The propagation of rayleigh waves over the surface of a non-homogeneous elastic body with an arbitrary form

THE PROPAGATION OF RAYLEIGH WAVES OVER THE SURFACE OF A NON-HOMOGENEOUS ELASTIC BODY WITH AN ARBITRARY FORM* V.M. BABICH and N.Ya. (Leningrad) (Received 5

October

RUSAKOVA

1963)

In this paper solutions are derived for dynamical equations of the theory of elasticity with a discontinuity only at the boundary. (The boundary is assumed to be free from stresses). These solutions generalize the well-known “plane” Rayleigh waves for the case of a nonhomogeneous elastic body of arbitrary form (see El], chapter XII). “Generalized” Rayleigh waves propagate over the surface in accordance with. Fermat’s principle (i. e. along surface rays), and for them an analogue of the well?known ray method of calculating the intensity of wave fronts can be developed (see [21, [31). A detailed account is given here of the results of note [41 and their generalization for the case of a non-homogeneous body.

1. Introduction The solutions

of the equations of the theory of elasticity (h+p)graddivu+pAu=pi$,

(l-1)

having the form

are call ed 1ongitudinal

??

Zh.

vych.

mat

and transverse plane waves respectively.

2, No. 4, 652-665.1962. 719

CJI and

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720

Babich

and

N.Ya.

Rusakova

y are constant factors, c is an arbitrary constant, with the components a and p, f(. . . ) is an arbitrary

(a, p) is a vector function.

Let f(z) be a function of the complex variable z, regular for - E ( arg z < TT+ E, IZI < r, where E and r are constants. Let, further, f(z) have a singularity for z = 0 (for f(z) one can take In z, zp, zy ln’ z etc.). If c < a and c < b, the complex vectors

ub=‘Ic)

(-i)l/$-$,

-$

_-

)/(t-;+gi i/g-;),

f-&&,0

will satisfy the equations of the theory of elasticity (for y > 0 in the neighbourhood of the point x = ct, y = 0) and have a singularity only where the argument f becomes zero, i.e. for x = ct and y = 0. Thus the vectors ua and ub have a singularity only at the boundary of the half-space, and this singularity is displaced with a velocity c. Let us require that u = ua + ul, be such that the stresses at the boundary of the half-space appear, i.e. the following equations be fulfilled

(1.3) y > 0 dis-

%w=P($+~)I,=o=o. Substituting expressions the system of equations

(1.2)

and (1.3)

in equations (1.4)

we obtain

(&-$P+~ij&&=o, --_I

1

2. C

__-

v

CB

P-5) 1 a2(P+($-g)+=o.

Since q~and y are not simultaneously equal to zero, the determinant of the system must be equal to zero:

Propagation

Equation (1.6) is called f c, where 0 < c < b < n.

of

Rayleigh

Rayleigh’s

721

waves

equation.

It has two real

roots

Solutions of the equation of the theory of elasticity which satisfy conditions (1.2), (1.3), (1.5), (1.6) are called plane Rayleigh waves. The construction given here is essentially the same as the construction in [ll. Our problem is to construct similar solutions in the general case of a surface of arbitrary form and a non-homogeneous medium. Petrovskii showed that for very broad assumptions regarding the discontinuity u this discontinuity will move along the surface with a velocity which can be found from equation (1.6). The “local U behaviour of this discontinuous vector and the vector (1.2), (1.3) is identical (with a proper choice of f). It is natural to try to construct in the general case a vector corresponding to the Rayleigh wave in the form of (1.3), and to take solutions that are a generalization of plane waves (1.2) for ua and ub. longitudinal The basic idea of this paper is that a so-called (see [31) can be taken for ua(ub). verse) ray solution 2. Longitudinal

and transverse

ray

(trans-

solutions

wber;e, as in 8 1, f(z) is a regular function. For Let fO(z) =f(z), -e
u=

of the equations

5 Uk(& Y, z)fh.(t-T(Z,

of the theory

of

(2-l)

y, 2)).

x=0

If series (2.1) is substituted in equations of the theory of elasticity in the general case of a non-homogeneous medium and the coefficients of for u0 7 0 fk are equated to zero, then there are only two possibilities

1) either u. 11VT,

(vr)” = f

2) or

(vq” = &,

uo _LVT*

3

a=

h+& P

;

b=l/$.

(Here V is the Hamil tonian). If the first

(second)

possibility

is realized,

the formal

solution

722

V.M.

Babich

and

N.Ya.

Rusakova

of (2.1) is called the longitudinal (transverse) ray solution. Re now write in detail the recurrent relations realized between the vectors uk in the case of the longitudinal as we11 as in the case of the transverse ray solutions. A detailed derivation of these relations is given in the review article [31. Following the notations of [31 we shall add the index a (or b) to all quantities connected with the longitudinal (or transverse) wave. 1. Longitudinal ray solution: Uka =

(VV = g,

2a2V%*Vra $

(Pk[a2AZa-

MtUia)

a2vza

0.

=

f

P

M('k-1,

u;,= -

P-2) (2.3)

a).

L ("k-l,

-

da 1. Vfa;

(pkvra,

i*VTaJ+

(A $- 2/J)V

+

2.

da +

L ('k-2,

a) -

a)

a2

(2.4)

AfCl

Transverse ray solution:

h7rb)2=

&,

Ukb

= db + db,

M (ukb)

-

M

d,b

dib 11vrb,

L @k--l,

(uk-1,

L

(%-2,

(2.5) (2.6)

b) 11 vrb;

b) -

=

dtb j_ vrb;

b)

b2

(2.7)

h+CL

Here u_,,~

=U-2,a

=U_l,b

(2.8)

=u-2,bzo, 3

M (u, .t) = (A + p) [(vu)

+

VT t-

v (qv)l

vi (uvr!

-I-

P[UfV + 2 s=1 2i hM7~) is] +

-I- iv/4 VT + h7PV~‘ UP )

Lu = @ + p) v (vu) + pAu + VA (vu) + i

.s=l

(u = (~1, 242,US)= i

We

s

-

the unit

(2.9)

vector

of

the

(VP

2 + (VP-ph)is (2.9’)

us.is.

axis

xS,

s = 1,

2.

3).

note that the vector ukb can be written in the form Ukb

=

‘$kEt

+

$knfir

(2.10)

Propagation

of

Rayleigh

waves

723

where E = <(x, y, 2, t) and ‘1 = q(n, y, 2, t) are a pair of non-co11 inear vectors perpendicular to vzb. Substituting (2.10) in (2.6) we obtain for the coefficients yk4 and vykrla system of equations similar to equation (2.3). The vectors .$ and q can be chosen in innumerable wars. In [31 the normal and binormal to the ray are taken for c and q. In the given problem it is more convenient to take for < and q vectors naturally connected to the geometry of the surface wave. The corresponding constructions will be given in 9 4.

3. Auxiliary considerations

of a geometric nature

Let a non-homogeneous elastic body be bounded by a sufficiently smooth surface s. We shall construct below solutions of the equations of the theory of elasticity with a discontinuity only along a smooth line moving over s. As shown by Petrovskii [51 the velocity of this line (normal) at a given point 1, is determined only by the values of the elastic constants at this point and is found only from equation (1.6). Now let 1 be the position of the discontinuity or (what is the same) the position of the Rayleigh wave front at the moment of time t = 0. Fermat’ s principle for a Rayleigh wave follows from Petrovskii’ s

theorem:

if

the extremal

of the

’ds/c (Al), s

where

c(M)

velocity at the point M, is drawn through every point 1, orthogonal to it, the geometrical position of the points

is the Rayleigh of the curve is such that

I

(3.1) gives the position of the Rayleigh wavefront at the moment of time t. Characterizing every point of the curve 1 by the parameter a and every point on the extremal by the parameter T, where

we introduce on the surface S an orthogonal network of coordinates (u, T) (cf. [31, § 2). Further, let n = n(a, T) be the normal to the surface directed inside the elastic body. The radius vector x of an arbitrary point in the neighbourhood of s can be written as x = r(z, Here r = F(T, a) is the parametric

a)+

(3.2)

vn.

equation

of the surface

s (the

“ray”

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V.M.

Babich

and

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Rusakova

coordinates a and T are taken as parameters), v is the distance from the surface s, while values v > 0 correspond to points of the elastic body. The parameters (T, a, v) form a curvilinear coordinate system in the neighbourhood of 13, which, generally speaking, will not be orthogonal. Where convenient we shall use the notation 71, 72, 73 instead of -r, a, V. gas we shall denote the coefficients of the first quadratic JQ gTT’ g,, Gauss form (obviously g:crr= 0). by bTT, b, and baa the coefficients of the second quadratic form. Let ds be the differential the formula

of the arc length.

For its square we have

ds2 = Gijdqhqi

(3.3)

(here and below summationover repeated indices is understood). Neglecting terms of the order v*, for Gij we can write the formulae

Gm= gas- 2vL

G,,

Gw = 0,

= -

2vbas,

G,, = 0,

G,: = g,, -

2vL (3.4)

G,, = 1.

Using (3.4) it is easy to write with the same accuracy the expressions for the tensor G’j (GijG. = Eh, where 6: is Kronecker’s symbol); lk

G‘%;+&.5, aa

G” =

& + 23, g%

Fonnulae (3.4)

and (3.5)

&a G"'

G"'

. 1,

= 2vb,,+ aa

= 0,

G” = 0,

5T

G"" = 1.

are very important for what follows.

1. Ray solutions with complex

eikonals 7, and TV*

Henceforth we shall assume that the surface s is analytical, the Lemdparameters A(%, y, z), P(X, y, z) and the density of the medium p = p(r, y, z) are also analytical functions of x, y, z. On the surface s the required discontinuous solutions have a singularity only for t = -r (T = I dslc). It is therefore natural to assume that on the

??

Complex eikonals and surface rays in connection the skin effect were considered earlier in [II].

with

the

theory

of

Propagation

of

Rayleigh

waves

725

surface s the vector u has the form

u=

-$

Uk

(z, y, 2) fll (t - T).

Ii=1

We look for the vector u in the form of a sum

Taking into account what has been mentioned just now we must write

In accordance with formulae (1.2) and the fact that the functions _fb(z) are defined only for - E < arg z < TI+ E we require that for v > 0 Im T, and Im 7b be positive. The equations of the eikonal for 7a and -rb and the last equation give

Formulae (4.2) and (4.3) and the equations of the (Vfb)2= l/b2 uniquely define T, and ‘b inside the for small v. Conditions (4.2) and (4.3) define the Cauchy’s problem for the equation of the eikonal. Kovalevskaya’s theorem this problem has a solution is unique in the region of analytical functions. In the coordinates ql, q2, can be written in the form

q3(‘1,

eikonal (vza)” = I/$; elastic body at least initial data in Because of Cauchyfor small v, and it

a, v) the equations

of the eikonal

(4.4) Using (4.2), (4.3) and (4.4) it is easy to find the values of the second derivatives T, and fb on the surface 1% We shall not give the expressions for these. We now pass on to the construction of the vectors c and ‘1 which appear in formula (2.10). Let a0 and v” be vectors having the contravariant coordinates (0, 1, 0) and (0, 0, 1) respectively in the system of coordinates (T, a, v). By definition we have

726

V.M. Babich

and N.Ya.

Rusakova

We shall explain why such a choice of vectors c and ‘1 is natural in the case of our problem. First, by scalar mu1tiplication of the expression (4.5) by vzb it can be easily seen that c and q are perpendicular to Vrb. Further, on the surface 17 itself the derivative &b/&x is equal to zero and therefore (c, ‘1)s = 0. Also,

(Ev8s= &,

(rl, q)s

=

bo,

a")#&

Because of this c and ?j near s (all our considerations apply only near s> are linearly independent. Moreover, in the plane case every vector perpendicular t0 vrb, iS CO11 inear t0 a VeCtOr having the COIUponentS in Cartesian coordinates. The vector s changes to dr+kz) (@da?/, this vector in the plane case, and the vector q changes to a constant vector perpendicular to the plane in which the vibrations occur. Such a choice of c and q spares us the necessity of special consideration of the plane case. Substituting the normal and binormal to the ray for c and q (see [31) would be less convenient, because the normal and the binormal have a complex expression in terms of the derivatives of vb and are not more graphic (the rays in this case are complex). We must now write the recurrent relations (2.5), (2.6) in a more specific form. For this uib must be replaced by the expression (2. lo), where c and q have the form (4.5), the vector obtained must be expanded into three non-coplanar vectors c, ‘1, vrb and the components along c and TVequated to zero. All these calculations are conveniently carried out in (T, a, v) coordinates. If the vector

6=1

in formula (2.9) is replaced by a vector having in the coordinate system qi (i = 1, 2, 3) the form G’iVtUBrj, where Zj = &/dqj, viu6 is a COvariant derivative of the vector u.~, G’j are the contravariant components of the metric tensor, expression (2.9) will be written in an invariant form. It is now easy to find the components of the vector M(&-g + qnq + $vvrb,rb)at least on the surface itself. The Christoffel symbols, necessary for calculating the covariant derivatives,

Tropagat

ion

of

Rayleigh

727

waves

can be easily found by using formulae (5.3). The determination of the scalar VU = div u also is not difficult, if the contravariant components of the vector u are given. After cumbrous calculations it is found that the relation be rewritten in a form similar to (2.3):

2baV*ktVzb +

$kE

[b%b

-

VP

(+)

v%b

-pL

(“k-l,b)

+

A]

+

i%@k,,

(2.6)

can

(4.6)

+

0 + 2b2V$‘k+b

ba +

M(“kb)

E =

o, A

$k&

-1-

&re

+

E

(M

(db)

Is

=

0,

L (Uk--l,b)

-

q)

=

0.

(4.7)

Below we shall simply write wkS instead of vk. The expressions for A, 8, C, D and E will not be required.

5. Fulfilment of the boundary conditions We thus have two formal solutions of the equations of the theory of elasticity ua and ub similar respectively to a plane longitudinal and plane transverse wave. Formulae (2.2)-(2.10), (4.5)-(4.7) give a complete system of recurrent relations for uka and ukb. We shall now require that the sum ua + ub satisfy absence of stresses on the surface s.

the conditions

of

The displacement vector u and the stress tensor aij in Cartesian coordinates are connected by the relations

and the condition of absence of stresses on the boundary is expressed by the equation oijvi = 0, where (v’, v2, v3) is the normal to the surface s. Substituting for u the sum ua + ub and replacing ug and ub by the series (2.1) we obtain the relation h [div

Uka

-

(V’huk+l,a)l

vi

+

(54

p

-

uk+l,i

(Vra,

Y)

I

+

*. * = 0

(k = - 1, 0, 1, 2, . . .),

where the dots denote the expression that will be obtained if a is replaced by b everywhere in the terms written out. Expression (5.1) should

V.I.

728

Rabich

and N.Ya.

Pusakova

be written in such a ww that it is true for any, not necessarily orthogonal, system of coordinates. It will be noted that the vector with contravariant components (0, 0, 1) in the coordinate system (7, a, v) is the unit normal vector to 1% This can be easily seen by changing from (T, a, v) to the Cartesian system of coordinates. The invariant form of equation (5.1) will obviously be

1 [div

Uka -

(V~&+l,a)l

-

(Uk+l,a,

Vi-/-@‘”[(v&kja

$

V”) VaTa

(VT=,V”)I + * * - = 0

-

Uk+l,s,a

Vjha) V3-

Here, as always, V;ukja denotes the covariant derivatives I$’ vi the contravariant components of the vector v”. It is more convenient to write equation (5.2) eyq2,q3

(ql=r,

q2=a,

Instead of 11~~and ukb we substitute Uka =

qkv%a

+

&,

(5.2)

1, 2,. . .).

U-l=_0 (i, k=i,O,

of the vector

in the coordinates

q3=v).

their expressions in (5.2)

Ukb =

$kt

+

$km?l

+

(see $2): (5.3)

Uokb.

For k = - 1 we ohtnin hcpo

+

a2$0

=

0,

0390

-I-

aavo

=

0,

=

0,

$o*

(5.4)

where

(5.5)

Equations (5.4) and (5.5) coincide with equations (1.5). of this system is equal to zero because of (1.6). The general solution of system (5.3) ‘PO =

i el = --bB

elx0,

2 9 ’

q.

e2=,i

The determinant

will have the form =

2

+,x0, i

7-h”

i

In (5.2) we write k = 0. Instead of Uka and llkb we substitute

their

Propagation

expressions (2.4)-(2.9’).

(5.3),

Equations (5.2)

Rayleigh

of

729

and replace uia and uib in accordance with formulae become

~lcpl+%~l+~l(cpcl9

$0) =

0,

ascpr+a4%+

$0) =

09

Az((Po9 an

+A3(%9

Here al, a2, n3, ag have the values differential operators of the first shall not write out because of their satisfying equations (5.7) may exist

11=a4z-&_4 Equation (5.8)

waves

=

(5.7)

0.

in (5.5), hi(Qo, vO) denote linear order, the expressions for which we complexity. In order that ‘pi and vi it is necessary and sufficient that

l2 = -

c= '

%I

a2

=

-

fi

1

(5.8)

I/ --_-L_ C2

in (T, a, v) coordinates can be written in the form

bl ‘3 + b2 ;$ + bs 2

+ b4 %

(Calculations show that neither ential operators Ai. )

&+@a

(5.9)

+ bsscpo + b&o = 0. nor

ag,jaa

enter the differ-

Differentiation with respect to T is differentiation over the direction tangential to s. Using this fact and formulae (5.6) we can rewrite (5.9) in the form

bl g + b2‘ 3 + (b3el+ b4e2)$$+ xo (b3‘2 + b42

+ be1+ he,)= 0. (5.10)

F’or form

k =

0 relations

(2.3)

and (4.6) on the surface S’have a similar

NO

%T

fc2

ag

The terms acp,/da and

k&k

+caxo

=

0,

ago/a a will

are equal to zero on S.

dla$+d2~+d3Xo=0. be absent, because &@a

(5.11) and

Excluding a’po/av and a$,$% from (5.10) and (5.11) we obtain after very cumbrous calculations an equation of the first order with respect

730

V.M.

Babich

and

N.Ya.

Rusakova

to x0: g

The coefficient

+

I&

= 0.

(5.22)

I! will be written out in detail

in the appendix.

Now let the function j(, be known on the initial line 1. Then equation (5.12) can be used to determine x0 everywhere on S, where the coordinate network (a, T) is regular. Formulae (5.6) give the values for q+ and q+, on the surface 3, and to determine ‘p,,, y,,, v0 inside the body Cauchy’s problem must be solved for equations (2.3), ( 13.6) and (4.7) for k = 0 with initial conditions (5.6) and w,,,, = 0 (the latter follows from (5.4)). The solution will exist because of Cauchy-Kovalevskaya’s theorem. For Uga and U,,b we now have formulae (2.2)) (2.5)) (2.10) (&, E I& G 0). Since equation (5.8)

(Pl=d+elXlP

is satisfied,

43 =

Si +

from equation (5.7) wh,

hl = -A*

we find

@opo, WV

where 9; and vy are partial solutions of the system (5.7). Writing k = 1 in (5.2) we obtain, generally speaking, an insoluble algebraic system of equations for the determination of I++ my*.Writing the conditions of solvability we shall obtain an equation for x1 similar to (5.12). Knowing the initial data for xr we determine x1 using this equation over the entire surface A’%Then using Cauchy-Kovalevskaya’s theorem and equations (2.3), (4.6) and (4.7) we shall find ‘p2, dye, y1 and then ula and Ulb etc. Thus all Uko and Ukb (k = 0, 1, . ..) csn be a etermined if the initial data for xk (k = 0, 1, . . .) are known.

6. Concluding remarks 1. Let it be required to find the nature of the behaviour of the displacement vector in the neighbourhood of the Rayleigh wavefront arising as a result of the action of a concentrated impulse applied to the boundary of the body s. This problem is solved by using the usual reason. ing of the rw method. Forst the field of surface rays must be constructed, i.e. the field of extremals of the integral

(b$, E 8 is the point where the concentrated impulse was applied). function t will then be known. Further, we write

The

Propagation

of

Rayleigh

maves

731

To find x,, the initial data must be taken from the exact solution of the corresponding problem for a homogeneous hal fspace in the threedimensional case and the ha1f-plane in the plane case (see, for example, [61, hl, M). We can now write

(The function f is here the same as in the corresponding problem for the ha1f-space or ha1f-plane. ) 2. It is easy to see that there is in fact a solution of the elasticity equations which behaves line u = I$, + ub in the neighbourhood of S and exactly satisfies the condition of absence of stresses on the boundary. To verify

this let us consider the sum u@) = i

U&k (t - %a)+

k=o

i

ukbfk (t -

fb)

k=o

and continue it in an infinitely differentiable manner over the entire region occupied by the elastic body. For example, u( n, can be mu1tiplied by the infinitely differentiable function ; equal to unity in the neighbourhood of those points where t = T, v = 0, and smoothly decreasin;: to zero. This function must be zero where uCn) has been defined. Now let R be sufficiently large. We write w = u(“)G + v(n, y, z, t). We shall require that the vector w satisfy the equations of the theory of elasticity and the conditions of absence of stresses at the boundary. Since u(“) is the partial sum of the ray series satisfying the boundary conditions correct up to terms of order O(f,), the vector v will satisfy the equations of the theory of elasticity with a sufficiently smooth right hand side and the stress vector tfV) = cJijvj corresponding to the vector will be sufficiently smooth. Yk now write v I!=0= vg,

r1 \I=,= v1,

where v,, and vl are sufficiently smooth vectors on which only one condition is imposed: they must be such that coordination conditions of a

732

V.M.

Babich

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sufficiently high order are satisfied. As is well-known, such a vector can be found and it will be sufficiently smooth. Then the vector v will satisfy the equations of the theory of elasticity and the condition of absence of stresses on the boundary (compare the simioar reasoning in 193). 3. If we do not take up the problem of obtaining ray series, and from the beginning confine ourselves to the derivation of finite ray sums, solutions can be constructed having the same nature of discontinuity on the surface as the partial sum of the ray series ufn) and in the nonanalytical case.

We have in fact used the analyticity of S, A, u and p only in connexion with the reference to Kovalevskaya s theorem while solving Cauchy’ s problem for equations of the eikonal and equations (2.3), (4.6) and (4.7). Instead of this, by using the initial data and the equations for Ta’ ‘b8 (?k* v’k# yk,-,* we can find sufficiently many normal derivatives of la, Tb# ‘pk, yyk, vkrl ontthe surface s. We now continue the functions ralS, 7b Is, ‘pklS, q~ k1.s, yhls inside the elastic body in such a way that the continued functions have normal derivatives of the kind defined. The continuation must be smooth. We shall not impose other restrictions on the continuation. Instead of II(~) we can now take the ray sum u( :) with such “unreal” continued T,, ‘be which the vector w is constructed according to the ?k, ‘ykp &-J* after ( formula w - u $3 + v. The other arguments are the same. 4. As follows from (6.1) the displacement u in the neighbourhood of the Rayleigh wavefront is proportional to the quantity x,,. We shall call x0 the complex intensity of the Rayleigh wave. We write equation (5.12) for x0 in detail:

Propagation

of

Rayle

igh

waves

It can be easily shown, using the elementary inequality (a>O, p>O), that ‘4 > 0 always. Also, R = l/4 A.

733

a + Q > 2 vr/‘crp

me shall write out the expressions for the remaining coefficients in the appendix. Here we shall only point out that the coefficients c and 19are purely imaginary, the coefficient E is a linear combination of dhlav, ap/dv and apldv and also purely imaginary, F is a 1 inear combination of ahfar, dplaz, dp/ar and F is a real coefficient. Solving equation (6.2) :ve obtain

i.e. the complex intensity is expressed in the form of a product of six factors, each of which characterizes the effect of one or other factor on the Rayleigh wave. The first factor characterizes the initial form of the Rayleigh wave, the second depends only on the internal geometry of the rays and shows that the amplitude of the Rayleigh wave varies with the divergence of the surface rays in the same way as in the case of the propagation of volume waves. The following three factors show that the curvature of the surface along and across the ray and the rate of change of A, IJ and p with the depth affect only the phase of the Rayleigh wave, but not its amplitude. The rate of change of A, CIand p along the ray, on the other hand, affects the amplitude only. We do not know of works where Rayleigh waves propagating along the surface of a non-homogeneous elastic body were separated from exact solu. tions. In the case of homogeneous bodies Rayleigh waves were separated in the problem of impulse interaction on the boundary of a half-space 163, [d, [d and a sphere [IO]. It can be easily seen that the law in geometrical optics regarding the reduction of amplitudf _ofthe Rayleigh wave (i.e. proportionality of (x,) to the quantity 1 /v’gaa ) is true in all these cases. APPENDIX Expressions for the coefficients

A, 5, C, D, E, F:

A>%

734

V.M.

Rabich

and

N.Ya.

Rusakova

B=+A;

-_-

)

1 --( I.9

CI<

+ G$

1

1 _-c2

ua )(

1’ b2 )

where

2b4 +a-

(

+

1 --+;+2(&s)tb2c+ b2


$$&-$)

e1

b2]&,

$+

Propagation

of

Rayle

igh

735

waves

Here

aTa

.p -

y--z

s’

(~Is=-ij/cg-$,

sI,=-if-$_$

),

Translated

by

R.

Feinstein

REFERENCES 1.

2.

B.D., Asymptotic solution of Keller. J. B., Lewis, R.M. and Seckler, some diffraction problems. Cormun. Pure and Appl. Math., 9, No. 2. 1956.

207-265, 3.

In the collection “Voprosy dinamicheskoi Alekseyev, AS. et al., teorii rasprostraneniya seismicheskikh voln” (Problems of the No. 5. 3-35, dynamical theory of propagation of seismic WaVSS). 1961.

4.

Babich,

5.

Petrovskii,

6.

Petrashen’, in.

7.

V. M., Dokl.

G.I.

A.A.

Ggurtsov,

K.I.

Shewakin,

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et al.,

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Ye.1. Univ.

in.

SSSR, Nauk

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SSSR,

A.A.

47,

, G.I.,

uchenye 149,

V.L.,

Zhdanova,

1263-1267. 4,

NO.

Leningrad. 1950.

Zhdanova,

6,

NO.

71-116,

and Fainshmidt, im.

137,

Zapiski

uchenye

135,

and Petrashen’ Univ.

Gosudarst. 9.

Akad.

I. G., Dokl.

Gosudarst. 6.

Differentsial’ nye i integral’ nye uravneniya (Differential and integral equations of Tekhgiz. Moscow, 1937.

Frank, F. and Mizes, R., matematicheskoi fiziki mathemat.ical physics).

156-161,

Uchenye

177, 26,

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11. Keller, J.

Appl.

J.B.

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10. Gel’ chinskii,

1945.

Gosudarst.

Zapiski 24,

1961.

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in wave 1956. univ.

in.

1956.

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and propagation.