shows a deformation asso iated with a length hange, this body is under stress. This means that in ea h point of the body a relation between the omponents of the stress tensor and the deformation tensor exists
pij = fij (ǫ11 , ǫ12 , . . . , ǫ33 ; a1 , a2 , . . . , an ). As indi ated, other independent parameters
an o
ur. Generally,
τ
with
pij
−∞ < τ < t.
at time
t
ak ,
(2.14)
su h as time and temperature,
an depend on the previous history at times
If, for example, a beam has suered extreme bending
in the past, its behaviour will be dierent. The general study of (2.14) and a
orresponding lassi ation of materials as elasti , plasti and vis o-elasti , is the topi of
rheology.
is su ient, namely that
ǫkl
et .
For seismology, generally the most simple form of (2.14)
pij
at a point depends only on the present values of
at that point. In this ase, from
ǫkl = 0
, it follows
pij = 0,
i.e., deformation
eases instantly if the stress eases. This means
pij = fij (ǫ11 , ǫ12 , . . . , ǫ33 ) fij (0, 0, . . . , 0) = 0. If these onditions hold, this state is alled deformation,
pij
is a linear fun tion of all
ideal elasti ity.
ǫkl
(2.15)
Under in nitesimal
(expansion of (2.15) at
ǫkl = 0
)
2.4.
27
STRESS-STRAIN RELATIONS
pij
=
cijkl ǫkl
cijkl
=
elasti ity onstants.
Linear elasti ity theory
(SC
twi e)
(2.16)
studies elasti pro esses in bodies under the following
assumptions:
1. The deformations are in nitesimal. 2. The stress-strain relations are linear.
The important assumption is 1.
The well-known
Hooke's Law,
for the stret hing of a wire or the shearing of a
ube, is a spe ial ase of (2.16). Equation (2.16) is, therefore, alled generalised Hooke's law.
Its range of appli ability has to be determined by experiments
or observation. The relation in the following sket h holds, for example, for the stret hing of a wire. Between unit of the ross se tion
p11
E=
orresponds to (2.16) (
A
and
B
the relation between for e per square
and the relative hange in length
ǫ11
is linear and
Young's modulus).
Fig. 2.13: Sket h for the stret hing of a wire.
Between
B
and
C
the relation is no longer linear but still orresponds to ideal
elasti behaviour, i.e., if Beyond
C
p11
is redu ed to zero, no deformation
ǫ11
remains.
irreversible deformation o
urs (plasti behaviour, ow of material).
Finally the wire ruptures.
28
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
The tensor of 4th degree cijkl has 81 (9 x 9 =) omponents. Due to the symmetry of the deformation and stress tensors, only 36 (6 x 6 =) omponents are independent from ea h other. Sin e the elasti deformation energy (= elasti energy per unit volume) is onserved, this number redu es further to 21 omponents (see, e.g., pg. 268-269 in Sommerfeld). This is the maximum number of elasti ity onstants an
anisotropi
body an have. For spe ial forms of anisotropy,
and espe ially for isotropy, this number redu es further. For isotropi bodies whi h do not have preferred dire tions, only two elasti onstants remain. The
stress-strain relations
(2.16) an then be written as
pij = λθδij + 2µǫij where
λ
and
µ
(2.17)
are Lamés elasti ity onstant and elasti ity parameter, respe -
tively (both of whi h an be spatially dependent),
θ = ǫ11 + ǫ22 + ǫ33 is the ubi dilatation, and 1 for i = j : is the Krone ker symbol δij = 0 otherwise
or unit tensor
2.4.2 Derivation of (2.17) We hoose the main axis system of the stress tensor as the oordinate system, whi h under isotropy is identi al to that of the deformation tensor. We, furthermore, have the main stress and deformation omponents main deformations
E1 , E2 , E3 ,
P1 , P2 , P3
and the
respe tively, whi h have a linear relation. In the
isotropi ase, this be omes
The oe ient of
E2
and
P1
=
aE1 + b(E2 + E3 )
P2 P3
= =
aE2 + b(E1 + E3 ) aE3 + b(E1 + E2 ).
E3
in the equation for
P1
have to be the same, sin e
for an isotropi body the main axes 2 and 3 ontribute equally to the main stress
P1 .
The same holds for the other two equations. From this, it follows that
Pi
where the onstants
µ,
a
= (a − b)Ei + b(E1 + E2 + E3 ) = 2µEi + λ(E1 + E2 + E3 ),
and
(2.18)
b have been repla ed by the Lamé parameters λ and
respe tively. This shows that (2.17) for the main axis oordinate system has
no shear omponent of the deformation tensor and no shear stress.
2.4.
29
STRESS-STRAIN RELATIONS
Using (2.4) for the transformation of tensor omponents, the stress and deformations omponents in any oordinate system an be given as
p11
=
a211 P1 + a221 P2 + a231 P3
p22 p12
= =
a212 P1 + a222 P2 + a232 P3 a11 a12 P1 + a21 a22 P2 + a31 a32 P3
p23
=
a12 a13 P1 + a22 a23 P2 + a32 a33 P3
. . .
. . .
. . .
ǫ11
= a211 E1 + a221 E2 + a231 E3 . . .
ǫ12
= a11 a12 E1 + a21 a22 E2 +a31 a32 E3
(2.19) For the dire tional osines it holds that
aik ail = δkl
(SC).
Using (2.18) in the left equation of (2.19) and using the equations on the right gives
p11
=
2µǫ11 + λ(E1 + E2 + E3 )
p22 p12
= =
2µǫ22 + λ(E1 + E2 + E3 ) 2µǫ12
p23
=
2µǫ23
. . .
The relations for shear stress already have the nal form; those for the normal stress an be brought to the nal form with the tensor invariants
ǫ11 + ǫ22 + ǫ33 . Expressing
ǫkl
E1 +E2 +E3 =
This on ludes the proof of (2.17). in terms of the derivative of the displa ement ve tor, (2.17) an
be written in Cartesian oordinates as
pii pij
∂u1 ∂u2 ∂u3 ∂ui + + ) + 2µ ∂x1 ∂x2 ∂x3 ∂xi ∂ui ∂uj = µ( + ) (i 6= j). ∂xj ∂xi
= λ(
(no
SC!)
Equation (2.17) also holds in urved, orthogonal oordinates, like ylinder and spheri al oordinates, respe tively, if the deformation tensor is given in these
oordinates ( ompare se tion 2.1.5).
pij
refers then to the oordinate surfa es
30
CHAPTER 2.
of the orresponding system.
FOUNDATIONS OF ELASTICITY THEORY
Finally, it should be noted that
pij
is usually
understood as a stress added to a pre-stressed on guration. The assumption that ro ks and material of the deep Earth are isotropi is often valid. The rystals whi h make up the ro k building minerals are, on the other hand, mostly anisotropi , but if they are randomly oriented, the material appears ma ros opi ally isotropi .
2.4.3. Additions Thermo-elasti stress-strain relations These are examples of relations in whi h stress not only depends on deformation, but also on other parameters, e.g., temperature (α = volume expansion
oe ient,
T − T0
= temperature hange)
2 pij = λΘδij + 2µǫij − (λ + µ)α(T − T0 )δij . 3
Relation between λ and µ and other elasti ity parameters E = Young's modulus σ = Poisson's ratio k = Bulk modulus τ = Rigidity
E=
µ(3λ + 2µ) λ+µ
τ =µ In ideal uids
0.5.
σ=
λ 2(λ+µ)
λ=
σE (1+σ)(1−2σ)
2 k =λ+ µ 3 E . µ= 2(1 + σ)
τ = µ = 0, there is no resistan e to shearing.
Then
k=λ
and
σ=
Within the framework of elasti ity theory, uids and gases behave identi-
ally, but the bulk modulus of uids is signi antly larger than that of gases. Their Poisson's ratio
σ
lies between -1 and 0.5; negative
pare Exer ise 2.8). For ro ks,
σ
is usually lose to 0.25;
σ values are rare ( omσ = 0.25 means λ = µ.
Exer ise 2.7 Derive the formula for k. k is de ned as the ratio a body is under pressure
Θ < 0.
p
p −Θ
in an experiment in whi h
from all sides and has the relative volume hange
Des ribe the deformation and the stress tensor and then the stress-strain
relation.
2.5.
31
EQUATION OF MOTION, BOUNDARY AND INITIAL ...
Exer ise 2.8 E
E
p11 Derive the formula for and σ . is de ned as the ratio ǫ11 and σ is the ǫ22 ratio − in an experiment, in whi h a wire or rod is under extension for e p11 ǫ11 in the 1-dire tion (ǫ11 = extension, − ǫ22 = perpendi ular ontra tion, ǫ33 =
?, p22 =?, p33 =?).
Pro eed as in exer ise 2.7. What is the meaning of
σ < 0?
2.5 Equation of motion, boundary and initial onditions 2.5.1 Equation of motion pij ,
Using (2.17) in the equation of motion (2.12), whi h depends on
this equa-
tion only depends on the omponents of the displa ement ve tor
ρ
If
λ
and
µ
∂ 2 ui ∂t2
=
∂ ∂xj (λΘδij
+ 2µǫij ) + fi h ∂ui ∂ ∂ (λΘ) + = ∂x ∂xj µ ∂xj + i
∂uj ∂xi
i
+ fi
.
(2.20)
are independent of lo ation (homogeneous medium) it follows that
2 ∂Θ ∂ 2 ui ∂ 2 ui ∂ ∂u3 ∂u1 ∂u2 ∂ ui ∂ 2 ui + fi +µ + + + + + ρ 2 =λ ∂t ∂xi ∂x21 ∂x22 ∂x23 ∂xi ∂x1 ∂x2 ∂x3 ∂ 2 ui ∂Θ (2.21) ρ 2 = (λ + µ) + µ∇2 ui + fi . ∂t ∂xi This is the
equation of motion for homogeneous media
In symboli notation
ρ
in Cartesian oordinates.
→ (Θ = ∇ · − u) ∂ 2 ui − → → → = (λ + µ)∇ ∇ · − u + µ∇2 − u + f. ∂t2
This is only valid for Cartesian oordinates. In Cartesian oordinates
(2.22)
→ u is the ve tor (∇2 u1 , ∇2 u2 , ∇2 u3 ). ∇2 −
→ → → u = ∇∇· − u −∇× ∇×− u. ∇2 − (Verify that in urved orthogonal oordinates (2.23)
de nes
and it is not identi al with the ve tor, whi h results from the on the omponents.) Inserting (2.23) in (2.22) gives
(2.23)
→ u, ∇2 − 2 appli ation of ∇ the ve tor
32
CHAPTER 2.
ρ
FOUNDATIONS OF ELASTICITY THEORY
→ ∂ 2− u − → → → = (λ + 2µ)∇ ∇ · − u − µ∇ × ∇ × − u + f. 2 ∂t
(2.24)
This form of the equation of motion is independent of the oordinate system. It is the starting point for the following se tion: a
ording to se tion 2.3,
− → f
ontains only the body for es whi h a t in addition to those of the for es at rest.
2.5.2 Boundary onditions On a surfa e in whi h at least one material parameter
ρ, λ
or
µ is dis ontinuous,
the stress ve tor, relative to the normal dire tion of this surfa e, is ontinuous. To show this, onsider a small at ir ular ylinder of thi kness 2d whi h en loses the boundary between the two media.
The sum of all for es a ting on the
ylinder (body for es in the interior and surfa e for es on its surfa e) has to be zero.
2d Medium 1 Medium 2
-P 2
n
P1
n -n
P2
Fig. 2.14: Cir ular ylinder of thi kness 2d en losing the boundary between two media.
In the limit
d → 0,
only the surfa e for es on the top and bottom surfa e
∆f ,
have to be onsidered
− → − → P1 ∆f + (−P2 )∆f = 0.
From this, it follows that
− → − → P1 = P2 .
tangential stress are ontinuous.
This means that at boundaries
normal and
For the displa ement, it holds that at a solid-solid boundary, all omponents are ontinuous (no sliding possible). At a solid-liquid or liquid-liquid boundary only the normal displa ement is ontinuous.
Example:
z = 0. ux , uy , uz , and the stresses are pxx , pyy , pzz , pxy , pxz , pyz . boundary onditions z = 0 for the dierent ombinations of half-spa es are A body onsists of two half-spa es, separated by a plane at
The displa ements are The
2.5.
− solid − uid uid − uid solid − rigid uid − rigid solid − va uum uid − va uum solid
solid
: : : : : : :
ux , uy , uz , pxz , pyz , pzz ontinuous uz , pzz ontinuous, pxz = pyz = 0 uz , pzz ontinuous ux = uy = uz = 0 uz = 0 pxz = pyz = pzz = 0 free surfa e pzz = 0
If at a surfa e with the normal ve tor
Pi ),
ni ,
the stress is not zero (stress ve tor
(r)
(r)
.
the stress ve tor in the body has to a quire this boundary value
pij nj = Pi . pij
33
EQUATION OF MOTION, BOUNDARY AND INITIAL ...
(2.25)
are the boundary values of the omponents of the stress tensor at the sur-
fa e, and they an be al ulated from (2.25).
Example:
P (t)
on a plane surfa e. For the ase of pressure
P(t)
n
....x=0 x Fig. 2.15: Pressure on a plane surfa e.
− → n = (−1, 0, 0) = (n1 , n2 , n3 ) − → P = (P (t), 0, 0) = (P1 , P2 , P3 ) Equation (2.25) yields
(r)
−pi1 = Pi
.
or
p11 = pxx = −P (t) p12 = pxy = 0 p13 = pxz = 0
for
x = 0.
Similarly, displa ements an be pres ribed on the surfa es of a body.
34
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
2.5.3 Initial onditions The initial onditions pres ribe the spatial distribution of ertain parameters, in our ase the displa ement
ui (x1 , x2 , x3 , t) and the parti le velo ity ∂ui /∂t for
t=0 ∂ui (x1 , x2 , x3 , 0) = f2 (x1 , x2 , x3 ). ∂t
ui (x1 , x2 , x3 , 0) = f1 (x1 , x2 , x3 ),
The general wave propagation solution is an initial and a boundary problem, i.e., in addition to the equation of motion, the boundary and initial onditions have to be satis ed. Normally in seismologi al appli ations
f1 = f2 = 0 ,
and
no spe ial initial onditions have to be satis ed. The main problems are then to onsider the boundary onditions.
2.6 Displa ement potentials and wave types 2.6.1 Displa ement potentials A ve tor
− → u
an, in general, be des ribed as
− → − → u = ∇Φ + ∇ × Ψ where
Φ
s alar potential
is a
and
− → Ψ
a
ve tor potential.
(2.26)
In our ase, where
− → u
is
− → u
is
a displa ement eld, both are alled displa ement potentials. (Do not onfuse them with the
Φ
is alled
given,
Φ
elasti
potential, i.e., the elasti deformation energy.)
ompression potential
and
− → Ψ
and
− → Ψ
shear potential.
If the ve tor
an be omputed ( ompare exer ise 2.9)
Z − → → u ·− r 1 dV 3 4π r Z − → → u ×− r 1 − → dV. Ψ= 3 4π r Φ=
Ve tor
− → r
(with absolute value r) points from the volume element
point where For
− → Ψ,
Φ
and
− → Ψ
dV
to the
are omputed. The integration overs the whole volume.
there is the additional requirement that
− → ∇ · Ψ = 0. − → Ψ
(2.27)
has to be determined in Cartesian oordinates.
(2.28)
2.6.
DISPLACEMENT POTENTIALS AND WAVE TYPES
∇Φ
in (2.26) is alled
ompressional part
of
− → u.
35
It is free of rotation and
url. A
ording to se tion 2.1.4, the volume elements suer no rigid rotation, but only a deformation, whi h, in general, onsists of a volume hange and a shear omponent (in the sense of exer ise 2.2; se tion 2.1).
In the main axis
system of the deformation tensor, only volume hanges o
ur ( ompression or dilatation).
The ontribution
− → ∇× Ψ
in (2.26) is alled
shear omponent.
It
is free of divergen e and sour e ontributions; the volume elements suer no volume hange, but shear deformation and rigid rotation. Similarly to (2.26), the body for e
− → f
in (2.24) an be split into
− → − → f = ∇ϕ + ∇× ψ. Do not onfuse the ve tor potentials
− → Ψ
and
(2.29)
− → ψ.
Using (2.26) and (2.29) in (2.24) gives
− →# ∂2 Ψ ∂2Φ − → − → = (λ + 2µ)∇ ∇2 Φ − µ ∇ × ∇ × ∇ × Ψ + ∇ ϕ + ∇ × ψ . ρ ∇ 2 +∇× ∂t ∂t2 "
(2.30) We try now to equate all the gradient terms and also, separately, the rotation terms of this equation.
If the resulting dierential equations an be solved,
(2.30) and, therefore, (2.24) are satis ed. This leads to
2 ∂ Φ ∇ ρ 2 − (λ + 2µ)∇2 Φ − ϕ =0 ∂t # " − → ∂2 Ψ − → − → = 0. ∇ × ρ 2 + µ∇ × ∇× Ψ − ψ ∂t Sin e the ontent of the square bra kets has to vanish
ϕ 1 ∂2Φ =− 2 2 α ∂t λ + 2µ − → → 2− 1 ∂ Ψ ψ − → −∇ × ∇ × Ψ − 2 = − β ∂t2 µ ∇2 Φ −
α2 =
λ + 2µ ρ
β2 =
µ . ρ
(2.31)
− → − → ϕ and ψ have to be determined from f using (2.27). If no − → body for es a t, ϕ = 0 and ψ = 0. The equation for Φ is an inhomogeneous − → wave equation. In Cartesian oordinates the omponents of Ψ give also inhoThe potentials
mogeneous wave equations due to (2.23) and (2.28). In other oordinates, the equations for the omponents of
− → Ψ
look dierent ( ompare exer ise 2.10).
36
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
Both simpli ations used are, as experien e has shown, justi ed. The problem
an, therefore, be solved either via (2.24) or (2.31). In more ompli ated ases, (2.31) is easier to solve. In this ase, the boundary onditions for displa ement and stress have to be expressed as those for
Φ
and
− → Ψ.
2.6.2 Wave types The general dis ussion of the dierential equations (2.31) shows that they have solutions whi h orrespond to
waves
(for details, see se tion 3.1). Perturbations
ompressional
in the ompressional part of the displa ement ve tor propagate as 1/2 with the velo ity α = ((λ + 2µ)/ρ) through the medium. Perturba1/2 . tions in the shear part propagate as shear waves with the velo ity β = (µ/ρ)
waves
Thus, we have found the two basi wave types, whi h an propagate in a solid medium. For ro ks, it usually holds that λ = µ. In this ase, it follows that α/β = 31/2 . In liquid or gases, only ompressional waves ( ) an
sound waves
propagate sin e
µ = 0.
Often ompressional waves are alled
alled
transverse waves.
longitudinal waves
and shear waves are
The displa ement ve tor in a longitudinal wave is par-
allel to the propagation dire tion and perpendi ular to it in a transverse wave. A ompressional wave is, in general, primarily longitudinally polarised, and a shear wave is primarily transversely polarised. The identi ation is, therefore, not fully valid.
There exist spe ial ases in whi h a ompressional wave is
transversal and a shear wave is longitudinal (see se tion 3.5.1 and exer ise 3.5 in hapter 3). The seismologi al names for ompressional and shear waves are
S-waves, respe tively.
P-waves
and
This indi ates that the P-wave is the rst wave arriving
at a station from an earthquake (P from primary), whereas the S-wave arrives later (S from se ondary). In a homogeneous medium, ompressional waves and shear waves are
de oupled,
i.e., they propagate independently from ea h other. This no longer holds for inhomogeneous media in whi h point to point.
λ, µ
and/or
ρ,
and, therefore,
α
and
β,
vary from
But in this ase, usually two wave types propagate through
the medium, and the travel times of their rst onsets are determined by the velo ity distribution of
α
and
β,
respe tively. The faster of the two waves is no
longer a pure ompressional wave but ontains a shear omponent. The slower wave is, orrespondingly, not a pure shear wave but ontains a ompressional
ontribution.
This be omes plausible if one approximates an inhomogeneous
medium by pie e-wise homogeneous media. Satisfying the boundary onditions at the interfa es between the homogeneous media usually requires, on
both sides,
the existen e of ompressional and shear waves. Details on this will be given in se tion 3.6.2.
Compressional and shear waves whi h are de oupled in ho-
mogeneous se tions of the medium, reate re e ted and refra ted waves of the other type, respe tively, at interfa es. This hange in wave type o
urs ontinuously in ontinuous media and is stronger the stronger the hanges in
α, β,
2.6.
DISPLACEMENT POTENTIALS AND WAVE TYPES
and/or
ρ
37
per wave length are. The theory for ontinuous inhomogeneous media
is mu h more ompli ated then the theory for pie e-wise homogeneous media. Media in whi h
α, β
and
ρ
depend only on
one
oordinate, e.g., depth, an,
for many seismologi al appli ations, be approximated by layers of homogeneous media. For su h on gurations, ee tive methods for the use of omputers exist.
Exer ise 2.9 Show (2.27) by omparing (2.26) with the equation
→ → → a =∇ ∇·− a −∇× ∇×− a ∇2 − and onsider, that the Possion equation nates) the solution
1 − → a =− 4π
Z
→ → ∇2 − a =− u
has (in Cartesian oordi-
1 − → u dV. r
Exer ise 2.10 Write (2.26) in ylindri al oordinates
(r, ϕ, z) under the ondition that the → ϕ- omponent of − u is zero (Ψr =
medium is ylindri ally symmetri , and the
Ψz = 0).
What is the form of (2.31) for vanishing body for es?
Exer ise 2.11 Show that in a liquid with onstant density ρ, but variable ompressional module 2 k and pressure p, satis es the wave equation ∇2 p = α12 ∂∂t2p with spatially varying 1/2 sound velo ity α = (k/ρ) . Hint: Derive from the equation of motion (2.12) without body for es, the equa→ u /∂t2 = −∇ p and apply then the divergen e operation i.e. (p = ρ∂ 2 −
tion
→ −k ∇ · − u ).
38
CHAPTER 2.
FOUNDATIONS OF ELASTICITY THEORY
Chapter 3
Body waves 3.1 Plane body waves The most simple types of waves an be derived, if for an unbounded medium (full-spa e), solutions of the equation of motion are determined whi h depend only on
one
spatial oordinate. For example, we look for a solution of (2.21)
or (2.24) in the form of depends only on form of on
x
x
− → u = (ux (x, t), 0, 0),
and the time
− → u = (0, uy (x, t), 0),
t
i.e.,
t.
− → u
i.e.,
− → u
points in
x -dire tion
and
Alternatively, we look for a solution in the
points in
y -dire tion and depends also only
and . In the rst ase, it follows from (2.21) for
∂ 2 ux 1 ∂ 2 ux = 2 , 2 ∂x α ∂t2
α2 =
fi = 0
λ + 2µ , ρ
and in the se ond ase,
1 ∂ 2 uy ∂ 2 uy = , ∂x2 β 2 ∂t2 These are
one dimensional wave equations.
β2 =
µ . ρ
In the following, we onsider the
general form
1 ∂2u ∂2u = . ∂x2 c2 ∂t2
(3.1)
The most general solution of this equation is
u(x, t) = F (x − ct) + G(x + ct), 39
(3.2)
40
CHAPTER 3.
where
F(x)
and
G(x)
BODY WAVES
are any twi e dierentiable fun tions ( he k that (3.2)
solves (3.1)). Another form is
x x u(x, t) = F t − +G t+ . c c
(3.3)
The rst and the se ond term in (3.2) and (3.3) have to be interpreted as waves propagating in the positive and negative the rst term in (3.3) for
x = x1
x -dire tion, respe tively.
For example,
an be written as
x1 = F1 (t). u(x1 , t) = F t − c For another distan e
x2 > x1
x2 − x1 x2 − x1 x2 x1 =F t− = F1 t − . u(x2 , t) = F t − − c c c c
t at distan e x2 the same ee ts o
ur as at distan e t − (x2 − x1 )/c. This orresponds to a wave whi h has x2 in the time (x2 − x1 )/c. The propagation velo ity is,
This means that for time
x1
at the earlier time
travelled from therefore,
.
x1
The
to
wavefronts
of this wave, i.e., the surfa es between perturbed
and unperturbed regions, are the planes
x=
onst. Therefore, these are
plane
waves. If G(x) in (3.2) or G(t) in (3.3) are not zero, two plane waves propagate in opposite dire tions. In the ase of
u = ux , we have a longitudinal wave (polarisation in u = uy , we have a transverse wave
of propagation); in ase of
perpendi ular to the dire tion of propagation). Harmoni waves an be represented as
the dire tion (polarisation
h x i u(x, t) = A exp iω(t − ) = A exp [i(ωt − kx)] c
A=
Amplitude (real or omplex), ω = angular frequen y ν = ω/2π = freT = 1/ν = period, k = ω/c= wavenumber and Λ = 2π/k = wave length. Between c, Λ and ν the well-known relation c = Λν holds. The use of the omwith
quen y,
plex exponential fun tion in the des ription of plane harmoni waves is more
onvenient than the use of the real sine and osine fun tions. In the following, only the exponential fun tion will be used.
3.2 The initial value problem for plane waves We look for the solution of the one-dimensional wave equation (3.1) whi h satis es the initial onditions
3.2.
THE INITIAL VALUE PROBLEM FOR PLANE WAVES
u(x, 0) = f (x)
41
for displa ement and
∂u (x, 0) = g(x) ∂t
for parti le velo ity.
This is an initial value problem of a linear
ordinary
dierential equation, e.g.,
the problem to determine the movement of a pendulum, if initial displa ement
t=0, it also holds that
and initial velo ity are given. We start from (3.2). For
F (x) + G(x)
= f (x)
(3.4)
′
= g(x).
(3.5)
′
−cF (x) + cG (x) Integrating (3.5) with respe t to
x, gives
F (x) − G(x) = −
1 c
Z
x
g(ξ)dξ.
(3.6)
−∞
From the addition of (3.4) and (3.6), it follows that
1 F (x) = 2
Z 1 x f (x) − g(ξ)dξ , c −∞
and from the subtra tion of these two equations that
G(x) =
1 2
Z 1 x f (x) + g(ξ)dξ . c −∞
From this, it follows that
1 1 u(x, t) = {f (x − ct) + f (x + ct)} + 2 2c
Z
x+ct
g(ξ)dξ. x−ct
This solution satis es the wave equation and the initial onditions ( he k). We will dis uss
two spe ial ases.
3.2.1 Case 1 g(x) = 0,
i.e., the initial velo ity is zero. Then
u(x, t) =
1 {f (x − ct) + f (x + ct)} . 2
42
CHAPTER 3.
BODY WAVES
Two snap shots (Fig. 3.1) for t=0 and for t>0, illustrate this result.
u(x,0) f(x)
t=0
x
ct
ct u(x,t)
f(x-ct)
f(x+ct)
t>0
x
Fig. 3.1: Snap shots of two plane waves.
Two plane waves propagate from the point of ex itation in both dire tions with the velo ity
t=0.
.
A pra ti al example is a stret hed rope with the form
f (x)
for
3.2.2 Case 2 f (x) = 0, i.e., the initial displa ement is zero. Furthermore, we assume g(x) = V0 δ(x). δ(x) is Dira 's delta fun tion (see appendix A.3). g(x) orresponds to an impulse at x = 0. V0 has the dimension of velo ity times length. Then
u(x, t) =
The sket h (Fig.
V0 2c
Z
x+ct
δ(ξ)dξ.
x−ct
3.2) shows the value of the integrand and the integration
interval for a xed point in time
t>0
and for a lo ation
x > 0.
3.3.
SIMPLE BOUNDARY VALUE PROBLEMS FOR PLANE WAVES
43
Fig. 3.2: Value of the integrand and the integration interval of u(x,t).
Only when the integration interval in ludes the point
ξ = 0,
does the integral
be ome non-zero. Then it always has the value of 1
u(x, t) =
H(t)
is the
V0 2c H(t
− xc )
for x>0
V0 2c H(t
+ xc )
for x<0
Heaviside step fun tion, H(t) = 0 for t < 0, H(t) = 1 for t ≥ 0.
The displa ement jumps at
t = |x| /c
from zero to the value
V0 /2c.
3.3 Simple boundary value problems for plane waves The simplest boundary value problem is to determine the displa ement within a half-spa e for a time dependent pressure half-spa e. Sin e the displa ement
− → u
P (t)
only has an
at the surfa e
x = 0
x- omponent, ux
of this
and sin e
x
is the only expli it spatial oordinate, the one-dimensional wave equation (3.1) for a plane ompressional wave is appli able
1 ∂ 2 ux ∂ 2 ux = 2 . 2 ∂x α ∂t2 The solution for the ase onsidered here is
x ux (x, t) = F t − , α
44
CHAPTER 3.
x
sin e a wave an only propagate in + -dire tion.
Fun tion
BODY WAVES
F (t)
has to be
determined from the boundary ondition that the stress ve tor adapts without jump to the imposed stress ve tor at the free surfa e
x = 0 ( ompare (2.25) and
in se tion 2.5.2)
pxx = −P (t)
and
pxy = pxz = 0
The stress-strain relation (2.17) gives rst, that
for
pxy
x = 0.
and
pxz
are zero everywhere
in the medium, and se ond that
pxx = (λ + 2µ) For
x = 0,
∂ux x x λ + 2µ ′ = −ραF ′ t − . =− F t− ∂x α α α
it follows that
−ραF ′ (t) = −P (t) and after integration
1 F (t) = ρα For this, we assumed that
Z
t
1 P (τ )dτ = ρα −∞
P (t) = 0
for
t < 0.
Z
t
P (τ )dτ. o
Then, the displa ement an be
written as
1 u(x, t) = ρα
Z
t−x/α
P (τ )dτ.
0
The displa ement is proportional to the time integral of the pressure on the surfa e of the half-spa e. If a short impulse
ux (x, t) = P0
P (t) = P0 δ(t)
a ts, it follows that,
P0 x . H t− ρα α
has the dimension pressure times time (see also appendix A, se tion A.3.1).
t = x/α, all points in the half-spa e are displa ed instantly by P 0 /ρα in +x-dire tion and remain xed in this position. For P 0 = 1 bar se = 9.81 Nse / m2 ≈ 106 dyn se / m2 , ρ = 3 g/ m3 and α = 6 km/se the displa e-
At the time
ment is approximately 0.5 m.
This boundary value problem is simple enough so that it ould be solved dire tly with the equation of motion (2.21) or (2.24). One ould have also worked with
3.4.
45
SPHERICAL WAVES FROM EXPLOSION POINT SOURCES
displa ement potentials
Φ
and
− → Ψ
and their dierential equations (2.31) (show).
Exer ise 3.1 The tangential stress
T (t)
a ts on the surfa e of a half-spa e.
What is the
displa ement in the half-spa e? Appli ation: The tangential stress on the rup6 2 ture surfa e of earthquakes is 50 bar = 50 · 10 dyn/ m (stress drop). What is 3 the parti le velo ity (on the rupture surfa e) for (ρ = 3 g/ m , β = 3.5 km/se )?
Exer ise 3.2 An elasti layer of thi kness H overlies a rigid half-spa e. Pressure
P (t)
a ts on
the top of the elasti layer. What is the movement in the layer? Examine the
ase
P (t) = P 0 δ(t).
Exer ise 3.3 Solve the stati problem of exer ise 3.2 ( onstant pressure
P1
on the surfa e).
3.4 Spheri al waves from explosion point sour es In the previous se tions, we onsidered in nitely extended waves.
They are
an idealisation, be ause they annot be produ ed in reality sin e they require in nitely extended sour es. The most simple wave type from sour es with -
spheri al waves, i.e., waves whi h originate at a point (point sour e ) and propagate in the full-spa e. Their wavefronts are spheres. In the most simple ase, the displa ement ve tor is radially oriented and also radially symmetri relative to the point sour e, i.e., the radial displa ement on
nite extension are
a sphere around the point sour e is the same everywhere. If a spheri al explosion in a homogeneous medium far from interfa es is triggered, the resulting displa ement has these two properties. Therefore, we all these
sour es.
explosions point
The results derived with the linear elasti ity theory an only be applied
to spheri al waves from explosions in the distan e range in whi h the prerequisites of the theory (in nitesimal deformation, linear stress-strain relation) are satis ed. In the
plasti zone, the shattered zone and the non-linear zone
(this is
a rough lassi ation with in reasing distan e from the entre of the explosion) these requirements are not met.
For a nu lear explosion of 1 Megaton TNT
equivalent (approximately mb = 6.5 to 7.0), the shattered zone is roughly 1 to 2 km wide. We plan to solve the following boundary problem: given the radial displa ement at distan e for
r = r1
from the point sour e
U (r1 , t) = U1 (t), we want to nd U (r, t)
r > r1 .
We start from the equation of motion (2.24) with
− → f = 0.
This is how the
problem is solved in appendix A (appendix A.2.2) using the Lapla e transform.
46
CHAPTER 3.
BODY WAVES
Here, more simply, the displa ement potential from se tion 2.6 will be used. In this ase, the shear potential is zero, sin e a radially symmetri radial ve tor
an be derived solely from the ompressional potential
Φ(r, t) =
Z
r
U (r′ , t)dr′ .
In spheri al oordinates (r, ϑ, λ), it holds that
∇Φ = For
Φ,
1 ∂Φ ∂Φ 1 ∂Φ ∂r , r ∂ϑ , r sin ϑ ∂λ
the wave equation with
∇2 Φ =
ϕ=0
= (U (r, t), 0, 0).
an by written a
ording to (2.31) as
∂ 2 Φ 2 ∂Φ 1 ∂2Φ 1 ∂ 2 (rΦ) + = 2 2 = 2 2 ∂r r ∂r r ∂r α ∂t ∂ 2 (rΦ) 1 ∂ 2 (rΦ) = 2 . 2 ∂r α ∂t2
(3.7)
In the ase of radial symmetry, the wave equation an be redu ed to the form of a one-dimensional wave equation for Cartesian oordinates for the fun tion
rΦ, 1 ∂2u ∂2u = , ∂x2 α2 ∂t2 . The most general solution for (3.7) is ,therefore,
Φ(r, t) =
r r o 1n F (t − ) + G(t + ) . r α α
This des ribes the superposition of two ompressional waves, one propagating outward from the point sour e and the other propagating inwards towards the point sour e. In realisti problems, the se ond term is always zero and
Φ(r, t) = Fun tion potential.
F (t)
is often alled the
1 r . F t− r α
ex itation fun tion
The wavefronts are the spheres
(3.8)
or redu ed displa ement
r = onst.
The potential as a
fun tion of time has the same form everywhere, and the amplitudes de rease with distan e as
1/r.
The radial displa ement of the spheri al wave onsists of
two ontributions with dierent dependen e on
r
3.4.
SPHERICAL WAVES FROM EXPLOSION POINT SOURCES
U (r, t) =
∂Φ 1 ′ r r 1 − . = − 2F t− F t− ∂r r α rα α
47
(3.9)
These two terms, therefore, hange their form with in reasing distan e. Generally, this holds for the displa ement of waves from a point sour e. The rst term in (3.9) is alled
near- eld term sin e it dominates for su iently small r. far- eld term and des ribes with su ient a
ura y the
The se ond term is the
displa ement for distan es from the point sour e whi h are larger then several F (t) = eiωt ). That means there the displa ement
wave lengths (show this for redu es proportional to
1/r.
From the boundary ondition
−
r = r1 ,
it follows that
1 ′ 1 r1 r1 − 2F t− = U1 (t). F t− r1 α α r1 α
We hoose the origin time so that
U1 (t) only begins to be non-zero for t = r1 /α.
It, therefore, appears as if the wave starts at time t = 0 at the point sour e r (r = 0). If U 1 t − 1 = U1 (t), it holds that U 1 (t) is already non-zero for t > 0. α r With τ = t − α1 , it follows that
1 1 ′ F (τ ) + 2 F (τ ) = −U 1 (τ ). r1 α r1
(3.10)
The solution of (3.10) an be found with the Lapla e transform (see se tion A.2.1.1 of appendix A). Sin e the riterion (A.16) of appendix A is satis ed for all physi ally realisti displa ements
U 1 (τ ),
1 r1
F (+0) of F (τ ) is zero. Therefore, F (τ ) ←→ f (s) and U 1 (τ ) ←→ u1 (s) gives
the initial value
transformation of (3.10)with
s 1 + α r1
f (s) = −u1 (s)
f (s) = −r1 α With
s+
α r1
−1
1 u1 (s) s + rα1
.
(3.11)
α
←→ e− r1 τ
(see appendix A, se tion A.1.4), and using on-
volution (see appendix A, equation A.7), the inverse transformation of (3.11) reads as
F (τ ) = −r1 α From this, it follows
Z
0
τ
U 1 (ϑ)e
− rα (τ −ϑ) 1
dϑ.
48
CHAPTER 3.
F ′ (τ ) = −r1 αU 1 (τ ) + α2 The radial displa ement for
U (r, t) = with the
r1 r
r > r1
Z
τ
BODY WAVES
α
U 1 (ϑ)e− r1 (τ −ϑ) dϑ.
0
then an be written using (3.9) as
Z τ 1 1 − α (τ −ϑ) − dϑ U 1 (τ ) + α U 1 (ϑ)e r1 r r1 0
retarded time τ = t − αr .
(3.12)
This solves the boundary problem.
Appli ations 1.
U1 (t) = U 0 δ t −
r1 α
The dimension of
U0
, i.e., U 1 (t) = U 0 δ(t). is time times length. Equation (3.12) is valid also in
this ase (see appendix A)
1 r1 1 − rα τ 1 U (r, t) = U 0 δ(τ ) + α e − H(τ ) . r r r1
τ=0 (t=r/α)
τ
Fig. 3.3: U(r,t) as a fun tion of time.
2.
U1 (t) = U0 H t −
r1 α
The dimension of
U0
U (r, t)
= = =
, i.e. U 1 (t) = U0 H(t). is length. From (3.12), it follows
h r α iϑ=τ α r1 1 1 1 r ϑ H(τ ) e − r1 τ U0 H(τ ) + α − e 1 r r r1 α ϑ=0 n r o r1 1 − rα τ U0 H(τ ) 1 + −1 1−e 1 r r n r1 r1 − rα τ o r1 U0 H(τ ) + 1− e 1 . r r r
3.5.
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
49
r1 r U0
r12 U0 r2 τ=0
τ
Fig. 3.4: U(r,t) as a fun tion of time.
Exer ise 3.4 r1 a ts in a spheri al avity with radius r1 . α What is the dierential equation for the ex itation fun tion F (t) (analogue to Pressure
P (t)
P (t) = 0 for t <
3.10)? In the ase of radial symmetry, the radial stress radial displa ement
U
prr
is onne ted to the
as (show)
prr = (λ + 2µ)
U ∂U + 2λ . ∂r r
Whi h frequen ies are preferably radiated (eigenvibrations of the avity)? This
an be derived / seen from the dierential equation without solving it ( ompare to the dierential equation of the me hani al os illator, see appendix A.2.1.1). Solve the dierential equation of
P (t) = P 0 δ(t − r1 /α).
3.5 Spheri al waves from single for e and dipole point sour es 3.5.1 Single for e point sour e A single for e in the entre of a Cartesian oordinate system a ting in z-dire tion with a for e-time law
K(t)
has the for e density ( ompare appendix A.3.3)
− → f = (0, 0, δ(x) δ(y) δ(z) K(t)) .
50
CHAPTER 3.
BODY WAVES
z K(t)
x
y
Fig. 3.5: Cartesian oordinate system with for e-time law
The separation
− → − → f = ∇ϕ + ∇ × ψ
ϕ(x, y, z, t) = =
− → ψ (x, y, z, t) = =
K(t).
is possible with the help of (2.27)
Z Z Z +∞ 1 1 (z − ζ)δ(ξ)δ(η)δ(ζ)K(t)dξdηdζ ′3 4π r −∞ K(t)z , r2 = x2 + y 2 + z 2 4πr3
(3.13)
Z Z Z +∞ 1 1 {−(y − η), x − ξ, 0} δ(ξ)δ(η)δ(ζ)K(t)dξdηdζ 3 4π r −∞ K(t) (3.14) (−y, x, 0). 4πr3
If (3.13) and (3.14) are used in the dierential equation (2.31) of the displa e-
0 and that for Ψx hold; the same is
− → Ψ = (Ψx , Ψy , Ψz )
Ψz = Ψy , due to (2.23) and (2.28), the following wave equations true for the ompression potential Φ
ment potentials, it follows that for the shear potential and
1 ∂2ϕ α2 ∂t2 1 ∂ 2 Ψx ∇2 Ψx − 2 β ∂t2 1 ∂ 2 Ψy ∇2 Ψy − 2 β ∂t2 ∇2 Φ −
= = =
K(t)z 4πρα2 r3 K(t)y 4πρβ 2 r3 K(t)x − . 4πρβ 2 r3 −
The solution of the inhomogeneous wave equation
3.5.
51
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
∇2 a −
1 ∂2a = f (x, y, z, t) c2 ∂t2
an for vanishing initial onditions, be written as
1 a(x, y, z, t) = − 4π
ZZZ
+∞
1 f r′
−∞
r′ dξdηdζ ξ, η, ζ, t − c
(3.15)
with
r′2 = (x − ξ)2 + (y − η)2 + (z − ζ)2 . Equation (3.15) is
Kir hho's Equation
for an in nite medium. It is the ana-
logue to the well-known Poisson's dierential equation whi h is also a volume integral over the perturbation fun tion ( ompare exer ise 2.9). Equation (3.15)
an also be omputed in non-Cartesian oordinates, something we now use.
Appli ation of the wave equation for Φ (r′ , ϑ, λ) relative to point P . λ is de ned, Cartesian oordinate system (x, y, z ).
We introdu e the spheri al oordinates see sket h, via an additional
Fig. 3.6: Additional Cartesian oordinate system (x, y, z ).
The
x-axis of
then in the
OP. The z -axis is OP. Then the z, ζ -axis has
of this oordinate system is identi al to the line
in the plane de ned by the
zandζ -axis
x − y − z -system
and the line
the dire tion of the unit ve tor
− → n = (cos γ, 0, sin γ) =
(
1 ) z2 2 z . , 0, 1 − 2 r r
52
CHAPTER 3.
Ve tor
− → r′′
from
O
to
Q
BODY WAVES
an be written in the same system as
− → r′′ = (r − r′ cos ϑ, r′ sin ϑ sin λ, r′ sin ϑ cos λ). These two ve tors are needed later. Equation (3.15) an for
Φ
then be written
as
Z
∞
and
r′′
1 Φ(x, y, z, t) = 16π 2 ρα2 We still must express
ξ r′′2
ζ
0
Z
π 0
Z
ζ K t− · r′′3 r′
2π 0
in terms of
r′ , ϑ
and
r′ α
· r′2 sin ϑ dλ dϑ dr′ .
λ
1 − → z z2 2 ′ ′′ ′ − → = r · n = (r − r cos ϑ) + 1 − 2 r sin ϑ cos λ r r = r2 + r′2 − 2rr′ cos ϑ
(rule
of osine).
This gives
Φ(x, y, z, t) =
1 16π 2 ρα2
Z ∞Z πZ 0
0
0
2π
z 1−
r′ r
cos ϑ + r′ 1 −
r3 1 +
r ′2 r2
r′ · r′ sin ϑ dλ dϑ dr′ . ·K t − α The part of the integrand with
λ.
cos λ does
z2 r2
21
sin ϑ cos λ
3 ′ − 2 rr cos ϑ 2
not ontribute to the integration over 2π . With a = r/r′ and
The other part has only to be multiplied by
Z
π
0
(1 − a cos ϑ) sin ϑ
(1 + a2 − 2a cos ϑ)
3 2
dϑ =
= =
= = =
1 + a2 − 2a cos ϑ 2a sin ϑdϑ
(1+a)2
1 + 21 (u − 1 − a2 ) 3
2au 2
(1−a)2
=
u du
Z
1 4a
Z
(1+a)2
(1−a)2
1 1
u2
+
1 − a2 3
u2
du
du
o(1+a)2 1 1 n 1 2u 2 + 2(a2 − 1)u− 2 4a (1−a)2 1 {1 + a − |1 − a|} 2a 1 1 1 (a + 1)(a − 1) · − + 2a 1+a 1−a 2 f or 0 < a < 1 0 f or a > 1
3.5.
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
53
it follows that
Z r z r′ ′ dr′ r K t − 4πρα2 r3 0 α Z αr z K(t − τ )τ dτ. 4πρr3 0
Φ(x, y, z, t) = = The wave equations for
Ψx
and
Ψy
are solved in a similar fashion. Therefore,
it is possible to write the potentials of the single for e point sour e as
Φ(x, y, z, t)
=
Ψx (x, y, z, t) = Ψy (x, y, z, t)
=
Ψz (x, y, z, t)
=
with r2
=
r α
r R y β − 4πρr3 0 K(t − τ )τ dτ r R β x K(t − τ )τ dτ 4πρr 3 0 0 2 2 2 x +y +z . z 4πρr 3
R
0
K(t − τ )τ dτ
Before we derive the displa ements, we hange to spheri al oordinates relative to the single for e point sour e
z
K(t)
P υ r
x
Fig. 3.7: Spheri al oordinates
λ
(r, ϑ, λ).
y
(3.16)
(r, ϑ, λ)
54
CHAPTER 3.
x
= r sin ϑ cos λ
y
= r sin ϑ sin λ
z
= r cos ϑ.
In spheri al oordinates, the shear potential has no and for the
λ- omponent
BODY WAVES
r- and ϑ- omponent (show),
it holds that
Ψλ
=
−Ψx sin λ + Ψy cos λ.
y
Ψy
Ψx λ x Fig. 3.8: x-y-plane of Fig. 3.7.
This gives
Φ(r, ϑ, t) Ψλ (r, ϑ, t)
=
cos ϑ 4πρr 2
=
sin ϑ 4πρr 2
This equation does not depend on
− → − → u = ∇Φ + ∇ × Ψ
λ.
R
r α
0
R
The
r β
0
K(t − τ )τ dτ
K(t − τ )τ dτ.
(3.17)
displa ement ve tor
an be written in spheri al oordinates (show) as
ur
=
∂Φ ∂r
uϑ
=
1 ∂Φ r ∂ϑ
uλ
=
0.
+
1 ∂ r sin ϑ ∂ϑ (sin ϑΨλ )
−
1 ∂ r ∂r (rΨλ )
(3.18)
3.5.
55
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
This shows that the
P -wave following from Φ is not purely longitudinal, but it (in uϑ ). Similarly, the S -wave following from
ontains a transverse omponent
Ψλ is not purely transverse sin e ur ontains a shear omponent. The rst term in uϑ and the se ond in ur are near eld terms ( ompare exer ise 3.5). Here we ompute only the far- eld terms of ur and uϑ (only dierentiation of the integrals in(3.17))
ur uϑ
≃
cos ϑ 4πρα2 r K
≃
− sin ϑ 4πρβ 2 r K
t−
t−
r α r β
(longitudinal P − wave) (transversal
The far- eld displa ements have, therefore, the form of the for e with
1/r.
and the
(3.19)
S − wave).
K(t) de reasing
The single for e point sour e has dire tionally dependent radiation,
far- eld radiation hara teristi s
are shown in Fig. 3.9.
Fig. 3.9: Far eld radiation hara teristi s of single for e point sour e.
The radiation hara teristi s (P- and S-waves) are ea h two ir les. Those for 2 2 the -waves have a radius whi h is α /β larger then those of the P-waves. If the
S radiation angle ϑ is varied for xed r, the displa ements ur are proportional
to the distan e
OP 2 .
OP 1 ,
and the displa ements
The sign of the displa ement
radiation ir le to the se ond. from that shown in Fig.
ur
uϑ
are proportional to the distan e
hanges in transition from the rst
P-
The full 3-D radiation hara teristi s follows
3.9 by rotation around the dire tion of the for e.
S -wave is radiated in P -wave is radiated (but
Within the framework of the far- eld equations (3.19), no the dire tion of the for e, and perpendi ular to it, no
ompare exer ise 3.5).
The pra ti al use of the single for e point sour e, a ting perpendi ular on the free surfa e, is that it is a good model for the ee t of a by
vibro-seis
and often also for
explosions
drop weight, ex itation
detonated lose to the surfa e.
A
omplete solution requires the onsideration of the ee ts of the free surfa e, but that is signi antly more ompli ated. Furthermore, the dieren es to the
56
CHAPTER 3.
full-spa e models for all
P -waves
and for
S -waves,
BODY WAVES
for radiation angles smaller
then 30 to 40 degrees, are small, respe tively.
Exer ise 3.5 Compute the omplete displa ement (3.18) using (3.17) and examine in parti 0 ular, the dire tions ϑ = 0 and ϑ = 90 . Whi h polarisation does the displa ement ve tor have, and at whi h times are arrivals to be expe ted? Compute for
K(t) = K0 H(t)
the stati displa ement
(t > r/β).
3.5.2 Dipole point sour es A for e dipole an be onstru ted from two opposing single for es whi h are
dipole with moment for whi h the line onne ting the for es is perpendi ular to the dire tion of the for e. The onne ting line for a "dipole without moment " points in the a ting on two neighbouring points. Fig. 3.10 shows, on the left, a
dire tion of the for e.
Fig. 3.10: Single ouple and double ouple onstru ted from single for es.
Two dipoles with moment for whi h the sum over the moments is zero (right in Fig. 3.10), are a good model for many earthquake sour es, i.e., in the ase where the spatial radiation of earthquake waves of su iently large wave length is similar to that of a double ouple model. The a tual pro esses a ting in the earthquake sour e are naturally not four single for es. Usually, the ro k breaks along a surfa e if the shear strength is ex eeded by the a
umulation of shear
shear rupture ).
stress (
Another possibility is that the shear stress ex eeds the
stati fri tion on a pre-existing rupture surfa e. Sour e models from single for es and dipoles are only
equivalent point sour es.
In the following, we derive the far- eld displa ement of the single ouple model and give the results for the double ouple model.
ouple (with
x0 6= 0)
in Fig.
We start from the single
3.11 and ompute rst from (3.19) the
P -wave
3.5.
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
displa ement of for e
K(t)
with
x0 )2 + y 2 + z 2 ux uy = uz
Cartesian
z 4πρα2 r 2 K
t−
z
omponents
r
α
57
cos ϑ = z/r, r2 = (x −
(x − x0 )/r y/r · z/r.
P(x,y,z) y r
K(t) υ
λ
x0 ε
x -K(t)
Fig. 3.11: Single ouple model.
The displa ements a tion shifted by
u′x , u′y , u′z
ǫ,
of for e
−K(t) with the two neighbouring points of ux , uy , uz
an be determined using the Taylor expansion of
at the sour e oordinate
x0
and trun ating after the linear term. This leads, for
example, to
u′x
∂ux = − ux + ǫ . ∂x0
The single ouple displa ement then follows by superposition
u′′x = ux + u′x = −
∂ux ǫ. ∂x0
To obtain the far- eld displa ement requires only the dierentiation of the for e
K(t − r/α) with respe t to r, and additional dierentiation ∂r/∂x0 = −(x − x0 )/r. The other terms with x0 ontribute only to the near eld, the amplitude of whi h de reases faster then 1/r. This leads to
u′′x = −
z r −1 −(x − x0 ) x − x0 ′ t − K ǫ . 4πρα2 r2 α α r r
58
The
CHAPTER 3.
y-
and
z -displa ement u′′x u′′y u′′z
As expe ted, the
are treated similarly. Therefore,
z(x−x0 ) ′ = − 4πρα t− 3 r3 K
P -displa ement
r
α
(x − x0 )/r y/r ǫ· z/r.
K(t),
(3.20)
of the single ouple is also longitudinal.
The for e dipole is de ned stri tly by the limit taneous in rease of
BODY WAVES
so that
ǫ → 0,
ombined with a simul-
lim K(t) ǫ = M (t)
ǫ→0 remains nite (but non-zero).
M (t)
is alled
moment fun tion
of the dipole
with the dimensions of a rotational moment. From (3.20) with
P -wave
z/r = cos ϑ
(x − x0 )/r = sin ϑ cos λ, r-dire tion is
and
it follows that the
displa ement of the single ouple in
ur = −
r cos ϑ sin ϑ cos λ ′ . t − M 4πρα3 r α
In on luding, we now assume that
uϑ =
x0 = 0.
For the
S -wave,
it follows similarly
sin ϑ sin ϑ cos λ ′ r . t − M 4πρβ 3 r β
As for the single for e, the azimuthal omponent is zero. The following shows the results for the single ouple and the radiation in the
z P y r
υ
λ x
Fig. 3.12: Single ouple in the x-z-plane.
x − z -plane (y = 0)
3.5.
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
ur
2ϑ cos λ ′ t− = − sin8πρα 3r M
uϑ
=
uλ
sin2 ϑ cos λ ′ 4πρβ 3 r M
t−
r β
59
r α
(3.21)
= 0.
z
S + - P - +
x
Fig. 3.13: Far eld displa ement of a single ouple.
0 The ratio of the maximum S -radiation (for ϑ = 90 ) to the maximum P √ 0 radiation (for ϑ = 45 ) is about 10, if α ≈ β 3. The radiation hara teristi s in planes other then
y = 0
cos λ. z=0
P.
Plane
x=0
is a
is one only for
follow from the one shown by multipli ation with
nodal plane
for
P-
as well as for
The far- eld displa ements for a double ouple in the
z P y r
λ x
Fig. 3.14: Double ouple in the x-z-plane.
the plane
x−z -plane are (see exer ise
3.6)
υ
S -radiation;
60
CHAPTER 3.
ur
=
uϑ
=
uλ
=
2ϑ cos λ ′ − sin4πρα t− 3r M
r α
2ϑ cos λ M′ t − − cos4πρβ 3r
r β
cos ϑ sin λ ′ 4πρβ 3 r M
t − βr .
one
The moment fun tion in (3.22) is that of
ouple. The radiation hara teristi s in the
BODY WAVES
(3.22)
of the two dipoles of the double
x − z -plane
are shown in Fig. 3.15.
z
+
- P
-
+
S x
Fig. 3.15: Far eld displa ement of a double ouple. The
P -radiation
of the double ouple has the same form as that of a single
S to P is x = 0 and
ouple but is twi e as large; the ratio of the maximum radiation of now about 5 (for
z = 0.
The
S -wave
An (in nitesimal) pla ement in
z -dire tion,
√ λ ≈ β 3). P -nodal
planes are the planes with
has no nodal planes, but only nodal dire tions (whi h?).
shear rupture,
x-dire tion
either in the plane
or in the plane
and perpendi ular to it. If, by using the
P
z = 0
with relative dis-
with relative displa ement in
radiates waves as a double ouple, i.e., (3.22) holds. A shear rup-
ture or earthquake, therefore, radiates no of the
x = 0
P -waves in the dire tion of its rupture
distributions of the signs of rst motion
wave, the two nodal planes have been determined, the two possible rup-
ture surfa es are found. The determination of the
P -nodal plane of earthquakes
3.5.
SPHERICAL WAVES FROM SINGLE FORCE AND DIPOLE ...
fault plane solution ),
(
61
is an important aid in the study of sour e pro esses as
well as the study of large-s ale te toni s of a sour e region. Often the de ision between the two options for the rupture surfa e an be made based on geologi al arguments.
The moment fun tion of an earthquake with a smooth rupture, is, to a good approximation, a step fun tion with non-vanishing rise time
M0 ,
the
moment
T
and nal value
of the earthquake (see Fig. 3.16). The far- eld displa ements
are then, a
ording to (3.22), one-sided impulses.
M M0
0
T
t
T
t
M’
0
Fig. 3.16: Moment fun tion and far- eld displa ement of a smooth rupture.
Propagation ee ts in layered media, e.g., the Earth's rust, an hange the impulse form. In reality, the displa ements look very often dierent, relative to the one shown here, due to ompli ated rupture pro esses.
Exer ise 3.6 Derive the double ouple displa ement
ur
in (3.22) from the orresponding single
ouple displa ement in (3.21). Use equation (3.20) in Cartesian oordinates.
62
CHAPTER 3.
BODY WAVES
3.6 Re e tion and refra tion of plane waves at plane interfa es 3.6.1 Plane waves with arbitrary propagation dire tion In se tions 3.1 to 3.3 plane waves travelling in the dire tion of a oordinate axis were used. In the following, we need plane waves with an arbitrary dire tion of propagation. They an be des ribed by the following potentials
Φ
=
− → Ψ =
− → !# − → x k A exp iω t − α " − → !# − → x k − → B exp iω t − n. β "
Their variation with time is also harmoni .
(3.23)
(3.24)
This assumption is su ient for
− → → n are onstant unit ve tors, B are onstant, k and − − → x is the lo ation ve tor, ω is the angular frequen y and i the imaginary unit. − → Φ and the omponents of Ψ satisfy the wave equation (please on rm) most on lusions.
∇2 Φ =
A
and
1 ∂ 2Φ , α2 ∂t2
∇2 Ψj =
1 ∂ 2 Ψj β 2 ∂t2
(Cartesian
oordinates).
Sin e, a
ording to (3.23) and (3.24), the movement at all times and lo ations is non-zero, the wavefronts an no longer be de ned as surfa es separating undisturbed-disturbed from disturbed regions. fronts as
We, therefore, onsider wave
− → → x k /c) with c = α or c = β . surfa es of onstant phase ω(t − −
These
surfa es are de ned by
d dt
− → − → x k t− c
!
= 0.
− → k,
whi h also gives the dire tion of propagation. The wavefronts move parallel with respe t to themselves with the phase → k multiplied by the wavenumber ω/α or ω/β , is alled the velo ity . Ve tor − wavenumber ve tor. The polarisation dire tion of the ompressional part They are perpendi ular to ve tor
" − → !# − → iω x k − → ∇ Φ = − A exp iω t − k α α
(3.25)
3.6.
63
REFLECTION AND REFRACTION OF PLANE WAVES ...
is longitudinal (parallel to
− → k)
and that of the shear omponent
" − → !# − → iω x k − → − → − ∇ × Ψ = − B exp iω t − k ×→ n β β
(3.26)
− → → → → (rot(f · − n) = f · ∇ × − n −− n × ∇ f ) is transversal (perpendi ular to k ). − → From (3.26), it follows, that for Ψ , without loss of generality, the additional − → − → − →
ondition of orthogonality of k and n an be introdu ed. (Separation of n in − →
omponents parallel and perpendi ular to k ).
3.6.2 Basi equations We onsider a ombination of two half-spa es whi h are separated by a plane at z = 0. The ombination is arbitrary (solid-solid, solid-va uum, liquid-liquid, ...). We use Cartesian oordinates as shown in Fig. 3.17.
Fig. 3.17: Two half-spa es in Cartesian oordinates.
The
y -axis
points out of the plane. The displa ement ve tor is
− → u = (u, v, w), and its omponents are
independent of y , i.e., we treat a plane problem in whi h
on all planes parallel to the
x − z -plane,
the same onditions hold. The most
simple way to study elasti waves, under these onditions, is to derive but not
v,
from potentials. Writing
− → − → u = ∇Φ + ∇ × Ψ by omponents,
u = v
=
w
=
∂Φ ∂Ψ2 − ∂x ∂z ∂Ψ3 ∂Ψ1 − ∂z ∂x ∂Φ ∂Ψ2 + , ∂z ∂x
u
and
w
64
CHAPTER 3.
it is obvious that for o
ur in
u and w.
BODY WAVES
v two potentials Ψ1 and Ψ3 are required, and these do not v , it is better to use dire tly the equation of motion (2.21)
For
without body for es, whi h under these onditions be omes a wave equation
∇2 v = The
The
1 ∂2v . β 2 ∂t2
basi equations, therefore, are, if Ψ instead of Ψ2
boundary onditions
∇2 Φ
=
∇2 Ψ
=
∇2 v
=
∇2
=
1 ∂2Φ α2 ∂t2 1 ∂2Ψ β 2 ∂t2 1 ∂2v β 2 ∂t2 ∂2 ∂2 + 2 2 ∂x ∂z
u =
∂Φ ∂x
−
∂Ψ ∂z
w
∂Φ ∂z
+
∂Ψ ∂x .
=
is used
on the surfa e
z=0
(3.27)
(3.28)
between the half-spa es requires
ontinuity of the stress omponents
pzz
=
pzx
=
pzy
=
∂w ∂w → λ∇ · − u + 2µ = λ∇2 Φ + 2µ ∂z ∂z ∂w ∂u µ + ∂x ∂z ∂v µ , ∂z
or
λ ∂2Φ α2 ∂t2
+ 2µ
pzz
=
pzx
2 ∂ Φ + = µ 2 ∂x∂z
pzy
= µ ∂v ∂z .
2 ∂ Ψ − ∂z2
∂2Φ ∂z 2
∂2 Ψ ∂x2
+
∂2Ψ ∂x∂z
(3.29)
Whi h of the displa ement omponents is ontinuous depends on the spe ial
ombination of the half-spa es.
3.6.
65
REFLECTION AND REFRACTION OF PLANE WAVES ...
v with Φ and Ψ exists via the boundary onditions and, u and w, it follows, that the S -waves, the displa ement of whi h is only horizontal (in y -dire tion: SH-waves ), propagate independently from the P -waves, following from Φ, and the S -waves, following from Ψ, that also have a verti al omponent (in z -dire tion: SV-waves ). If a SH-wave impinges on an Sin e no onne tion of
therefore, with
interfa e, only re e ted and refra ted If, on the other hand, a and refra ted
SH-waves
P − (SV −)wave
o
ur, but no
P- or SV-waves.
intera ts with an interfa e, re e ted
SV −(P −)waves o
ur, but no SH -waves o
ur.
These statements
hold, in general, only for the ase of an interfa e between two solid half-spa es.
SH- nor SV-waves propagate; in a va uum a rigid half-spa e,
In liquids, neither
or no waves propagate at all. Correspondingly, the situation is even more simple if su h half-spa es are involved. The
de oupling
of
P-SV-
SH-waves
and
z = onst
the simple ase of an interfa e
holds for plane problems not only in
between two homogeneous half-spa es,
but also in the more ompli ated ase of an inhomogeneous medium, as long as density, wave velo ity, and module are only fun tions of
x
z.
and
One on-
sequen e of this de oupling is that in the following, re e tion and refra tion of
P - and SV-waves an
SH-waves. S-wave of arbitrary polarisation in its
be treated independently from that of the
Furthermore, it is possible to disse t an
SV- and SH- omponent and to study
their respe tive re e tion and refra tion
independently from ea h other. In ea h ase, we assume for the in ident plane wave a potential the form of (3.23) or (3.24), respe tively, (in the se ond ase
y - omponent Ψ).
In ase of a
SH-wave, we assume that v
an equation in the form of (3.23) with is part of the dire tion ve tor
− → k
β
instead of
α.
Fig. 3.18: In ident plane wave and angle of in iden e
− → k = (sin ϕ, 0, cos ϕ).
− → Ψ
Φ
or
− → Ψ
in
has only the
an be des ribed by
The angle of in iden e
ϕ
ϕ.
(3.30)
66
CHAPTER 3.
For the re e ted and refra ted wave an ansatz is made with
A and B, respe tively, and dierent
dire tion ve tors
− → k.
BODY WAVES
dierent amplitudes
The relation between
the new dire tion ve tors and (3.30) is via Snell's law. The relation between the displa ement amplitudes of the re e ted and the refra ted wave with the in ident wave, is alled
re e tion oe ient
and
refra tion oe ient,
respe -
tively, and it depends on the angle of in iden e and the material properties in the half-spa es. for
Rpp , Rps , Bpp , Bps , Rss , Rsp , Bss , Bsp will be the oe ients and bss those for the SH-waves. The rst index indi ates
P-SV-waves, rss
the type of in ident wave, the se ond the re e ted and refra ted wave type, respe tively. We dis uss, in the following, only relatively simple ases, for whi h illustrate the main ee ts to be studied.
3.6.3 Re e tion and refra tion of SH-waves Re e tion and refra tion oe ients The displa ement
v0
of the in ident
SH-wave in y -dire tion is
cos ϕ sin ϕ x− z . v0 = C0 exp iω t − β1 β1
v0
(3.31)
v1 ϕ
ϕ1
z=0 ϕ2
µ1,ρ1 ,β1 µ2,ρ2,β2
x
v2 z Fig. 3.19: In ident, re e ted and dira ted SH-waves at a plane interfa e.
The
ansatz for the re e ted and refra ted SH-wave as plane waves with re e tion ϕ1 and the refra tion angle ϕ2 , respe tively, and the same frequen y as
angle
the in ident wave is
re e tion
:
v1
=
refra tion
:
v2
=
cos ϕ1 sin ϕ1 x+ z C1 exp iω t − β1 β1 sin ϕ2 cos ϕ2 C2 exp iω t − x− z . β2 β2
(3.32)
(3.33)
3.6.
The unknowns are the angles and the refra tion oe ient The
67
REFLECTION AND REFRACTION OF PLANE WAVES ...
ϕ1 and ϕ2 , the bss = C2 /C0 .
re e tion oe ient
rss = C1 /C0
boundary onditions require at z = 0 the ontinuity of displa ement (that is
a reasonable requirement) and ontinuity for the normal and tangential stresses. This leads to
v0 + v1 = ∂ µ1 ∂z (v0 + v1 ) = The stress omponents
pzz
and
pzx
v2 2 µ2 ∂v ∂z
for
z = 0.
(3.34)
are zero everywhere, sin e no
P- and/or SV-
wave o
ur. Insert (3.31), (3.32) and (3.33) into (3.34). From the rst boundary
ondition this leads to
sin ϕ sin ϕ1 sin ϕ2 C0 exp iω t − x +C1 exp iω t − x = C2 exp iω t − x . β1 β1 β2 (3.35)
We plan to nd solutions
C1
and
and
v2
C2
v1
and
v2
of the problem, for whi h the amplitudes
are independent of lo ation, sin e only then an we be sure that
are solutions of the orresponding wave equation.
C1
and
C2
v1
be ome
only independent of lo ation if in (3.35)
sin ϕ1 sin ϕ2 sin ϕ = = , β1 β1 β2
(3.36)
sin e only then the exponential term an be an elled. Equation (3.36) is the well-known
Snell's Law
angle of in iden e
ϕ
whi h states that the re e tion angle
and that for the refra tion angle
ϕ2
ϕ1
is equal to the
is
β2 sin ϕ2 . = sin ϕ β1 With (3.35), this leads to
C2 − C1 = C0 . The se ond boundary ondition in (3.34) gives
cos ϕ cos ϕ2 cos ϕ1 µ1 iω − C0 + C1 = −µ2 iω C2 . β1 β1 β2 With
ϕ1 = ϕ
and
µ1,2 /β1,2 = ρ1,2 β1,2 ,
it follows that
(3.37)
68
CHAPTER 3.
BODY WAVES
ρ1 β1 cos ϕ(C1 − C0 ) = −ρ2 β2 cos ϕ2 C2 or
ρ2 β2 cos ϕ2 C2 + C1 = C0 . ρ1 β1 cos ϕ
(3.38)
From (3.37) and (3.38) follow the re e tion and refra tion oe ients
C1 C0 C2 = C0
rss =
=
bss
=
ρ1 β1 cos ϕ − ρ2 β2 cos ϕ2 ρ1 β1 cos ϕ + ρ2 β2 cos ϕ2 2ρ1 β1 cos ϕ . ρ1 β1 cos ϕ + ρ2 β2 cos ϕ2
(3.39)
(3.40)
With (3.36), this leads to
12 β22 2 cos ϕ2 = (1 − sin ϕ2 ) = 1 − 2 sin ϕ . β1 1 2
2
For
perpendi ular
in iden e (ϕ
rss =
In this ase,
rss
and
two half-spa es. For
bss
(3.41)
= 0)
ρ 1 β1 − ρ 2 β2 ρ 1 β1 + ρ 2 β2
and
bss =
2ρ1 β1 . ρ 1 β1 + ρ 2 β 2
impedan es ρ1 β1 and ρ2 β2 of the = π/2), rss = −1 and bss = 0. The
depend only on the
grazing
in iden e (ϕ
absolute value of the amplitude of the re e ted wave is never larger then that of the in ident wave; that of the refra ted wave an be larger if for If
ρ 2 β2 < ρ 1 β1
(e.g.,
ϕ = 0).
rss
is negative, this means that in one point of the interfa e the displa ement
−y -dire tion, if the displa ement ve tor +y -dire tion. For impulsive ex itation (see also
ve tor of the re e ted wave points in of the in ident wave points in
later), this means that the dire tion of rst motion of the in ident and the re e ted wave are opposite. The following gure shows
β1 /β2 > 1
and
ρ1 = ρ2 .
|rss |
as a fun tion of
ϕ
for dierent velo ity ratios
3.6.
REFLECTION AND REFRACTION OF PLANE WAVES ...
Fig. 3.20:
|rss |
as a fun tion of
ϕ
69
for dierent velo ity ratios.
Total re e tion If
β2 < β1 , as in Fig. 3.20, cos ϕ2 is real for all angles of in ident ϕ, the same rss and bss . Total re e tion,, i.e., |rss | = 1, is then only possible for
is true for
grazing in iden e. If
β2 > β1 , cos ϕ2
is only real as long as
ϕ =< ϕ∗ = arcsin
β1 . β2
ϕ∗ is the riti al angle (or limiting angle of total re e tion ). A
ording to (3.41), ϕ = ϕ∗ is onne ted to the ase with grazing propagation of the wave in the se ond half-spa e (ϕ2 = π/2). ϕ > ϕ∗ , cos ϕ2 be omes imaginary, or, to be more exa t, negative imaginary ω and positive imaginary for negative ω , sin e only then v2 for z → +∞ remains limited. rss and bss be ome omplex. v1 and v2 still solve
If
for positive
the wave equations and satisfy the boundary onditions, even when posing the
ansatz (3.32) and (3.33) have not expli itly been hosen in omplex form. The re e tion oe ient an then be written as
= exp −2i arctan ab
rss
=
a−ib a+ib
a
=
ρ1 β1 cos ϕ
b
=
−ρ2 β2
β22 β12
sin2 ϕ − 1
12
ω |ω|
.
(3.42)
70
CHAPTER 3.
BODY WAVES
It has the absolute value 1 and a phase that depends on the angle of in iden e.
Its sign hanges with the sign of the frequen y.
Values of
3.21 for a few ombinations.
Fig. 3.21:
|rss |
as a fun tion of
ϕ
β1 / sin ϕ.
are shown in Fig.
for dierent velo ity ratios.
The refra ted wave propagates for lo ity
|rss |
ϕ > ϕ∗
parallel to the interfa e with the ve-
Its amplitude is not only ontrolled by
by the exponential term whi h depends on
z.
bss , but is also ontrolled
The amplitude of the refra ted
wave de ays, therefore, exponentially with in reasing distan e from the interfa e
inhomogeneous
(
or
boundary layer wave ). "
|ω| v2 = bss C0 exp − β2
It follows that (please he k)
β22 sin2 ϕ − 1 β12
21 # sin ϕ x . z exp iω t − β1
Other ases The treatment of the re e tion of plane
liquids
P -waves
at an interfa e between two
gives similar results to the one dis ussed above (see also exer ise 3.9).
If the interfa e between two
solid half-spa es
is onsidered, the omputational
eort is signi antly larger, sin e now re e ted and refra ted
SV -waves have to
be in luded. We, therefore, skip the details. The absolute value of the re e tion
oe ient
Rpp
is shown in Fig. 3.22.
3.6.
| R pp |
| R pp |
α1 sin ϕ∗ = α −
β 2 < α 1 < α2
0
π/2
ϕ∗
0
ϕ
ϕ = 0, Rpp =
|Rpp |
for
SV -wave SV -wave
1 sin ϕ∗∗ = α β−2
α 1 < β 2 < α2
2
0
ϕ∗
0
Fig. 3.22: Absolute value of the re e tion oe ient
For
71
REFLECTION AND REFRACTION OF PLANE WAVES ...
ϕ∗∗
π/2
ϕ
Rpp .
ρ2 α2 −ρ1 α1 ρ2 α2 +ρ1 α1 ( ompare also exer ise 3.9).
ϕ∗ < ϕ < π/2
is smaller then 1 for two reasons. First, the re e ted
also arries energy; se ond, for the ase on the left of Fig.
all
3.22, a
propagates in the lower half-spa e for ϕ, and, similarly, for the ∗∗ ∗∗
ase on the right of Fig. 3.22, for ϕ < ϕ . ϕ < ϕ is the whi h exists only for
ϕ∗∗ = arcsin
For angles
ϕ
larger then
total re e tion o
urs. transported in the
se ond riti al angle
α1 < β2 < α2
ϕ∗∗ ,
α1 α1 > ϕ∗ = arcsin . β2 α2
the se ond energy loss is no longer possible, and
The re e ted energy is then, to a smaller part, also
SV -wave.
Some numeri al results for re e tion and refra tion oe ients for a
P-SV- ase
are given in Fig. 3.23 (model of the rust-mantle boundary (Moho) with α1 = 6.5km/sec, β1 = 3.6km/sec, ρ1 = 2.8g/cm3, α2 = 8.2km/sec, β2 = 4.5km/sec, ρ2 3.3g/cm3).
=
72
CHAPTER 3.
BODY WAVES
Fig. 3.23: Absolute value of re e tion and refra tion oe ients and
Rpp , Rps , Bpp
Rss .
Transition to impulsive ex itation The transition from the harmoni ase, treated up to now, to the impulse ase,
an be done with the (3.31), the
SH-wave
Fourier transform
v0 = F
( ompare appendix A.1.7). Instead of
sin ϕ cos ϕ t− x− z β1 β1
3.6.
73
REFLECTION AND REFRACTION OF PLANE WAVES ...
may impinge on the interfa e, and we disse t integral in
partial vibration
F (t) F (ω)
Z +∞ 1 F (ω)eiωt dω 2π −∞ Z +∞ = F (t)e−iωt dt
F (t)
with the aid of the Fourier
=
F (t).
Fourier transform of
−∞
We then study, as before, re e tion and refra tion of the
partial waves
1 sin ϕ cos ϕ dv0 = F (ω) exp iω t − x− z dω 2π β1 β1 and then sum the re e ted partial waves to derive the re e ted
v1 =
1 2π
Z
+∞
−∞
SH -wave
cos ϕ sin ϕ x+ z dω. rss F (ω) exp iω t − β1 β1
(3.43)
As long as the re e tion oe ient rss is frequen y independent (whi h is the β2 < β1 or for ϕ < ϕ∗ with β2 > β1 ), it an be moved before the
ase for
integral, thus, yielding
v1 = rss F
sin ϕ cos ϕ t− x+ z . β1 β1
The re e ted impulse has, in this ase, the same form as the in ident impulse. The amplitude ratio of the two impulses is equal to the re e tion oe ient.
rss , β2 > β1 . Then
a
ording to (3.42), be omes dependent from One then has to pro eed dierently.
ω
We disse t
for
ϕ > ϕ∗
rss
into real and
with
imaginary parts
rss
=
R(ϕ) = I(ϕ)
=
R(ϕ) + iI(ϕ) a2 − b 2 a2 + b 2 2ab − 2 a + b2
ω |ω|
(b
for
ω > 0).
A
ording to (3.43), it holds that
1 v1 = R(ϕ)F (τ ) + I(ϕ) 2π
Z
+∞
−∞
iω F (ω)eiωτ dω |ω|
(3.44)
74
CHAPTER 3.
BODY WAVES
with
τ =t−
sin ϕ cos ϕ x+ z. β1 β1
I(ϕ) in (3.44) (iω/ |ω|) · F (ω).
The fun tion, with whi h
is multiplied, an be written, here via
its Fourier transform, as
This is a simple
lter
of fun tion
F (τ ) ω in +900
( ompare general omments on lters in appendix A.3.4). Ea h frequen y
F (τ ) keeps its amplitude, but its phase is hanged. The phase hange is 0 for ω > 0 and −90 for ω < 0. This orresponds to a Hilbert transform and is shown in appendix B. The fun tion with whi h I(ϕ) in (3.44) is multiplied is, therefore, the Hilbert transform FH (τ ) of F (τ )
1 FH (τ ) = P π
P
Z
+∞
−∞
1 F (t) dt = t−τ π
Z
+∞
−∞
ln |t| F ′ (τ − t)dt.
indi ates the main value (without the singularity at
form of
FH (τ )
t = τ ),
(3.45)
and the se ond
follows from the rst by partial integration. Thus,
v1 = R(ϕ)F (τ ) + I(ϕ)FH (τ ).
(3.46)
Due to the se ond term in (3.46), the form of the re e ted wave is from that of the in ident wave. Fig. 3.24 shows the 0 waves and angle of in iden e ϕ from 0 to 90 .
SH-
For pre- riti al angles of in iden e
ϕ < ϕ∗ = 480 ,
results
dierent
of the re e tion of
the re e tion has the form of
the in ident wave with positive and negative signs. Beyond the riti al angle, 0 in the range of total re e tion, impulse deformations o
ur until at ϕ = 90 the in ident wave form appears again, but with opposite sign ( orresponding to a 0 0 re e tion oe ient rss = −1). The phase shift of rss at ϕ = 55 is about ±90 , with the onsequen e that
R(ϕ) ≈ 0.
The re e tion impulse for this angle of
in iden e is, therefore, lose to the Hilbert transform
FH (τ )
of
F (τ )
(the exa t
Hilbert transform is an impulse that is symmetri with respe t to its minimum).
3.6.
REFLECTION AND REFRACTION OF PLANE WAVES ...
Fig. 3.24: Re e tion of
SH-waves for dierent angle of in iden e ϕ.
75
76
CHAPTER 3.
BODY WAVES
Exer ise 3.7: Whi h sign does the re e tion oe ient
rss
have in Fig. 3.20 and Fig. 3.21, in
the regions where it is real?
Exer ise 3.8: Determine the angle of in iden e for whi h
rss
Brewster angle),
is zero (
give the onditions under whi h this a tually happens ( ompare 3.24).
ϕ ≈ 400
and
in Fig.
Exer ise 3.9 Compute the re e tion and refra tion oe ients for a plane surfa e between two liquids and for a plane harmoni longitudinal wave under angle of in iden e
ϕ impinges. Give, qualitatively, the trend of the oe ients for ρ1 = ρ2 with α1 > α2 and α1 < α2 . Hint: Use an ansatz for the displa ement potential in the form of (3.31) to (3.33) and express the boundary onditions via potentials as dis ussed in se tion 3.6.2.
3.6.4 Re e tion of P-waves at a free surfa e Re e tion oe ients P-waves from a free surfa e is of pra ti al imP -waves from earthquakes and explosions propagate
The study of the re e tion of portan e for seismology.
through the Earth and impinge at the seismi station from below. tal and verti al displa ement are modi ed by the free surfa e. re e ted
P-
and
S -waves
Horizon-
Furthermore,
are re e ted downwards and re orded at larger dis-
tan es, sometimes with large amplitudes. It is, therefore, useful and ne essary to know the re e tion oe ient of the Earth's surfa e. For the moment, we negle t the layered nature of the rust in our model, thus, only giving a rst approximation to reality. Based on the omments given at the end of se tion 3.6.2, we sele t the following ansatz for the potentials
in ident
P − wave Φ0
re e ted
sin ϕ cos ϕ A0 exp iω t − x− z α α
(3.47)
=
sin ϕ1 cos ϕ1 A1 exp iω t − x+ z α α
(3.48)
=
sin ϕ′1 cos ϕ′1 B1 exp iω t − x+ z . β β
(3.49)
P − wave Φ1
re e ted
=
SV − wave Ψ1
3.6.
REFLECTION AND REFRACTION OF PLANE WAVES ...
Fig. 3.25: In ident
77
P-wave and re e ted P- and S-wave .
The boundary onditions at z = 0 require vanishing normal and tangential stress pzz = pzx = 0. No boundary onditions for the displa ement exist. With (3.29) − → and Φ = Φ0 + Φ1 (Ψ = Ψ1 = y − omponent of Ψ ), it follows that
2µ ∂ 2 ∂ 2 Ψ1 1 ∂2 (Φ + Φ ) + (Φ + Φ ) + 0 1 0 1 α2 ∂t2 λ ∂z 2 ∂x∂z 2 ∂ ∂ 2 Ψ1 ∂ 2 Ψ1 2 (Φ0 + Φ1 ) + − ∂x∂z ∂x2 ∂z 2
=
0
z=0
=
0
z = 0. (3.51)
(3.50)
As in the last se tion, Snell's law follows from the boundary onditions
From this, it follows that
sin ϕ1 sin ϕ′1 sin ϕ = = . α α β β · sin ϕ < ϕ. ϕ1 = ϕ and ϕ′1 = arcsin α
(3.52)
With (3.47), (3.48), (3.49) and
µ β2 µ ρβ 2 = = = λ λ + 2µ − 2µ ρα2 − 2ρβ 2 α2 − 2β 2 (3.50) leads to
1 (A0 + A1 ) (iω)2 α2 " 2 # 2β 2 iω iω iω ′ ′ + 2 (A0 + A1 ) cos ϕ + B1 − sin ϕ1 cos ϕ1 = 0. α − 2β 2 α β β Then
78
CHAPTER 3.
BODY WAVES
2α2 β 2 sin ϕ′1 cos ϕ′1 cos2 ϕ A0 + A1 + 2 − B1 (A0 + A1 ) = 0. α − 2β 2 α2 β2 with
1+
2β 2 cos2 ϕ = α2 − 2β 2 = = =
and
γ=
α2 β2
>2
2 2β 2 α 2 − 1 + cos ϕ α2 − 2β 2 2β 2 2 α 2β 2 2 − sin ϕ α2 − 2β 2 2β 2 2 α β2 2 − 2 sin ϕ α2 − 2β 2 β 2
γ − 2 sin2 γ γ−2
, it follows that
γ − 2 sin2 ϕ 2γ sin ϕ′1 cos ϕ′1 (A0 + A1 ) − B1 = 0. γ−2 γ−2 From this
(γ − 2 sin2 ϕ)
1 B1 A1 − 2 sin ϕ(γ − sin2 ϕ) 2 = 2 sin2 ϕ − γ. A0 A0
(3.53)
Equation (3.51) then gives
iω iω iω iω − cos ϕ + 2A1 − sin ϕ cos ϕ 2A0 − sin ϕ α α α α 2 2 iω iω +B1 − sin ϕ′1 − B1 cos ϕ′1 =0 β β or
sin2 ϕ′1 − cos2 ϕ′1 2 sin ϕ cos ϕ (A0 − A1 ) + B1 = 0. 2 α β2 Equation (3.52) then gives
2 sin ϕ cos ϕ
B1 A1 + γ − 2 sin2 ϕ = 2 sin ϕ cos ϕ. A0 A0
(3.54)
3.6.
79
REFLECTION AND REFRACTION OF PLANE WAVES ...
From (3.53) and (3.54), it follows that the amplitude ratios are
1 2 4 sin2 ϕ cos ϕ γ − sin2 ϕ 2 − γ − 2 sin2 ϕ A1 = 2 1 A0 4 sin2 ϕ cos ϕ γ − sin2 ϕ 2 + γ − 2 sin2 ϕ
(3.55)
4 sin ϕ cos ϕ γ − 2 sin2 ϕ B1 = 1 2 . A0 4 sin2 ϕ cos ϕ γ − sin2 ϕ 2 + γ − 2 sin2 ϕ
To derive displa ement amplitudes (that is how the oe ients
(3.56)
Rpp
and
Rps
in
se tion 3.6.2 were de ned) from the ratios of potential amplitudes given here,
P -wave is PP -re e tion
we use (3.25) and (3.26). The displa ement amplitude of the in ident
iω − iω α A0 ; that of the re e ted P -wave is − α A1 .
This then gives the
oe ient (see also (3.55))
Rpp =
A1 . A0
(3.57)
Equation (3.26) gives the displa ement amplitude of the re e ted
− iω β B1 .
Thus, the
PS -re e tion oe ient is (see also (3.56)) Rps =
Rpp
and
Rps are real
and
is always positive. For and only a
α B1 . β A0
and
ϕ=
π 2,
Rpp = −1
Fig. 3.26: Re e tion and refra tion oe ients of
ϕ.
(3.58)
frequen y independent for all angles of in ident ϕ. Rps
ϕ=0
P -wave is re e ted.
of in iden e
SV -wave as
and
Rps = 0,
respe tively,
P-waves for dierent
angles
80
CHAPTER 3.
The meaning of the
negative signs
BODY WAVES
in the re e tion oe ients be omes lear, if
the displa ement ve tor of the in ident and re e ted waves are represented via (3.25) and (3.26)
− → u0
=
− → u1
=
− → u ′1
=
sin ϕ cos ϕ iω − → k0 ∇ Φ0 = − A0 exp iω t − x− z α α α iω sin ϕ cos ϕ − → ∇ Φ1 = − A0 Rpp exp iω t − x+ z k1 α α α sin ϕ′1 cos ϕ′1 iω − →′ − − → x+ z k1×→ n. ∇ × Ψ = − A0 Rps exp iω t − α β β
Fig. 3.27: Polarity of re e ted
Rpp < 0,
P- and SV-waves.
therefore, means that if the displa ement of the re e ted
P -wave in a
− → = 0) points in the dire tion of − k 1 , the in ident wave − → points in the dire tion of k 0 . For Rps < 0, the displa ement of the re e ted SV − →′ → wave would, for su h an in ident wave, be pointing in the dire tion of − k 1 × − n. point on the interfa e (z
These onne tions be ome more obvious if we go from the harmoni ase to the
impulsive ase
( ompare se tion 3.6.3, transition to impulse ex itation).
For the problem studied, the re e tion oe ients are frequen y independent. Therefore, the re e ted waves have always the same form as the in ident wave
− → u0 − → u1
sin ϕ cos ϕ − → = F t− x− z k0 α α cos ϕ sin ϕ − → x+ z k1 = Rpp F t − α α
(3.59)
(3.60)
3.6.
REFLECTION AND REFRACTION OF PLANE WAVES ...
In the ase
− → u ′1
=
ϕ′1
=
sin ϕ′1 cos ϕ′1 − →′ → Rps F t − x+ z k1×− n β β β sin ϕ . arcsin α
(3.61)
Rpp < 0, if the rst motion of the in ident P -wave is dire ted towards z = 0, this also holds for the re e ted SV -wave and the re e ted
the interfa e
P -wave;
81
otherwise, the rst motion of the re e ted
the interfa e. Fig. 3.28 shows the ase for
P -wave
points away from
Rpp < 0.
Fig. 3.28: De nition of the rst motion of re e ted
P- and SV-waves.
Displa ements at the surfa e Finally, we ompute the resulting displa ement at the free surfa e (z=0) in whi h the three waves (3.59), (3.60) and (3.61) superimpose.
Horizontal displa ement u = u = fu (ϕ)
and
=
(positive in
Rps cos ϕ′1 ] F
[(1 + Rpp ) sin ϕ + sin ϕ fu (ϕ)F t − x α
=
sin ϕ x t− α
1 4γ sin ϕ cos ϕ γ − sin2 ϕ 2 2 1 4 sin2 ϕ cos ϕ γ − sin2 ϕ 2 + γ − 2 sin2 ϕ
Verti al displa ement
w
x -dire tion):
(positive in
z -dire tion):
[(1 − Rpp ) cos ϕ +
Rps sin ϕ′1 ] F
sin ϕ t− x α
(3.62)
82
CHAPTER 3.
w fw (ϕ)
BODY WAVES
sin ϕ = fw (ϕ)F t − x α =
2γ cos ϕ γ − 2 sin2 ϕ 2 . 1 4 sin2 ϕ cos ϕ γ − sin2 ϕ 2 + γ − 2 sin2 ϕ
The ampli ation fa tors (or transfer fun tions of the surfa e) respe tively, are given in Fig. 3.29 for the ase
(3.63)
fu (ϕ) and fw (ϕ),
γ = 3.
Fig. 3.29: Transfer fun tions of the free surfa e.
linearly polarised wave with the apparent velo ity α/ sin ϕ propapolarisation angle ǫ, (see Fig. 3.30), is not identi al to the angle of in iden e ϕ. ǫ is also alled the apparent angle of in iden e. Therefore, a
gates at the surfa e. The
ǫ = arctan
u w
12 2 sin ϕ γ − sin ϕ . = arctan γ − 2 sin2 ϕ 2
3.6.
REFLECTION AND REFRACTION OF PLANE WAVES ...
Fig. 3.30: Polarisation angle
ǫ
and angle of in iden e
ε
ϕ.
ε=ϕ
π 2
ε(ϕ)
π 2
O Fig. 3.31: Qualitative relationship between
ǫ
83
and
ϕ
.
1
π
= arctan
ǫ′ (0)
=
2
ǫ
ϕ
2 1
γ2
2 (γ − 1) 2 γ−2
!
.
In ident SV -wave SV -wave, instead of the P -wave onsidered up until now, impinges on the P -wave is re e ted for angles of in iden e ϕ > ϕ∗ = arcsin αβ , but only an SV -wave (|Rss | = 1) is re e ted. This follows from onsiderations ∗ similar to that for an in ident P -wave. For ϕ < ϕ , the displa ement at the ∗ free surfa e is linearly polarised, but for ϕ > ϕ , it is polarised ellipti ally. If a
free surfa e, no
84
CHAPTER 3.
BODY WAVES
SV
This property is observed: -waves from earthquakes for distan es smaller 0 then about 40 are ellipti ally polarised, but are linearly polarised for larger distan es.
Fig. 3.32: Polarisation of
SV -waves from earthquakes .
3.6.5 Re e tion and refra tion oe ients for layered media Matrix formalism In the last two se tions, we studied the re e tion and refra tion of plane waves at
one interfa e.
The re e tion and refra tion oe ients depend, then, mainly on
the properties of the half-spa es and the angle of in iden e. Only if the riti al angle is ex eeded, a weak frequen y dependen e o
urs: the sign of the phase (the oe ients be ome omplex) is ontrolled by the sign of the frequen y of the in ident wave ( ompare se tion 3.6.3). The frequen y dependen e be omes mu h more pronoun ed when the re e tion and refra tion of plane waves in
layered media is onsidered (two or more interfa es). Then, interferen e phenomena o
ur and for spe ial frequen ies (or wave lengths) onstru tive or destru tive interferen es o
ur. a (sub-parallel)
generally,
Here, we will study the re e tion and refra tion of
liquid layers between two liquid half-spa es.
P -waves
from a pa ket of
The orresponding problem for
SH -
waves in solid media an be solved similarly. There is a lose similarity between
P -waves
SH -waves in layered solid media. The P-SV-waves in solid media (possibly with interspersed liquid layers)
in layered liquid media and
treatment of
is, in prin iple, the same, but the derivation is signi antly more ompli ated. In all these approa hes, a
matrix formalism
is used, whi h is espe ially ee tive
for implementing on omputers. We hoose the annotation of the liquid-layered medium as given in Fig. 3.33.
3.6.
85
REFLECTION AND REFRACTION OF PLANE WAVES ...
Fig. 3.33: Liquid-layered medium with n layers. The displa ement potential
Φj
in the
j -th
layer
(j = 1, 2, . . . , n)
satis es the
wave equation
∂ 2 Φj 1 ∂ 2 Φj ∂ 2 Φj + = 2 . 2 2 ∂x ∂z αj ∂t2 Solutions of this equation, whi h an be interpreted as harmoni plane waves, have the form
exp [i (ωt ± kj x ± lj z)] with
kj2 + lj2 = ω/α2j ,
where
kj
is the
horizontal, lj
respe tively.
We assume positive frequen ies
wavenumbers
kj .
ω
the
verti al wavenumber,
and non-negative horizontal
Then, we an disregard the sign + of
kj x,
sin e it orre-
sponds to waves whi h propagate in -x-dire tion. This is not possible for our sele tion of the in ident wave, (see Fig.
3.33).
The two signs of
lj z
z
have to
be kept, sin e in all layers (ex ept the n-th) waves propagate in + - and in
-z -dire tion.
We then ome to the
Φj
=
z1 Bn
potential ansatz
(3.64)
=
Aj exp [i (ωt − kj x − lj (z − zj ))] +Bj exp i ωt − kj′ x + lj′ (z − zj )
z2 = 0,
=
0.
(3.66)
kj2 + lj2 = kj′2 + lj′2 =
ω2 α2j
(3.65)
86
CHAPTER 3.
BODY WAVES
In (3.64), we have assumed, for the moment, that the wavenumbers of the waves propagating in +z- and -z-dire tion are dierent. Furthermore, we have repla ed
z
by
z − zj .
This does not hange the meaning of the terms but simpli es the
omputations. The part
A1 exp [i (ωt − k1 x − l1 z)] of Φ1
( ompare, e.g., (3.47)). This means that of in iden e
ϕ
will be interpreted as in ident
k1
P -wave
and l1 are onne ted with the angle
as
k1
=
ω α1
l1
=
ω α1
sin ϕ cos ϕ.
B1 exp [i (ωt − k1′ x + l1′ z)] of Φ1 is ered half-spa e z > 0. We want to ompute the refra tion oe ient Bpp (again de ned The part
amplitudes )
Rpp
=
B1 A1
Bpp
=
α1 αn
·
The boundary onditions for the interfa es
(3.67)
the wave re e ted from the laythe re e tion oe ient as the ratio of the
An A1 .
Rpp
and
displa ement
(3.68)
z = z 2 , z3 , . . . , zn
require ontinuity pzz = λ∇2 Φ =
of the verti al displa ement ∂Φ/∂z and of the normal stress ρ∂ 2 Φ/∂t2 . For z = zj , this gives
∂Φj ∂z
∂Φj−1 ∂z
=
and
ρj
∂ 2 Φj ∂t2
=
ρj−1
From the rst relation, it follows that (the phase term
∂ 2 Φj−1 ∂t2 .
eiωt
is negle ted in the
following sin e it an els out),
−lj Aj exp [−ikj x] + lj′ Bj exp −ikj′ x The se ond relation gives
ρj Aj exp [−ikj x] + ρj Bj exp −ikj′ x Both equations hold for
= −lj−1 Aj−1 exp [i (−kj−1 x − lj−1 dj−1 )] ′ ′ ′ + lj−1 Bj−1 exp i −kj−1 x + lj−1 dj−1 .
= +
ρj−1 Aj−1 exp [i (−kj−1 x − lj−1 dj−1 )] ′ ′ ρj−1 Bj−1 exp i −kj−1 x + lj−1 dj−1 .
j = 2, 3, . . . , n, and dj−1 = zj −zj−1 (d1 = 0).
x
As before,
we require that the exponential terms depending on must an el, leading to ′ kj = kj′ = kj−1 = kj−1 . This, then, gives (with (3.67))
3.6.
87
REFLECTION AND REFRACTION OF PLANE WAVES ...
′ kn′ = kn = kn−1 = kn−1 = . . . = k1′ = k1 =
This is an alternative form of
lj′
Snell's law.
With (3.65), this leads to
! 12
! 21 α2j 2 . 1 − 2 sin ϕ α1
ω2 − k12 α2j
= lj =
ω sin ϕ. α1
ω = αj
(3.69)
If sin ϕ > α1 /αj , lj is imaginary (and even negative imaginary ), only then for j = n is the amplitude of the potentials limited for z → ∞. This leads to the following system of equations, whi h onne ts Aj and Bj with Aj−1 and Bj−1 , respe tively
Aj − Bj
=
Aj + Bj
=
lj−1 Aj−1 e−ilj−1 dj−1 − Bj−1 eilj−1 dj−1 lj ρj−1 Aj−1 e−ilj−1 dj−1 + Bj−1 eilj−1 dj−1 . ρj
In matrix form, this an be written as (please he k)
Aj Bj
where
mj
e−ilj−1 dj−1 lj−1 ρj + lj ρj−1 = −l 2lj ρj j−1 ρj + lj ρj−1 Aj−1 · Bj−1 Aj−1 = mj · Bj−1 is the
(−lj−1 ρj + lj ρj−1 )e2ilj−1 dj−1 (lj−1 ρj + lj ρj−1 )e2ilj−1 dj−1
(3.70)
layer matrix.
Repeated appli ation of (3.70) gives
An Bn
= = =
On omputers, the produ t
M
mn · mn−1 · . . . · m3 · m2 A1 M B1 A1 M11 M12 . B1 M21 M22 of the layer matri es
qui kly and e iently. First, the angular frequen y
A1 B1
mn to m2 an be determined ω and the angle of in iden e
88
ϕ
CHAPTER 3.
are given; then, the
lj 's
BODY WAVES
are determined with (3.69), and the matri es are
multiplied. This gives the elements of
An = M11 A1 + M12 B1
M.
and
From
Bn = M21 A1 + M22 B1
with (3.66), it follows that
M21 B1 =− A1 M22 The re e tion oe ient
Rpp
and
M12 M21 An = M11 − . A1 M22 Bpp
of the layered
M12 M21 M11 − . M22
(3.71)
and the refra tion oe ient
medium, therefore, an be written a
ording to (3.68) as
Rpp = −
M21 M22
and
α1 αn
Bpp =
Two homogeneous half-spa es In this very simple ase, it follows (with
M = m2 =
1 2l2 ρ2
d1 = 0)
l 1 ρ2 + l 2 ρ1 −l1 ρ2 + l2 ρ1
that
−l1 ρ2 + l2 ρ1 l 1 ρ2 + l 2 ρ1
and, therefore, a
ording to (3.71)
With
Rpp
=
Bpp
=
l1 =
ω α1
−l2 ρ1 + l1 ρ2 l 2 ρ1 + l 1 ρ2 α1 (l1 ρ2 + l2 ρ1 )2 − (l2 ρ1 − l1 ρ2 )2 2l1 ρ1 α1 . = α2 2l2 ρ2 (l2 ρ1 + l1 ρ2 ) α2 l2 ρ1 + l1 ρ2
cos ϕ
and l2
=
ω α2
refra tion), it follows that
Rpp
=
Bpp
=
( ompare with exer ise 3.9). For
1−
α22 α21
12 sin2 ϕ =
ω α2
cos ϕ2 (ϕ2 =angle
ρ2 α2 cos ϕ − ρ1 α1 cos ϕ2 ρ2 α2 cos ϕ + ρ1 α1 cos ϕ2 2ρ1 α1 cos ϕ ρ2 α2 cos ϕ + ρ1 α1 cos ϕ2 ϕ = 0 (→ ϕ2 = 0),
it follows that
of
3.6.
89
REFLECTION AND REFRACTION OF PLANE WAVES ...
2ρ1 α1 . ρ2 α2 + ρ1 α1
(3.72)
These are equations that also hold for an interfa e between two
solid half-spa es.
Rpp =
ρ2 α2 − ρ1 α1 ρ2 α2 + ρ1 α1
and
Bpp =
Lamella in full-spa e We limit our study here to
verti al re e tions
from a lamella.
α1 ,ρ1
z 2= 0
x
α2,ρ2
d 2= d z
α3=α1 ,ρ3= ρ1
Fig. 3.34: Lamella of thi kness d. In this ase,
n = 3, l1 = l3 = ω/α1
and l2
= ω/α2 .
Then with (3.70) and
d1 = 0,
d2 = d
m2
=
α2 2ρ2
=
α2 2α1
−iω αd
α1 e m3 = 2α2 with
γ=
ρ1 α1 ρ2 α2 and
γ′ = d
e−iω α2 M = m3 m2 = 4γ We now ompute the
ρ2 α2 ρ1 α1
2
=
ρ2 α1 − αρ21
+ +
1+γ −1 + γ
1 + γ′ −1 + γ ′
ρ1 α2 ρ1 α2
− αρ21 + αρ12 ρ2 ρ1 α1 + α2 −1 + γ 1+γ
d
(−1 + γ ′ )e2iω α2 2iω d (1 + γ ′ )e α2
d
(1 + γ)2 − (1 − γ)2 e2iω α2 2iω d 1 − γ 2 + (γ 2 − 1)e α2
re e tion oe ient Rpp
d
γ 2 − 1 + (1 − γ 2 )e2iω α2 2iω d −(1 − γ)2 + (1 + γ)2 e α2
(a
ording to (3.71))
d
=
M21 1 − e−2iω α2 (1 − γ 2 )(1 − e2iω α2 ) − = = R0 d 2iω −2iω αd M22 2 (1 − γ)2 − (1 + γ)2 e α2 1 − R02 e
=
1−γ ρ2 α2 − ρ1 α1 . = 1+γ ρ2 α2 + ρ1 α1
with
R0
!
1 γ . This leads to
d
Rpp
(3.73)
!
.
90
R0 ,
CHAPTER 3.
BODY WAVES
a
ording to (3.72), is the re e tion oe ient of the interfa e
relatively small re e tion oe ients the Earth
(|R0 | < 0.2),
R0 ,
z = 0.
For
whi h are typi al for dis ontinuities in
one an write as a good approximation
−2iω αd 2 Rpp = R0 1 − e .
Dis ussion of Rpp
(3.74)
Rpp , in the form of (3.73) or (3.74), is zero for angular frequen ies ω , for whi h 2ω αd2 is an even multiple of π . With the frequen y ν and the wave length Λ in the lamella (α2 = νΛ), the ondition for destru tive interferen e is 1 3 d = , 1, , 2, . . . Λ 2 2
(3.75)
The lamella has to have a thi kness of a multiple of the half wave-length so
that in re e tion destru tive interferen e refra tions show onstru tive interferen e. A
ording to (3.74), multiple of
π.
Rpp
is maximum
Then
o
urs with
Rpp = 0.
(|Rpp | = 2 |R0 |)
if
2ω αd2
In this ase
is an uneven
d 1 3 5 7 = , , , ,... Λ 4 4 4 4 In this ase, the waves interfere
for refra tion.
The periodi ity of
Rpp ,
(3.76)
onstru tively for re e tion
and
destru tively
visible in (3.75) and (3.76), holds generally so
α2 π Rpp ω + n = Rpp (ω), d
n = 1, 2, 3, . . .
impulP -wave has the verti al
To on lude, we dis uss how the re e tion from a lamella looks for an
sive ex itation.
w0 = F t − displa ement w1 of
displa ement verti al
We assume that the verti ally in ident
w1 = with
Rpp (ω)
z α1
and that
F (ω)
is the spe trum of
F (t).
The
the re e ted wave is then ( ompare se tion 3.6.3)
1 2π
Z
+∞
Rpp (ω)F (ω)e
iω t+ αz
1
−∞
dω
(3.77)
from (3.73). In pra tise, integral (3.77) is omputed numeri ally,
sin e fast numeri al methods for
Fourier analysis
the spe trum from the time fun tion (
F (ω)
from
exist and omputation of
F (t))
and
Fourier synthesis,
i.e., omputation of the time fun tion from its spe trum (w1 from its spe trum
3.6.
REFLECTION AND REFRACTION OF PLANE WAVES ...
Rpp (ω)F (ω)eiωz/α1 ).
form (FFT).
Su h numeri al methods are known as
91
Fast Fourier trans-
Insight into the pro esses o
urring during re e tion, the topi of this hapter, 2
an be a hieved as follows: we expand (3.73) (whi h due to R0 < 1 always
onverges)and get
Rpp (ω)
∞ X n 2d 2d −iω α −iω α 2 2 R02 e = R0 1 − e n=0
−iω 4d 2d −iω α 2 3 α2 2 − R e = R0 − R0 1 − 0 1 − R0 e −iω 6d α2 − ... −R05 1 − R02 e R02
(3.78)
Substitution of this into (3.77) and taking the inverse transform of ea h element gives
w1
z z 2d 2 = R0 F t + (3.79) − R0 1 − R0 F t + − α1 α1 α2 z z 4d 6d − R03 1 − R02 F t + − R05 1 − R02 F t + − ... − − α1 α2 α1 α2
The rst term is the
re e tion from the interfa e z = 0.
expe ted, is the re e tion oe ient
R0
Its amplitude, as
of this interfa e.
The se ond term
des ribes a wave whi h is delayed by twi e the travel time through the lamella, thus, orresponding to the
re e tion from the interfa e z = d.
the expe ted size; the re e tion oe ient of this interfa e is
Its amplitude has
−R0 .
The produ t
z = 0 for waves travelling in +z− 2 and −z−dire tion, 2ρ1 α1 /(ρ2 α2 + ρ1 α1 ) and 2ρ2 α2 /(ρ1 α1 + ρ2 α2 ), is 1 − R0 . In
of the re e tion oe ients of the interfa e
multiple re e tions within the lamella (with three and ve re e tions, respe tively). The terms in (3.79) orrespond to the rays shown in Fig. 3.35.
the same way, the third and fourth term of (3.79) an be interpreted as
z=0 z =d Fig. 3.35: Re e ted and multiple re e ted rays in a lamella.
Equation (3.79) is a de omposition of the re e ted wave eld in (in nite many) ray ontributions. It is fully equivalent to (3.77).
92
CHAPTER 3.
The approximation (3.74) for
BODY WAVES
Rpp (ω) orresponds to the trun ation of the exn = 0 and, therefore, the limitation on the the interfa es z = 0 and z = d, respe tively (and
pansion in (3.78) after the term for two
primary re e tions
negle ting
R02
from
relative to 1).
Exer ise 3.10 Show that for the refra tion oe ient in (3.71), it holds that
Bpp =
α1 detM αn M22
with
detM =
l 1 ρ1 . l n ρn
Apply this formula in the lamella, in ases in whi h (3.75) and (3.76) hold.
Exer ise 3.11 The
P -velo ity
α2 > α1 .
of the lamella is larger then that of the surrounding medium:
Does then total re e tion o
ur? Dis uss this qualitatively.
3.7 Re e tivity method: Re e tion of spheri al waves from layered media 3.7.1 Theory The results of se tion 3.6.5 an, with relative ease, be extended to the ex itation by spheri al waves. For simpli ation of representation, we again assume that we deal, at the moment, only with
P -waves in liquids.
Fig. 3.36: Explosive point sour e over liquid, layered medium.
3.7.
93
REFLECTIVITY METHOD: REFLECTION OF ...
The spheri al waves are ex ited by an explosion point sour e lo ated at hight
h
above the layered medium.
The displa ement potential of this sour e for
harmoni ex itation is ( ompare se tion 3.4)
Φ1e = with
R2 = r2 + (z + h)2 .
z -axis,
Φj
in the
t− αR
1
(3.80)
Be ause of the symmetry under rotation around the
r j -th layer is
ylindri al oordinates
potential
1 iω e R
and
z
are used.
The wave equation for the
1 ∂Φj 1 ∂ 2 Φj ∂ 2 Φj ∂ 2 Φj + = . + ∂r2 r ∂r ∂z 2 α2j ∂t2
(3.81)
Elementary solutions of this equation are (please he k)
J0 (kr) exp [i (ωt ± lj (z − zj ))]
with
( ompare se tion 3.6.5 for notation).
ω2 k 2 + lj2 = 2 , lj = αj J0 (kr)is
the
ω2 − k2 α2j
Bessel fun tion
! 12
(3.82)
of rst kind
and zeroth order ( ompare appendix C).
−ikj x i(ωt±lj (z−zj )) Equation (3.82) is an analogue to the solutions e ·e of the wave 2 2 2 2 2 2 2 equation ∂ Φj /∂x + ∂ Φj /∂z = (1/αj )∂ Φj /∂t dis ussed in the last hapter. In (3.82), the index
k
j
of the horizontal wavenumber
k
has been dropped, sin e
is a parameter over whi h one an integrate (furthermore, it was shown in
se tion 3.6.5 that all
kj 's
are identi al).
With (3.82), the fun tions
Z
∞
f (k)J0 (kr)ei(ωt±lj (z−zj )) dk
(3.83)
0
are also solutions of (3.81) if the integral onverges.
Thus, we ome to the
potential ansatz
Φj =
Z
0
∞
o n J0 (kr) Aj (k)ei(ωt−lj (z−zj )) + Bj (k)ei(ωt+lj (z−zj )) dk.
(3.84)
Note the lose relation of (3.84) to (3.64). Whether this ansatz a tually has a
rstly, if Φ1e se ondly, if Φj
solution, depends
from (3.80) an be represented in the integral
form (3.83) and
in (3.84) satis es the boundary onditions for
z = z2 , z3 , . . . zn .
94
CHAPTER 3.
BODY WAVES
The rst requirement is satis ed sin e the following integral representation is
Sommerfeld integral, ompare appendix D)
valid (
1 iω e R
t− αR
1
=
Z
∞
J0 (kr)
0
k i(ωt−l1 |z+h| e dk. il1
(3.85)
We, therefore, an interpret the rst part
Z
∞
J0 (kr)A1 ei(ωt−l1 z) dk
(3.86)
0
of
Φ1
(with
z1 = 0)
as the in ident wave (see also se tion 3.6.5)
Φ1r ).
part is the re e ted wave
Φ1e
(the se ond
We have to ompare (3.85) and (3.86) for
lo ations in whi h the spheri al wave passes on in iden e at the interfa e
−h < z ≤ 0. A1 (k) = (k/il1 )e−il1 h . i.e., for
The
|z + h| = z + h
In this ase,
boundary onditions
for the interfa es an be taken from se tion 3.6.5. The
potentials (3.84) are dierentiated under the integral. from the boundary onditions is only satis ed
The identity following
for all r,
if the integrands are
identi al. This leads to the same system of equations for in se tion 3.6.5, i.e., (3.70). wavenumbers
lj
Aj (k)
ϕ).
k
and
and
Bj (k)
as
In ontrast to the previous se tion, the verti al
have to be onsidered now as fun tions of
angle of in iden e
z = 0,
and the omparison gives
ϕ
k
(and not of the
are onne ted via
k=
ω sin ϕ. α1
(3.87)
Rpp = B1 /A1 = −M21 /M22 has ω and angle of in iden e introdu ed via (3.87): Rpp = Rpp (ω, k).
Following se tion 3.6.5, the re e tion oe ient
been omputed as a fun tion of the angular frequen y
ϕ;;
then the dependen e on
The se ond part of
Φ1r
Φ1 ,
= =
k
an be
the re e ted wave, an then be written as
Z
∞
Z0 ∞ 0
J0 (kr)A1 (k)Rpp (ω, k)ei(ωt+l1 z) dk k J0 (kr)Rpp (ω, k)ei(ωt+l1 (z−h)) dk. il1
The orresponding verti al displa ement is
∂Φ1r = eiωt w1r (r, z, ω, t) = ∂z
Z
∞
kJ0 (kr)Rpp (ω, k)eil1 (z−h) dk
0
and the horizontal displa ement (with
J0′ (x) = −J1 (x))
(3.88)
3.7.
95
REFLECTIVITY METHOD: REFLECTION OF ...
∂Φ1r = eiωt u1r (r, z, ω, t) = ∂r
∞
Z
0
−k 2 J1 (kr)Rpp (ω, k)eil1 (z−h) dk. il1
(3.89)
The integrals in (3.88) and (3.89) are best omputed numeri ally, espe ially,
solid
in the ase of many layers. For the re e tion oe ient and
w1r
and
u1r
Rpp (ω, k)
ompressional part of the re e tion from the shear part, similar results hold, whi h only
des ribe only the
layered half-spa e
z ≥ 0.
For the
now ontain the re e tion oe ients The transition to
media, (3.88) and (3.89) also hold, but
is more ompli ated than for liquid media
Rps (ω, k).
impulse ex itation Φ1e =
1 F R
R t− α1
instead of (3.80) is relatively simple (see se tion 3.6.3). If of
F (t),
F (ω)
is the spe trum
it holds that
Φ1e =
1 2πR
Z
+∞
F (ω)e
−∞
iω t− αR
1
dω.
The orresponding displa ements of the re e ted wave are
W1r (r, z, t) U1r (r, z, t) with
w1r
=
from (3.88) and
u1r
1 2π
R +∞ −∞
F (ω)
w1r (r, z, ω, t) u1r (r, z, ω, t)
dω
(3.90)
from (3.89). The integrals in (3.88) and (3.89),
Fourier transforms of the displa ement. The numeri al omputation of (3.88), (3.89) and (3.90) is alled the Re e tivity method ; it is a pra ti al approa h for the omputation of theoreti al seismograms of body waves. With it, the amplitudes of body waves from explosions and earthquakes an be studied, thus, progressing beyond the more lassi al travel time interpretation. multiplied by
F (ω),
are, therefore, the
3.7.2 Re e tion and head waves An example for theoreti al seismograms is given in Fig. 3.37 (from K. Fu hs: The re e tion of spheri al waves from transition zones with arbitrary depthdepended elasti moduli and density. Journ. of Physi s of the Earth, vol. 16, Spe ial Issue, S. 27-41, 1968). It is the result for a simple model of the rust, assumed to be homogeneous.
The point sour e and the re eivers are at the
Earth's surfa e; the in uen e of whi h has been negle ted here. The transition
96
CHAPTER 3.
Mohorovi£i¢-zone,
of the rust to the upper mantle ( order
dis ontinuity,
short
BODY WAVES
Moho )
is a rst
i.e., the wave velo ities and the density hange abruptly
(for a dis ontinuity of 2nd order these parameters would still be ontinuous, but their derivative with depth would have a jump).
Fig. 3.37: Syntheti seismogram for re e tion and refra tion from a 1st order dis ontinuity (from K. Fu hs, 1968, Journ. of Physi s of the Earth).
re e tion from the Moho. For distan es from the
riti al point r∗ = 74.91 km, orresponding to the riti al angle of in iden e ϕ∗ , the rst onset is the head wave with the apparent velo ity The dominant wave is the sour e beyond the
of 8.2 km/se . Its amplitude de ays rapidly with in reasing distan e, and its ∗ form is the time integral of the re e tion for r < r . For pre- riti al distan e
r,
the form of the re e tion is pra ti ally identi al to that of the in ident wave.
At the riti al point, it begins to hange its form. This was already dis ussed in se tion 3.6.3 in terms of the properties of the re e tion oe ient for plane waves (this holds for
P-
and
SH-waves, respe tively).
impulse form is roughly opposite to that for sin e the re e tion oe ient to
−1
Rpp
For large distan es, the
r < r∗ .
This is also expe ted,
for the angle of in iden e
ϕ = π/2
is equal
(for liquids, this follows from the formulae given in se tion 3.5.6). The
amplitude behaviour of the re e tion is relatively similar to the trend of the absolute value
|Rpp |
of the re e tion oe ient, if
|Rpp |
is divided by the path
3.7.
97
REFLECTIVITY METHOD: REFLECTION OF ...
length and if one onsiders the verti al omponent (see, e.g., Fig.
3.22 and
orresponding equations). The main dis repan ies are near the riti al point. A
ording to
|Rpp |,
the re e ted wave should have its maximum dire tly at the
riti al point, whereas in reality, it is shifted to larger distan es. This shift is larger, the lower the frequen y of the in ident wave.
Fig. 3.38: Re e tion amplitude versus oset as a fun tion of frequen y.
The onsideration of this shift is important when determining the riti al point from observed re e tions, e.g., in re e tion seismi s.
3.7.3 Complete seismograms Fig. 3.39 shows the
SH -seismograms
potential of the re e tivity method.
This shows omplete
for a pro le at the surfa e of a realisti Earth model.
The
sour e is a horizontal single for e at the Earth's surfa e, a ting perpendi ular to the pro le. The dominant period is 20 se . The most pronoun ed phases are the dispersive Love waves (for surfa e waves, see hapter 4), whose amplitudes are mostly lipped. (mantle wave
S
and
The propagation paths of the largest body wave phases
SS, ore re e tion S S
and dira tion at the ore
Sdif f
are
also sket hed.) A detailed des ription of the re e tivity method is given in G. Müller: The re e tivity method: A tutorial, Journ. eophys,., vol. 58, 153-174, 1985.
98
CHAPTER 3.
Fig. 3.39: Complete
SH -seismograms for a
BODY WAVES
pro le at the surfa e of a realisti
Earth model.
3.8 Exa t or generalised ray theory - GRT We ontinue the hapter on elasti body waves with the treatment of re e tion and refra tion of
ylindri al waves
radiated from a
line sour e
and re e ted
and refra ted at a plane interfa e whi h is parallel to the line sour e.
This
problem is more simple and less pra ti al than the ase onsidered in se tion 3.7 of a point sour e over a layered medium. On the other hand, we will learn a totally dierent way of treating wave propagation whi h leads to relatively simple
analyti al
(and not only numeri ally solvable) results. This is the main
aim of this se tion. This method, originally developed by Cagniard, de Hoop and Garvin (see, e.g., W.W. Garvin: Exa t transient solution of the buried line sour e problem, Pro . Roy. So . London, Ser. A, vol 234, pg. 528-541, 1956),
an also be applied for layered media and be modi ed for point sour es.
In
that form it is, similar to the re e tivity method, usable for the omputation of theoreti al body-wave seismograms in the interpretation of observations.
3.8.
99
EXACT OR GENERALISED RAY THEORY - GRT
Again, we limit ourselves to treat the problem of a
liquid
model (see Fig. 3.40),
sin e we an then study the main ideas with a minimum of omputation.
Fig. 3.40: Explosive line sour e in a liquid medium.
We work with the displa ement potentials
(Φ1e =
in ident, Φ1r
=
re e ted
P − wave)
Φ1 = Φ1e + Φ1r in half-spa e 1 Φ2 in half-spa e 2. The three
and
potentials satisfy the wave equations
∇2 Φ1e,r =
1 ∂ 2 Φ1e,r , α21 ∂t2
∇2 Φ2 =
1 ∂ 2 Φ2 . α22 ∂t2
(3.91)
s2 ϕ2 , α22
(3.92)
The Lapla e transform of these equations gives
∇2 ϕ1e,r =
s2 ϕ1e,r α21
and
∇2 ϕ2 =
where ϕ1e , ϕ1r and ϕ2 are the transforms of Φ1e , Φ1r and Φ2 , respe tively, and s is the transform variable (see appendix A). We assume that the P -wave starts at time t =0 at the line sour e. Therefore, the initial values of Φ1e , Φ1r and Φ2 , and their time derivatives for
t= +0, are zero outside the line sour e.
The time
derivatives have to be onsidered in the se ond derivative with respe t to
t
in
(3.91).
3.8.1 In ident ylindri al wave First, we have to study the in ident wave. Sin e the line sour e is explosive and has, therefore, ylindri al symmetry around its axis, it holds that
∇2 ϕ1e = with
R2 = x2 + (z + h)2 .
∂ 2 ϕ1e 1 ∂ϕ1e s2 + ϕ1e = ∂R2 R ∂R α21
(3.93)
The solution of (3.93), whi h an be interpreted as a
ylindri al wave in +R dire tion, is
100
CHAPTER 3.
BODY WAVES
R 1 ϕ1e = f (s) K0 ( s), s α1
(3.94)
where f (s) is the Lapla e transform of an arbitrary time fun tion K0 ( αR1 s) is one of the of zeroth order.
modi ed Bessel fun tions
Proof: Using the substitution x = Rα
s 1
F(t)
and
, (3.93) an be expressed as the dierential
equations of the modi ed Bessel fun tion
x2
dy d2 y +x − (x2 + n2 )y = 0. 2 dx dx
In the ase onsidered here, linear solutions,
K0 (x)
and
n =0.
The dierential equation has two independent
I0 (x),
respe tively. For real
x, Fig.
3.41 shows their
qualitative behaviour.
3 I0 (x)
2 1 K 0 (x) 0 1
0
Fig. 3.41: Behaviour of linear solutions
This shows that only for
x → ∞.
K0 (x)
Referen e :
(
K0 (x)
x
3
2
and
I0 (x).
is a possible solution, sin e
I0 (x)
grows in nitely
M. Abramovitz and I.A. Stegun: Handbook of Mathe-
mati al Fun tions, H. Deuts h, Frankfurt, 1985). Taking the inverse Lapla e transform of (3.94) in the time domain, and using the orresponden e
1 K0 s
f (s) •−◦ F (t) (F (t) ≡ 0 for t < 0) R 0 for t < R/α1 s •−◦ cos h−1 ( αR1 t ) for t > R/α1 α1
with a typi al behaviour of the solution given in Fig. 3.42;
3.8.
101
EXACT OR GENERALISED RAY THEORY - GRT
t
R/α 1 Fig. 3.42: Behaviour of the solution. the potential an be written as
Φ1e =
Z
t
R/α1
By varying
F(t),
F (t − τ ) cos h−1 (
α1 τ )dτ R
(t ≥ R/α1 ).
(3.95)
the ylindri al wave an be given dierent time dependen-
ies. Equation (3.95) is the analogue to the potential
Φ1e =
1 RF
spheri al wave from an explosive point sour e. In the following, we treat the ee ts an be studied.
spe ial ase F (t) = δ(t),
t−
R α1
of a
for whi h all important
If realisti ex itations have to be treated, the results
for the potentials and displa ements of the re e ted and dira ted waves, respe tively, derived with time dependent realisti
F(t).
For
F (t) = δ(t),
have to be onvolved with
F (t) = δ(t) Φ1e = cosh−1 (
α1 t ), R
and the orresponding radial displa ement in R-dire tion is
UR =
∂Φ1e =− ∂R
t R
t2
−
Fig. 3.43: Displa ement in R dire tion.
R2 α21
1/2
(t >
R ). α1
(3.96)
102
CHAPTER 3.
BODY WAVES
The orresponding point sour e results are
Φ1e
=
UR
=
1 R H t− R α1 1 R R 1 ∂Φ1e − . δ t− = − 2H t − ∂R R α1 Rα1 α1
Fig. 3.44: Displa ement in R dire tion.
3.8.2 Wavefront approximation for UR If we write (3.96) as
UR =
R t+
R α1
−t 1/2
t−
R α1
1/2 ,
R and onsider values of t near α , we nd the approximation 1
UR ≈
−1
1/2
(2Rα1 )
·
1 t−
R α1
1/2 .
(3.97)
t is to R/α1 , and, therefore, wavefront approximation. It is more a
urate for large R, and it is, therefore, also the far- eld approximation of the ylindri al wave. Within the framework of the wavefront approximation (3.97), the impulse form
This approximation is more a
urate the loser this is alled
of the ylindri al wave is independent from R, and its amplitude is proportional −1/2 to R . Both statements be ome espe ially obvious if (3.97) is onvolved with realisti ex itation fun tions integrable.
F (t).
The singularity in (3.96) and (3.97) is
3.8.
103
EXACT OR GENERALISED RAY THEORY - GRT
3.8.3 Re e tion and refra tion of the ylindri al wave The oordinates most appropriate for the study of re e tion and refra tion are the Cartesian oordinates
x
and
z.
Equation (3.92), then, takes the form
s2 ∂2ϕ ∂2ϕ + = 2 ϕ. 2 2 ∂x ∂z α 2 Appropriate elementary solutions have the form cos(kx) exp(±imz) with k + 2 2 2 m = −s /α . From these elementary solutions, more ompli ated solutions in integral form (similar to se tion 3.7.1) an be onstru ted
ϕ=
Z
∞
f (k) cos(kx)e±imz dk,
0
rstly, the potential (3.94) of the in ident se ondly, the boundary onditions for z = 0. Spe i ally, we use the
with whi h we an try to satisfy wave, and
following ansatz
ϕ1e
=
ϕ1r
=
ϕ2
m1,2
For
K0
R∞ 0
R∞ 0
R∞
=
0
s2 = −i k 2 + 2 α1,2
R α1 s
A1 (k) cos(kx)e−im1 z dk B1 (k) cos(kx)eim1 z dk A2 (k) cos(kx)e−im2 z dk
!1/2
(negative
(z > −h)
(3.98)
imaginary for positive radi ands).
in (3.94) an integral representation, similar to (3.85) for the spher-
i al wave, an be found. With this
ϕ1e
(with
f(s)=1, sin e F (t) = δ(t)) it follows
that
ϕ1e
1 = s
Z
0
∞
1
k2
A omparison with
+
s2 α21
ϕ1e
1/2
1/2 # s2 2 cos(kx) exp − |z + h| k + 2 dk. α1 "
from (3.98) for
A1 (k) =
z>-h
gives
e−im1 h . ism1
(3.99)
104
CHAPTER 3.
The boundary onditions for
BODY WAVES
z= 0 are ( ompare se tion 3.6.5)
∂ ∂Φ2 (Φ1e + Φ1r ) = , ∂z ∂z
ρ1
∂ 2 Φ2 ∂2 (Φ + Φ ) = ρ . 1e 1r 2 ∂t2 ∂t2
The Lapla e transform gives
∂ϕ2 ∂ (ϕ1e + ϕ1r ) = , ∂z ∂z
ρ1 (ϕ1e + ϕ1r ) = ρ2 ϕ2 .
From (3.98), it follows that
m1 (A1 (k) − B1 (k)) ρ1 (A1 (k) + B1 (k))
= =
m2 A2 (k) ρ2 A2 (k),
and from this
B1 (k) = Rpp (k)A1 (k)
and
A2 (k) = Bpp (k)A1 (k)
. Thus,
ρ2 m 1 − ρ1 m 2 ρ2 m 1 + ρ1 m 2
Rpp (k) = The potentials
ϕ1r
ϕ2
and
ϕ1r ϕ2
= =
and
Bpp (k) =
2ρ1 m1 . ρ2 m 1 + ρ1 m 2
(3.100)
are, therefore,
Z
∞
Z0 ∞
A1 (k)Rpp (k) cos(kx)eim1 z dk A1 (k)Bpp (k) cos(kx)e−im2 z dk.
0
w and u of the verti al and horizontal displa ement W ∂ϕ U, respe tively, an, in general, be written as w = ∂ϕ ∂z and u = ∂x
The Lapla e transforms and
and spe i ally
w1r = u1r w2 = u2
R∞ 0
R∞ 0
Rpp (k) sim1 Bpp (k) sim1
im1 cos(kx) e−im1 (h−z) dk −k sin(kx) −im2 cos(kx) e−i(m2 z+m1 h) dk. −k sin(kx)
(3.101)
3.8.
EXACT OR GENERALISED RAY THEORY - GRT
105
transformed ba k.
This is impossible
These Lapla e transforms must now be
with (A.9) in appendix A. One, rather, uses an approa h that is based in transforming (3.101) several times with fun tion theory methods until the integrals are of the form
w u The inverse
W (t) U (t)
= Z(t)
=
R∞ 0
Z(t)e−st dt.
an then be identi ed
(3.102)
dire tly.
positive real s, i.e., we do not onsider the whole onvergen e half-plane of the Lapla e
An important limitation has to be mentioned rst: we only onsider transform, but only the positive real axis.
This simpli es the omputations
signi antly, without limitation of its generality, sin e the Lapla e transform is an analyti al fun tion.
It is, therefore, determined in the whole onvergen e
half-plane by its values on the real axis, where the integral (3.101) is real. With
cos(kx) = Re(e−ikx ) w1r = u1r w2 = u2
Re Re
R∞ 0
R∞ 0
and
sin(kx) = Re(ie−ikx ),
(3.101) an be written as
m1 e−i(kx+m1 (h−z)) dk −k −m2 Bpp (k) e−i(kx+m1 h+m2 z) dk sm1 −k
Rpp (k) sm1
The next step is a hange of the integration variables
u=
m1,2
= =
with
m1,2
u -axis.
!1/2 2 1/2 s −i −s2 u2 + 2 = −is −u2 + α−2 1,2 α1,2 1/2 −s u2 − α−2 = −sa1,2 1,2 1/2
.
(3.104)
The transformed integration path is, therefore, in the sheet of the of the square root
the sheet with
The
gives
a1,2 = u2 − α−2 1,2
plane
(3.103)
ik s
so that the integration path is now along the positive imaginary transformation of the square root
.
a1,2 ,
a1,2 ≃ −u).
in whi h
a1,2 ≃ u
for
|u| → ∞
Riemann
holds (and not in
Introdu ing (3.104) in (3.100), gives
106
CHAPTER 3.
ρ2 a1 − ρ1 a2 ρ2 a1 + ρ1 a2
Rpp (u) =
With
k = −isu
w1r u1r w2 u2
and
dk = −isdu,
= Re
Z
and
= Re
Z
2ρ1 a1 . ρ2 a1 + ρ1 a2
(3.105)
it follows from (3.103)
+i∞
Rpp (u)
0
Bpp (u) =
BODY WAVES
+i∞
Bpp (u)
0
−i − au1
i aa21 − au1
e−s(ux−ia1 (h−z)) du
(3.106)
e−s(ux−ia1 h−ia2 z) du.
(3.107)
These expressions already have a ertain similarity with (3.102) sin e s only o
urs in the exponential term. The next step is, therefore, a new hange in the integration variable in (3.106)
t = ux − ia1 (h − z)
(3.108)
t = ux − ia1 h − ia2 z.
(3.109)
in (3.107)
From both equations,
u
has to be determined as a fun tion of
be inserted in (3.106) and (3.107), respe tively.
t
and has to
This will be dis ussed later
in more detail. At the same time, the integration path has to be transformed a
ordingly. For
u = 0,
it follows from (3.108) and (3.109), respe tively, that
t(0) = t(0) =
t0 t0
The transformed integration paths
C1
= =
h−z α1 h z α1 + α2 .
(3.110)
C2 (for u → +i∞, they
(for (3.106)) and
spe tively, start on the positive real t-axis. For
(3.107)), reapproa h an
asymptote in the rst quadrant whi h passes through the entre of the oordinate system and has the slope
tan γ = x/(h + z)
tan γ = x/(h − z)
(for the ase (3.109)).
(for the ase (3.108)) and
In (3.108),
z
is always negative in
(3.109) always positive. The transformed integration paths are shown in Fig. 3.45.
3.8.
107
EXACT OR GENERALISED RAY THEORY - GRT
Fig. 3.45: Transformed integration paths.
In the ase of the re e ted wave (left in Fig. 3.45), in the ase of the refra ted wave,
C1
C1
is part of a hyperbola;
is part of a urve of higher order whi h is
similar to a hyperbola. We then get
w1r u1r
w2 u2
= Re
Z
Rpp (u(t))
(
Bpp (u(t))
(
C1
= Re
Z
C2
The last step is now to deform the paths
Cau hy's integral
C1
−i
)
du −st e dt dt
(3.111)
ia2 (u(t)) a1 (u(t)) − a1u(t) (u(t))
)
du −st e dt. dt
(3.112)
− a1u(t) (u(t))
and
C2
towards the real axis a -
ording to . The path C1,2 an now be repla ed by the path ′ C 1,2 + C1,2 in Fig. 3.46 if no poles of the integrands in (3.111) and (3.112) are lo ated inside the two paths.
Im t
C1,2 C’1,2 C1,2 t0 Fig. 3.46: Integration paths in the omplex plane.
Re t
108
CHAPTER 3.
BODY WAVES
bran h points
This is satis ed be ause the only singularities of a1 and a2 are the −1 and u = ±α2 , respe tively, and these bran h points are integrable ′ singularities and not poles. Finally, the ontribution of the urve C1,2 goes to
u = ±α−1 1
zero if its radius be omes in nite. Thus,
w1r u1r
Z
=
"
∞ h−z α1
+
Re Rpp (u(t)) h−z α1
Z
0
w2 u2
Z
=
∞
− a1u(t) (u(t))
[0] e−st dt "
Z
+ αz 2 h z α1 + α2
)
−i
# du −st e dt dt (3.113)
Re Bpp (u(t))
h α1
+
(
(
ia2 (u(t)) a1 (u(t)) − a1u(t) (u(t))
)
#
du −st e dt dt
[0] e−st dt.
(3.114)
0
In these expressions, only the real part of the square bra kets has to be onsid−st ered sin e e is real and the integration is only over real . The addition of the
t
se ond integral with vanishing ontribution was only done for formal reasons, to allow integration over
t
from 0 to
∞ a
ording to (3.102).
Equation (3.113) and
(3.114) have, therefore, the standard form of a Lapla e transform, from whi h the original fun tion an be read
dire tly.
W1r and U1r of (h−z)/α1 . This is
The displa ements
the re e ted wave are, therefore, zero between the time 0 and not surprising sin e
(h − z)/α1
is the travel time from the sour e perpendi ular
down to the re e ting interfa e and ba k to level
z
of the sour e. This time
is, therefore, smaller, or at most equal, to the travel time of the rst re e ted onsets at this point. For
W1r U1r with
u(t)
t > (h − z)/α1
"
= Re Rpp (u(t))
h/α1 + z/α2
are zero, and for
W2 U2 u(t)
(
−i
− a1u(t) (u(t))
)
du dt
#
(3.115)
from (3.108).
Similarly, the displa ements
with
it holds that
W2 and U2 of the dira ted wave t > h/α1 + z/α2 , it holds that "
= Re Bpp (u(t))
(
ia2 (u(t)) a1 (u(t)) − a1u(t) (u(t))
)
du dt
#
for
0 ≤ t <
(3.116)
from (3.109).
All that is needed to al ulate these relatively simple solutions of (3.108) and (3.109), with respe t of
u
algebrai fun tions, is the
as a fun tion of real times
3.8.
109
EXACT OR GENERALISED RAY THEORY - GRT
t > t0
du dt . In the ase of
(from (3.110)) and the knowledge of the derivative
(3.108), this is very easy sin e here
u(t) =
x 2t R x 2t R
R = x2 + (h − z)2
− +
1/2
h−z 2 R h−z i 2 R
u
t21 − t2
an be given expli itly
21
for
1 2 2
t2 − t1
for
≤ t ≤ t1 =
R α1
(3.117)
t > t1 .
is the distan e of the sour e point from its mirror image,
i.e., from the point with the oordinates
a
ording to
h−z α1
x = 0
and
z = +h; t1 = R/α1
is,
Fermat's prin iple, the travel time of the a tual re e tion from the
interfa e. The urve of
u(t)
is given in Fig. 3.47. The derivative
omputed dire tly from (3.117). It has a singularity at
Fig. 3.47: The path of
u(t)
du/dt
in the omplex plane.
In the ase of the refra ted wave,
u(t)
has to be omputed numeri ally with a
similar urve as for the re e ted wave (Fig. 3.47).
Fig. 3.47: The path of
u(t)
an be
t = t1 .
for the refra ted wave in the omplex plane.
110
CHAPTER 3.
The numeri al omputations of a large eort.
u(t),
BODY WAVES
and its derivative, are possible without
In the following se tion, we fo us on the re e ted wave using
(3.115) and (3.117).
3.8.4 Dis ussion of re e ted wave types We assume that the
P -velo ity
in the lower half-spa e is larger than that of
α2 > α1 . First, we onsider u(t1 ) = x/(Rα1 ) < α−1 2 . Sin e x/R = sin ϕ (ϕ= ∗ ∗ means that sin ϕ < α1 /α2 = sin ϕ (ϕ = riti al angle
the upper half-spa e ontaining the line sour e; re eivers
P (x, z)
for whi h
angle of in iden e), this
of in iden e), and this implies
pre- riti al in iden e of the ylindri al wave.
Fig. 3.49: Sket h for line sour e and its mirror point.
In this ase,
−1 u(t) for t < t1 is smaller than α−1 2 and, thus, even smaller than α1 .
a1 (u(t)) and a2 (u(t)), a
ording to (3.104), are positive imaginary, Rpp (u(t)), a
ording to (3.105), is real. Sin e du/dt is real for the times
Therefore, and
onsidered, it follows with (3.115) that the real part of the square bra kets for all
t < t1
is zero. For
t > t1 , u(t)
be omes omplex, and the real part is non-zero.
Not surprisingly, the displa ement, therefore, starts at If
α1 < α2 ,
t = t1 .
this argument holds for arbitrary re eiver lo ations in the upper
half-spa e. If
u(t1 ) = x/(Rα1 ) > α−1 2 ,
the
head wave
angle of in iden e ϕ is larger than the angle ϕ∗ ,
as the rst onset. In this ase, a2 (u(t)) be omes u(t2 ) = α−1 2 . The same does not hold for a1 (u(t)). Thus, Rpp (u(t)) has non-zero real and imaginary parts for t > t2 , and, −1 in (3.108), it therefore, (3.115) is already non-zero for t > t2 . Putting u = α2 follows that and we expe t a
real at the time t2 whi h is de ned via
t2 =
x −2 21 + (h − z)(α−2 1 − α2 ) < t1 . α2
3.8.
111
EXACT OR GENERALISED RAY THEORY - GRT
This is the arrival time of the head wave as expe ted a
ording to Fermat's prin iple for the ray path from Q to P(x,z) in Fig. 3.50.
Fig. 3.50: Path of head wave from sour e Q to re eiver P.
Considering this ase at time
t = t1 , we expe t,
due to the sudden hange in the
u(t) propagates, signi ant hanges in the displa ement (3.115), re e tion proper gives a signi ant signal, and this is something
urve on whi h i.e., that the
whi h indeed an be observed. Our derivation has shown that the head wave, and also the re e tion, an be derived from the potential
Φ1r , i.e., no separate des ription was ne essary for the
head wave. If we had studied solid media, we, possibly, ould have identi ed
P
a se ond arrival whi h is an additional interfa e or boundary wave (
to
S
onversion) ( ompare, e.g., the work by Garvin quoted earlier). In se tion 3.7 (see head wave in Fig.
3.37), we en ountered a similar situation in that the
head wave was in luded in the solution. Both methods (re e tivity method in se tion 3.7 and GRT this se tion) give a omplete solution unless simpli ations for numeri al reasons are introdu ed.
Fig. 3.51: Sket h of the displa ement (in horizontal and verti al dire tion) for pre- riti al (left) and post- riti al (right) in iden e, respe tively.
112
For
CHAPTER 3.
t → ∞,
BODY WAVES
the limit of the displa ement is non-zero, as for the ase of the in-
ident wave ( ompare (3.96)). The singularities at
t = t1
are always integrable.
Therefore, onvolution with a realisti ex itation fun tion
F(t)
is always possi-
ble. On the right side of Fig. 3.51, the displa ement starts before the re e tion proper arriving at
t1 .
Fermat's prin iple, therefore, does not give exa tly the
arrival time of the rst onset in the ase where the re e tion is not
arrival.
the rst
3.9.
113
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
Fig. 3.52: Theoreti al seismograms (verti al displa ement) for a rustal model. α1 = 6, 4km/s, α2 = 8, 2 km/s, ρ1 = 3, 0 g/ m3 , ρ2 = 3.3 g/ m3 ,
Parameters: h
= 30
km, z
= −30
km.
On the left, the exa t seismograms (GRT) and
on the right a wavefront approximation is shown.
From G. Müller:
Exa t
ray theory and its appli ation to the re e tion of elasti waves from verti ally inhomogeneous media, Geophys. Journ. R.A.S. 21, S. 261-283, 1970.
For realisti
F(t),
this dis repan y is usually small. A full example using the
theory of this se tion, is given in Fig. 3.50b. For a des ription of the dierent wave types
et ., see also se tion 3.7.2.
In the ase of a layered medium with
more than one interfa e, the wave eld an
be broken into separate ray ontributions, as was done for the lamella in se tion 3.6.5. For ea h ray ontribution, a formula of the type of (3.115) or (3.116) an be given, whi h an ontain head or boundary wave ontributions. This is the reason for the name "exa t or generalised ray theory". Another ommon name is the "Cagniard-de Hoop-method".
Exer ise 3.12: Give wavefront approximations for the re e ted wave and head wave, i.e., expand (3.115) around the arrival times wave), respe tively.
t1
(of the re e tion) and
Distinguish between
slowly
(of the head
t = t1 and t = t2 , respe tively, and t − t1 , t1 − t and t − t2 , respe tively.
an be repla ed by their values for varying terms, whi h depend on
t2
varying ontributions, whi h
rapidly
3.9 Ray seismi s in ontinuous inhomogeneous media With the re e tivity method and the GRT, we have dis ussed wave-seismi methods, whi h if applied in
ontinuous inhomogeneous
media (in our ase ver-
ti ally inhomogeneous media), require a segmentation in homogeneous regions (in our ase homogeneous layers).
Wave-seismi methods for ontinuous in-
homogeneous media, without this simpli ed representation of real media, are often more ompli ated ( ompare example in se tion 3.10). In this se tion, we will now sket h the
ray-seismi (or ray-opti al) approximation
of the wave the-
ory in inhomogeneous media. We will also show that it is the
approximation
high frequen y
of the equation of motion (2.20) for the inhomogeneous elasti
ontinuum. We restri t our dis ussion again to a simpli ed ase, namely the propagation of
SH -waves
in a two-dimensional inhomogeneous medium.
velo ity
β
and shear modulus
µ
displa ement omponent is in the
depend
The
y -dire tion where density ρ, S only on x and z. The only non-zero
sour e is assumed to be a line-sour e in the
y -dire tion v = v(x, z, t).
114
CHAPTER 3.
BODY WAVES
3.9.1 Fermat's prin iple and the ray equation Fermat's prin iple states that the travel time of the (SH -)wave from the sour e Q to an arbitrary re eiver P along the seismi ray is an extremum and, therefore, stationary, i.e., along ea h in nitesimally adja ent path between P and Q (dashed in Fig. 3.53) the travel time is either larger or smaller.
Fig. 3.53: Ray with extremum path and in nitesimally adja ent ray.
In most ases, the travel time along a seismi ray is a minimum, but there are also ases, where it is a maximum (e.g., the body waves PP, SS, PKKP). If we des ribe an arbitrary path from
{x = x(p), z = z(p)},
ds = We onsider now
many
p = p1
p = p2
at
Q
and
P
to
Q
via a
the element of the ar length
"
dx dp
2
+
Q P, respe tively.
su h paths from at
dz dp
2 # 21
to
s
parameter representation
an be written as
dp.
(3.118)
P. They all have the same value p annot be identi al to
Therefore,
s ; p ould, for example, be the angle between the line onne ting the oordinate x- or z-axis. The seismi ray is the
entre to the point along the way and the path for whi h
T =
Z
p2
β
−1
(x(p), z(p))
p1
"
dx dp
2
+
dz dp
2 # 12
dp =
Z
p2
F
p1
dx dz x, z, , dp dp dp
dz dx = x′ , = z′ dp dp is an extremum.
The determination of the seismi ray has, therefore, been
redu ed to a problem of
equations
al ulus of variations.
∂F d ∂F =0 − ∂x dp ∂x′
and
This leads to the
∂F d ∂F = 0. − ∂z dp ∂z ′
Euler-Lagrange
3.9.
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
115
This gives then, for example,
1 ∂ x +z 2 ∂x ′2
Division by
′2
(x′2 + z ′2 )1/2 ,
square bra ket with
1 β(x, z)
d 1 x′ = 0. − dp β(x, z) (x′2 + z ′2 ) 12
multipli ation of nominator and denominator of the
dp, and use of (3.118) gives d ds
1 dx β ds
∂ = ∂x
1 . β
(3.119)
d ds
1 dz β ds
∂ = ∂z
1 . β
(3.120)
Similarly,
Equations (3.119) and (3.120) are the the parameter representation of the ray. With
dierential equations of the seismi ray in
{x = x(s), z = z(s)} where s
is now the ar length
dz dx = sin ϕ, = cos ϕ ds ds (ϕ= angle of the ray versus the
z -dire tion), it follows that
sin ϕ β cos ϕ d ds β d ds
=
∂ ∂x
=
∂ ∂z
1 β . 1
(3.121)
β
x
1 0
Q
ϕ z
dx ds dz
11 00 00 11 00 11
P
11 00
Fig. 3.54: Ray in x-z oordinate system.
These two equations an now be onverted into another form of the ray equation (show)
116
CHAPTER 3.
1 ∂ϕ = ds β This dierential equation for the ray path. The inverse of
sin ϕ
ϕ(s) is dϕ/ds
∂β ∂β − cos ϕ ∂z ∂x
BODY WAVES
.
(3.122)
well suited for numeri al omputations of
−1 ∂β ∂β ds = β sin ϕ − cos ϕ r= dϕ ∂z ∂x is the
radius of the urvature of the ray.
(3.123)
The ray is urved strongly (r is smaller)
where the velo ities hange strongly (∇β large).
Spe ial ases a)
β=
onst.
From (3.122), it follows that
b)
β = β(z)
dϕ/ds = 0.
(no dependen e on
The ray is straight.
x)
The rst equation in (3.121) gives, after integration,
sin ϕ = q = const β
Snell's law)
along the whole ray (
sour es and re eivers at the level apex
S. The ray parameter q
velo ities
β(0)
where
z=0
q
is the
(3.124)
ray parameter
is onne ted to the take-o angle
at the sour e and the turning point
q=
ϕ
zs
z Fig. 3.55: Ray in 1-D medium.
β(zs ),
00000 11111 P 00 11 00 11 11 00 S 1 0
ϕ0 ,
the seismi
respe tively, via
sin ϕ0 1 = . β(0) β(zs )
11111∆ Q 00000 z=0 0 1
of the ray. For
the ray is symmetri with respe t to its
x
3.9.
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
zs
A turning point depth distan e
∆
117
β(z) < β(zs ) for all z < zs . For z = 0, it holds (using (3.124))
is only possible if
in whi h the ray reappears at level
that
∆(q) = 2
Z
S
dx = 2
Z
zs
Z
tan ϕdz = 2q
0
Q
zs 0
β −2 (z) − q 2
21
dz.
(3.125)
The ray's travel time is
T (q) = 2
Z
S
Q
ds =2 β
Z
0
zs
dz =2 β cos ϕ
Z
0
zs
− 1 β −2 (z) β −2 (z) − q 2 2 dz.
Equations (3.126) and (3.127) are parameter representations of the
urve of the model.
(3.126)
travel time
An example for ray paths and travel time urves in a model
with a transition zone is given in Fig. 3.56.
Fig. 3.56: Ray paths and travel time urves in a model with a transition zone.
The
slope
of the travel time urve is (show)
dT = q. d∆
)
β = a + bx + cz
(linear dependen e from
In this ase, it follows from
(3.127)
x
and
z)
118
CHAPTER 3.
BODY WAVES
c sin ϕ − b cos ϕ dϕ = ds β and by dierentiation with respe t to
s (ϕ and β are fun tions of s
x
via
and
z)
c cos ϕ + b sin ϕ dϕ c sin ϕ − b cos ϕ d2 ϕ = (b sin ϕ + c cos ϕ) = 0. − ds2 β ds β2 This implies
dϕ/ds
is a onstant and, therefore, the urvature radius along the
The ray is, therefore, a ir le, or a se tion of it.
whole ray.
from (3.123) if
β
Its radius
r
ϕ0 .
to the take-o angle
follows
ϕ
equal
M, the entre of the ir le, an be found from
as
is hosen identi al to the value at the sour e Q and
Q
shown in Fig. 3.57.
Fig. 3.57: Ray paths in a model with a linear velo ity law in x and z. The travel time from
T =
Z
P
Q
with An
ds = β
Z
Q ϕ1
ϕ0
to
P
is
dϕ = c2 + b c sin ϕ − b cos ϕ
δ = arctan(b/c).
arbitrary
velo ity law
β(x, z),
1 2 −2
i h tan (ϕ12−δ) h i ln tan (ϕ0 −δ) 2
given at dis rete points
(xn , zn )
in a model,
an linearly be interpolated pie e-wise between three neighbouring points. Then the laws derived here an be applied. The ray then onsists of several se tions of
ir les, and at the transition between two regions with dierent linear velo ity laws, the tangent to the ray is ontinuous.
A orresponding travel time and
plotting program is, for many pra ti al appli ations, already su ient.
Exer ise 3.13 a) Show that point M is on the line
β = 0.
b) Derive the formula for T given above.
3.9.
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
119
3.9.2 High frequen y approximation of the equation of motion The equation of motion for inhomogeneous, isotropi media (2.20) an for
SH -
wave propagation in two-dimensional media without volume for es, be simpli ed to
∂ 2v ∂ = ∂t2 ∂x
ρ
∂v ∂ ∂v µ + µ . ∂x ∂z ∂z
(3.128)
For the time harmoni ase we use the ansatz
v(x, z, t) = A(x, z) exp [iω (t − T (x, z))] . This is an ansatz for expe t that amplitude
(3.129)
high frequen ies sin e only for su h frequen ies an we A(x,z) and travel time fun tion T(x,z) to be frequen y
independent in inhomogeneous media. In homogeneous media far from interfa es, this is true for all frequen ies as long as one is a few wavelengths away from the sour e. Using (3.129) in (3.128), and sorting with respe t to powers of
ω,
it follows that
2
( "
ω A µ
+iωA
∂T ∂x
2
+
∂T ∂z
2 #
−ρ
)
2 ∂µ ∂T ∂ T ∂2T ∂µ ∂T ∂ ln A ∂T ∂ ln A ∂T + + +µ + + 2µ ∂x ∂x ∂z ∂z ∂x2 ∂z 2 ∂x ∂x ∂z ∂z 2 ∂µ ∂A ∂µ ∂A ∂ A ∂2A (3.130) − = 0. + + +µ ∂x ∂x ∂z ∂z ∂x2 ∂z 2
For su iently high frequen ies, the three terms of this equation are of
dierent
magnitudes. To satisfy (3.130), ea h term has then to be zero independently,
espe ially the rst two terms
2 2 ∂T 1 ∂T + = ∂x ∂z β2 ∂ ln A ∂T ∂µ ∂T ∂ ln A ∂T ∂µ ∂T 2µ = − + − − µ∇2 T. ∂x ∂x ∂z ∂z ∂x ∂x ∂z ∂z
(3.131)
(3.132)
Eikonal equation, and it ontains on the right side the S -velo ity β = (µ/ρ)1/2 . After solving the Eikonal, T is transport equation (3.132), and ln A and A are determined.
Equation (3.131) is the lo ation-dependent inserted in in the
This, in prin iple, solves the problem. The frequen y-independent third term
120
CHAPTER 3.
in (3.130) will usually not be zero with
A
from (3.132).
BODY WAVES
Solution (3.129) is,
therefore, not exa t, but be omes more a
urate, the higher the frequen y. We still have to derive the onditions under whi h the rst two terms of (3.130) are indeed of dierent order and, therefore, the separation into (3.131) and (3.132) is valid.
ea h single ea h single sum-
The ondition follows from the requirement that
summand in the se ond term has to be small with respe t to mand in the rst term, for example,
2 ∂µ ∂T ω ≪ ω 2 µ ∂T . ∂x ∂x ∂x
From (3.131), it follows roughly
With
µ = ρβ 2 ,
it follows
|∂T /∂x| = 1/β .
Thus,
β ∂µ µ ∂x ≪ ω. β ∂ρ ∂β ≪ ω. + 2 ρ ∂x ∂x
(3.133)
Similar relations follow from the other summands in (3.130). Usually, the required onditions are formulated as follows: the high frequen y approximations (3.131) and (3.132) are valid for frequen ies whi h are large with respe t to the
velo ity gradients
ω ≫ |∇β| =
"
∂β ∂x
2
+
∂β ∂z
2 # 21
.
(3.134)
Equation (3.133) shows also, that density gradients have also an in uen e. Equation (3.134) an be expressed even more physi ally: the relative hange of the velo ity over the distan e of a wavelength has to be smaller then
2π
(show).
Example We solve (3.131) and (3.132) in the simplest ase of a plane in z-dire tion with the assumption that
ρ, β
and
µ
SH -wave propagating z. Ansatz
depend only on
(3.129) then simpli es to
v(z, t) = A(z) exp [iω (t − T (z))] . The solution of (3.131) is
(3.135)
3.9.
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
1 dT = , T (z) = dz β(z) where
T (z)
is the
Z
z
0
121
dζ β(ζ)
S-wave travel time, with respe t to the referen e level z = 0.
Equation (3.132) an be written as
1 dµ µ dβ 2µ d ln A =− + 2 . β dz β dz β dz Thus,
−1 d ln (ρβ) 2 d ln β d ln µ = = − dz dz dz 12 ρ(0)β(0) . = A(0) ρ(z)β(z)
d ln A dz
1 2
A(z) The amplitudes of the impedan e
ρβ .
SH -wave
vary, therefore, inversely proportional to the
The nal solution of (3.135) is
ρ(0)β(0) v(z, t) = A(0) ρ(z)β(z)
21
Z exp iω t −
0
z
dζ β(ζ)
.
From these results we on lude that, in the ase onsidered, an frequent
SH -wave propagates without hanging its form.
(3.136)
impulsive, high
Exer ise 3.14 Vary the velo ity
z,
but via a
exa tly via
β
and the density
ρ
not
step somewhere in between. the SH -refra tion oe ient
ontinuously
from depth 0 to depth
Then the amplitudes an be derived (3.40).
Show that (3.136) gives the
same results if the relative hange in impedan e is small with respe t to 1. Hint: Expansion in
both
ases.
3.9.3 Eikonal equation and seismi rays From (3.129), it follows that surfa es of onstant phase are given by
t − T (x, z) = const. In the impulse ase, these surfa es are the wavefronts separating perturbed and unperturbed regions. This is why the term wavefront is used also here. Complete dierentiation with respe t to
t
gives
122
CHAPTER 3.
BODY WAVES
− → dx ∂T dx ∂T dz + = ∇T · = 1, ∂x dt ∂z dt dt where
− → dx/dt = (dx/dt, 0, dz/dt)
(3.137)
is the propagation velo ity of the wavefront.
− → dx/dt and the ve tor ∇ T , whi h is perpendi ular to the wavefront, are parallel, sin e a
ording → − to the Eikonal equation (3.131) |∇ T | = 1/β , it holds that dx/dt = β . This The obvious interpretation of (3.137) for isotropi media is that
means that the wavefronts propagate perpendi ular to themselves with the lo al velo ity
The
β.
orthogonal traje tories of the wave are de ned as seismi rays.
We still have
to show that they are the rays de ned via the Fermat's prin iple. We demonstrate this by showing that the dierential equations of the seismi ray, (3.119) and (3.120), also follow from the Eikonal equation. As before, we des ribe the ray via its parameter representation
s.
Ve tor
− → dx/ds = (dx/ds, 0, dz/ds)
{x = x(s), z = z(s)}
with the ar length
is a unit ve tor in ray dire tion for whi h,
using the statements above, we an write
− → dx = β∇ T = β(∂T /∂x, 0, ∂T /∂z). ds
(3.138)
Instead of (3.119), we, therefore, have
d ds
1 dx β ds
=
d ds
∂T ∂x
.
Using (3.138) and the Eikonal equation on the right side, we derive
d ds
1 dx β ds
= = =
2 ∂ 2 T dx ∂ 2 T dz ∂ 2 T ∂T ∂ T ∂T + = β + ∂x2 ds ∂x∂z ds ∂x2 ∂x ∂x∂z ∂z " 2 2 # β ∂ ∂T 1 ∂T β ∂ = + 2 ∂x ∂x ∂z 2 ∂x β 2 β (−2) ∂β 1 1 ∂β ∂ . =− 2 = 2 β 3 ∂x β ∂x ∂x β
That is identi al to (3.119) and a similar derivation holds for (3.120). following is the
ray equation in ve tor form,
The
whi h is also valid in the three-
dimensional ase
d ds
− →! 1 1 dx =∇ . β ds β
(3.139)
3.9.
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
This shows that
mi s.
123
ray seismi s is a high frequen y approximation of wave seiskinemati aspe ts of wave
We have, until now, limited the dis ussion on
propagation, i.e., on the dis ussion of wave paths, travel times and phases.
Dynami
parameters, espe ially amplitudes, were not dis ussed ex ept in the
simple example in se tion 3.9.2. The following se tion gives more details on this aspe t. Before doing this, we give the form of (3.139) whi h is often used in numeri al
al ulations, espe ially in three dimensions. equation of 2nd order for 1st order for
− → x
and the
the absolute value
1 β)
− → x
The
single
ordinary dierential
(3.139) is repla ed by a system of
→ p = slowness ve tor −
− →
two
equations of
1 dx β ds (ve tor in ray dire tion with
− → − → dx dp 1 → = β− p, =∇ . ds ds β Ee tive numeri al methods for the solution of systems of ordinary dierential equations of 1st order exist, e.g., the Runge-Kutta-method.
3.9.4 Amplitudes in ray seismi approximation Within the framework of ray seismi s developed from Fermat's prin iple, amplitudes are usually omputed using the assumption that the energy radiated into a small ray bundle, remains in that bundle. This assumption implies that no energy exits the bundle sideways via dira tion or s attering and no energy is re e ted or s attered ba kwards. This is only valid for high frequen ies. In the following, we derive a formula whi h des ribes the hange of the displa ement amplitude along a ray radiated from a line sour e in a two dimensional inhomogeneous medium. The medium shall have no dis ontinuities. We onsider (see Fig. 3.58) a ray bundle emanating from a line sour e a width of the point
dl(M ) at the
P.
referen e point
M
lose to
1 0 Q 0 1 dl(M)
11 00 00 11 M
Q
and a width of
x z
dl(P)
P 0 1
Fig. 3.58: Ray bundle emanating from the sour e Q.
Q
with
dl(P ) near
124
CHAPTER 3.
Equation (3.129) holds for the displa ement in
M
and
BODY WAVES
P, or in real form
v = A sin [ω(t − T ] . Our aim is the determination of the amplitude ration
A(P )/A(M ).
We rst de-
termine the energy density of the wave, i.e., the sum of kineti and potential en1 1 2 2 2 2 ergy per unit volume. The kineti energy density is ρv 2 ˙ = 2 ρω A cos [ω(t − T )]. Averaged over the period 2π/ω , the potential and kinemati energy density have 1 1 2 2 2 the same value 4 ρω A sin e the average of cos x is identi al to 2 . Then the energy density averaged over a period an be written as
∆E 1 = ρω 2 A2 . ∆V 2 Consider a ube with the volume
∆V = dl dy ds; its ross se tion dl dy is perds is exa tly 1 wavelength β2π/ω .
pendi ular to the ray bundle and its length The energy
∆E =
1 2 2 ρω A ∆V = πωρβA2 dl dy 2
ontained within this ube ows per period through the ross se tion
∆E
the ray bundle. Sin e no energy leaves the bundle,
M. From this the amplitude ratio
at
P
dl dy
of
is the same as at
follows as
1 A(P ) ρ(M )β(M )dl(M ) 2 . = A(M ) ρ(P )β(P )dl(P ) As in (3.136), impedan e hanges o
ur along the ray.
(3.140)
The square root of
the hange in the ross se tion is the important parameter for the amplitude variation. In the most general three-dimensional ase, the ross se tion
surfa e
of the
three-dimensional
dl
has to be repla ed by
ray bundle.
Equation (3.140) an be approximated by tra ing su iently many rays through the medium using the methods dis ussed previously, and then determining their perpendi ular distan es (or ross-se tion surfa es in three dimensions). These methods are based mainly on the solution of the ray equation (3.139), whi h is also alled the equation of the
kinemati ray tra ing.
A more stringent approa h
to al ulate (3.140) is based on dierential equations whi h are dire tly valid for the ross se tion of a ray bundle; they are alled equations of the
ray tra ing.
dynami
Their derivation annot be treated here; for details, see the book
of ervený, Molotov and P²en ík (1977). For point
P
on a horizontal pro le, e.g., at
z = 0,
a loser look at (3.140) and
Fig. 3.59 helps in understanding the physi al meaning.
3.9.
RAY SEISMICS IN CONTINUOUS INHOMOGENEOUS MEDIA
d∆
∆
P 0 1 0 ϕ1
11 00 Q 00 11
a
1M 0
ϕ0
125
x,∆ d∆ dl(P)=cos ϕ d ϕ d ϕ 0 0
dl(M)=ad ϕ 0
z Fig. 3.59: Ray paths in an inhomogeneous medium.
The horizontal distan e of
Q
to
dl(P )
P.
P
is
∆(ϕ0 ) with the take-o angle ϕ0
The distan e of the referen e point
M
from
Q
is
a.
of the ray from
With
dl(M )
and
from Fig. 3.59, it follows from (3.140) that
21 1 A(P ) a ρ(M )β(M ) 2 . · = d∆ A(M ) ρ(P )β(P ) cos ϕ dϕ 0
(3.141)
P is on, or lose to, the neighturning points of the travel time urve on the horizontal pro le
This expression shows that problems o
ur if bourhood of
( ompare, e.g., the travel time urve in Fig. 3.56). At these points,
d∆/dϕ0
hanges its sign, and that an happen either with a ontinuous or non- ontinuous pass through a zero. In the rst ase, in nite amplitudes o
ur in ond ase, the amplitudes be ome non- ontinuous. and nonphysi al.
P ; in the se -
Both ases are unrealisti
Equations (3.141) and (3.140), respe tively, an, therefore,
only be used at some distan e from the turning points ( austi s) of the travel time urves. Unfortunately, this means that the points with some of the largest amplitudes annot be treated properly under these assumptions; more sophisti ated methods (like the WKBJ method, to be dis ussed later, or the Gaussian Beam method) must be employed. Despite this disadvantage, the formulae given above (and their orresponding equations in three dimensions) are very useful in seismologi al appli ations. They an be easily extended to in lude refra tions and re e tions at dis ontinuities.
This requires the determination of hanges in the ross se tion of
the ray bundle at dis ontinuities, and the in lusion of re e tion and refra tion
oe ients. A further problem of the energy ansatz used in this se tion is that it only gives the amplitudes of a seismi ray but no information on its phase hanges that o
ur in
addition
always su ient to add
to the phase hanges in the travel time term. It is not
exp [iω(t − T )] to (3.140) and (3.141), respe tively, e.g.,
on retrograde travel time bran hes in the ase that the velo ities are only a
126
CHAPTER 3.
fun tion of
z.
BODY WAVES
The WKBJ method and the Gaussian Beam method solve this
problem by tra king an additional parameter, the KMAH index named after Keller, Maslov, Arnold and Hormander, whi h ounts the austi s en ountered along the ray. In the following se tion, the WKBJ theory for verti al inhomogeneous media, whi h avoids some of the ray theory problems dis ussed, is presented; it ontains more wave seismi elements.
Exer ise 3.15 Use the ray parameter
q instead of the take-o angle ϕ0 in the amplitude formula
(3.141) in the ase of a verti ally inhomogeneous medium and then use (3.127). What is the relation between the amplitudes and the travel time urve
T (∆)?
WKBJ method
3.10
Now we will onsider total re e tion at a verti ally inhomogeneous medium using the WKBJ method.
3.10.1 Harmoni ex itation and re e tion oe ient We onsider a medium whose velo ity for
z > 0
β(z)
in the medium. A
for
z≤0
is
β(0),
i.e., onstant and
ontinuous fun tion of z, i.e., no dis ontinuities exist plane SH -wave may propagate obliquely in the lower half-
an be any
z < 0 with the horizontal wavenumber k = ω sin ϕ/β(0) (ray parameter q = sin ϕ/β(0) (ϕ = angle of in iden e). Note the dieren e to the example in
spa e
hapter 3.9.2 with verti al propagation. Then the ray seismi s of the verti ally inhomogeneous medium, se tion 3.9.1, suggests inserting
∂T /∂x = q = constant
in the Eikonal equation (3.131) ( om-
pare with (3.127)). This then, gives
T (x, z) = qx +
Z
0
z
−2 1 β (ζ) − q 2 2 dζ,
travel time of the S-wavefront from the interse tion of the origin x,z ). The rest of the dis ussion is as in se tion 3.9.2., and leads to
and, thus, the to the point (
v0 (x, z, t) = s(ζ)
=
1 Z z µ(0)s(0) 2 A(0) exp iω t − qx − s(ζ)dζ (3.142) µ(z)s(z) 0 1 −2 β (ζ) − q 2 2 . (3.143)
3.10.
q
127
WKBJ METHOD
is the horizontal and
s
is identi al to (3.136).
S -wave.
is the verti al
slowness
Equation (3.142) is the
q= 0, (3.142) WKBJ -approximation of the
of the wave. For
It is a useful high frequen y approximation, as long as
β −1 (z) > q ,
i.e., as long as the seismi ray whi h an be asso iated with the wave
propagating horizontally.
is not
If the velo ity, e.g., with in reasing depth de reases or −1 , (3.142) is appli able . if it in reases, but does not rea h the value q
for all z
For ases of interest and a medium with in reasing velo ities for in reasing −1 depth, a depth zs is rea hed where β(zs ) = q . At this depth, where the ray propagates horizontally, (3.142) diverges. Equations (3.129), (3.131) and (3.132) are
insu ient
for the des ription of the wave eld near the turning point of
rays. If (3.142) is onsidered for
z > zs
with
β(z) > β(zs ),
i.e., the velo ity
ontinues to in rease, a stable result an, again, be obtained. The integral in the exponential term from
zs
to
z
is imaginary, thus, giving an exponential
z SH -wave de reases as expe ted.
de ay of the amplitudes with in reasing , i.e., below the ray's turning point the amplitude of the
For
z < zs ,
the wave eld is
insu iently des ribed by (3.142) sin e (3.142) represents only the downward
propagating in ident SH -wave. A similar equation an be given for the re e ted SH -wave upward propagating from the turning point
µ(0)s(0) v1 (x, z, t) = RA(0) µ(z)s(z)
21
Z exp iω t − qx +
z
s(ζ)dζ 0
.
(3.144)
That this wave propagates upwards an be seen from the positive sign before the integral in the exponent.
R is, as an be seen by the sele tion z=0
(3.144), the amplitude ratio wave; in other words, the
v1 (x, 0, t)/v0 (x, 0, t)
in (3.142) and
of the re e ted to the in ident
re e tion oe ient of the inhomogeneous half-spa e
is z>0.
Its determination requires a quantitative onne tion of the whole eld
v0 + v1 z > zs .
To tie these two solutions together is, as mentioned before, not possible
for
z < zs
with the already mentioned exponentially de aying eld for
with the high frequen y approximation of the equation of motion used until now. The required onne tion be omes possible with another high frequen y approximation of (3.128), namely a wave equation with depth dependent velo ity
∇2 V v
= =
∂2V ∂x2 1 1
2
+ ∂∂zV2 = V
1 ∂2V β 2 (z) ∂t2
µ 2 (z)
)
.
(3.145)
This high frequen y approximation is valid under ondition (3.134), as an be shown by inserting in (3.128). For plane waves, it follows from the ansatz
V (x, z, t) = B(z) exp [iω(t − qx)]
128
CHAPTER 3.
via (3.145) an ordinary dierential equation for
BODY WAVES
B(z)
B ′′ (z) + ω 2 β −2 (z) − q 2 B(z) = 0.
This equation has now to be solved for large ω . ω 2 f (z)B(z) = 0 in the neighbourhood of a zero of
alled For
(3.146)
The solutions of
B ′′ (z) +
f(z) and for large ω is generally WKBJ -solution after the authors - Wentzel, Kramers, Brillouin, Jereys.
z < zs ,
the previously dis ussed superposition of (3.142) and (3.144) of the
in ident and re e ted wave of
v
results. For
z
z > zs
the exponentially de aying
is in the immediate neighbourhood of zs 2 2 has to be examined in more detail. We approximate the oe ient ω s (z) 2 of B(z) (with from (3.143)) and get, with s (zs ) = 0, β(zs ) = q −1 and β ′ (zs ) > 0,
solution follows.
The ase that
s(z)
linearly
B ′′ (z) − 2ω 2 q 3 β ′ (zs )(z − zs )B(z) = 0.
(3.147)
This equation an, with the substitution,
1 y(z) = 2ω 2 q 2 β ′ (zs ) 3 (z − zs )
be transformed into the dierential equation of the
(3.148)
Airy fun tions
C ′′ (y) − yC(y) = 0. The solution of interest to us,
C(y) = Ai(y),
is dis ussed in appendix E (more
on Airy fun tions an be found in M. Abramovitz and I.A. Stegun: Handbook of Mathemati al Fun tions, H. Deuts h, Frankfurt, 1985). The depths
z < zs
Ai(y). From Fig. E.2, it follows that the transition from the os illatory solution B(z) of (3.147) with z < zs to the exponentially damped solution for z > zs is without singularity. This then, also holds for the displa ement v, in ontrast to what one would (z
> zs )
orrespond to arguments
y < 0 (y > 0)
of
expe t from (3.142) and (3.144). The os illatory behaviour of
v0
and the re e tion
v1 ,
B(z)
for
z < zs
indi ates that the in ident wave
build a standing wave with nodes of the displa ement
at depths whi h orrespond to the zeros of the Airy fun tion. The re e tion
oe ient R in (3.144) is now determined in su h a way, that the superposition of (3.142) and (3.144), in the term that depends on z, is identi al to the Airy fun tion. Due to the high frequen y assumption, the asymptoti form of Ai(y) for large negative y an be used
Ai(y) ≃ π Furthermore, for
z < zs
− 12
− 41
|y|
sin
π 2 23 |y| + 3 4
.
(3.149)
3.10.
129
WKBJ METHOD
v0 + v1
h i1 Rz µ(0)s(0) 2 = A(0) µ(z)s(z) exp iω t − qx − 0 s s dζ Rz Rz Rz · exp iω z s s dζ + R exp 2iω 0 s s dζ · exp −iω z s s dζ .
(3.150)
The
z -dependen e
v0 + v1
of
is given by the urved bra ket.
It will now 2 2
be determined in approximation. With the approximation (3.147) 2ω 2 q 3 β ′ (zs )(zs − ζ), it follows
ω
with
y = y(z)
Z
zs
s dζ z
1 = ± 2ω 2 q 3 β ′ (zs ) 2
Z
zs
z
ω s (ζ) =
1
(zs − ζ) 2 dζ
1 2 3 = ± 2ω 2 q 3 β ′ (zs ) 2 (z − zs ) 2 3 2 3 = ± |y| 2 3 = ±Y
from (3.148).
The positive (negative) sign holds for positive
(negative) frequen ies. Thus, for the urved bra kets in (3.150)
{· · ·} = e±iY + Ze∓iY with the abbreviation
Z Z = R exp 2iω
0
With
Z = ±i
for
ω
{· · ·} = =
zs
s dζ .
(3.151)
<0 (>0), it follows that
1 π (1 ± i)(cos Y + sin Y ) = 2 2 (1 ± i) sin Y + 4 1 π 2 23 |y| + 2 2 (1 ± i) sin , 3 4
and, therefore, the required agreement with the main term in (3.149). in (3.151) gives then the
Rz ω exp −2iω 0 s s(ζ)dζ = i |ω| 1 s(ζ) = β −2 (ζ) − q 2 2 β(0) β(zs ) = q −1 = sin ϕ. R
Z = ±i
re e tion oe ient in the WKBJ-approximation
(3.152)
130
CHAPTER 3.
total re e tion ).
Its absolute value is 1 (
i.e., a onstant phase shift of
±π/2
for
BODY WAVES
It des ribes only the
ω < 0 (ω > 0)
phase shifts,
is added to the phase
shifts due to the travel time in the exponential term of
R.
Compared to the
re e tion oe ients of layered media, derived earlier without approximation; the form of (3.152) is
simple.
It is only valid for su iently high frequen ies
( ondition (3.134)) and for angle of in iden e
ϕ with
total re e tion. Re e tion
oe ients of the type of (3.152) are useful in seismology but even more so for the propagation of sound waves in o eans or the propagation of radio waves in the ionosphere ( ompare, e.g., Budden (1961) and of Tolstoy and Clay (1966)). The
re e ted SH-wave
observed at the oordinate entre follows, then, by in-
serting (3.152) in (3.144)
v1 (0, 0, t) = A(0)i
Z zs ω s(ζ)dζ . exp iω t − 2 |ω| 0
(3.153)
Then
τ (q) = 2
Z
zs
s(ζ)dζ
0
is the
delay time,
, i.e., the time between the interse tion of the in ident and
the re e ted wave with the oordinate entre. This time delay orresponds to the ray segments
AC, BD, or OP.
Fig. 3.60: Constru tion of a austi from the envelopes of rays.
Note that the wavefronts are
z< 0 are the wavefronts plane.
Fig.
urved
3.60 shows also that the line
for
z>0;
z = zs
only in the homogeneous region
is the envelope of all rays.
Su h
envelopes are alled austi s, and they are hara terised by large energy on entrations. Within the amplitude approximation formula (3.140), and due to
3.10.
131
WKBJ METHOD
dl(P)= 0
z = zs , in nite amplitudes would result ± π2 , as dis ussed before, an be interpreted
for a point P on the austi
there. The additional phase shift of
physi ally as the ee t of the strong intera tion of ea h ray with its neighbouring rays in the vi inity of the austi . More ompli ated austi s o
ur in verti ally
inhomogeneous media, if the in ident wave is from a point or line sour e, reπ spe tively. In su h ases, the phase shift per austi en ountered, is ± . More 2
ompli ated austi s are en ountered in two and three dimensional media.
3.10.2 Impulsive ex itation and WKBJ-seismograms If an
impulsive wave,
produ ing a displa ement
v0 (0, 0, t) = F (t)
at the oor-
dinate entre, instead of a harmoni wave is in ident, it follows from (3.153)
F(t)
that the orresponding re e tion is the time delayed Hilbert transform of ( ompare se tion 3.6.3)
v1 (0, 0, t) = FH (t − τ (q)). This means that the re e tion for or horizontal slowness
q)
all
(3.154)
angles of in iden e
ϕ
(or ray parameters
have the same form ex ept for the time delay
τ (q).
This is, therefore, dierent from the results for a dis ontinuity of rst order in
hapter 3.6.3 (there the impulse form hanged in the ase of total re e tion also with the angle of in iden e When
ylindri al waves
ϕ).
are onsidered, the prin iple of superposition is used.
line sour e in the oordinate entre, is represented by many plane waves with radiation angles ϕ from 0 to π/2. This orresponds to positive values of q. The
The ylindri al wave, assumed to originate from an isotropi ally radiating
re e tions are superimposed similarly. First, (3.154) is generalised for arbitrary
x>0 v1 (x, 0, t) = FH (t − τ (q) − qx) = FH (t) ∗ δ (t − τ (q) − qx) . Then these plane waves are integrated over
WKBJ-seismogram
at distan e
v(x, 0, t) = FH (t) ∗ The
Z
0
π 2
x
ϕ
from 0 to
π/2
and, thus, the
from the line sour e is derived
δ (t − τ (q) − qx) dϕ = FH (t) ∗ I(x, t).
(3.155)
impulse seismogram I(x, t) an now be derived numeri ally via I(x, t) ti
Usually, the
P = i δ(t − ti )∆ϕi = τ (qi ) + qi x , qi =
sin ϕi β(0) .
ϕi are hosen equidistant (∆ϕi = ∆ϕ = const).
(3.156)
The delta fun tions
are shifted from the times ti to their immediate neighbouring time points
I(x, t)
132
CHAPTER 3.
BODY WAVES
and possibly amass there, i.e., in the dis retised version of I(x, t), multiples of ∆ϕ o
ur there. I(x, t) is then onvolved with FH (t). The most time- onsuming part is the omputation of the delay time τ (qi ); on the other hand, e ient rayseismi methods exist for that. In omparison to the re e tivity method and the GRT, the WKBJ-method is signi antly faster. There are also other numeri al realisations of this method than (3.155) and (3.156). WKBJ-seismograms have
other phase relations and impulse forms
pe ted from (3.152) and (3.154), respe tively.
than ex-
This is due to the summation
of many plane waves. Pro-grade travel time bran hes (see Fig. 3.56) show no phase shift, i.e., the impulse form of the in ident ylindri al wave is observed there. Phase shifts and impulse form hanges only o
ur on retro-grade travel time bran hes. Furthermore, the seismogram amplitudes are nite in the vi inity of the turning points of travel time urves, i.e., the WKBJ-method is valid at austi s. WKBJ-seismograms for a simple rust-mantle model and a line sour e at the Earth's surfa e are shown in Fig. 3.61. The omputations were performed with a program for
SH -waves; even so, the velo ity model (Fig.
3.62) is valid for P-
waves. An a ousti P-wave omputation would give, in prin iple, the same result for
pressure.
The travel time urve of the re e tion
PM P
from the rust-mantle
boundary (Moho) is retrograde and the travel time bran h of the refra ted wave
Pn
from the upper mantle is prograde. Times are redu ed with 8 km/s.
Fig. 3.61: WKBJ-seismograms for a simple rust-mantle model and a line sour e at the Earth's surfa e.
3.10.
133
WKBJ METHOD
Fig. 3.62: Velo ity model used in Fig. 3.61.
The impulse forms of these waves are as expe ted: radiated wave, whereas
PM P
Pn
has the form of the
is roughly the Hilbert transform of it. At distan es
smaller than the riti al distan e ( a. 100 km), the amplitudes in rease strongly. At the riti al point, whi h is lo ated on a austi within the rust, the wave eld remains nite. In seismologi al appli ations of WKBJ-seismograms, their approximate nature, due to the high frequen y approximation (3.152) for the re e tion oe ient, should be kept in mind.
This approximation is insu ient in regions of the
Earth where wave velo ity and density hange rapidly with depth, e.g., at the
ore-mantle boundary or at the boundary of the inner ore of the Earth.
134
CHAPTER 3.
BODY WAVES
Chapter 4
Surfa e waves 4.1 Free surfa e waves in layered media 4.1.1 Basi equations In addition to the body waves that penetrate to all depths in the Earth, another type of wave exists whi h is mostly limited to the neighbourhood of the surfa e of the Earth alled
surfa e waves.
These waves propagate along the surfa e of the
Earth, and their amplitudes are only signi ant down to the depth of a few wave lengths. Below that depth, the
displa ement is negligible.
Be ause surfa e waves
are onstrained to propagate lose to the Earth's surfa e, their amplitude de ay as a fun tion of sour e distan e is smaller than for body waves, whi h propagate in three dimensions.
This is why surfa e waves are usually the dominating
signals in the earthquake re ord. Another signi ant property is their
dispersion,
i.e., their propagation velo ity is frequen y dependent. Therefore, the frequen y within a wave group varies as a fun tion of time ( ompare example in 4.1.4).
These are some of the main observations and explanations for surfa e waves. The rst s ientists to study surfa e waves (Rayleigh, Lamb, Love, Stoneley et al.) found the theoreti al des riptions explaining the main observations. The fundamental tenet of this approa h is the des ription of surfa e waves as an
eigenvalue problem but omitting the sour e of the elasti waves.
We onsider a
layered half-spa e with parameters as given in Fig. 4.1 with a free surfa e. 135
136
CHAPTER 4.
SURFACE WAVES
Fig. 4.1: Layered half-spa e with free surfa e.
We work with Cartesian oordinates
x,y,z
and assume
independen e from y.
Then, the equations from se tion 3.6.2 an be used whi h have been derived to des ribe re e tion and refra tion. The separation of the displa ement into
P-, SV- and SH - ontributions holds as before, as does the fa t, that the P-SVSH-waves. Surfa e waves of the P-SV-type are alled Rayleigh waves. They are polarised in the x-z-plane
ontributions propagate independently from the
The potentials
Φ
horizontal displa ement
u
=
∂Φ ∂x
−
∂Ψ ∂z
verti al displa ement
w
=
∂Φ ∂z
+
∂Ψ ∂x .
and
Ψ,
in ea h layer, satisfy the wave equation
∇2 Φ = Surfa e waves of the
(4.1)
1 ∂2Φ , α2 ∂t2
∇2 Ψ =
1 ∂ 2Ψ . β 2 ∂t2
(4.2)
SH -type are alled Love waves. They are polarised in y v in ea h layer the following wave equation
dire tion, and for the displa ement holds
∇2 v =
1 ∂2v . β 2 ∂t2
The boundary onditions are given in se tion 3.6.2. The
(4.3)
ansatz for Φj , Ψj
in the j-th layer of the model for harmoni ex itation (ω
Φj Ψj
=
Aj (z) Bj (z)
> 0)
and
x i Aj (z) = exp [i (ωt − kx)] exp iω t − Bj (z) c h
vj
is
(4.4)
4.1.
137
FREE SURFACE WAVES IN LAYERED MEDIA
and
vj
h x i = Cj (z) exp [i (ωt − kx)] . = Cj (z) exp iω t − c
(4.5)
Consider the onditions for whi h a plane wave exists, whi h propagates in xdire tion with the is
?
phase velo ity , where
Then the fun tions
Aj (z), Bj (z)
and
is identi al in all layers. How large
Cj (z)
exist, so that
An (z) Bn (z) lim = 0. z→∞ Cn (z)
This problem is an
(4.6)
eigenvalue problem, and and the wavenumber k = ω/c, eigenvalues. In many ases (for xed ω ), a
respe tively, are the orresponding nite number
(≤ 1)
of eigenvalues exist. The problem is omparable to that of
natural os illations (or free os illations) of nite et .) ( ompare exer ise 4.1).
determining the frequen ies of bodies (beams, plates, bodies
Here we are only interested in the ase where the eigenvalues are real (>0). This has the largest pra ti al appli ation. The orresponding surfa e waves are alled
normal modes.
There exist also waves whi h an be des ribed with omplex
k = k1 − ik2 (k1,2 > 0).
These surfa e waves are alled
their amplitude de reases exponentially with is
exp(−k2 x).
leaking modes
k:
sin e
Their phase velo ity
ω/k1 .
The ansatz with plane waves negle ts the in uen e of ex itation. This, then, leads to a major simpli ation of the problem. Su h surfa e waves are alled in ontrast to
for ed
free
surfa e waves whi h are ex ited by spe i sour es. The
analogy to the free and for ed resonan es of limited bodies is also helpful here in the ontext of ex itation. The treatment of free surfa e waves is an important requirement for the study of for ed surfa e waves ( ompare se tion 4.2). We will soon show that the dispersive properties of both wave types are identi al. Sin e this property depends on the medium, they an be used to determine medium parameters. This is why the study of the dispersion of free surfa e waves is of great pra ti al importan e.
Exer ise 4.1 P
The radial os illations of a liquid sphere with -velo ity α are des ribed by Φn (r, t) ∼ (eiωn t/r ) sin(ωn r/α) (n=1,2,3...). Determine the eigen
the potential frequen ies
ωn
from the ondition that the surfa e of the sphere at
r=R is stress
free (prr from exer ise 3.4); give the radial displa ement. Where are the nodal planes?
138
CHAPTER 4.
SURFACE WAVES
4.1.2 Rayleigh waves at the surfa e of an homogeneous half-spa e The half-spa e (z>0) has the velo ities
α and β
for
P - and S -waves, respe tively.
Inserting the ansatz
Φ = A(z) exp [i (ωt − kx)] into the wave equation (4.2) with equations for
A(z) and B(z), e.g.,
and
Ψ = B(z) exp [i (ωt − kx)]
∇2 = ∂ 2 /∂x2 + ∂ 2 /∂z 2 ,
A′′ (z) + k 2
(4.7)
gives the dierential
c2 − 1 A(z) = 0. α2
The general solution of this equation is
A(z) = A1 e Due to (4.6),
δ
−ikδz
+ A2 e
ikδz
with
δ=
has to be purely imaginary. Then,
δ , a rst statement on the phase possible: c < α. It also holds that
the properties of be omes
c2 −1 α2
A2 = 0
21
.
has to hold. From
velo ity of the Rayleigh wave
A(z) = A1 e−ikδz , and, similarly, it follows that
B(z) = B1 e−ikγz with
γ = c2 /β 2 − 1
1/2
(negative imaginary). This further limits
: c < β.
The potential ansatz (4.7) an now be written as
Φ = A1 exp [i (ωt − kx − kδz)] , Inserting the
Ψ = B1 exp [i (ωt − kx − kγz)] .
boundary onditions pzz = pzx = 0 for z= 0 with pzz
(3.29), it follows that
−
ω2 λ = A1 + 2 −k 2 δ 2 A1 − k 2 γB1 α2 µ −2k 2 δA1 + −k 2 + k 2 γ 2 B1 =
0 0.
and
pzx
(4.8)
from
4.1.
139
FREE SURFACE WAVES IN LAYERED MEDIA
Division by
−k 2
λ/µ = (α2 − 2β 2 )/β 2
and use of
c2 α2 − 2β 2 +2 α2 β2
c2 −1 α2
gives
A1 + 2γB1 c2 2δA1 + 2 − 2 B1 β
= 0 = 0.
This leads to
− 2 A1 + 2γB1 2 −2δA1 + βc 2 − 2 B1 c2 β2
0
= =
0.
This system of equations only has non-trivial solutions minant is zero.
This leads to an equation for :
In the range of interest
c2 −2 β2
0 < c < β,
c2 −2 β2
2
2
A1
(4.9)
and
B1 ,
if its deter-
+ 4δγ = 0.
we have
1 1 c2 2 c2 2 1− 2 . =4 1− 2 β α
Squaring this gives
c4 β 2 c2 β2 c2 c6 = 0. − 8 4 + 24 − 16 2 − 16 1 − 2 β2 β6 β α β2 α Solution
c=0
(4.10)
is not of interest, therefore, only the terms in the bra ket have
to be examined. For
c = 0,
it is negative, and for
c = β , positive. Therefore, at β . The eigenvalue problem
least one real solution of (4.10) exists between 0 and
has, thus, a solution, i.e., along the surfa e of a homogeneous half-spa e a wave
an propagate, the amplitudes of whi h de ay with depth. In this simple ase
no dispersion
o
urs and
In the spe ial ase
λ=µ
ω. √ α = β 3), c = 0, 92β .
is independent of
(i.e.,
wave is only slightly slower than the We now examine the
displa ement
S -wave.
In general, the Rayleigh
of the Rayleigh wave
u = −ikA1 e−ikδz + ikγB1 e−ikγz exp [i (ωt − kx)] .
140
CHAPTER 4.
With
γB1 = − c2 /2β 2 − 1 A1
u = −ikA1 e
−ikδz
e
+
−ikδz
+
SURFACE WAVES
(from (4.9)), it follows
c2 −ikγz exp [i (ωt − kx)] −1 e 2β 2
(4.11)
c2 − 1 e−ikγz =: a(z). 2β 2
Similarly,
w = −ikδA1 e−ikδz − ikB1 e−ikγz exp [i (ωt − kx)] .
With
B1 = δA1 c2 /2β 2 − 1 "
+
e−ikδz +
A1
−ikδz
(from (4.9)), it follows
c2 −1 2β 2
w = −ikδA1 e
We assume that
−1
−1
e
−1
e−ikγz =: b(z).
c2 −1 2β 2
−ikγz
#
exp [i (ωt − kx)]
(4.12)
is positive real and onsider the real parts of (4.11) and
(4.12)
u = w = For
z= 0,
kA1 a(z) sin(ωt − kx) −|δ|kA1 b(z) cos(ωt − kx).
it holds that a(0)>0 and b(0)<0. In the ase of
u
and
w,
show the
behaviour given in Fig. 4.2 (e.g., for x = 0). The displa ement ve tor des ribes an
ellipse
with
retro-grade
motion.
Fig. 4.2: Behaviour of u and w.
4.1.
141
FREE SURFACE WAVES IN LAYERED MEDIA
Fig. 4.3: Theoreti al seismograms for an explosive point sour e in a homogeneous half-spa e and re orders at its surfa e ( omputed with the GRT for point sour es, ompare se tion 3.8).
For su iently large
|γ| < |δ|
z,
the se ond term in
a(z)
and
b(z)
dominates due to
so that both fun tions are negative there. The displa ement ve tor,
again, des ribes an ellipse but now with
prograde dire tion.
retro-grade to pro-grade motion o
urs at the depth where this is the ase at about
z = 0, 2Λ
This depth is, therefore, a
where
The transition from
a(z)= 0.
Λ = 2πc/ω = 2π/k
For
λ = µ,
is the wavelength.
nodal plane of the horizontal displa ement.
142
CHAPTER 4.
SURFACE WAVES
Ellipti al polarisation of the displa ement ve tor and the existen e of nodal planes of the displa ement omponents, are also hara teristi s of free Rayleigh waves in
layered
media, with the additional feature of dispersion.
The Rayleigh wave is, therefore, for ed. indi ate the theoreti al arrival times
r/
The arrows above the seismograms where
is the phase velo ity of the
free Rayleigh wave. Fig. 4.4 shows a hodograph of a point at the surfa e, i.e., its tra e during the passage of a Rayleigh wave whi h has roughly ellipti al form.
r=32 km z=0 u
w Fig. 4.4: Hodograph at a point at the surfa e (see Fig. 4.3).
The results of the theory of the free Rayleigh wave are, therefore, relatively well
on rmed.
Exer ise 4.2 Does the homogeneous half-spa e have free Love waves?
4.1.3 Love waves at the surfa e of a layered half-spa e Matrix formalism and mode on ept We now study
Love waves, i.e., waves of the SH-type, for example, surfa e waves
in layered media. First, we dis uss the general ase of arbitrarily many layers and give the
k = k(ω)
numeri al
method, with whi h the
dispersion relation c = c(ω) or
an be determined; then the ase of a single layer over a half-spa e is
dis ussed in more detail. We start from the basi equations in se tion 4.1.1 and use the ansatz (4.5) for Love waves in the wave equation (4.3) for the displa ement in this, we derive, in analogy with (4.8), in the j-th layer
y -dire tion.
By
4.1.
143
FREE SURFACE WAVES IN LAYERED MEDIA
vj = Dj exp [i (ωt − kx + kγj (z − zj ))] + Ej exp [i (ωt − kx − kγj (z − zj ))] ,
(4.13)
where
Dj
and
Ej
are now onstants and
c2 −1 βj2
γj =
We postulate, that
γj
! 12
.
is positive real or negative imaginary, depending on its
radi and being positive or negative, respe tively. In the half-spa e has to be negative imaginary due to (4.6), i.e.,
c < βn ,
(j = n), γn
and
Dn = 0.
(4.14)
z = z1 , z2 , . . . , zn ontinuity of the tangenµ∂v/∂z and for z = z2 , z3 , . . . , zn ontinuity of the displa ement v. ∂v1 /∂z = 0 for z = z1 = 0, it follows that
The boundary onditions require for tial stress From
E1 = D1 . For
(4.15)
z = zj (j ≥ 2) with vj = vj−1 and µj ∂vj /∂z = µj−1 ∂vj−1 /∂z , the following Dj and Ej with dependen e on Dj−1 and Ej−1 , an be derived
equations for
Dj + Ej
=
Dj − Ej
=
Dj−1 eikγj−1 dj−1 + Ej−1 e−ikγj−1 dj−1 µj−1 γj−1 Dj−1 eikγj−1 dj−1 − Ej−1 e−ikγj−1 dj−1 . µj γj
As in se tion 3.6.5, this an be expressed in matrix form
Dj Ej
=
1 ikγj−1 dj−1 e 2
1 + ηj 1 − ηj
(1 − ηj )e−2ikγj−1 dj−1 (1 + ηj )e−2ikγj−1 dj−1
or
Dj Ej
Dj−1 Ej−1
(4.16)
=
ηj
=
layer matrix
µj−1 γj−1 . µj γj
mj
Dj−1 Ej−1
Su
essive appli ation of (4.16) leads to
144
CHAPTER 4.
Dn En
M
is the produ t of the layer matri es. With (4.14) and (4.15), the equation for
= mn ·mn−1 . . . m3 ·m2
D1 E1
=M
SURFACE WAVES
D1 E1
=
M11 M12 M21 M22
D1 E1
or k as a fun tion of ω and the layer parameters an be given as a dispersion equation M11 + M12 = 0.
(4.17)
This equation is ordinarily solved numeri ally. For this, a value of interval from 0 to
βn
within the
M11 + M12 is omputed via the multifun tion of ω in the relevant frequen y range.
is hosen, and then
pli ation of the layer matri es as
Finally, their zeros are determined. shifted zeros are determined,
et .
Then,
is varied and the orresponding
If zeros exist, their lo ation depends on the
S -velo ity and the density as a fun tion of depth. Ea h zero gives one bran h of the dispersion urve of the phase velo ity ci (ω) (see Fig. 4.5).
Fig. 4.5: Dispersion urves of the phase velo ity.
Ea h bran h has a
(lower) uto frequen y νi = ωi /2π.
Theoreti al dispersion
urves are omputed as a fun tion of frequen y, period or wavenumber, respe tively (in the last ase the frequen y is xed). Experimentally determined urves are mostly given as a fun tion of period. The wave behaviour in the half-spa e, orresponding to a ertain bran h of the dispersion urve, is alled
mode.
For Love waves, these are
normal modes
of
.
4.1.
145
FREE SURFACE WAVES IN LAYERED MEDIA
SH -type
the
that propagate undamped. The on ept of modes also holds for
Rayleigh waves and damped surfa e waves. Modes are lassi ed by their order: 1st mode, 2nd mode,
et .
Often the rst mode is also alled
and the numbering starts after it. dier by their number of
fundamental mode
Besides their dispersive properties, modes
node surfa es.
±1)
This number is (up to
identi al
with their order. Whi h modes o
ur in reality, depends on the frequen y range in whi h the sour e radiates, as well as its depth ( ompare se tion 4.2).
For
earthquakes usually, only a few modes ontribute, and often only the fundamental mode ontributes to the surfa e waves. In Fig.
3.39, the Love waves
onsist mainly of the fundamental modes.
Spe ial ase n=2 In the ase of a single layer over a half-spa e, the dispersion equation (4.17) an be written as
e−2ikγ1 d1 =
with If
2 γ1,2 = (c2 /β1,2 − 1)1/2 . γ2
β1 > β2 , γ1
η2 + 1 µ1 γ1 + µ2 γ2 = η2 − 1 µ1 γ1 − µ2 γ2
is negative imaginary and
(4.18)
c < β2 .
is also negative imaginary. Then, the right side of (4.18) is real
and larger than 1. The left side is also real, but smaller than 1. Therefore, no real solution The
S -velo ity
layer, i.e.,
of (4.18) exists in this ase. in the half-spa e, therefore, has to be larger than that of the
β2 > β1 .
In this ase, we an ex lude values of
with the same arguments as for and
β2 .
In this ase
γ1
β1 > β2 .
between 0 and
This leaves values for
between
β1 β1
is positive real, and both sides in (4.18) are omplex.
The absolute value of both sides is 1. Thus, the mat hing of the phases gives the
dispersion equation of the Love waves
−2kγ1 d1 = −2 arctan From this, it follows with
µ2 |γ2 | . µ1 γ1
ω = kc
µ2 c−2 − β2−2
µ1 β1−2 − c−2
12 12
h 1 i = tan ωd1 β1−2 − c−2 2 .
This trans endent equation is solved by sele ting
(4.19)
with β1 < c < β2 and inver-
sion of the radi and, thus, giving the orresponding ω (or the the orresponding ω ′ s). For a dis ussion, we introdu e a new variable
general
x
146
CHAPTER 4.
x(c)
= ωd1 β1−2 − c−2
c(x)
=
ωd1
ω 2 d2 β2 1
1
−x2
21 .
12
µ1
(4.20)
The left side of (4.19) an then be written as
µ2 c−2 − β2−2
SURFACE WAVES
21
1 µ2 2 2 −2 ω d1 β1 − β2−2 − x2 2 = f (x, ω). 1 = µ1 x β −2 − c−2 2 1
Equation (4.19) an then be expressed (see also Fig. 4.6) as
f (x, ω) = tan x.
Fig. 4.6:
f (x, ω)
f (x, ω)
and tan x.
is real between
x= 0 and its zero x0 = ωd1 β1−2 − β2−2
This zero moves to the right as a fun tion of se tions of in
f (x, ω)
with
tan x.
ω
The interse tion
21
.
(4.21)
and reates, thus, more inter-
xi = xi (ω)
gives, substituting
(x) from (4.20), the dispersion relation of the i-th mode (i = 1, 2, . . .) ci (ω) =
ωd1 ω 2 d21 β12
−
12 .
x2i (ω)
(4.22)
4.1.
147
FREE SURFACE WAVES IN LAYERED MEDIA
The i-th mode o
urs only if be omes
νi =
x0 > (i − 1)π .
uto frequen y
i−1 ωi = 1 . 2π 2d1 β1−2 − β2−2 2
The i-th mode exists only for frequen ies mode,
With (4.21), its
ν > νi .
(4.23)
The rst mode (fundamental
i= 1) exists, due to ν1 = 0, at all frequen ies.
The phase velo ity of ea h mode at its uto frequen y is most simply derived from the fa t that at this point the tangent is zero in (4.19)
ci (ωi ) = β2 . For
ω → ∞, xi → (i−1/2)π and the tangent in (4.19) approa hes ∞. lim ci (ω) = β1 .
ω→∞
(4.24) Therefore,
(4.25)
An upper-limiting frequen y does not exist, and the velo ities of the layer and the half-spa e are the limiting values of the phase velo ity. A al ulated example for the dispersion urves of the rst three Love modes of a rust-mantle model, onsisting of a half-spa e with a layer above, is given in Fig. 4.7.
Fig.
4.7:
model.
Dispersion urves of the rst three Love modes of a rust-mantle
148
CHAPTER 4.
In addition to the urves of the phase velo ities
,
the
SURFACE WAVES
group velo ities U
are
shown
U=
dω dc dc c c =c+k =c−Λ = = ω dc dk dk dΛ 1 − c dω 1 + Tc
(Λ= wavelength,
T = period).
dc dT
.
(4.26)
The group velo ity ontrols, as we will see, the
propagation of an impulse from the sour e to a re eiver, i.e., ea h frequen y travels with its group velo ity from the sour e to the re eiver,
not
with its
phase velo ity. Phase and group velo ities an be determined from observations (see se tion 4.1.4 and 4.1.5). Thus, both an be used for interpretation.
Nodal planes and eigen fun tions Finally, we examine the
nodal planes
of the Love modes for the ase just dis-
ussed, i.e., the surfa es on whi h the displa ement is zero. From (4.13) with (4.14) and (4.15), it follows that
v1 v2 where
v1
and
v2
= =
2E1 cos(kγ1 z) exp [i (ωt − kx)] E2 exp (−ikγ2 (z − d1 )) exp [i (ωt − kx)]
are for the layer and the half-spa e, respe tively. These expres-
sions also hold for ea h individual mode, and the dispersion relation (4.22) has to be used. The relation between
E1
E2
and
is
E2 = 2E1 cos(kγ1 d1 ); E1
an be
hosen arbitrarily. This means that at the surfa e
z = 0,
the
maximum displa ement
is always
observed, and that nodal planes an only o
ur in the layer, but not in the half-spa e. Their position is determined by the zeros of the osine. For the i-th mode they an be derived via the equation
kγ1 z = ω β1−2 − c−2 i where
Ni
21
z = (2n − 1)
is their number determined by
For the lower frequen y limit
ω = ωi ,
(i − 1)π
π 2
(n = 1, 2, . . . , Ni ),
z ≤ d1 .
from (4.23), it follows due to (4.24)
π z = (2n − 1) . d1 2
This is satis ed for
z=
2n − 1 d1 2i − 2
with
(4.27)
n = 1, 2, . . . , Ni = i − 1.
4.1.
For
149
FREE SURFACE WAVES IN LAYERED MEDIA
ω = ωi ,
therefore,
i−1
nodal planes exist. Their spa ing is
d1 (i = 3, 4, . . .). i−1
∆z =
i= 1) has no nodal plane, neither for ω = ω1 = 0 nor for nite
The rst mode (
ω > 0. The other extreme on the frequen y s ale of ea h mode is dis ussion of the behaviour of
lim ω
ω→∞
β1−2
−
ci (ω)
c−2 i
12
for
ω→∞
xi (ω) = = lim ω→∞ d1
Equation (4.27) leads to the fa t that all satisfy the relation
ω = ∞.
From the
(see (4.25)), it follows that
z ≤ d1
1 π i− . 2 d1
have to be determined whi h
1 z π i− π = (2n − 1) . 2 d1 2 These are the values
z=
For
ω = ∞,
therefore,
i
2n − 1 d1 2i − 1
with
n = 1, 2, . . . , Ni = i.
nodal planes exist with the spa ing
∆z =
d1 i − 12
(i = 2, 3, . . .).
The hange relative to the situation where
ω = ωi
holds, is rst, the de rease in
the spa ing of the nodal planes, se ond a general move to shallower depth and nally, the addition of the i-th nodal plane
z = d1 .
This means that for
the half-spa e remains at rest.
ω=∞
Fig. 4.8 shows quantitatively the amplitude behaviour of the rst three modes as a fun tion of depth for the rust-mantle model used for Fig. 4.7. The period is also indi ated. Su h amplitude distributions are alled
eigen fun tions
modes. They follow from the z-dependent part of the displa ement dis ussed above.
v1
of the
and
v2
150
CHAPTER 4.
SURFACE WAVES
Fig. 4.8: Quantitative amplitude behaviour of the rst three modes as a fun tion of depth for the rust-mantle model used for Fig. 4.7.
Exer ise 4.3 Derive the dispersion equation for free Rayleigh waves in a liquid medium onsisting of a layer over a half-spa e and ompare this to (4.19). Sket h a gure similar to Fig. 4.6. What is the dieren e, espe ially for the rst mode?
4.1.4 Determination of the phase velo ity of surfa e waves from observations In the last se tion, we saw how the phase velo ity of a Love mode in a layered half-spa e an be determined if the half-spa e is known. In the same way (but with more ompli ations), the same an be done for Rayleigh waves. We now dis uss the derivation of the phase velo ity from only
one
mode is present.
observations. We assume that lters have to be used to
If that is not the ase,
separate the dierent modes. Sin e these are sometimes ompli ated methods, they are not dis ussed here. An overview, and appli ations for surfa e waves,
an be found, e.g., in Aki and Ri hards, Dahlen and Tromp, and Kennett. We work here with plane surfa e waves, i.e., we negle t the sour e term. The method for the determination of the phase velo ity thus derived, is su iently a
urate for pra ti al purposes.
The
modal seismogram
of any displa ement
omponent at the Earth's surfa e an be written for propagation of the mode in
x -dire tion as
4.1.
151
FREE SURFACE WAVES IN LAYERED MEDIA
1 u(x, t) = 2π
Z
+∞
−∞
x dω. A(ω) exp iω t − c(ω)
(4.28)
This is a superposition of the previously dis ussed harmoni surfa e waves with
c(ω) u(x, t).
the aid of the Fourier integral. derived from the re ordings of mograms for dierent at (arbitrary)
x= 0.
Φ(ω) = arg A(ω),
x
are dierent.
is the phase velo ity of the mode to be Sin e
A(ω)
is frequen y dependent, the seis-
is the spe trum of the displa ement
The amplitude spe trum is
i.e.,
|A(ω)|
and the phase spe trum
A(ω) = |A(ω)|eiΦ(ω) . We assume that
u(x, t)
x = x1
is known for
and
x = x2 > x1
and apply a
Fourier analysis to the seismograms
u(x1,2 , t) = where
G1,2 (ω)
1 2π
Z
+∞
G1,2 (ω)eiωt dω,
(4.29)
−∞
is the spe trum of the seismogram for
x = x1,2 .
The omparison
of (4.29) with (4.28) gives
G1,2 (ω) = |G1,2 (ω)|eiϕ1,2 (ω) = x1,2 x1,2 = |A(ω)| exp i Φ(ω) − ω . A(ω) exp −iω c(ω) c(ω) If the time
t= 0 is identi al for both seismograms, and, if possible, jumps of ±2π ϕ1,2 (ω) G1,2 (ω)...)
have been removed from the numeri ally determined phases
−π and +π ), the phases of the top ( |A(ω)|...) an be mat hed and give
between bottom
ϕ1,2 (ω) = Φ(ω) − ω Subtra ting
ϕ1 (ω)
from
ϕ2 (ω),
(observation
(usually and the
x1,2 . c(ω)
the unknown phase spe trum
Φ(ω)
of
u(0, t)
an els and the following result for the phase velo ity is left
c(ω) =
(x2 − x1 )ω . ϕ1 (ω) − ϕ2 (ω)
(4.30)
For pra ti al appli ations of this method, the surfa e waves have to be re orded at two stations whi h are on a great ir le path with the sour e. In ase
three
stations are available, this requirement an be ir umvented by onstru ting a triangle between the stations. In both approa hes, the phase velo ities derived are representative for the region
between
the stations.
152
CHAPTER 4.
SURFACE WAVES
Fig. 4.9: Example for seismograms of Rayleigh waves from L. Knopo, et al., 1966. The tra es have been shifted.
The interpretation based on the phase velo ity from Fig. 4.9 is given in Fig. 4.10. It shows short period group velo ity observations from near earthquakes as well as phase-velo ity measurements for the region of transition (Central Alps to northern Foreland, Fig. 4.9) (from L. Knopo, St. Müller and W.L. Pilant: Stru ture of the rust and upper mantle in the Alps from the phase velo ity of Rayleigh waves, Bull. Seism. So . Am.
Fig. 1966.
56, 1009-1044, 1966).
4.10: Short period group velo ity observations from L. Knopo, et al.,
4.1.
153
FREE SURFACE WAVES IN LAYERED MEDIA
4.1.5 The group velo ity Seismograms with dispersion, as shown in Fig. variation of frequen y with time, so that
one
4.9, often have a very slow
frequen y an be asso iated with
a ertain time. If this is done for two dierent distan es
x1
and
x2 ω
ir le through the sour e), and if the times, for whi h frequen y are t1 (ω) and
t2 (ω),
(on a great is observed
it follows that
U (ω) =
x2 − x1 . t2 (ω) − t1 (ω)
(4.31)
U (ω) is the velo ity with whi h this frequen y, or a wave group with a small frequen y band ∆ω around the frequen y ω , propagates. U is, therefore, alled the group velo ity. The theory of surfa e waves from point sour es in se tion 4.2 gives the even simpler formula seismogram;
r
U (ω) = r/t(ω), whi h only requires one t(ω) is relative to the time
is the distan e from the sour e, and
when the wave started (sour e time).
In pra ti e, this seismogram is ltered
in a narrow band with the entral frequen y
ω.
The arrival time
t(ω)
is at
the maximum of the envelope of the ltered seismogram. The group velo ity
an, therefore, in prin ipal be determined without di ulty from observations. Another question is, how the group velo ity is onne ted with the phase velo ity and, thus, with the parameters of the Earth, i.e., the velo ity of
P - and S -waves
and density, as a fun tion of depth. To study this relation, we start from the des ription of the modal seismogram (4.28) and express it using the wavenumber
u(x, t) =
For su iently large
x
1 2π
Z
k = ω/c
as
+∞
−∞
A(ω) exp [i (ωt − kx)] dω.
(4.32)
t
and, therefore, also large , the phase
ϕ(ω) = ωt − kx is
rapidly varying ompared to A(ω).
ω ϕ(ω)
has hanged by
2π ,
whereas
This means, for example, that for hanging
A(ω)
is pra ti ally un hanged, and the
ω-
interval, therefore, does usually not ontribute to the integral (4.32). This is espe ially true if
ω = ω0 ,
ϕ(ω) an be approximated linearly. This no longer holds for ϕ(ω) has an extremum (see Fig. 4.11), i.e., when it be omes
for whi h
stationary.
154
CHAPTER 4.
Fig. 4.11: Extremum of This frequen y
ω0
SURFACE WAVES
ϕ(ω).
depends on
t
and follows from
ϕ′ (ω0 ) = t − x
dk |ω=ω0 = 0. dω
(4.33)
The frequen y ω0 , whi h satis es (4.33), dominates at time t in the modal seismogram. Sin e for plane surfa e waves lo ation and time origin are arbitrary, (4.33) an be written for two distan es and
t2 (ω0 ),
lo ity
x1
and
x2 and orresponding times t1 (ω0 )
respe tively. Subtra tion gives the
basi formula for the group ve-
x2 − x1 dω = U (ω0 ) = |ω=ω0 . t2 (ω0 ) − t1 (ω0 ) dk ω
and
k
are onne ted via the phase velo ity
c = ω/k .
(4.34)
Using this, the expli it
group velo ity (4.26) an be derived dire tly from (4.34) (see exer ise 4.4). The arguments sket hed here are the entral ideas of the
phase.
method of stationary
We will use it later to al ulate integrals of the form (4.32) approxi-
mately; here, it was only used to demonstrate that it is the group velo ity whi h determines the sequen e and possible interferen e in the modal seismogram. To make this statement more obvious, we onsider an arbitrary mode of the
Rayleigh waves of a liquid half-spa e with a layer at the top.
Its dispersion urves
for phase and group velo ity look like those in Fig. 4.12 ( ompare exer ise 4.3).
Fig. 4.12: Dispersion urves for a liquid half-spa e with a top layer.
4.1.
155
FREE SURFACE WAVES IN LAYERED MEDIA
Instead of (4.31), or the left of (4.34), we use
U (ω) = r/t(ω),
whi h refers to
the sour e, to interpret the urve of the group velo ity. From the trend in
Un , we on lude that for the mode onsidered, the frequen ies ωn arrive at an arbitrary
in the neighbourhood of the lower limiting frequen y distan e
r
from the sour e.
This assumes that su h frequen ies are a tually
ex ited at the sour e. Their group velo ity is is
r/α2 .
α2 ,
and their group travel time
For later times, whi h are still smaller than
r/α1 ,
the frequen y of the
arriving os illations slowly in reases, orresponding to the steep trend in the
urve of
Un .
greater then
This wave train is alled the
r/α1
there are
two
frequen ies,
fundamental wave. ω′
and
ω ′′ ,
At later times
whi h ontribute to the
seismogram. This has the ee t that the higher frequen y waves (water waves) ride on the fundamental waves. The frequen ies of the two waves approa h ea h
r/Unmin .
other for in reasing time and be ome identi al at time
Here,
is the minimal group velo ity. The orresponding wave group is the and it onstitutes the end of the modal seismogram.
Unmin
Airy phase,
Exa t omputations of
modal seismograms, dis ussed later, on rm these qualitative statements. The example in Fig. 4.13 shows the behaviour of pressure of the fundamental mode (from C.L. Pekeris: Theory of propagation of explosive sound in shallow water, Geol. So . Am. Memoir No. 27, 1948).
Fig. 4.13: Pressure of the fundamental mode from C.L. Pekeris, 1948. 3 3 1500 m/s , α2 = 1650 m/s, ρ1 = 1 g/ cm , ρ2 = 2 g/ cm , d1 = 20 m.
The dispersion of the fundamental wave of the example in Fig.
α1
=
4.13, shown
for a sour e distan e of 9200 m , i.e., de reasing group velo ity with in reasing frequen y (in rease of frequen y with time), is alled
regular dispersion.
For the
water wave, the group velo ity grows with the frequen y (frequen y in reases with in reasing time); this is alled
inverse
dispersion.
The notation regular
and inverse dispersion should not be onfused with the expressions
normal
and
156
CHAPTER 4.
anomal
SURFACE WAVES
dispersion, whi h express, that the group velo ity is larger or smaller
than the phase velo ity, respe tively. Fig. 4.14 is a sket h to demonstrate the basi propagation properties of a dispersive wave train. It assumes that the sour e radiates an impulse with onstant spe trum in the frequen y band only regular dispersion.
ω1 ≤ ω ≤ ω2
and that the medium produ es
The larger the propagation distan e, the longer the
wave train be omes. At the same time, the amplitudes de rease (not shown in Fig. 4.14).
Fig. 4.14: Basi propagation properties of dispersive wave trains.
Constant frequen ies with a slope of
dr/dt
o
ur on
straight lines
through the origin (r
e.g., a ertain maximum or an interse tion with zero, are
lines,
= 0, t = 0)
Constant phases, situated on urved
that is identi al to the group velo ity.
and the frequen y varies along these urves.
The lo al slope
dr/dt
of
these urves is the phase velo ity for the dominating frequen y at this time.
Exer ise 4.4 Derive (4.26) for the group velo ity. How does abnormal dispersion, respe tively?
dc/dω
behave for normal and
4.1.
157
FREE SURFACE WAVES IN LAYERED MEDIA
Exer ise 4.5 a) What is the form of the most general phase velo ity velo ity
U (ω)
c(ω) for whi h the group
is onstant? Interpret the orresponding seismogram (4.28).
c1 (ω) and 1/U = dk/dω =
b) What is the most general onne tion between phase velo ities
c2 (ω) with identi al d(ω/c)/dω .
group velo ities
U1 (ω)
U2 (ω)?
and
Use
4.1.6 Des ription of surfa e waves by onstru tive interferen e of body waves Up to this point, surfa e waves have been treated for the most part theoreti ally, namely based on an ansatz for the solution of dierential equations. Input in these equations have been the on entration of the wave amplitude near the surfa e, propagation along the surfa e and dispersion. We have not rea hed a physi al understanding how these waves an be onstru ted. In this se tion, we will show for the simple example of Love waves in a half-spa e with one layer at the top, that surfa e waves an basi ally be understood as arising from onstru tive interferen e of body waves whi h are re e ted between the interfa es. We onsider SH -body waves whi h propagate up and down in the layer with an angle of in iden e and re e tion
ϕ.
The re e tion at the surfa e is loss
free; the re e tion oe ient a
ording to (3.39) equals +1. During re e tion at the lower boundary of the layer (z = d1 ), energy loss through re e tion ∗ o
ur, as long as ϕ < ϕ = arcsin(β1 /β2 ). In this ase, the amplitude of the re e ted waves de rease with the number of re e tions at the lower boundary. ∗ If on the other hand, ϕ > ϕ , the re e tion oe ient at the lower interfa e has the absolute value of +1 (see (3.42)).
Thus, no wave in the lower half-
spa e exists transporting energy away from the interfa e and the amplitude of ea h
single
multiple re e tion is preserved.
The wave eld in this layer is
then basi ally ontrolled by the interferen e of
ertain values of
ϕ
all
multiple re e tions.
For
there will be onstru tive interferen e and for other values
there will be destru tive interferen e, respe tively. We try to determine those
ϕ
whi h show onstru tive interferen e. To a hieve this, we approximate the
momentary wave eld pi ture of Fig. 4.15, lo ally, by plane parallel wavefronts with a orresponding angle of in iden e
ϕ
(see Fig. 4.16).
Fig. 4.15: Pi ture of momentary wave eld.
158
CHAPTER 4.
SURFACE WAVES
Fig. 4.16: Approximation of Fig. 4.15 by plane, parallel wavefronts. This approximation is better at larger distan es from the sour e. The limitation on
plane
free
waves means that we onsider
normal modes.
The phases of neighbouring wavefronts 1, 2, 3 are and
Φ 3 = Φ1 + ǫ 1 + ǫ 2 ,
A and B, respe tively.
re e tions in
the phase dieren e travel time
respe tively, where
ωs/β1
Φ3 − Φ1
s=2
SH -waves.
follows for
ǫ1
and
To ensure that wave 1 and 3 are
2π .
in phase,
With
onstru tive interferen e
2ωd1 cos ϕ + 2nπ, β1
ǫ2
Φ1 (arbitrary), Φ2 = Φ1 + ǫ1 ǫ2 are the phase shifts of the
d1 sin ϕ = 2d1 cos ϕ, tan ϕ
we derive the following ondition for
The phase shifts;
and
has to be equal to the phase dieren e due to the
plus a multiple of
ǫ1 + ǫ2 =
ǫ1
n = 0, 1, 2, . . . .
(4.35)
are the phases of the re e tion oe ients for plane ϕ > ϕ∗ = arcsin(β1 /β2 ), it
Sin e we only onsider post- riti al
ǫ1
from (3.42) and with
ω>0
12 2 β2 2 sin ϕ − 1 −ρ β b 2 2 β12 ǫ1 = −2 arctan = −2 arctan . a ρ1 β1 cos ϕ
For the re e tion at
B,
it holds that
ǫ2 = 0 ,
sin e a
ording to (3.39), the
re e tion oe ient at the free surfa e is always +1. Substituting all of the above into (4.35), we get an equation for in iden e
ϕ,
whi h produ e a for given
ρ 2 β2 arctan
β22 β12
sin2 ϕ − 1
ρ1 β1 cos ϕ
12
ω,
those
angles of
onstru tive interferen e
ωd1 cos ϕ + nπ, = β1
n = 0, 1, 2, . . .
(4.36)
4.1.
159
FREE SURFACE WAVES IN LAYERED MEDIA
In this equation, we introdu e the apparent velo ity
c=
β1 sin ϕ
(4.37)
horizontal
with whi h the wavefronts propagate in a dire tion. With cos ϕ = 2 and ρ1,2 β1,2 = µ1,2 , we derive by reversing (4.36), an equation
β1 (β1−2 − c−2 )1/2 for
µ2 c−2 − β2−2 µ1 β1−2 − c−2
21
h 1 i −2 −2 2 . 12 = tan ωd1 β1 − c
This equation is identi al with the dispersion equation (4.19) of Love waves. We would have found the same equation, if we had onsidered waves whi h propagate upwards (and not downwards) in Fig. 4.16. The superposition of both
verti al wavefronts.
is not only an apparent velo ity, but also the phase velo ity of this
groups of waves gives, for reason of symmetry, a wave with Therefore,
resulting wave. We also see that the Love waves in the half-spa e with a top layer are produ ed by
onstru tive interferen e of body waves
whi h have a
post- riti al
angle of
in iden e. For these angles of in iden e, no energy is lost from the layer into the half-spa e. The energy remains in the layer whi h a ts as a perfe t
guide.
wave
This is generally true for normal modes of Love and Rayleigh waves in
horizontally layered media, in whi h ase that normal modes exist. From this, we an also on lude that the phase velo ity of normal modes an, equal to the
S -velo ity of the half-spa e under the layers
at most, be
c ≤ βn . If it were larger, energy would be radiated into the half-spa e in the form of an
S -wave. Leaking modes
also o
ur by onstru tive interferen e of body waves.
In this ase, the angles of in iden e are larger than
βn .
pre- riti al,
and the phase velo ity is
Thus, radiation into the lower half-spa e o
urs and the wave
guide is not perfe t. Finally, it should be noted that the explanation of surfa e waves via onstru tive interferen e of body waves annot be applied to the
Rayleigh waves.
fundamental mode of
The Rayleigh wave of the homogeneous half-spa e, for example,
exists without additional dis ontinuities at the surfa e. No simple explanation exists for the fundamental mode of Rayleigh waves.
Exer ise 4.6 Determine the dispersion urves for a liquid layer whose boundaries are (1) both free, (2) both rigid, (3) one rigid and one free, respe tively. Use the arguments of se tion 4.1.6 and ompare with the solution of the orresponding eigenvalue problem. Give the group velo ity and sket h the pressure-depth distributions.
160
CHAPTER 4.
SURFACE WAVES
4.2 Surfa e waves from point sour es 4.2.1 Ideal wave guide for harmoni ex itation Expansion representation of the displa ement potentials We study the propagation of mono hromati sound waves from an explosive point sour e in a liquid layer with a free surfa e situated above a
rigid
half-
spa e.
Fig. 4.17: Explosive point sour e in a liquid layer with a free surfa e atop a rigid half-spa e.
This is an ideal wave guide sin e no waves an penetrate the half-spa e. For su h a s enario, the key features of surfa e waves from point sour es an be studied without too mu h mathemati al eort. For the displa ement potential
Φ
in the layer, we assume the following integral
ansatz, using the analogy to (3.84) and applying (3.85) for the potential of the iωt spheri al wave from the sour e. In the following, the time-dependent term e is omitted
Φ
=
Z
∞
ω2 − k2 α2
J0 (kr)
0
l
J0 (kr)
=
12
k −il|z−h| e + A(k)e−ilz + B(k)eilz dk il
(4.38)
.
is the Bessel fun tion of rst kind and zeroth order,
k
and
zontal and verti al wavenumber, respe tively. The square root
l
are the hori-
l is, as in se tions
3.6.5 and 3.7, either positive real or negative imaginary. It an be shown that
Φ
is a solution of the wave equations in ylindri al oordinates. The rst term in (4.38) is the wave from the sour e, the se ond and third orrespond to the waves
4.2.
161
SURFACE WAVES FROM POINT SOURCES
z -dire tion,
propagating in positive and negative
respe tively.
A(k)
and
B(k)
follow from the boundary onditions for the interfa es
z=0:
stress
pzz = ρ
z=d:
normal displa ement
∂2Φ = −ρω 2 Φ = 0 ∂t2
or
Φ=0
∂Φ = 0. ∂z
This gives
A(k) + B(k)
=
A(k) − e2ild B(k)
=
k − e−ilh il k − eilh . il
The solution of this system of equation is (please he k)
k cos [l(d − h)] il cos(ld) k sin(lh) −ild e . l cos(ld)
A(k)
= −
B(k)
=
Inserting them into (4.38) gives
0≤z≤h:
Φ
=
h≤z≤d:
Φ
=
∞
sin(lz) cos [l(d − h)] dk l cos(ld) Z0 ∞ sin(lh) cos [l(d − z)] 2 kJ0 (kr) dk. l cos(ld) 0 2
Z
kJ0 (kr)
(4.39)
(4.40)
Before these expressions are solved with methods from omplex analysis, it should be noted that an ex hange of sour e and re eiver does not hange the value of
Φ.
Displa ement and pressure are also the same for this ase. This is
an example for
re ipro ity relations, whi h is important in the theory of elasti
waves. The poles
kn
of the integrand in (4.39) and (4.40) are determined via
dln = d This gives
ω2 − kn2 α2
12
π = (2n − 1) , 2
n = 1, 2, 3 . . .
162
CHAPTER 4.
kn =
ω2 (2n − 1)2 π 2 − 2 α 4d2
12
SURFACE WAVES
.
(4.41)
The in nite number of poles are situated on the real axis between
+ω/α and
−ω/α
and
on the imaginary axis, respe tively. The number of poles on the real
axis depends on
ω.
Due to these poles, the integration path in (4.39) and (4.40)
have to be spe i ed in more detail.
We hoose path
C1
in Fig.
4.18 whi h
ir umvents the poles in the rst quadrant.
Fig. 4.18: Integration path
C1
whi h ir umvents the poles in the rst quadrant.
In the following, we dis uss only (4.39) in detail. Equation (4.40) an be solved similarly. We use the identity
J0 (kr) =
where
(1)
H0 (kr)
fun tions )
and
(2)
H0 (kr)
i 1 h (1) (2) H0 (kr) + H0 (kr) , 2
Hankel
are Bessel fun tions of the third kind (=
and zeroth order (Appendix C, equations (C.2) and (C.3), respe -
tively). Then,
Φ=
h i sin(lz) cos [l(d − h)] (1) (2) dk. k H0 (kr) + H0 (kr) l cos(ld) C1
Z
Using relation (C.6) from appendix C,
(1)
(2)
H0 (x) = −H0 (−x)
(4.42)
4.2.
163
SURFACE WAVES FROM POINT SOURCES
the rst part of the integral in (4.42) an be written as
−
Z
(2)
C1
kH0 (−kr)
sin(lz) cos [l(d − h)] dk = l cos(ld)
(2)
C2
C2
where u=-k is used. The integration path
C1
Z
uH0 (ur)
sin(lz) cos [l(d − h)] du l cos(ld)
is point-symmetri al to the path
with respe t to the oordinate entre, but it goes from
k
this in (4.42) and with onsistent use of
Φ=
Z
(2)
C
kH0 (kr)
Fig. 4.19: Integration path
C
sin(lz) cos [l(d − h)] dk = l cos(ld)
from
real axis.
Integration path
C,
−∞
−∞
to 0. Inserting
as integration variable, gives
to
+∞
Z
I(k)dk.
ir umventing the poles on the
therefore, is, as indi ated in Fig. 4.19, from
and ir umventing the poles on the real axis. Despite the non-uniqueness of the square root fun tion of
k.
The reason for this is that
sign of the square root of
l
does
not
e.g., if the half-spa e is not rigid, more ompli ated (introdu tion of Now we apply the
I(k)
l
in (4.43),
I(k)
−∞
to
+∞
is a unique
l
is an even fun tion of , thus, the
matter. For more ompli ated wave guides,
I(k)
is not unique and the theory be omes
bran h uts ).
remainder theorem on the losed integration path shown in C and a half ir le with in nite radius. The
Fig. 4.20 whi h onsists of path
only
(4.43)
C
singularities in luded are the poles of
I(k).
164
CHAPTER 4.
Fig. 4.20: Integration path
SURFACE WAVES
C
and half ir le with in nite radius.
Z
= −2πi
Then
Z L
+
L
C
∞ X
ResI(k)|k=kn .
n=1
indi ates the lo kwise integration in the lower plane of Fig. 4.20, and ea h
term in the sum is the residue of If the
asymptoti representation
I(k)
at the pole
k = kn .
of the Hankel fun tion is used, it follows for
large arguments (Appendix C, equation (C.4))
(2) H0 (kr)
≃
2 πkr
12
e−i(kr− 4 ) . π
(4.44)
We see that their values on the semi- ir le in the lower half-plane, where a negative imaginary part, be omes zero (for R going to
∞).
k
has
The orresponding
integral also goes to zero, and we have found a representation of the potential
Φ
as an in nite sum of residuals. The determination of the residue of the quotient
f1 (k)/f2 (k)
if
kn
at the lo ation
is a pole of
rst
kn
with
f2 (kn ) = 0
is done with the formula
f1 (k) f1 (kn ) Res , = ′ f2 (k) k=kn f2 (kn)
order. In our ase,
kn
follows from (4.41) and is either
positive real or negative imaginary. The orresponding
ln = Furthermore, it holds here, that
l
is
(2n − 1)π . 2d
f2 (k) = cos(ld),
f2′ (kn ) = d
kn sin(ln d). ln
f2 (kn ) = cos(ln d) = 0
and
4.2.
165
SURFACE WAVES FROM POINT SOURCES
Finally,
cos [ln (d − h)] = cos(ln d) cos(ln h) + sin(ln d) sin(ln h) = sin(ln d) sin(ln h). Thus,
Res I(k)|k=kn = We, therefore, get the following
sion, for whi h eiωt Φ=−
1 (2) H (kn r) sin(ln z) sin(ln h). d 0
representation of the potential Φ as an expan-
has now to be added again for ompleteness
∞ h 2πi X πz i (2) πh sin (2n − 1) H0 (kn r)eiωt . sin (2n − 1) d n=1 2d 2d
This expression not only holds for
0 ≤ z ≤ h
but also for arbitrary depth,
sin e a
ording to (4.40) the same expansion an be found. displa ement omponents
an be derived.
(4.45)
From (4.45) the
∂Φ/∂r and ∂Φ/∂z and the pressure p = −pzz = ρω 2 Φ
For the ideal wave guide the eld an be onstru ted
solely
from the ontribu-
tions from the poles, ea h of whi h represents a mode, as will be shown later. For ompli ated wave guides, ontributions in the form of urve integrals in the
omplex
k -plane have to be
added to the pole ontributions. These additional
ontributions orrespond mostly to body waves.
Modes and their properties Ea h term in (4.45) represents a
mode.
This is only a de nition, but it ts well
into the mode on ept introdu ed in the previous se tions for
free surfa e waves. r, we an use
If we onsider, for example, the terms in (4.45) for large distan es (4.44)
(|kn r| > 10)
√ π ∞ h −2 2πiei 4 X πz i 1 πh i(ωt−kn r) Φ= . sin (2n − 1) sin (2n − 1) 1 e d 2d 2d 2 (k r) n n=1
(4.46)
kn . Their numω . They orrespond to waves with ylindri al wavefronts whi h propagate in +r-dire tion with the frequen y dependent phase
The most important terms in (4.46) are those with positive ber is nite and in reases with velo ity
− 1 ω (2n − 1)2 π 2 α2 2 cn (ω) = =α 1− . kn 4d2 ω 2
(4.47)
166
CHAPTER 4.
If the eigenvalue problem for
SURFACE WAVES
free surfa e waves in the same wave guide is solved n-th free normal mode
( ompare exer ise 4.6), it follows for the
h πz i iω Φn = A sin (2n − 1) e 2d with
cn (ω)
t− cnx(ω)
,
(4.48)
from (4.47). Furthermore, the terms in (4.46) and (4.48) agree, that
des ribe the
z-dependen e agree. It, therefore, makes sense to name the single n-th for ed normal mode, if kn > 0. The dieren e
terms in (4.45) and (4.46) the
with respe t to (4.48) is in the amplitude redu tion proportional to the addition of a term that depends on the sour e depth the
ex itation fun tion of the mode.
h.
r−1/2
and in
This term is named
If the the sour e is lo ated at a nodal plane
of the free mode (4.48), the ex itation fun tion is zero, and the mode is not ex ited. Maximum ex itation o
urs, if the sour e is at a depth where the free mode has its maximum. From the omparison of the free and the for ed normal modes, the importan e of the study of free modes be omes obvious. It des ribes the dispersive properties and the amplitude-depth distributions (eigen fun tions) of the for ed normal modes and, therefore, their most important property.
ompli ated wave guides.
The terms in (4.46) with negative imaginary
kn
This also holds for more
are not waves but represent
os illations with amplitudes that de rease exponentially in
r -dire tion.
They
only ontribute to the wave eld near the sour e, where (4.45) has to be used for ompleteness.
The far- eld is dominated by normal modes.
The number of
nodal planes of the n-th mode is n, and their spa ing is 2d/(2n−1)
(n = 2, 3 . . .).
The potential
have a node for
z=0
Φ,
horizontal displa ement
and a maximum for
z = d,
The opposite is true for the verti al displa ement
∂Φ/∂r
and pressure
p
respe tively (see Fig. 4.21).
∂Φ/∂z .
Fig. 4.21: Modes and nodal planes, n = 1, 2, 3, 4.
The phase velo ity (4.47) of the n-th mode an be written as
4.2.
SURFACE WAVES FROM POINT SOURCES
167
ω 2 − 21 n cn (ω) = α 1 − ω
(4.49)
with the lower frequen y limit
ωn =
(2n − 1)πα . 2d
In nitely high phase velo ities an o
ur. A
ording to (4.26), the group velo ity is
Un (ω) = α 1 −
ω 2 12 n
ω
.
(4.50)
Fig. 4.22: Group and phase velo ities. An important property of the ideal wave guide with a rigid and a free interfa e is that the angular frequen ies and waves length
Λ > 4d,
ω < ω1 = πα/2d
respe tively)
(or the frequen ies
ν < α/4d
annnot propagate undamped.
This no
longer holds for the ideal wave guide with two rigid walls ( ompare exer ise 4.6). In this ase, an additional fundamental mode exists, in addition to the modes dis ussed before, but with dierent limiting frequen ies. That mode an o
ur at all frequen ies, and its phase velo ity is frequen y independent and equal to
α.
4.2.2 The modal seismogram of the ideal wave guide In this se tion, we will ompute the orresponding arbitrary summand in (4.45), exa tly. method of stationary phase to do this.
modal seismogram
for an
In the next se tion, we will use the
168
CHAPTER 4.
SURFACE WAVES
The potential (4.45) orresponds to time harmoni ex itation, i.e., for the potential of the explosion point sour e
Φ0 = it holds that
F (t) = eiωt .
1 F R
R , t− α
(4.51)
From this mode of ex itation, we now will move to
the ex itation via a delta fun tion, F (t) = δ(t). Multiplying (4.45), without the iωt fa tor e , with the spe trum of the delta fun tion F (ω) = 1, gives the Fourier transform of the displa ement potential. Finally, the result is transformed ba k into the time domain. These modal seismograms an then be onvolved with realisti ex itation fun tions stood for
F (t),
but the basi features an already be under-
F (t) = δ(t).
For this, we onsider the n-th mode in the expansion (4.45). Its Fourier transform for ex itation via a delta fun tion is, ex ept for geometry fa tors, equal (2) 2 2 1/2 /α. We now use the Lapla e to H n (ω) = iH0 (kn r) with kn = (ω − ωn ) transform ( ompare se tion A.1.7)
(2)
hn (s) = H n (−is) = iH0 and the relation
(2)
H0 (−ix) =
h r 1 i −i s2 + ωn2 2 α
2i K0 (x) π
between the Hankel fun tion and the modi ed Bessel fun tion
K0 (x) (see se tion
3.8). This gives then
hr 1 i 2 hn (s) = − K0 s2 + ωn2 2 . π α
The original fun tion an then be found in tables of the Lapla e transform. It r r is zero for t < α , and for t > α it holds that
Hn (t) = −
2 · π
cos ωn t2 − t2 −
r2 α2
21
r2 α2
12
.
Thus the n-th mode of the potential an be written as
Φn =
0
4 d
for
h
cos πz sin (2n − 1) πh sin (2n − 1) 2d 2d
2
ωn t
r2 −α 2 2
r t2 − α 2
21
12 i
for
t< t>
r α r α. (4.52)
4.2.
169
SURFACE WAVES FROM POINT SOURCES
That is a normal mode for
all n sin e the delta fun tion ontains arbitrarily high
frequen ies, ensuring that the lower limiting frequen y of ea h normal mode an be ex eeded. For simpli ity, we limit the dis ussion in the following to the potential All on lusions also hold for displa ement and pressure.
Φn .
The seismogram in
t = r/α with a singularity that is integrable. Then the 1/t, for times large ompared to r/α, while os illating. The most important feature of Φn is its frequen y modulation or dispersion. The frequen ies de rease from large values to the limiting frequen y ωn of the mode
Fig. 4.23 starts at time
amplitudes de rease with
onsidered. The dispersion in the example shown is, therefore, inverse.
Fig. 4.23: Seismogram showing frequen y modulation (dispersion).
What we have learned about the group velo ity in se tion 4.1.5 an be on rmed with (4.52). We rst ask whi h frequen ies
ω
dominate at a ertain time
t0
in
the modal seismogram. Outside the singularity, the dis ussion an be limited
f (t) near t = t0 to be able cos [f (t)] in the neighbourhood of t = t0 by the mono hromati os illation cos [ϕ0 + ω(t0 )t]. Here ϕ0 is a phase that is independent of t, and ω(t0 ) is the instantaneous angular frequen y required. This leads
to the osine fun tion in (4.52). We plan to linearise to approximate the fun tion
to
cos [f (t)] ≈ cos [f (t0 ) + f ′ (t0 )(t − t0 )] . From this, it follows that
ω(t0 ) = f ′ (t0 ).
If applied to (4.52), it follows
− 21 r2 2 ω(t0 ) = ωn t0 t0 − 2 . α
170
CHAPTER 4.
From this, we derive the quotient of frequen y
ω(t0 )
SURFACE WAVES
r/t0 , i.e., the velo ity with whi h a wave group
propagates from the sour e to the re eiver, and we get (with
ω0 = ω(t0 )) " 2 # 21 r ωn = Un (ω0 ) =α 1− t0 ω0 with
Un (ω0 )
from (4.50), i.e., exa tly the
group velo ity
of the n-th mode. We,
therefore, on rm the statement from se tion 4.1.5: that ea h frequen y that is radiated from the sour e propagates to the re eiver with the group velo ity. The
omplete seismogram
in the wave guide is produ ed by onvolving the
modal seismogram (4.52) with a realisti ex itation fun tion
F (t), the spe trum
of whi h has an upper limiting frequen y, and sum. Only those normal modes (4.52) ontribute signi antly to the seismogram whi h have lower limiting frequen ies that are smaller than the upper limiting frequen y of
F (t).
Often the
response of hydro-phones and seismometers, together with the dissipative me hanisms in the wave guide, redu e the number of modes. In pra tise usually only a few modes ontribute to the observed surfa e waves.
4.2.3 Computation of modal seismograms with the method of stationary phase The omputation of modal seismograms is only possible
exa tly
guides (with rigid and/or free boundaries, respe tively).
for ideal wave
In the following, an
approximation is dis ussed and demonstrated, whi h gives the modal seismogram for the far- eld form of a normal mode of the type of (4.46). This is the
method of stationary phase
mentioned before.
Multiplying a normal mode in (4.46) with the spe trum fun tion
F (t)
F (ω)
of the ex itation
in (4.51), then transforming ba k into the time domain, gives
the modal seismogram as a Fourier integral. To avoid integration over negative frequen ies, we use the fa t that
real
fun tions
f (t)
annot only be represented
as
Z
+∞
1 f (t) = Re π
∞
f (t) =
1 2π
f (ω)eiωt dω
−∞
but also as
This gives the modal seismogram as
Z
0
f (ω)eiωt dω.
4.2.
171
SURFACE WAVES FROM POINT SOURCES
Φn = Re
(
) √ Z ∞ π h πh πz i 1 F (ω) i(ωt−kn r) −2 2iei 4 √ √ √ e sin (2n − 1) sin (2n − 1) dω . πd 2d 2d r 0 kn (4.53)
The approximate omputation of the integral over on the fa t that at times
t
ω
is based, as in se tion 4.1.5,
to be onsidered, the phase
ϕ(ω) = ωt − kn r is usually
rapidly varying
ompared to fun tion
tribute little to the integral in (4.53).
(4.54)
F (ω).
Su h frequen ies on-
This is dierent for frequen ies with
stationary phase values. Su h a frequen y
ω0
follows from the equation
dkn =0 ϕ (ω0 ) = t − r dω ω=ω0 ′
and depends on
t.
This means that the frequen y
ω0 ,
for whi h the group
velo ity is
dω r Un (ω0 ) = = , dkn ω=ω0 t
t. From this follows the prin iple of determining the group velo ity from an observed modal seismogram. For a given time t, relative to the sour e time, the moment frequen ies and the orresponding group velo ities, using t and sour e distan e r, are determined. The sour e time and epi entre of the earthquake,
dominates the modal seismogram at time
therefore, have to be known. This gives a pie e of the group velo ity dispersion
urves. One has now to verify this pie e of the urve via forward modelling. The asso iation of a ertain frequen y to a ertain time, ne essary here, is in prin iple not unique, but the error asso iated with it an be estimated. With this method, applied to surfa e waves of earthquakes, several important results on the stru ture of the Earth were found, for example, the average rustal thi kness in dierent parts of the Earth is shown in the dierent bran hes in Fig. 4.10. A disadvantage of this method is that the result is only an average over the
whole
region between sour e and re eiver. Therefore, today several stations
are used in the interpretation of the phase velo ity ( ompare se tion 4.1.4). The
omputation of the modal seismogram requires then the following additional
steps: for given time t, we expand the phase (4.54) at the frequen y
ω0 ,
whi h
is determined by
Un (ω0 ) =
r t
(4.55)
172
CHAPTER 4.
= ϕ(ω0 ) + 21 ϕ′′ (ω0 )(ω − ω0 )2
ϕ(ω) ϕ′′ (ω0 ) An
0
=
important requirement
∞
Z
F (ω) iϕ(ω) √ e dω kn
≈ ≈
SURFACE WAVES
r
2 (ω ) Un 0
is that
dUn dω (ω0 )
ϕ′′ (ω0 ) 6= 0.
.
(4.56)
Then,
ω0 +∆ω
1 ′′ F (ω) 2 √ exp i ϕ(ω0 ) + ϕ (ω0 )(ω − ω0 ) dω 2 kn ω0 −∆ω Z F (ω0 )eiϕ(ω0 ) ω0 +∆ω i ′′ p ϕ (ω0 )(ω − ω0 )2 dω. exp 2 kn (ω0 ) ω0 −∆ω
Z
Here, we limited our dis ussion to the neighbourhood of the frequen y
ϕ(ω) is stationary. The other frequen ies 1/2 x = (ω − ω0 ) 12 |ϕ′′ (ω0 )| , we get ω0 +∆ω
Z
exp
ω0−∆ω
i ′′ 2 ϕ (ω0 )(ω − ω0 ) dω 2
=
≈
with
2 ′′ |ϕ (ω0 )|
12 Z
+∆ω
2 |ϕ′′ (ω0 )|
12 Z
+∞
2π |ϕ′′ (ω0 )| 1 R +∞ R +∞ π 2 2 2 cos x dx = sin x dx = . 2 −∞ −∞ =
ω0 , where
do not ontribute signi antly. With
−∆ω
|ϕ′′ (ω0 | 2
|ϕ′′ (ω0 )| 2
2
1/2
e±ix dx
−∞
12
π
e±i 4
The positive and the negative sign in the exponential term hold, if
ϕ′′ (ω0 ) >
0 and < 0, respe tively. The extension of the limits √ of the integration to x = ±∞ is possible, sin e they are proportional to r and r is very large.
Furthermore, signi ant ontributions to the integral ome only from relatively small values of x ( a.
|x| ≤ 5).
Putting all this together, the modal seismogram
for the ideal wave guide in the approximation given by the method of stationary ′′ phase (with ϕ (ω0 ) > 0) an be written as
Φn
= ·
√ π h πz i πh −2 2iei 4 √ sin (2n − 1) sin (2n − 1) Re 2d 2d πd ) 1 2 2π π F (ω0 )eiϕ(ω0 ) ei 4 . ′′ rkn (ω0 ) |ϕ (ω0 )|
2
1/2 e±ix dx
(
(4.57)
4.2.
173
SURFACE WAVES FROM POINT SOURCES
Next, one uses
ϕ(ω0 ) =
ω0 t − kn (ω0 )r, " 2 # 12 ωn ω0 1− , α ω0 " 2 # 21 ωn Un (ω0 ) = α 1 − ω0
kn (ω0 ) = r t
=
−1/2 ω0 = ω0 (t) = ωn t t2 − r2 /α2 from (4.57). After some al ulations, and for the assumption F (ω0 ) = 1, whi h orresponds to the ex itation fun tion F (t) = δ(t), the following modal seismogram and
ϕ′′ (ω0 )
from (4.56) and deletes
an be derived (please on rm)
Φn =
0
4 d
for
h
cos πz sin (2n − 1) πh 2d sin (2n − 1) 2d
ωn t2 − 2 t2 − r 2 α
12 i r2
α2
21
t<
for
t>
r α r α. (4.58)
We, therefore, get the stringent results of (4.52). This is surprising, onsidering the approximations used. From this we an draw the general on lusion that the method of stationary phase is a good approximation for normal modes even for more ompli ated wave guides.
dUn dω (ω0 ) = 0 and with stationary values of the group velo ity (whi h do not o
ur for ideal wave guides), the For frequen ies
ω0
with
ϕ′′ (ω0 ) = 0,
i.e., with
expansion in (4.56) has to be extended by one additional term. The treatment of the al ulations following is, therefore, slightly dierent (see, for example, Appendix E). It leads to the
behaviour of Airy phases
and shows that they
are usually the dominating parts of the modal seismograms ( ompare also Fig. 4.13).
4.2.4 Ray representation of the eld in an ideal wave guide In the last two se tions we have learned that the wave eld in an ideal wave guide is omposed of for ed normal modes. Furthermore, we found in se tion 4.1.6, that free normal modes are omposed of multiple re e ted plane body waves in the wave guide. This raises the question, an the eld of a point sour e in an ideal wave guide also be represented by the superposition of multiple re e tions? In other words, in this ase is there also a ray representation of the wave eld? In addition, it is interesting to see if mode and ray representations of the wave eld are then also equivalent.
174
CHAPTER 4.
SURFACE WAVES
We rst examine the re e tion of the spheri al wave
1 Φ0 = F R0
R0 t− α
(4.59)
at the interfa e of the wave guide in the neighbourhood of the point sour e, e.g., the free surfa e.
Q1 000000 111111 0000000000 1111111111 h 0000000000 1111111111 z=0 0000000000 r 1111111111 0000000000 1111111111 R1 h 111111 000000 000000000 111111111 0000000000 1111111111 1 0 000000000 111111111 0 1 0000000000 1111111111 Q0 000000000 111111111 0 1 R0
z=d
P(r,z)
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 0 1 z
Fig. 4.24: Re e tion of a spheri al wave at the interfa e of the wave guide in the neighbourhood of the point sour e.
The potential of the re e tion is
−1 Φ1 = F R1 R1
is the distan e between
P (r, z)
R1 . t− α
and the
image sour e Q1 .
(4.60)
As long as the
re e tion from the lower (rigid) interfa e of the wave guide has not rea hed
Φ0 + Φ1 and, Φ0 + Φ1 satisfy, therefore, for su h times, the 2 2 surfa e z = 0 (pzz = ρ∂ (Φ0 + Φ1 )/∂t ).
the surfa e, the potential in the neighbourhood of the surfa e is therefore, zero on the surfa e.
ondition of no stress at the
Similarly, if we onsider the re e tion of the spheri al wave from interfa e
z = d,
Q0
at the
the potential of the wave re e ted there an be written as
1 F Φ2 = R2
R2 , t− α
(4.61)
R2 now has to be determined for a new image sour e with the z - oordinate d + (d − h) = 2d − h. That Φ0 + Φ2 satis es the boundary onditions ∂(Φ0 + Φ2 )/∂z = 0 for z = d (zero normal displa ement), an be seen easily, sin e for where
points in that interfa e
4.2.
175
SURFACE WAVES FROM POINT SOURCES
R0
= R2
∂R0 ∂z
=
∂R2 ∂z
=
z − h R0
d−h R0 z=d d−h ∂R0 2d − h − z =− =− − . R2 R2 ∂z z=d =
With the two previously onsidered re e tions of the spheri al wave originating from
Q0
at the interfa es of the wave guide, boundary onditions an only be
satis ed for ertain times, e.g., only as long as the re e tions
Φ1
and
Φ2
have
rea hed the opposite interfa e, respe tively. Sin e they are of the same form as
Φ0 ,
higher order re e tions an be onstru ted in the same way. Ea h re e tion
and multiple re e tion seems to ome from an image sour e, whi h was reated by the appli ation of multiple mirror images of
Q0
at the interfa es (Fig. 4.25).
The sign of the orresponding potential is negative if the number of re e tions at the surfa e is odd, otherwise it is positive. Ea h image sour e orresponds to a
ray
from the sour e to the re eiver whi h has undergone a ertain number of
re e tions.
Fig. 4.25: Image sour es for multiple re e tions in the wave guide.
176
CHAPTER 4.
The ray on ept an, without di ulties, be generalised for
SURFACE WAVES
solid
media (in lud-
ing S-waves), but this is not true for the on ept of the image sour e. This is why, in general, and also in the ase presented here, we speak of a
tation of the wave eld.
Φ
ray represen-
It an be expressed as
∞ X
1 Rj1 Rj2 1 + F t− F t− (−1) − Rj1 α Rj2 α j=0 1 1 Rj3 Rj4 + − F t− F t− Rj3 α Rj4 α
=
j
(4.62)
with
2
2 Rj1
=
(2jd + h + z) + r2
2 Rj2
=
2 Rj3
(2jd − h + z)2 + r2
=
2 Rj4
=
2
(2(j + 1)d − h − z) + r2 2
(2(j + 1)d + h − z) + r2 .
Only those terms in (4.62) are non-zero, for whi h the argument is positive, and for whi h
F (t)
at
t = 0
is not zero.
The number of su h terms is nite and
in reases with time. For the ideal wave guide, the determination of the ontribution of a ray is simple, sin e it follows the same time law as the exiting spheri al wave. For other wave guides, methods like those presented in se tion 3.8 have to be used. The resulting numeri al eort is then signi antly greater and seems only justi ed if not too many rays have to be summed up, but that is ne essary for large horizontal distan es from the sour e where the paths of many rays be ome very similar. In this ase, the representation of the wave eld as a sum of only a few normal modes is
signi antly more e ient.
The ray representation is most suited for
su h distan es from the sour e where the typi al normal mode properties of the wave eld have
not yet
developed.
Finally, we will show that (4.62) for tively, are
F (t) = eiωt
and (4.39) and (4.40), respe -
two dierent representations of the same wave eld.
dis ussion rst to the ase
h ≤ z ≤ d.
If
F (t) = eiωt
We limit our
is inserted into (4.62),
and the Sommerfeld integral (3.85) for a spheri al wave is used for ea h term, iωt it follows (the fa tor e is again omitted)
Φ
=
Z
0
∞
J0 (kr)
∞ kX (−1)j [− exp (−il |2jd + h + z|) il j=0
4.2.
177
SURFACE WAVES FROM POINT SOURCES
+ exp (−il |2jd − h + z|)
+ exp (−il |2(j + 1)d − h − z|) − exp (−il |2(j + 1)d + h − z|)] dk. z ≥ h, the ontributions are exp(−i2ljd) an be separated If
Φ
=
Z
0
∞
equal to the arguments
everywhere.
Then
∞ X k j J0 (kr) −e−2ild [− exp (−il (h + z)) il j=0
+ exp (−il (−h + z))
+ exp (−il (2d − h − z)) − exp (il (2d + h − z))] dk. The expansion in the rst square bra ket has a sum of −ild
an be extra ted giving se ond square bra ket, e
Φ
=
1/(1 + e−2ild ).
From the
∞
k [− exp (il (d − h − z)) 2il cos(ld) 0 + exp (il (d + h − z)) + exp (−il (d − h − z)) Z
J0 (kr)
− exp (−il (d + h − z))] dk.
The remaining square bra ket is equal to
−2i sin [l (d − h − z)] + 2i sin [l (d + h − z)] = 4i cos [l (d − z)] sin (lh) . Thus,
Φ=2
Z
0
∞
J0 (kr)
k cos [l(d − z)] sin(lh) dk, l cos(ld)
(4.63)
whi h agrees with (4.40). If sour e and re eiver are ex hanged in (4.62), the potential of a single ray is un hanged sin e it depends only on the path travelled. Therefore, ex hanging
z
with
h
in (4.63) gives the potential for
Thus, the proof of the identity of (4.62) (for
0 ≤ z ≤ h whi h leads to (4.39). F (t) = eiωt ) with (4.39) and (4.40),
respe tively, is omplete. Finally, we would like to reiterate ( ompare se tion 4.2.1) that a representation of the wave eld
by normal modes alone
is only possible for ideal wave guides
178
CHAPTER 4.
SURFACE WAVES
whi h have upper and lower boundaries that are ompletely re e ting for all angles of in iden e. In other media, additional ontributions (body waves, leaky modes) o
ur whi h are not due to the poles in the omplex plane like the normal modes.
Exer ise 4.7 Study the polarisation of the displa ement ve tor of the se ond free normal
n
mode of the ideal wave guide ( =2 in (4.48)) as a fun tion of depth.
Exer ise 4.8 An explosive point sour e is lo ated at depth half-spa e. The displa ement potential
Φ
h below the free surfa e of a liquid
is the sum of the potentials (4.59) of
the dire t wave and (4.60) for the re e tion. Give an approximation for
Φ whi h
holds under the following onditions (dipole approximation) : a) The dominant period of the ex itation fun tion travel time
h/α
b) The distan e
F (t)
is mu h larger than the
from the sour e to the surfa e.
r
to the re eiver is mu h larger than
Introdu e spheri al oordinates
R
and
ϑ
h.
relative to the point
( ompare Fig. 4.24). What is the result, if the surfa e is not free but rigid?
r = 0,
and
z=0
Appendix A
Lapla e transform and delta fun tion A.1 Introdu tion to the Lapla e transform A.1.1 Literature Spiegel, M.R.
: Lapla e Transformation, S haum, New York, 1977
Riley, K.F., Hobson, M.P. and Ben e, J.C.
: Mathemati al methods for
physi s and engineering, A omprehensive guide, Cambridge University Press, Cambridge, 2nd edition, 2002
A.1.2 De nition of the Lapla e transform The Lapla e transform asso iates a fun tion transforms a fun tion
f (s) =
Z
0
F (t)
=
f (s) =
∞
F (t)
into the fun tion
f (s) with f (s).
the fun tion
F (t),
or it
e−st F (t)dt = L {F (t)}
original fun tion image fun tion
Symboli notation:
(Lapla e − transform,
f (s) •−◦ F (t) (•−◦ = symbol t = s =
real variable
σ + iω
(of
abbreviated L
of asso iation) time)
omplex variable
179
− transform)
180
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
A.1.3 Assumptions on F(t) 1. F(t) is usually a real fun tion 2.
F (t) ≡ 0
3.
F (t)
for
t < 0 (satis ed
for many physi al parameters - ausality)
should be integrable in the interval
[0, T ],
and for
t> T
it should
hold that
|F (t)| < eγt
with real
γ.
f (s) of F (t) s > γ ( onvergen e half-plane ). All limited fun tions (α > 0), sin βt et . have an L-transform but also non-limited −1/2 n , t and eαt (n, α > 0). Note assumption 2. Many fun tions fun tions as t −1 t2 in physi s also have an L-transform. The fun tions t and e do not have an These are su ient onditions for the existen e of the L-transform
s
for omplex −αt as, e.g., e
with Re
L-transform.
A.1.4 Examples a)
0 for t < 0 H(t) = Heaviside step fun tion (unit step) 1 for t ≥ 0 Z ∞ ∞ 1 1 f (s) = e−st dt = − e−st 0 = for Re s > 0 ( onverg. half-plane) s s 0 1 H(t) ◦−• s F (t)
=
b)
F (t) f (s)
= =
Z
0
eδt · H(t) For
δ=0
◦−•
0 eδt ∞
for for
t<0 t≥0
e−(s−δ) dt = −
1 s−δ
∞ 1 1 e−(s−δ)t = , s−δ s−δ 0
transition to the L-transform of
Re s > δ
H(t)
)
sin at 1 . · H(t) ◦−• 2 a s + a2 Tables of many more orresponden es an be found in the literature given.
A.1.
181
INTRODUCTION TO THE LAPLACE TRANSFORM
A.1.5 Properties of the Lapla e transform Similarity theorem a > 0 F (at) ◦−•
Z
∞
e
−st
F (at)dt =
0
Therefore,
Z
∞
e
s −a at
0
d(at) 1 F (at) = a a
Z
∞
0
1 s F (at) ◦−• f . a a
Thus, only the L-transform of
F (t)
s
e− a τ F (τ )dτ
(A.1)
has to be known.
Example: A
ording to se tion A.1.4
et · H(t) ◦−•
1 . s−1
With the similarity theorem, it follows that
eat · H(t) ◦−•
1 a
s a
1 1 = , −1 s−a
i.e., the result of the dire t omputation in se tion A.1.4.
Displa ement theorem
Fig. A.1: Displa ement theorem.
F (t − ϑ) ◦−•
Z
F (t − ϑ) ◦−• e
∞
0 −ϑs
e
−st
F (t − ϑ)dt =
f (s)
Z
∞
e−s(τ +ϑ) F (τ )dτ = e−ϑs f (s)
0 (A.2)
Damping theorem (α arbitrary omplex) e
−αt
F (t) ◦−•
Z
∞
e−(s+α)t F (t)dt = f (s + α)
0
e−αt F (t) ◦−• f (s + α)
(A.3)
182
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
Dierentiation theorem F ′ (t) ◦−•
∞
Z
e−st F ′ (t)dt = e−st F (t)|∞ 0 +s
0
Z
∞
e−st F (t)dt
0
The rst term is zero at its upper limit due to the assumption 3 from hapter A.1.3. Thus,
F ′ (t) ◦−• sf (s) − F (+0)
F (+0)
=
lim t→0 t>0
right
F(t) is the limit from the
side.
(A.4)
Generalisation:
F ′ (t) F ′′ (t) . . .
F
(n)
◦−• sf (s) − F (+0) ◦−• s2 f (s) − sF (+0) − F ′ (+0)
(A.5)
(t) ◦−• s f (s) − s F (+0) − s F (+0) (n−2) (n−1) − . . . − sF (+0) − F (+0) n
n−1
n−2
′
Integration theorem G(t) =
Z
0
t
1 F (τ )dτ ◦−• f (s) s
(A.6)
From this, it follows that
G′ (t) = F (t) ◦−• f (s) − G(+0) = f (s).
Convolution theorem Z
0
t
F1 (τ )F2 (t − τ )dτ ◦−• f1 (s)f2 (s)
The integral is alled onvolution of Furthermore, it holds that
F1
with
F2 ,
symboli notation
(A.7)
F1 ∗ F2 .
A.1.
183
INTRODUCTION TO THE LAPLACE TRANSFORM
Z
0
t
F1 (τ )F2 (t − τ )dτ =
Z
t
0
F1 (t − τ )F2 (τ )dτ
or
F1 ∗ F2 = F2 ∗ F1 , i.e., the onvolution is ommutative. Further elementary properties of the L-transform are that it is rstly homogeneous and linear, i.e., it holds that
a1 F1 (t) + a2 F2 (t) ◦−• a1 f1 (s) + a2 f2 (s), and se ondly, that from
F (t) ≡ 0
it follows that
f (s) ≡ 0
and vi e versa.
An important property, whi h follows from the de nition of the L-transform, is
lim
Re s→+∞ Only
then
is a fun tion
f (s)
f (s) = 0.
(A.8)
an L-transform and an be transformed ba k (see
next hapter).
A.1.6 Ba k-transform −1
F (t) = L
1 {f (s)} = 2πi
Z
c+i∞
ets f (s)ds
(A.9)
c−i∞
Fig. A.2: Convergen e half-plane of Lapla e inverse-transform. The integration path is parallel to the imaginary axis and has to be situated in the onvergen e half-plane of the integration path, the left.
f (s)
f (s),
otherwise,
is arbitrary. To the right of
annot have singularities, but it an have them to
The integration path an be deformed in a
ordan e with Cau hy's
integral law and the remainder theorem.
184
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
A.1.7 Relation with the Fourier transform A ommon representation of the Fourier transform
F (ω) =
+∞
Z
F (ω)
of a fun tion
F (t)e−iωt dt.
F (t)
is
(A.10)
−∞
F (ω)
is also alled the
omplex spe trum
of
F (t); ω
is the angular frequen y.
The inverse-transform is given by
1 F (t) = 2π
Z
+∞
F (ω)eiωt dω.
(A.11)
−∞
This equation an be interpreted as the superposition of harmoni os illations. This is the reason why the Fourier transform is often used in physi s. If the behaviour of a system, whi h an be des ribed by linear dierential equations, is known for harmoni ex itation, its behaviour for impulsive ex itation an be determined via (A.11).
To do this, the ex itation has to be broken into its
spe tral omponents a
ording to (A.10). Then, the problem is solved for ea h spe tral omponent, and, nally, all spe tral solutions are superimposed via (A.11). Su h an approa h is used in se tion 3.6.3 in the study of the re e tion of impulsive waves at an interfa e. It is often less physi al, but often more elegant and simple, to use the Ltransform. The onne tion between
F (t)
that are zero for
t<0
F (ω)
and
f (s)
is very lose for fun tions
( ausality)
F (ω) =
Z
∞
e−iωt F (t)dt = f (iω),
0
i.e., the Fourier transform is also the L-transform on the imaginary axis of the
omplex s-plane.
1 In an alternative representation of the Fourier transform, the fa tor 2π is not in (A.11) but in (A.10). Then the Fourier transform F (ω) is equal to f (iω)/2π .
A.2 Appli ation of the Lapla e transform A.2.1 Linear ordinary dierential equations with onstant
oe ients Dierential equation
L(Y ) = F (t) ≡
Y (n) + an−1 Y (n−1) + an−2 Y (n−2) + . . . + a1 Y ′ + a0 Y = F (t) (A.12) 0 for t < 0
A.2.
185
APPLICATION OF THE LAPLACE TRANSFORM
Initial onditions
(n−1)
Y (+0) = Y0 , Y ′ (+0) = Y0′ , . . . , Y (n−1) (+0) = Y0 Y (t) ◦−• y(s), F (t) ◦−• f (s) + . . . + a1 s + a0 y(s) = f (s)
L-transform of (A.12) with
n
s + an−1 s
n−1
and (A.5)
+ sn−1 + an−1 sn−2 + . . . + a2 s + a1 Y0 + sn−2 + an−1 sn−3 + . . . + a3 s + a2 Y0′ +... (n−2)
+ (s + an−1 ) Y0 (n−1)
+Y0
.
The polynomials an be written as
pi (s) =
n−i X
ak+i sk , i = 0, 1, 2, . . . , n, an = 1.
k=0 Therefore,
n
X pl (s) (l−1) f (s) + Y . p0 (s) p0 (s) 0
y(s) =
(A.13)
l=1
The right side ontains the L-transform of the known fun tion
F (t),
some poly-
nomials, the oe ients of whi h are known and the known initial values of the fun tion
Y (t) to be solved for.
If it is possible to determine the inverse transform
on the right side of (A.13), the problem is solved. In the ase presented here, this is not di ult.
Before this is done, we dis uss the omparison with the
standard method to solve linear ordinary dierential equations with onstant
oe ients. First, in the standard method the homogeneous dierential equation is solved generally (i.e., it ontains n undetermined oe ients); then a spe ial solution of the inhomogeneous dierential equation is determined, e.g., by guessing or by variation of the onstants.
This is then the general solution of the inho-
mogeneous dierential equation from whi h the n onstants an be determined via the initial onditions. It is not ne essary to nd a general solution if the L-transform is used. Here, the solution that orresponds to the initial onditions is determined dire tly. That is the great advantage of this method. This advantage is even greater for the solution of partial dierential equations with boundary and initial onditions and is why the L-transform is widely used. This is espe ially true in ele troni s. One onsequen e of this is that extensive tables with inverse transforms for many L-transforms exist. The inverse-transform of (A.13) an be split into two steps (but that is not a ne essity).
186
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
(n−1)
f (s) 6= 0, Y0 = Y0′ = . . . = Y0
1.
= 0.
This orresponds to the solution
of the inhomogeneous dierential equation with zero initial values (See se tion A.2.1.1).
f (s) = 0,
2.
initial value
6= 0.
This orresponds to the solution of the
homogeneous dierential equation with non-zero initial onditions (See se tion A.2.1.2).
The sum of solution 1 and 2 is the inverse transform of (A.13) to be determined (See se tion A.2.1.3).
A.2.1.1 Inhomogeneous dierential equations with zero initial values This ase is also of pra ti al interest sin e in many ases in whi h a system is zero up to time
t=0
Y (t) ≡ 0
(i.e.,
this ase,
for
y(s) = 1/p0 (s)
Sin e
for
exists
n≥1
t < 0),
the initial values are zero. In
f (s) . p0 (s)
is always an L-transform ( ompare (A.8)), the inverse
1 •−◦ Q(t). p0 (s) Q(t)
is the
Green's fun tion
then gives the solution
Y (t) =
Z
t
0
The determination of
of the problem.
The onvolution theorem (A.7)
Y (t) F (t − τ )Q(τ )dτ =
Q(t)
Z
t
0
F (τ )Q(t − τ )dτ.
is, therefore, the remaining task. To that end, we
introdu e an expansion into partial fra tions of that the zeros
αk
of
p0 (s)
are all
dierent
1/p0 (s)
under the assumption
n
X dk 1 = p0 (s) s − αk k=1
where
dk
is the residue of
dk = lim
s→αk
(A.14)
1/p0 (s)
at the lo ation
s − αk = lim s→αk p0 (s)
αk
1 p0 (s)−p0 (αk ) s−αk
=
1
. p′0 (αk )
A.2.
187
APPLICATION OF THE LAPLACE TRANSFORM
Thus,
n
X 1 1 1 = · ′ p0 (s) p0 (αk ) s − αk k=1
.
Inverse-transform, with a result from se tion A.1.4, gives
Q(t) = H(t) ·
n X eαk t . ′ p0 (αk )
(A.15)
k=1
αk is real, the orresponding summand in Q(t) is also real. If αk is omplex, α1 with α1 = α∗k (the onjugate omplex value to αk ) exists as part of the other zeros, sin e p0 (s) has real oe ients. Then,
If
an
∗
eαk t eαk t + ′ ∗ ′ p0 (αk ) p0 (αk )
∗
= =
Q(t)
eαk t e(αk t) + ′ ′ p0 (αk ) p0 (αk )∗ αk t ∗ eαk t e eαk t = 2Re ′ + . ′ ′ p0 (αk ) p0 (αk ) p0 (αk )
is, therefore, always real.
Relation to the usual solution method The determination of the zeros of
p0 (s)
is ompletely identi al to the determi-
nation of the zeros for the hara teristi equation
p0 (λ) = 0 of the homogeneous
dierential equation. The eort involved is, therefore, the same. For the usual method, the additional eort of nding a spe ial solution of the inhomogeneous dierential equation and the determination of n onstants in the solution of the homogeneous dierential equation from the zero initial onditions is needed. It is also interesting to see under whi h onditions on
F (t)
the initial values of
the solution
Y (t) =
Z
0
t
F (τ )Q(t − τ )dτ
are indeed equal to zero. One an show that
sk+1 s→∞ p0 (s)
Q(k) (+0) = lim and thus,
(k = 0, 1, . . . , n − 1),
188
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
Q(+0) = Q′ (+0) = . . . = Q(n−2) (+0) = 0, If this is used during the dierentiation of
Y (t),
Q(n−1) (+0) = 1.
it follows that
Y (+0) = Y ′ (+0) = . . . = Y (n−2) (+0) = 0, and under the ondition that
Z
lim t→0 t>0 it also holds that
Y (n−1) (+0) = 0.
t
F (τ )dτ = 0
(A.16)
0
This means that due to the fa t that a phys-
i al fun tion in general satis es (A.16) (as long as they have a de ned start), the assumption of zero initial values is most often satis ed. Equation (A.14) with (A.15) is then the solution of the problem. An ex eption an be found in exer ise A.2.
Appli ation example The dierential equation of the me hani al resonator an be written as
1 Y¨ + 2αω0 Y˙ + ω02 Y = K(t) m with
Y (t) = K(t) =
displa ement from zero a ting for e ( = 0 for t < 0 )
m α
= =
ω0 ω
= eigen frequen y of the undamped resonator 1 = ω0 (1 − α2 ) 2 eigen frequen y of the damped
We hoose
α<1
mass damping term
(α = 1 :
aperiodi limit)
resonator ase ).
(
The L-transform of the dierential equation leads to
y(s) =
k(s) mp0 (s)
resonator
.
A.2.
APPLICATION OF THE LAPLACE TRANSFORM
p0 (s)
=
α1
=
α2 ′ p0 (s) ′ p0 (α1 )
= = =
189
s2 + 2αω0 s + ω02 = (s − α1 )(s − α2 ) 1 −ω0 α + i(1 − α2 ) 2 = −αω0 − iω −αω0 + iω 2s + 2αω0
−2iω = −p′0 (α2 ).
Thus, the Green's fun tion an be written as
Q(t) = = Q(t) = Q(+0) =
1 −αω0 t−iωt −e + e−αω0 t+iωt · H(t) 2iω e−αω0 t iωt e − e−iωt · H(t) 2iω 1 −αω0 t e sin ωt · H(t) ω 0, Q′ (+0) = 1.
The solution of the dierential equation is, therefore,
Y (t) = =
(t ≥ 0)
Z t 1 K(t − τ )e−αω0 τ sin ωτ dτ ωm 0 Z t 1 K(τ )e−αω0 (t−τ ) sin ω(t − τ )dτ. ωm 0
(A.17)
If the polynomial p0 (s) has several zeros, an extension into partial fra tions of 1/p0 (s) is also possible, but it looks dierent as if only simple zeros were present. Thus, the orresponding Green's fun tion Q(t) looks dierent ( ompare also the usual method of solution). Equation (A.14) is also valid in this ase.
Exer ise A.1 Give the solution of the inhomogeneous equation
L(Y ) = ω02 Y0 H(t), where
H(t)
is the Heaviside step fun tion.
Exer ise A.2 Solve the dierential equation of the me hani al seismograph
¨ L(Y ) = −X
(X(t) = ground
displa ement)
190
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
with the aid of the method of the variation of the onstants and with the Ltransform. Assume
X(t) ≡ 0
for
t < 0,
Derive the initial onditions for
Y (+0) = 0, Y˙ (+0) = −V0 . In both exer ises,
X(+0) = 0,
Y (t)
˙ X(+0) = V0 .
from physi al prin iples and show that
L(Y ) = Y¨ + 2αω0 Y˙ + ω02 Y, α < 1.
A.2.1.2 Homogeneous dierential equations with arbitrary initial values de ay of os-
This ase has also a pra ti al appli ation sin e it des ribes the
illations
of physi al systems. The important points an be learned from the
following exer ise.
Exer ise A.3 Solve the dierential equation of the eigen os illation of a me hani al resonator with
L(Y ) = 0, Y (+0) = Y0 , Y˙ (+0) = 0 using the L-transform, and with the solution L(Y ) of exer ise A.1 as done above.
with the initial onditions
ompare the solution
A.2.1.3 Inhomogeneous dierential equations with arbitrary initial values We superimpose the solutions of se tion A.2.1.1 and se tion A.2.1.2. This means that
Y (t)
onsists of the two ontributions
Y (t) = Y1 (t) + Y2 (t). Y1 (t)
is the solution of the homogeneous dierential equation that satis es the
initial onditions
(n−1)
Y1 (+0) = Y0 , Y1′ (+0) = Y0′ , . . . , Y1 Y2 (t)
(n−1)
(+0) = Y0
.
is the solution of the inhomogeneous dierential equation with zero initial
values
A.2.
(n−1)
Y2 (+0) = Y2′ (+0) = . . . = Y2 Y (t)
191
APPLICATION OF THE LAPLACE TRANSFORM
(+0) = 0.
satis es the dierential equation and the given initial onditions, and is,
therefore, the solution of the problem. For physi al problems, the initial onditions
always
have to be derived from
physi al prin iples, for example: 1. Me hani al resonator: For
t < 0, Y = Y0 (t)
Due to the
Y¨ + 2αω0 Y˙ + ω02 Y =
is given. At time
ontinuity requirement
t=0
1 m K(t) the for e
K(t)
begins to a t.
it must, therefore, hold that
Y (+0) = Y0 (−0), Y˙ (+0) = Y˙ 0 (−0). The resonator starts at time
t=0
(A.18)
with the initial values whi h onne t
ontinuously to the previous values. The ontribution of pla ement
Y (t)
eigen resonan e, whi h ontinues the os illation tion
Y2 (t),
Y1 (t)
to the dis-
has the initial value given in (A.18) and is, therefore, an
Y0 (t).
The for ed os illa-
given in (A.17), with zero initial values, is then superimposed
on that os illation.
¨ Y¨ + 2αω0 Y˙ + ω02 Y = −X rest until the time t = 0. Due
2. Me hani al seismograph: The ground may be at
of ontinuity, it follows that
to the
requirement
˙ Y (+0) = 0, Y˙ (+0) = −X(+0). Homogeneous equation:
˙ Y¨1 + 2αω0 Y˙ 1 + ω02 Y1 = 0, Y1 (+0) = 0, Y˙ 1 (+0) = −X(+0) L-transform:
˙ s2 + 2αω0 s + ω02 y1 (s) = −X(+0)
Similar to se tion A.2.1.1, the eigen resonan e an be written as
Y1 (t) = −
˙ X(+0) e−αω0 t sin ωt · H(t). ω
Inhomogeneous equation:
¨ Y2 (+0) = Y˙ 2 (+0) = 0 Y¨2 + 2αω0 Y˙ 2 + ω02 Y2 = −X, L-transform:
˙ s2 + 2αω0 s + ω02 y2 (s) = − s2 x(s) − X(+0)
The for ed resonan e, therefore, is (
Y2 (t) = − The omplete solution
1 ω0
Z
0
t
t ≥ 0)
¨ )e−αω0 (t−τ ) sin ω(t − τ )dτ. X(τ
Y (t) = Y1 (t) + Y2 (t)
is the same as in exer ise A.2.
192
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
A.2.2 Partial dierential equations We now use an example to demonstrate the main points dis ussed so far. We will examine, unlike in se tion 3.4, the propagation of a ompressional wave from an explosive point sour e. The starting point is the equation of motion of the elasti ontinuum without body for es.
→ u ∂ 2− → → = (λ + 2µ)∇ ∇ · − u − µ∇× ∇ × − u ∂t2 µ=Lamé's parameters).
ρ (ρ=density,
λ
and
In our problem, for whi h we use spheri al oordinates, the displa ement only a radial omponent
U,
∂U + ∂r 2 ∂ U ∂r2
→ ∇·− u = → ∇∇ · − u = α2 = (λ + 2µ)/ρ
− → u has
and the only spatial oordinate is the distan e
from the explosive point sour e. In this ase,
With
(A.19)
→ ∇×− u
r
is zero and it holds that
2 U r 2 ∂U 2 + − 2 U, 0, 0 . r ∂r r
(=velo ity of the ompressional waves), it follows from
(A.19) that
2 ∂U 1 ∂2U 2 ∂2U + U − = 0. − ∂r2 r ∂r r2 α2 ∂t2 The
boundary onditions
assumed are that for
r = r1
(A.20)
the displa ement is pre-
s ribed as
U (r1 , t) = U1 (t). The
initial onditions
are
U (r, 0) = U1 (t),
(A.21)
whi h shall be zero for
∂U (r, 0) = 0. ∂t
t < 0, has r = r1 .
to start smoothly, so that the initial
onditions are also satis ed for L-transform then gives
u(r, s) =
Z
0
u1 (s) =
Z
0
∞
∞
e−st U (r, t)dt e−st U1 (t)dt
(A.22)
A.2.
∂U L = ∂r 2 ∂ U = L ∂r2
Z
∞
e
−st ∂U
0
∂ dt = ∂r ∂r
∂2 u(r, s). ∂r2
Z
∞
e−st U dt =
0
With this and (A.20), equation (A.20) leads to an for
193
APPLICATION OF THE LAPLACE TRANSFORM
ordinary
∂ u(r, s) ∂r
dierential equation
u(r, s) d2 u 2 du + − dr2 r dr
s2 2 + 2 2 r α
u = 0.
(A.23)
For ordinary dierential equations, the L-transform leads to an algebrai equa-
t
tion (polynomials). For partial dierential equations in whi h, together with , only
one
additional oordinate o
urs (the ase studied here), the L-transform
leads to ordinary dierential equations. whi h, in addition to
t i, more than one
For partial dierential equations in
oordinate o
urs, partial dierential
equations are derived. In ea h ase, the dependen e on We hange the variables in (A.23) to
x=
t
is eliminated.
rs α
d2 u s2 d2 u = 2 · 2. 2 dr dx α
du s du = · , dr dx α Thus, (A.23) be omes
x2
du d2 u + 2x − x2 + 2 u = 0. 2 dx dx
This is a spe ial ase of the dierential equations of the
fun tions
x2
(A.24)
modi ed spheri al Bessel
dy d2 y + 2x − x2 + n(n + 1) y = 0. dx2 dx
Compare, e.g., M. Abramovitz and I.A. Stegun: Handbook of Mathemati al Fun tions, H. Deuts h, Frankfurt, 1985. In our ase,
n = 1,
and the solution of (A.24), whi h has the properties (A.8)
of L-transforms, is
1 u(x) = x
1 1+ e−x · F (s), x
x=
rs . α
(A.25)
As will be ome lear in the following, the integration onstant
F (s) is important.
r = r1 ,
has to be ome the
We now spe ify (A.25) for
i.e.,
x = r1 s/α,
thus,
u(x)
194
APPENDIX A.
known L-transform
LAPLACE TRANSFORM AND DELTA FUNCTION
u1 (s)
of the displa ement
U1 (t)
given at the limit
r = r1
(see (A.21))
α u1 (s) = r1 s From this,
F (s)
r1 s α 1+ e− α F (s). r1 s
an be derived. Therefore, (A.25) an be written as
u (r, s) =
α r−r r1 1 + rs − α1s u1 (s) α e r 1 + r1 s
" s + rα1 + αr − r1 = · r s + rα1
α r1
#
u1 (s) · e−
r−r1 α
(A.26)
s
.
The term in the square bra ket an now be given as
u1 (s) + α
1 1 − r r1
u1 (s) •−◦ U1 (t) + α s + rα1
1 1 − r r1
i h α − t U1 (t) ∗ e r1 · H(t) .
In the last step, the onvolution theorem (A.7) was applied. If the displa ement theorem (A.2) is used, the inverse transform of (A.26) follows as
# " Z t− r−r1 r−r α r1 r − r1 1 1 − rα (t− α 1 −ϑ) U1 (ϑ)e 1 dϑ . U (r, t) = U1 t − +α − r α r r1 0 (r − r1 )/α refers here not to the explosion point sour e, but to r = r1 , from whi h the wave starts at time t = 0. The retarded time is, therefore, τ = t − (r − r1 )/α, and the arrival of the wave at ea h re eiver with r > r1 follows from τ = 0. Then The retardation the sphere
Z τ r1 1 1 − rα (τ −ϑ) 1 dϑ . U (r, t) = U1 (τ ) + α − U1 (ϑ)e r r r1 0
A.3 The delta fun tion δ(t) A.3.1 Introdu tion of δ(t) We examine the result (A.17) for the me hani al resonator
A.3.
δ(T )
THE DELTA FUNCTION
Y (t) =
for the for e
1 ωm
K(t) = Iδǫ (t),
Z
0
t
195
K(τ )e−αω0 (t−τ ) sin ω(t − τ )dτ,
where
I
is a onstant with the dimension of for e
time (=dimension of an impulse) and
δǫ (t) =
0
1 ǫ
0
for for for
×
t<0 0ǫ
is a square fun tion as shown in Fig. A.3.
δε ( t ) 1/ε1 1/ε2 1/ε3 ε1 ε2 ε3
Fig. A.3: Representation of
The area under the urve
ǫ,
δǫ (t)
δǫ (t)
always the same impulse
I
as square fun tions.
is always equal to 1. Therefore, independent of
is transfered. Then the displa ement for
be written as
Yǫ (t) = =
t
Z ǫ I e−αω0 (t−τ ) sin ω(t − τ )dτ ωmǫ 0 Z t I e−αω0 u sin ω u du. ωmǫ t−ǫ
t>ǫ
an
196
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
Fig. A.4: Behaviour of the term under integral in The mean value theorem for integrals gives (0
Yǫ (t) =
arbitrary)
lim Yǫ (t) =
where
< λ < 1):
I −αω0 (t−λǫ) e sin ω(t − λǫ). ωm
The next step is the transition to
ǫ→0
Yǫ (t).
ǫ → 0.
(A.27)
For (A.27) follows the result (t is
I −αω0 t I e sin ωtH(t) = Q(t), ωm m
Q(t) is the Green's fun tion of the dierential equation of the me hani al ǫ → 0 means
resonator ( ompare se tion A.2.1.1). For the for e, the transition
that the impulse is transfered to the resonator in shorter and shorter time. It is physi ally plausible, that this time small ompared to the de ay time
then
is not important, if it is su iently
(αω0 )−1 and the eigenperiod 2π/ω of the eigen
resonan e of the resonator. Therefore, it makes sense to adopt the limiting ase
ǫ=0
also for the for e, i.e.,
lim K(t) = I lim δǫ (t) = Iδ(t),
ǫ→0 where
δ(t)
is the
ǫ→0
delta fun tion δ(t) = lim δǫ (t).
(A.28)
ǫ→0
Other names are
impulse fun tion
or
unit impulse.
It is obvious that
δ(t)
an-
not be treated as a standard fun tion. On the other hand, it would be wrong to study the fun tion
ontrary,
δ(t)
δ(t)
separated from the ordinary fun tions
has to be understood as a series of
the mathemati al point of view,
tributions,
δ(t)
is part of the
{δǫ (t)}
with
generalised
δǫ (t). On the ǫ → 0. From
fun tions or
dis-
for whi h extensive theories and literature exist. For our purposes,
A.3.
THE DELTA FUNCTION
δ(T )
197
the physi al approa h to the delta fun tion given, will be su ient. The definition of
δ(t)
as a series of ordinary fun tions is the basis of an exa t theory
that an also be understood by non-mathemati ians; see, e.g., Riley, K.F., M.P. Hobson and S.J. Ben e: Mathemati al methods for physi s and engineering, A omprehensive guide, Cambridge University Press, Cambridge, 2nd edition, 2002.
A.3.2 Properties of δ(t) Due to (A.28)
δ(t) = 0
for
t 6= 0.
Furthermore, it follows from the properties of
Z
δǫ (t),
that
+∞
δ(t)F (t)dt = F (0).
−∞
The delta fun tion, therefore, is the value of
Z
F (t)
at
t = 0;
su h that
+∞
δ(t)dt = 1.
−∞
G(t)δ(t) = G(0)δ(t). If G(0) = 0, then G(t)δ(t) ≡ 0. δ(t − τ ), whi h is displa ed by τ , it holds that
Furthermore, it holds that For the delta fun tion
δ(t − τ ) = 0
for
t 6= τ
and
Z
+∞
−∞
The de nition of
δ(t)
δ(t − τ )F (t)dt = F (τ ).
is not only possible with the series
{δǫ (t)}, the fun tions {δn (t)} with
of whi h are dis ontinuous. An alternative option is the series
δn (t) =
n 12 π
2
e−nt .
198
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
δn( t )
t Fig. A.5: Representation of
δn (t).
In this ase,
δ(t) Z
=
lim δn (t)
n→∞
+∞
δn (t)dt
= 1.
−∞
With the fun tions
δn (t), the derivatives of the the delta fun tion an be de ned
as
δ (k) (t) = lim δn(k) (t). n→∞
δ’n( t )
t
Fig. A.6: Derivative of the delta fun tion
It holds that
δ (k) (t) = 0
for
t 6= 0.
δn (t).
Furthermore, it follows that
A.3.
THE DELTA FUNCTION
Z
+∞
−∞
δ (k) (t − τ )F (t)dt
δ(T )
= =
after
k
199
lim
n→∞
Z
+∞
−∞ k
lim (−1)
n→∞
δn(k) (t − τ )F (t)dt Z
+∞
−∞
δn (t − τ )F (k) (t)dt,
partial integrations. This gives
Z
+∞
δ (k) (t − τ )F (t)dt = (−1)k F (k) (τ ).
−∞
(A.29)
onne tion between the delta fun tion and the step fun tion H(t) from se tion A.1.4. We onsider the fun tion
Finally, we dis uss the
Hǫ (t) =
Z
t
δǫ (τ )dτ =
−∞
0 t ǫ
1
for for for
t<0 0≤t≤ǫ t>ǫ
H ε (t)
1
ε Fig. A.7: Fun tion
t
Hǫ (t).
This means that
δǫ (t) = Hǫ′ (t), and in the limit
ǫ→0 δ(t) = H ′ (t).
The delta fun tion is the
derivative of the step fun tion.
have been a hieved with the de nition of It should also be mentioned that
dimension of time.
H(t)
The same result ould
δ(t) with the use of the fun tions δn (t).
is dimensionless, but δ(t) has the inverse
200
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
A.3.3 Appli ation of δ(t) 1. The option to des ribe
impulses of for e
(and, similarly, of stress and ur-
rent) will not be dis ussed now sin e that was the topi of the introdu tion of this appendix. 2. A
point mass m density
(or similarly a point harge) an be des ribed by the
following
ρ = m δ(x)δ(y)δ(z). This holds be ause
ρ=0
for
be des ribed via
ZZZ
+∞
−∞
3. The
ρ dx dy dz = m ·
Z
(x, y, z) 6= (0, 0, 0),
+∞
−∞
δ(x)dx ·
Z
and the whole mass an
+∞
−∞
δ(y)dy ·
Z
+∞
δ(z)dz = m.
−∞
harge distribution of a point-like dipole an be des ribed by the line density σ(x) (e.g., Coulomb per meter) on the x -axis as
following
σ(x) = M δ ′ (x), sin e we de ne
σ(x)
M >0
(dimension :
by the series
harge * length
),
{M δn′ (x)}.
M δ’ n (x) + x -
Fig. A.8: Charge distribution of a point-like dipole. The transition
n → ∞
gives then two in nitely large opposite point
harges, whi h are in nitely lose to ea h other.
+ x=0
x
Fig. A.9: Charge distribution of a point-like dipole for two in nitely large opposite point harges, whi h are in nitely lose to ea h other.
A.3.
THE DELTA FUNCTION
δ(T )
201
The moment of su h an arrangement relative to
Z
+∞
xσ(x)dx = M
−∞
Z
+∞ −∞
x=0
is
xδ ′ (x)dx = −M.
The sign is orre t, sin e the ve tor of the moment points from the negative
spatial harge
to the positive harge. The dimension is also orre t. The ′ of the dipole would be σ(x, y, z) = M · δ (x) · δ(y)
density
Coulomb per ubi meter). 4.
· δ(z)
(e.g.,
The role of the delta fun tion for the solution of inhomogeneous linear ordinary dierential equations. We start with
L(Y ) = Y (n) + an−1 Y (n−1) + . . . + a1 Y ′ + a0 Y = δǫ (t).
(A.30)
The solution is, a
ording to se tion A.2.1.1,
Y (t) = Yǫ (t) =
Z
0
with
Q(t)=Green's
t
δǫ (τ )Q(t − τ )dτ,
(A.31)
fun tion, and it satis es the initial onditions
Yǫ (+0) = Yǫ′ (+0) = . . . = Yǫ(n−1) (+0) = 0. The transition
ǫ→0
in (A.30) and (A.31) gives
L(Y ) = δ(t)
(A.32)
with the solution
Y (t) = Q(t). The Green's fun tion of a system, whi h an be des ribed by a linear ordinary dierential equation, is also the solution of the inhomogeneous equation, whi h has the delta fun tion as the term of perturbation. Expressed
the Green's fun tion is the response fun tion of a perturbation of the system by the delta fun tion (impulse response).
dierently,
The initial values of
Q(t)
are
Q(+0) = Q′ (+0) = . . . = Q(n−2) (+0) = 0, Q(n−1) (+0) = 1, and are, therefore, dierent from those of the fun tions a onsequen e of the transition
ǫ → 0.
Yǫ (t).
This is
If (A.32) has, therefore, to be
202
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
solved dire tly, one has to use whi h a tually hold for L-transform of
Q(t).
always
δ(t) L {δ(t)} = f (s) =
δ(t) is a generalised fun tion, Re s → ∞ ( ompare (A.8).
Sin e for
zero initial values and not those,
In that situation, it is useful to know the
∞
Z
e−st δ(t)dt = 1.
0
it does
not
hold here that
lim f (s) = 0
Now we an give the general solution of the initial value problem for a
t = 0
system whi h is at rest up to the time
new interpretation.
arbitrary way, a
Y (t) =
Z
t
0
F (t − τ )Q(τ )dτ =
Z
t 0
F (τ )Q(t − τ )dτ,
is derived by the onvolution of the solution of the system by
F (t) = δ(t)
and is then ex ited in an
The solution (A.14), namely
(A.33)
Q(t) for the spe ial ex itation F (t).
with an arbitrary perturbation
A.3.4 Duhamel's law and linear systems In (A.33), we hoose
F (t) = H(t)
Y (t) = YH (t)
=
(step fun tion). In this ase,
Z
t
Q(τ )dτ
(step
response)
0
YH′ (t)
= Q(t).
Integrating by parts, it follows from (A.33)
Y (t) = F (t − τ ) Thus, due to
+
Z
t 0
F ′ (t − τ )YH (τ )dτ.
YH (+0) = 0 Y (t)
This is
t YH (τ )|0
= F (+0)YH (t)
+
= F (+0)YH (t)
+
Rt 0
Rt 0
F ′ (t − τ )YH (τ )dτ
(A.34)
F ′ (τ )YH (t − τ )dτ
Duhamel's law, whi h des ribes how solutions for arbitrary F (t) an be
determined from those for
F (t) = H(t).
A.3.
THE DELTA FUNCTION
δ(T )
203
Generalisation If we use
Q(t) = Yδ (t) Y (t) =
in (A.33), it follows that
Z
t
0
The relation between
F (t − τ )Yδ (τ )dτ =
Yδ (t)
and
YH (t)
t
Z
0
F (τ )Yδ (t − τ )dτ.
(A.35)
is
Yδ (t) = YH′ (t). Equations (A.34) and (A.35) ontain the statement that the response of a system has to be known only for very spe ial ex itations like the delta and the step fun tions. From this, the solution for arbitrary ex itation an be given. This is not only true for systems whi h follow linear
ordinary partial
but also for systems whi h an be des ribed by
dierential equations, dierential equations
simultaneous dierential equations as long as they are linear and have time independent oe ients. One requirement for this to hold is that the or systems of
system is at rest in the beginning.
The perturbation an, depending on the
problem, have a dierent form (e.g., for e, temperature, displa ement
et .),
indi ated in Fig. A.10.
output fun tion
input fun tion or
or
perturbation fun tion
response fun tion
δ(t) H(t) F (t)
Yδ (t) = Q(t) (impulse response) Rt YH (t) = 0 Q(τ )dτ (step response) Y (t) a
ording to (A.34) or (A.35)
Fig. A.10: Linear system with input and output.
Transition into the frequen y domain using the Fourier transform The Fourier transform of
F (t), Q(t)
and
Y (t)
is
F (ω), Q(ω)
F (ω) Z +∞ F (t) Q(t) e−iωt dt. = Q(ω) −∞ Y (t) Y (ω)
and
Y (ω)
as
204
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
The lose relation with the L-transform has been dis ussed in se tion A.1.7. Therefore, it holds, as for the Fourier transform
Y (ω) = F (ω) · Q(ω), Y (ω) of the input fun tion Y (t) is the produ t of F (ω) of the input fun tion F (t) with Q(ω) of the Green's
i.e., the Fourier transform the Fourier transform fun tion
Q(t). Q(ω) is alled the transfer
fun tion
of the linear system or lter.
Separation into absolute value and phase gives
Q(ω) = A(ω) = ϕ(ω) = A(ω)
A(ω)ei ϕ(ω) amplitude hara teristi s of the system phase hara teristi s of the system
des ribes the ampli ation or de rease of the ir ular frequen y
spe tively, and
ϕ(ω)
ω,
re-
des ribes the phase shift. A mono hromati os illation as
input
F (t) = a sin ωt, has the output
Y (t) = A(ω)a sin (ωt + ϕ(ω)) . The transfer fun tion of the system has, therefore, a very physi al meaning, and it is thus, used widely.
Exer ise A.4 A sphere of mass
m
drops from the height
h1
on the mass
M
(verti al-)resonator, is re e ted there and rea hes the height
of a me hani al
h2 .
No additional
intera tions between the two masses follow. Determine the displa ement of mass
M:
1. Using the homogeneous dierential equation
L(Y ) = 0
and the orre-
L(Y ) =?
and zero initial
sponding initial onditions. 2. Using the inhomogeneous dierential equation values.
It holds that
L(Y ) = Y¨ + 2αω0 Y˙ + ω02 Y,
α < 1.
A.3.
THE DELTA FUNCTION
δ(T )
205
Exer ise A.5 The ground displa ement of the me hani al seismograph (dierential equation
¨) L(Y ) = −X
is given by
a) X(t) =
t2 H(t) , 2
b)
X(t) = tH(t),
c)
X(t) = H(t).
Determine, in ea h ase, the displa ement
Y (t) and dis uss the relation between
the three ases.
A.3.5 Pra ti al approa h for the onsideration of non-zero initial values of the perturbation fun tion F(t) of a linear problem For the me hani al seismograph, the perturbation fun tion of the dierential
X(t), and we noti e Y (t) of the mass of the seismometer ˙ , respe tively ( ompare exer ise values X(+0) and X(+0)
equation is the se ond derivative of the ground displa ement that the initial values of the displa ement depends on the initial
A.2 and se tion A.2.1.3). prin iples.
This onne tion had to be derived from physi al
Cases exist in whi h this is di ult.
Therefore, we would like to
have an approa h, that onsiders the initial values of the perturbation fun tion. In the following, we de ne,
ontrary to the usage up to now,
the perturbation
fun tion as the fun tion, the single (or higher) derivatives of whi h o
ur in the dierential equation as inhomogeneities (we onsider an arbitrary linear system).
¨ . Now F (t) = X(t) for the me hani al seismograph and not F (t) = −X(t)
then solve the equation for up to
su iently
F (t) = Fn (t),
high orders. Then, one an assume that the initial values of
the orresponding solution
Yn (t)
Yn (t) = Therefore, we know the fun tion
i -th
are also zero. Thus,
Z
t 0
Yn (t)
an be written as
Fn(i) (t − τ )G(τ )dτ.
order of the derivative of
G(t).
Given a series of fun tions
F (t)
We
for whi h the initial values are zero,
(A.36)
Fn (t) (i ≥ 1)
and the
Fn (t) whi h onverge versus the perturbation fun tion
to be determined (with non-zero initial values),
206
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
lim Fn (t) = F (t).
n→∞
Compare also the omments to the de nition of the delta fun tion in hapters A.3.1 and A.3.2. Then, it follows that
lim Fn′ (t) =
F ′ (t) + F (+0)δ(t)
lim Fn′′ (t) =
F ′′ (t) + F (+0)δ ′ (t) + F ′ (+0)δ(t)
n→∞ n→∞
. . .
lim Fn(i) (t) =
F (i) (t) +
n→∞
δ (0) (t)
i−1 X
F (j) (+0)δ (i−j−1) (t).
(A.37)
j=0
is here equal to
δ(t).
Fig. A.11: F(t) and its derivative as a fun tion of time.
The general solution
Y (t) = =
Y (t)
for arbitrary initial values of
F (t)
is
lim Yn (t)
n→∞
Z
0
t
F (i) (t − τ )G(τ )dτ +
i−1 X
F (j) (+0)
Z
0
j=0
t
δ (i−j−1) (t − τ )G(τ )dτ,
if (A.37) is used in(A.36). Now
Z
0
t
δ
(i−j−1)
(t − τ )G(τ )dτ
= u=t−τ
Z
t 0
δ (i−j−1) (u)G(t − u)du
A.3.
THE DELTA FUNCTION
δ(T )
207
di−j−1 (−1) G(t − u) dui−j−1 i−j−1 u=0 d (−1)i−j−1 G(t − u) (−1)i−j−1 dti−j−1 u=0
= with (A.29)
i−j−1
=
G(i−j−1) (t).
=
The general solution of the problem for arbitrary initial values of the perturbation fun tion
F (t)
Y (t) =
is then
i−1 X
F (j) (+0)G(i−j−1) (t) +
Z
0
j=0
t
F (i) (t − τ )G(τ )dτ.
(A.38)
Appli ation Me hani al seismograph:
¨ Y¨ + 2αω0 Y˙ + ω02 Y = −X
The assumption that the ground displa ement and allows us to put the initial values of
Yn (t)
Xn (t)
starts su iently smooth
equal to zero
Yn (+0) = Y˙ n (+0) = 0. The dierential equation is then solved under this assumption, most easily with the L-transform ( ompare exer ise A.2)
Yn (t) = − Compare with (A.36):
1 ω
Z
0
t
¨ n (t − τ )e−αω0 τ sin ωτ dτ. X
Fn (t) = Xn (t), i = 2, G(t) =
1 − e−αω0 t sin ωt · H(t). ω
The general solution is, a
ording to (A.38),
Y (t) =
˙ X(+0)G′ (t) + X(+0)G(t) +
Z
t
0
=
˙ X(+0)G (t) + X(+0)G(t) + ′
Z
0
This result (with
X(+0) = 0)
exer ise mentioned above.
t
¨ − τ )G(τ )dτ X(t ¨ )G(t − τ )dτ. X(τ
was derived dire tly, ex ept for the sign, in the
208
APPENDIX A.
LAPLACE TRANSFORM AND DELTA FUNCTION
Appendix B
Hilbert transform B.1 The Hilbert transform pair The
Hilbert transform H(x) of the real fun tion h(x) is de ned by the following
integral (x and
ξ
are real)
1 H(x) = P π
P
is the
main value
+∞
Z
−∞
h(ξ) dξ. ξ−x
of the integral, i.e., the singularity
(B.1)
ξ=x
of the integrand
has been ex luded
P
Z
+∞
..dξ = lim
ǫ→0
−∞
The
inverse Hilbert transform
x−ǫ
Z
..dξ +
−∞
Z
+∞
x+ǫ
..dξ .
an be written as (proof follows)
1 h(x) = − P π
Z
+∞
−∞
H(ξ) dξ. ξ−x
(B.2)
Although this is dierent from the Lapla e and the Fourier transform, the two
orresponding fun tions Some
analyti al
h(x)
and
H(x)
have the
same argument.
Hilbert transform pairs are shown in Fig. B.1. 209
210
APPENDIX B.
HILBERT TRANSFORM
Fig. B.1: Analyti al Hilbert transform pairs.
B.2 The Hilbert transform as a lter Equation (B.1) is a onvolution integral
Pε (x) +ε
−ε
Fig. B.2: Form of
x
Pǫ (x).
H(x) =
Z
+∞
−∞
h(ξ) · P
− π1 x−ξ
1 dξ = h(x) ∗ P − πx
B.2.
THE HILBERT TRANSFORM AS A FILTER
1 P − = πx Pǫ (x)
=
211
lim Pǫ (x) ǫ→0 for |x| < ǫ 0 1 − πx otherwise .
Therefore, it holds for the Fourier transforms
H(ω) Z +∞ H(x) h(x) · e−iωx dx, = h(ω) 1 −∞ P − πx P (ω)
(B.3)
and a
ording to se tion A.3.4,
H(ω) = h(ω) · P (ω). The Hilbert transform is, therefore, a
linear lter.
(B.4)
The Fourier transform and
its inverse an be ee tively al ulated with the method of the Fast Fourier transform. It is, therefore, advantageous to perform the Hilbert transform in the frequen y domain via (B.4). To be able to do this, one needs the
fun tion P (ω) of the Hilbert transform. 1 P (ω) = − P π
Z
+∞
−∞
transfer
From (B.3), it follows that
1 −iωx e dx, x
P (0) = 0.
(B.5)
We ompute this integral with methods from omplex analysis by deforming the integration path to a semi- ir le with in nite radius in the upper (lower)
x -half-plane for ω < 0 (ω > 0), respe tively,
Fig. B.3: Integral path of
P (ω)
in the omplex plane.
212
APPENDIX B.
P
Z
+∞
−∞
e−iωx dx = x
Z
Z
=
HILBERT TRANSFORM
e−iωx e−iωx dx ± πiRes x x x=0
U/L
e−iωx dx ± πi, x
where the upper (lower) integration path and the + (-) sign for have to be hosen, respe tively.
(B.6)
ω<0
(ω > 0)
Note that the rst term on the right of the
rst equation has to be integrated along the real axis (ex luding the pole), and the residual in the se ond term is identi al to 1. The integration in the se ond equation is then along the upper (lower) half ir le U (L), respe tively. With the new variable
ϕ
on the half ir les, it follows that
x = Reiϕ , dx = Rieiϕ dϕ. This leads to
Z
U/L
e−iωx dx x
=
i
=
i
Z
0
exp [−iωR (cos ϕ + i sin ϕ)] dϕ
±π Z 0
±π
→ 0
for
exp [ωR sin ϕ − iωR cos ϕ] dϕ R → ∞,
sin e
ω sin ϕ < 0.
Equation (B.6), therefore, redu es to
P and the transfer fun tion
Z
+∞ −∞
P (ω)
P (ω) = i sign ω
e−iωx dx = ±πi, x
in (B.5) be omes the simple expression
with
−1 sign ω = 0 +1
for for for
ω<0 ω=0 ω > 0.
(B.7)
If the Hilbert transform is onsidered as a lter of the original fun tion, it follows from (B.4) with (B.7) that the frequen y 0 is suppressed (P (0)
other frequen ies remain un hanged in their amplitude ( P (ω)
ω 6= 0 only phase shifts ±900 results, respe tively. At
are produ ed. With
ω > 0 (ω < 0)
= 0), but all = 1 for ω 6= 0).
a phase shift of
In lter theory, the Hilbert transform is an
lter with removal of the average.
The pra ti al omputation of the Hilbert transform requires three steps:
H(x)
of
h(x),
all-pass
therefore,
B.2.
213
THE HILBERT TRANSFORM AS A FILTER
1. Computation of the Fourier transform
h(ω)
2. Multipli ation with the transfer fun tion 3. Ba k transformation of
of
h(x)
P (ω)
H(ω).
If the Hilbert transform is applied
twi e, it follows in the frequen y domain that 2
g(ω) = h(ω) · P (ω) = −h(ω), and, therefore,
g(x) = −h(x).
The original fun tion
h(x) is, therefore, obtained,
if the sign of the se ond Hilbert transform is reversed. This proves (B.2) for the inverse Hilbert transform.
h(0) = 0, i.e., in ases for whi h the h(x) is zero. The third example in Fig. B.1 is su h a ase. Equation holds if h(0) 6= 0. This is shown in the se ond example of Fig. B.1
This proof only holds for ases in whi h integral over (B.2) also
and an be on rmed with methods from omplex analysis.
The numeri al Hilbert transform, with (B.4) and (B.7) frequen y
ω = 0,
is
sometimes not treated properly. For numeri al reasons, it is assumed that due
P (0) = 0 the integral of the Hilbert transform is always zero. This is not R +∞ h(0) = −∞ h(x)dx is not nite or not orre tly de ned. The rst ase o
urs, if, e.g., h(x) is a step fun tion. The se ond ase o
urs, e.g., during 2 2 the inverse-transformation of the Hilbert transform h(x) = −ax/(a + x ), the de ay of whi h with in reasing |x| is proportional to −1/x and, therefore, not to
true, if
strong enough.
In su h ases, a onstant shift of the numeri al result in the
ordinate dire tion is often su ient. The frequen y
ω=0
is the only frequen y
for whi h the Hilbert transform omputed numeri ally an then deviate from the exa t result.
214
APPENDIX B.
HILBERT TRANSFORM
Appendix C
Bessel fun tions In the following, only the most important equations and properties of Bessel fun tions with
integer order
are listed. More details an be found, e.g., in M.
Abramovitz and I.A. Stegun (1985), or in Riley, K.F., M.P. Hobson and S.J. Ben e (2002). The
dierential equation
of the Bessel fun tion of integer order
n = 0, 1, 2, . . . is
x2 y ′′ + xy ′ + x2 − n2 y = 0.
(C.1)
The two linearly independent solutions of this equation are
y
=
y
=
Jn (x) =
n − order Bessel fun tion of se ond kind and n − th order Yn (x) = ′ or N eumann s f unction of n − th order. Bessel fun tion of rst kind and
Representation as a series Jn (x)
=
∞ X
k=0
Yn (x)
=
(−1)k x n+2k k!(n + k)! 2
n−2k n−1 x 1 X (n − 1 − k)! 2 2 0, 577216 + ln Jn (x) − π 2 π k! x k=0
∞ 1 X (−1)k (Φk + Φk+n ) x n+2k − π k!(n + k)! 2 k=0
Φl
=
l X s=1
1 s 215
216
APPENDIX C.
The graphi representation for
x≥0
BESSEL FUNCTIONS
is shown in Fig. C.1.
Fig. C.1: Bessel and Neumann fun tions.
Neumann's fun tions have a singularity at The
x = 0.
Hankel fun tions, or Bessel fun tions of the third kind, are de ned as Hn(1) (x) Hn(2) (x)
(1)
Hn (x)
and
= Jn (x) + iYn (x) = Jn (x) − iYn (x) (2)
Hn (x)
Hankel fun tion of rst kind Hankel fun tion of se ond kind
(C.2)
.
(C.3)
are linearly independent. The general solution of (C.1) is,
therefore, (with the arbitrary onstants A,B,C,D) either
y
=
AJn (x) + BYn (x)
=
CHn(1) (x) + DHn(2) (x).
or
y
Analogies to the dierential equations of the trigonometri fun tions (equation of os illation) y ′′ + n2 y = 0 or their well-known solutions tively,
cos nx
and
sin nx,
and
einx
and
e−inx ,
respe -
217
Bessel fun tions
Trigonometri fun tions
Jn (x) Yn (x) (1) Hn (x) (2) Hn (x)
cos nx sin nx inx e = cos nx + i sin nx e−inx = cos nx − i sin nx.
Asymptoti representation for x ≫ 1 Jn (x)
≃
Yn (x) (1) Hn (x) (2) Hn (x)
≃
≃
≃
1
2 2 πx 1 2 2 πx 1 2 2 πx 1 2 2 πx
cos x −
nπ 2 nπ 2
−
π 4 π 4
sin x − − nπ exp i x − 2 − π4 π exp −i x − nπ 2 − 4
Re ursion formulae (Zn = Jn , Yn , Hn(1) or Hn(2) ) 2n x Zn (x)
=
Zn−1 (x) + Zn+1 (x)
Zn′ (x)
=
n x Zn (x)
Zn′ (x)
=
− nx Zn (x) + Zn−1 (x)
(n = 1, 2, 3, . . .) (n = 0, 1, 2, . . .) (n = 1, 2, 3, . . .)
− Zn+1 (x)
(C.4)
(C.5)
Spe ial ases of the se ond and third re ursion formulae in (C.5) are then used
J0′ (x)
= −J1 (x)
J1′ (x)
= J0 (x) −
1 J1 (x). x
Up until now, we have onsidered the variable assume
|x| ≫ 1,
omplex
x
x
as real and positive. If we also
all formulae given also hold for
omplex, x
and for (C.4). If
are used, the following relations are often useful
Hn(1) (−x) = Hn(2) (−x) = with spe ial ase
−e−nπi Hn(2) (x) −enπi Hn(1) (x)
n=0 (1)
H0 (−x) (2) H0 (−x)
= −H02 (x) (1) = −H0 (x).
(C.6)
218
APPENDIX C.
BESSEL FUNCTIONS
Appendix D
The Sommerfeld integral We onsider a time harmoni explosion point sour e at the origin of a ylindri al
oordinate system. Its ompressional potential
1 iω(t− Rα ) e R
R2 = r 2 + z 2
solves the wave equation and an, therefore, be onstru ted from more elementary solutions of the wave equation in ylindri al oordinates (as long as
ylindri al symmetry is maintained); see also dis ussion in se tion 3.7 leading to (3.83)
1 iω(t− Rα ) e R
= e
l
To determine
g(k),
iωt
Z
g(k)kJ0 (kr)e−il|z| dk
(D.1)
0
=
ω2 − k2 α2
12
we onsider (D.1) at
1 −iω r α = e r and use then the
∞
Z
(positive
real or
negative imaginary).
z=0
∞
g(k)kJ0 (kr)dk
(D.2)
0
Fourier-Bessel transform g(k) =
Z
0
G(r)
=
Z
∞
G(r)rJ0 (kr)dr
(D.3)
g(k)kJ0 (kr)dk.
(D.4)
∞
0 219
220
g(k)
APPENDIX D.
is the Fourier-Bessel transform of
g(k)
THE SOMMERFELD INTEGRAL
G(r),
and
G(r)
is the inverse Fourier-
Bessel transform of . Equation (D.2) has the form of (D.4), therefore, e−iωr/α /r. Therefore, (D.3) an be used to ompute
g(k) =
Z
∞
g(k)
G(r) =
r
e−iω α J0 (kr)dr
0
=
Z
0
∞
Z ∞ r r J0 (kr)dr − i sin ω J0 (kr)dr. cos ω α α 0
With
Z
∞
Z
∞
0
0
r J0 (kr)dr cos ω α r J0 (kr)dr sin ω α
it follows that
or simply
g(k) =
0 − 12 = ω2 k2 − α 2 − 12 ω2 2 2 − k α = 0
− 1 −i ω22 − k 2 2 α g(k) = − 12 k 2 − ω22 α Z
0
∞
0
for
k>
for
0α
for
for
0
for
k>
1 il . Inserted into (D.1), this gives the
1 −iω R α = e R
for
ω α
ω α ω α
ω α
ω α,
Sommerfeld integral
k J0 (kr)e−il|z| dk. il
(D.5)
Appendix E
The omputation of modal seismograms E.1 Numeri al al ulations The treatment of point sour es in wave guides with
arbitrary
(horizontal) layer-
ing leads to the following general far- eld form for the eld values (displa ement, pressure, potential
et .)
of a normal mode
1
N (t) = r− 2 Re
Z
0
M (ω)
∞
M (ω) exp [i (ωt − kr)] dω.
onsists, in prin ipal, of fa tors that des ribe the
ex itation fun tion
(E.1)
sour e spe trum,
the
of the mode (depending on sour e depth, sour e orientation
ω ) and their eigen fun tion (amplitude-depth distribuk(ω) = ω/c(ω) ontains the dispersion information of the −1/2 (4.53) is a simple spe ial ase of (E.1) with M (ω) ∼ k (ω) .
and, in general, also on tion). Wavenumber mode. Equation
Integrals of the form (E.1) an be solved e iently with the help of the
Fourier transform.
Fast
In the ase of the ideal wave guide, for whi h the modal
seismograms omputed analyti ally are given in Fig. 4.3, the following numeri al result is derived for the potential (after a low-pass lter, whi h has de ayed to zero at the Nyquist frequen y). 221
222
APPENDIX E.
THE COMPUTATION OF MODAL SEISMOGRAMS
Fig. E.1: Modal seismogram for the ideal wave guide ( ompare to Fig. 4.23).
They agree very well with the analyti al seismograms in Fig. 4.33. If one is only interested in the study of dispersion on horizontal pro les, most times it is su ient to onsider in
M (ω) only the
es the studies, sin e then only the theory of the determination of
sour e spe trum. This simpli-
free
surfa e waves is needed (for
k(ω)).
E.2 Method of stationary phase The appli ation of the method of stationary phase in integrals of the type (E.1) has been des ribed in se tion 4.2.3. Here, it is shortly outlined again, sin e the results are needed as the basis for the treatment of the Airy phases in se tion E.3. It should be noted that today the results of this and the next se tion are not of great importan e in the numeri al omputation of modal seismograms, sin e the Fast Fourier transform mentioned in se tion E.1 is more suited for that purpose. Here, analyti al rules for the amplitude de ay of surfa e waves with in reasing distan e an be derived; this is an important addition to purely numeri al methods. The phase to
ω
ϕ(ω) = ωt − k(ω)r
in (E.1) has the following derivatives with respe t
(U = group velo ity)
ϕ′
=
ϕ′′ ϕ′′′
= =
r U −rk ′′ = rU −2 U ′ −rk ′′′ = r U −2 U ′′ − 2U −3 U ′ . t − rk ′ = r −
(E.2) (E.3) (E.4)
E.2.
223
METHOD OF STATIONARY PHASE
Stationary phase values follow from
ϕ′ (ω0 ) = 0
and are, therefore, determined
by
r U (ω0 ) = . t
(E.5)
U ′ (ω0 ) 6= 0,
(E.6)
Then
ϕ(ω)
an be approximated near
ω0
by
1 ϕ(ω) = ϕ0 + ϕ′′0 (ω − ω0 )2 2 (ϕ0 = ϕ(ω0 ), ϕ′′0 = ϕ′′0 (ω0 )).
The modal seismogram an then be written in the
stationary phase approximation
≃
x=
2
ω0 +∆ω
1/2
(ω − ω0 ).
21
|ϕ′′0 |
as
1 ′′ 2 dω r Re M (ω) exp i ϕ0 + ϕ0 (ω − ω0 ) 2 ω0 −∆ω ( ) 21 Z +∞ 2 − 21 iϕ0 ix2 signϕ′′ 0 r Re M (ω0 )e dx e |ϕ′′0 | −∞ Z
− 12
N (t) =
with
(E.7)
With (E.3), we nally derive
(E.8)
(U0 = U (ω0 ), U0′ =
U ′ (ω0 ), k0 = k(ω0 ) = ω0 /c(ω0 ))
U0 N (t) = r
2π |U0′ |
io n h π . Re M (ω0 ) exp i ω0 t − k0 r + sign U0′ 4
Equation (E.9) holds under the requirement (E.6). Then ne ted via (E.5) and` this produ es the
ω0 , t
frequen y modulation
and
mode. Its amplitude is also time dependent; this is mostly due to U0 and U0′ ( ).
also partially due to
amplitude modulation
r
(E.9)
are on-
of the normal
M (ω0 (t)) but
r
If we onsider the amplitudes of the normal mode as a fun tion of distan e , we −1 see that they de ay with r as long as (E.9) holds. This statement on erns the −1/2 amplitudes in the ; spe tral amplitudes de ay with r a
ording
time domain
to (E.1).
224
APPENDIX E.
THE COMPUTATION OF MODAL SEISMOGRAMS
E.3 Airy phases For realisti wave guides, one or several frequen ies exist for whi h the
velo ity
group
is stationary. In the following, we assume that ω0 is su h a frequen y. ′ It also holds that U (ω0 ) = 0, (E.6) is, therefore, violated and (E.9) no longer holds. A su ient approximation of the phase is, in this ase,
1 (ω − ω0 )3 ϕ(ω) = ϕ0 + ϕ′0 (ω − ω0 ) + ϕ′′′ 6 0
(E.10)
instead of (E.7). From (E.2) and (E.4), it follows that
ϕ′0 = t −
r , U0
−2 ′′ ϕ′′′ 0 = rU0 U0 .
(E.11)
no longer stationary at ω0
The phase is
but has a turning point there. The ϕ′0 hanges from negative > r/U0 and, thus, the se ond term in
third term in (E.10) has now to be onsidered sin e values
t < r/U0
to positive values for
t
(E.10) is not ne essarily dominant. In analogy to (E.8), the following approximation of the modal seismogram for times near
approximation )
N (t) =
≃ =
with
x=
Airy phase
an be derived (
Z ω0 +∆ω 1 r− 2 Re M (ω) ω0 −∆ω 1 3 dω (ω − ω ) · exp i ϕ0 + ϕ′0 (ω − ω0 ) + ϕ′′′ 0 6 0 Z +∞ 1 x3 r− 2 Re H · b · exp i ϕ′0 · b · x + sign ϕ′′′ dx 0 3 −∞ Z ∞ x3 ′′′ ′ − 12 cos sign ϕ0 · ϕ0 · b · x + 2r Re {H} · b · dx 3 0
|ϕ′′′ 0 | 2
r/U0
31
(ω − ω0 ), H = M (ω0 )eiϕ0
The integral an be expressed by the
1 Ai(z) = π whi h is shown in Fig. E.2.
Z
0
∞
and
b=
Airy fun tion
2 |ϕ′′′ 0 |
x3 dx, cos zx + 3
31
.
E.3.
225
AIRY PHASES
Ai(z)
0.5
-8
-4 -6
-2
0
2
4
z
-0.5
Fig. E.2: Airy fun tion.
With (E.11), the end result for the
N (t) =
2
2U0 |U0′′ |
Airy phase
an be written as
13
Re M (ω0 ) exp [i(ω0 t − k0 r] r " 1 # sign U0′′ 2U02 3 r ·Ai . t− 1 |U0′′ | U0 r3 5 6
This is a mono hromati os illation with frequen y
0),
If
ω0
(following from
the amplitude of whi h is modulated by the Airy fun tion.
sign U0′′ > 0,
i.e., if we are at a group velo ity
minimum,
t ).
U ′ (ω0 ) =
the modal seis-
mogram looks qualitatively like that in Fig. E.3 (the argument fun tion in reases with
(E.12)
z
of the Airy
226
APPENDIX E.
THE COMPUTATION OF MODAL SEISMOGRAMS
N(t)
t
t=r/u
0
Fig. E.3: Modal seismogram.
The seismogram
ends
with strong amplitudes in the neighbourhood of the the-
oreti al arrival times of the Airy phase. Fig. 4.13 gives quantitative results for a liquid wave guide; in the range of the Airy phases, the seismogram has been
omputed with the theory if this hapter. If
sign U0′′ < 0
t
(i.e., we are near a group velo ity
maximum ), z
de reases for in-
reasing , and the Airy fun tion is sampled from right to left. The seismogram,
starts with large amplitudes. −5/6 The amplitudes of the Airy phases de rease with r as a fun tion of distan e therefore,
r, i.e., they de rease slower then given in (E.9). This is the reason why the Airy phase dominates for in reasing distan es.
Index Airy fun tions, 128, 224
group velo ity, 148, 152, 154, 155,
Airy phase, 155, 222, 224
157, 171, 173, 225 GRT, 98, 111, 113, 132, 141
Brewster angle, 76 head wave, 95, 96, 110
austi , 125, 130, 133
ompressional module, 37
impedan e, 68, 121
ompressional potential, 46, 219
inhomogeneous wave, 35
ompressional waves, 36, 46 Kir hho 's equation, 51
riti al angle, 69, 71, 84, 96, 110
riti al point, 96, 133
Lamé's parameter, 28, 192
ubi dilatation, 17, 18
leaky mode, 178 deformation, 12
longitudinal waves, 36, 40, 76
deformation tensor, 13, 17, 20, 35
Love waves, 136, 142, 144, 145, 157,
delay time, 130, 132
159
dipole, 49, 56, 178, 201 method of stationary phase, 154, 167,
dispersion, 142
172, 222
dispersion urve, 144, 154, 171
modal seismogram, 150, 154, 157,
dispersion equation, 145, 150, 159
167, 173, 221, 224
displa ement potential, 34, 45, 46,
moment fun tion, 58, 61
50, 76, 85, 93, 99, 160, 168, 178
near eld, 55, 57
double ouple, 56, 59
nodal plane, 59, 60, 141, 148, 166 normal mode, 4, 26, 137, 144, 158,
eigen fun tion, 148, 149, 166, 221
159, 166, 169, 173, 176, 221
Eikonal equation, 119, 121
normal stress, 23, 86
equation of motion, 25, 26, 31, 39, 119
phase velo ity, 62, 137, 142, 144,
ex itation fun tion, 46, 102, 113, 221
147, 150, 151, 154, 156, 159,
ex itation fun tions, 113
165, 167, 171 far- eld, 55, 59, 166
Poisson's ratio, 30
far- eld approximation, 102
polarisation, 40, 56, 62, 65, 142
Fermat's prin iple, 109, 112, 114, radiation hara teristi , 55, 59
123
ray equation, 114, 122, 124 ray parameter, 116, 126, 131
generalised ray theory, 98 227
228
INDEX
Rayleigh waves, 136, 138, 142, 145, 150, 154, 159 re ipro ity, 161 redu ed displa ement potential, 46 re e tion oe ient, 66, 67, 69, 70, 73, 74, 80, 86, 88, 89, 94, 95, 126, 128, 157 Re e tivity method, 92, 95, 97, 132 refra tion oe ient, 66, 71, 84 retarded time, 48, 194 rotation tensor, 13 seismi ray, 114, 121, 122, 125 SH-wave, 65, 72, 84, 96, 97, 113, 119, 121, 126, 130, 136, 157 shear modulus, 113 shear potential, 34, 46, 54 shear wave, 36 single ouple, 56, 58, 61 slowness, 123, 127, 131 Snell's law, 66, 67, 77, 87, 116 Sommerfeld integral, 94, 176, 219, 220 stress, 21 stress tensor, 21, 23, 26, 28, 30 stress ve tor, 26, 32, 33, 44 stress-strain relation, 26, 27, 30, 44, 45 SV-wave, 65, 84 tangential stress, 23, 32, 45, 67, 143 total re e tion, 69, 71, 74, 126, 130 transport equation, 119 transversal waves, 36 water wave, 155 wave equation, 35, 37, 40, 43, 46, 50, 53, 62, 64, 67, 85, 93, 99, 127, 136, 138, 142, 219 wave guide, 159, 160, 165, 167, 170, 172, 173, 176, 221, 224, 226 wavefront approximation, 102 wavenumber, 40, 62, 85, 86, 94, 126, 137, 153, 160, 221 WKBJ method, 126 WKBJ-approximation, 127, 129
