Appl. Sci. Res. 23

J a n u a r y 1971

R E F L E C T I O N AND D I F F R A C T I O N OF SH WAVES D U E TO A L I N E SOURCE IN A H E T E R O G E N E O U S MEDIUM M. L . G H O S H Dept. of Mathematics Bengal Engineering College, Howrah-3, West Bengal, INDIA

Abstract

The fieId due to a line source of h a r m o n i c S H waves e m b e d d e d in a semi infinite m e d i u m whose density and rigidity v a r y e x p o n e n t i a l l y w i t h d e p t h is derived in the integral form. The displacement due to diffraction at a n y p o i n t in t h e shadow zone is o b t a i n e d and, b y t h e saddle p o i n t m e t h o d of e v a l u a t i o n of the integral, the field at a n y p o i n t in the illuminated region is also found. Finally, geometrical i n t e r p r e t a t i o n is given to t h e different rays a r r i v i n g in the illuminated as well as in the shadow zone.

Nome~zclature b C

shear w a v e v e l o c i t y on the free surface wave velocity H(~l)(p), H(~2)(p) H a n k e l ' s function of the first and second kind respectively k F o u r i e r t r a n s f o r m p a r a m e t e r w i t h respect to x v t h e displacement V fourier t r a n s f o r m of v w i t h respect to x X grazing angle a,/~ small positive constants

/*0 /*is O 00 o)

2~

positive constant, - - 2 coefficient of rigidity at t h e free surface coefficient of rigidity values of/~i at t h e saddle p o i n t (i = 1, 2, 3, 4) the d e n s i t y of t h e m e d i u m the d e n s i t y of the m e d i u m at the free surface f r e q u e n c y of v i b r a t i o n

--

3 7 3

--

374

M.L. GtlOSt~

§ 1. Introduction

The problem of propagation of/Love and Rayleigh waves in an inhomogeneous medium has been studied by many authors. Dere~ siewicz Ell derived the secular equation governing the phase velocity of Love waves in a homogeneous elastic crust resting on an inhomogeneous elastic substratum. Hudson E2~ studied extensively the free Love wave propagation in a heterogeneous layer of finite depth. Wilson E3~ considered th~ propagation of Love and Rayleigh waves in a medium in which the elastic properties are exponential functions of the depth. The shear vibration due to a transverse surface traction on a semi infinite elastic solid with similar properties have also been considered b y Mitra E4I. But the study of the displacement due to a point source embedded in a heterogeneous medium are few. Hook E51 derived the expression for the P, SV and SH displacement generated b y an impulsive point source buried in an inhomogeneous medium of a particular type, which leads to the separation of the vector wave equation. Pekeris E6] obtained the field of the diffraction of sound waves from a point source in a liquid medium of constant velocity gradient. Benmenahem ET] studied the diffraction of elastic waves from a surface source in a heterogeneous medium. Vlaar ES] also studied the propagation of waves due to the presence of an SH point source in the interior of a piecewise continuously stratified half space. But, unfortunately, he missed a term in the basic differential equation of displacement of SH waves. In this paper, we have derived the field due to a source of disturbance emitting SH waves embedded in a heterogeneous medium. Following Wilson ~31 and Mitra E4~, the rigidity and density are taken exponentially increasing with the depth where the shear wave velocity is assumed to have an exponentially decreasing character leading to the formation of a shadow zone. The displacement in the illuminated region arising out the direct and the reflected wave and the field due to diffraction in the shadow region together with the geometrical interpretation of the different rays as well are derived. § 2. Formulation of the problem

The x-axis is taken on the free surface and the z-axis vertically downwards into the medium. The source of disturbance emitting SH waves is taken at tile point (0, z0). The displacement v at any

R E F L E C T I O N A N D D I F F R A C T I O N OF SH W A V E S

375

point in the medium, excepting the source point, satisfies the equation

~2v

d# 3v

O2v

02v

Assuming the solution of the above equation in the form v = e -i~t o~ V(k,z) eikxdk --oo

we see that V(k, z) satisfies the equation d~F d# # ~ z 2 + dz

dV dz + (pc°2-/~k2) ~ = 0 "

(zl)

The modulus of rigidity # and density p are assumed to vary in accordance with the equations #=#0e

~z and

p:p0e

~z

where x/~[p=be-vZ, b = x/#0~0 and y = ( f i - - a ) / 2 which is a positive quantity. By the substitution ~ = evZ, (2.1) takes the form d~ 2 +

-}- y ~ J d~ -}-

y2b2

y2~

~ = 0.

(2.2)

Next the substitution V = ~-~/2~ff reduces the equation further to dgU ~d~@

1 dU ( ~ d~ -}- p 2 _

v2)_ ~2 U : 0

(2.3)

where o~

P-- yb

and

a2 k2 4Y 2-+~y~x-:~"

l"wo independent solutions of (2.3) are H~a)(fl~) and H~2)(fl~) where H~ 1) and H~2) are the Hankel functions of the first and second kind respectively. When z approaches infinity, ~ also approaches infinity and for large values of ~, fhe asymptotic values of H~l)(p~) and H~)(p~)

376

M.L. GHOSH

are given b y

H~e)(P~) ~ ( ~ ) ~ e x p [ - - i (

p~

vrc 2

~/2

)]

.

Therefore e -i°~t H~l)(p~) represents a downgoing wave for z approaching infinity. We shall now determine the field due to a source at the point (0, z0). For z > z0, the solution Ve(z) of (2.2) to be t a k e n must be of the form V2(z) = BH~)(p~) ~-o,/2~, since it gives the downgoing wave. For 0 ~< z < z0, the solution Fl(z), which is a linear combination of ~-~/2~H~l)(p~) and ~-~'/~2"H~)(p~) and satisfies the condition of vanishing stress on the free surface z = 0, is of the form

pyH~2)'(P) -- °'/2H~)(P) Vl(z) = A H~)(p~) -- pyH~X),(p) _ ~ / 2 H ~ ( p )

H~l)(p~)] ~-~/2r.

The dash in the above expression denotes the differentiation with respect to p. Therefore the solution due to a source of disturbance at (0, Zo) is given b y

vl(z) =

~(zo) Y~(zo) ~l(z) W

=/toe 2

X

W (0 ~< z < zo) and V2(z) =

W

:#oe

~

X

(z >zo)

(2.4)

REFLECTION AND DIFFRACTION OF SFI WAVES

377

where So = eYZoand W is the Wronskian:

#(z)Eg,(z) G ( z ) -

G(z) gi.(z)? =

#,/r~I),(p)PvH~e)'(P)--a/2H~2)(P)__

=/*°e~ZEe-~Z/2{H~2)(P~)--

a/2 H~I)(p) /_/~l)(p~)} X

d d × dTz (e-~'/~ H~'(p~)) -- e-~'/~ H~'(p~) ~zz x

~o~ [H(2 ) d = ~o T L " (p~) ~

=#o

~o~ 4i ~ × r~p~

H~'(P~) -- H ( ' ( ~

1

d

4i#o rc ;""

Therefore,

4iy

pyH~l)'(p) --

k

a/2 H~l)(p)

]

O~Z~Zo and C~

7¢ e~- (zo-z) X I

pTH~2)'(p)- c¢/2 H~2)(p) H~i)(p~o)J H~l)(ib~) (z > zo).

V(x, z)

(2.5)

So the expression for the displacement in the region 0 <~ ~< z G z0 due to a source of disturbance at (0, zo) is (omitting the time factor e -i~t)

378

M.L. GHOSH

e~- (zo-z) V(x, z) -

4i 7

×

~I ×

pTH~)'(p)- ~/2 H~2)(p) H~l)(p}) ] ~)(P~)

-

~

- ~/2 H ~ ' ( p )

X

--co

X H~l)(p~o)ei~x dk.

(2.6)

§ 3. Diffracted waves

To study the diffracted wave at any point in the region (0 <~ z <~ z0) within the shadow zone, the residues of the integral (2.6) must be considered. Here the discussion will only be confined to the case when p is large, which occurs for large values of the frequency. For large values of p, writing

pyH~2)'(p) -- ~/2 H~)(p) ~ pyH~2)'(p) and pyH~l)'(p)

-- 0~/2 H~I)(#) ~ pyH~l)'(p)

(2.6) can be reduced to the form ~e

V ( x , z) -

2

(zo-z)

4i7

×

co

× f[H~)(p~)

H~I),(p)H~)'H~l)(p~,lH~l'(p~o)eikxdk. (P)

(3.1)

--co

For the evaluation of (3.1), the contribution from the zeros of well as the contribution from the branch points given b y v = 0 i.e. k = ~:i~/2 are to be considered. In order to evaluate the integral for positive values of x, we construct a contour in the upper half of the complex k plane as shown in Fig. 1. Since H~l)(z) and H~)(z) satisfy the relations

H~l)'(p) as

R~(z)

= e'~ H~'(z)

a~(~)

= e -i,~ HT)(~)

and

REFLECTION

AND DIFFRACTION

E

OF SH WAVES

379

B

b -g-

C

F

Fig. 1.

PV

The

A

path of integration in the complex k plane.

the sum of the contributions of the integrals along BC and DE, on two sides of the branch cut vanishes. So the value of (2.6) is equal to the sum of the residues from the poles given by the zeros of H~ll'(p). For large values of p, the asymptotic expansion of H~l)(p) for different values of v have been thoroughly discussed by Striefer and Kodis E9]. They have shown that for asymptotic values of H~l)(p) and H~2)(p), the complex v plane can be broken up in the following regions as shown in Fig. 2.

% % %

AN£

$:

\ \ zz

\x t

Fig. 2. The different regions for different asymptotic expressions of the cylindrical functions.

The equations of the curves separating the regions are defined by ImE(p ~ - v2)~ - v cos -1 v/p] = O.

380

M, L. GHOSH

In the regions I and II exp i(p 2 -- v2)~ + iv cos -1

--

--

it:/4

P V

Imcos -1-<0

in r e g i o n I

P V

I m cos -1 - - > 0

in region II

P and in the region I I I exp [ - - ( v 2 H~l)(p) = - - i ( 2 )

~

p2) ~ + v cosh-1 p ]

(v2 - p2)~

whereas on the solid curve separating the regions I and I I I sin (p2 _ v2)~- _ v cos -1 - - -- ~/4

P

H~l)(p)-- 2i ( 2 ) ~

(p~ -

v~)~

V

I m cos -1 - - < 0.

P So on the b o u n d a r y of the regions I and III, we can write 2 H~l)(p) = 2i l 7~p sin

sin Yl

(Im a < 0)

where Yl =

p(sin ~ -- ~ cos ~) -- ~/4

and cos

~

=

v/p.

Therefore, on the same boundary, the asymptotic expression of

H~l)'(p) for large values of p m u s t be of the form 1 2 H~l)'(p)

=

2i A -

s i n (x s i n )]1.

~P

R E F L E C T I O N A N D D I F F R A C T I O N OF SH W A V E S

381

This shows t h a t the zeros of H~l)'(p) in the u p p e r half of the complex v plane occur on the solid curve separating the regions I and I I I . We shall consider only those roots Vm of Ha)'Pb~ = 0, of which the Vm \ Y ! i m a g i n a r y parts are small since values of Vm with small i m a g i n a r y parts will h a v e a significant c o n t r i b u t i o n to the residue of (3.1) as will be clear from the later development. I t appears from Fig. 2 t h a t only for [vml ~ 15 the i m a g i n a r y p a r t of Vm will be small. Now, since / 2 sin a = 2i ~/ - cos Yl,

H~l)'(p)

~p

the zeros of H~l)'(p) are given b y cos Ylm = 0, from which we obtain Ylm = ± (m + ½) T:. Since [vm[ ~- p, the relation cos am = vm/p shows t h a t am should be small and since I m cos -z vm/P < O, O~mshould lie in the fourth q u a d r a n t . So the relation Ylm = p(sin am -- am cos am) -- ~/4 can be written a p p r o x i m a t e l y as Ylm

3

4

which gives am = (3/p)* (Yam @ x/4) ~. T a k i n g Ylm = (m + ½) ~, we obtain =(3~

~'

\~-1

~m

(4m +

3)~e ±~/~.

And taking Ylm = - - ( m + ½) ~, am =

\4p)

(4m + 1)~ e ~i/3.

Since am should be in the f o u r t h q u a d r a n t , we take

~m

=(3 \ 4p /y (4m +

(3.2)

and Ylm = --(m + ½) r~. Now ~'m =

P COS a m

,~,p 1 _ a ~ 2

= p

1 + 1(4~+

1)~\~-)

e ~/~

382 since Ivml ~ P,

M.L. GHOSH

Ivml

is large and therefore,

(3.3) as a is small. So

(3.4) The position of the roots of H~1)'(p) = 0 and their values being now clearly understood, we n e x t proceed to evaluate (3.1) from the consideration of the residues which give

(3.5) Using

and the relation

Therefore,

N e x t using

H~l)'(p)

= 0, we obtain

REFLECTION AND DIFFRACTION OF SH WAVES

383

and the asymptotic expressions of H~,o o )(p~) and H~,o (1)(P~0) for large values of p given by

(i)(p~) = 2i i . =p sin2 ~z H~.,

elY2

(1) ~ 2 H~,, (P~0) = 2i =p sin ~0

eiyo

and

where (~

y2=p~

2

I

p~72

-1

fl cos

~m

"~

~)--x/4

_.co. and YO ' ~

1

P'X/~"2 - - 1 - - 'Pro COS- 1 ~0

=/4)

we finally obtain

V(x, z)

2=i7 e~- (zo-~)

p __

2=i 7

ey

vm

Z km~

(zo-z)

e i(y0 + Y~ q- kmx )

x/sin c~2 sin c~o

X

P Vm

x >2

e i[p(V~°~f-1+v'fi~l-1)-v'dc°s-ll/¢°+c°s-~l/~)+kmx-r:/2]

(3.6)

The expression (3.6) for the displacement at the point P2(x, z) in the shadow region may be physically interpreted from the following geometrical consideration. If X and C are the grazing angle and wave velocity respectively at any point on the limiting ray, then it is known that cos X/C remains constant on that ray. But at A (Fig. 3) where the limiting ray touches the free surface, the grazing angle is zero and the wave

384

M.L. GHOSH . . . . .

3

R. . . .

A

x~

i Az

,

I.SHADOW

i zo=

[

,

I I

I I

G

o

• X

%

a

Fig. 3. Picture showing the reflected and diffracted rays. velocity is b. Therefore, on the limiting ray cos X

1

C

b

i.e.

cosX=e

-vz.

Now using the relation --(dz/dx) = tan X, we obtain x~

0

0

f a x = -- f c °es X d-z - -vf z

d -- z-- _ _ x/1 --e-2V z

x/1 - - c o s 2 X 0

Zo

go

which gives 1 X 1

=

- -

1

1

cos-l(e-VZo) = - - cos -1. - - y ~o

(3.7)

where X l is the distance of the point A where the limiting ray touches the free surface to the origin of coordinates. We also can show t h a t 1 X2

=

--

1 COS - 1 -

(3.8)

where x2 is the horizontal distance from the point P on the limiting ray to the point A, the point P being at a depth z below the free surface. It is, of course, obvious from the geometry t h a t x2 is also the horizontal distance of the point P2(x, z) on the diffracted ray from the point A2 where the diffracted ray leaves the free surface. Again, the times t 1 and t2 taken by the disturbance to travel the portions of the p a t h O1A and A2P2 respectively of the diffracted r a y OA1A~P2 are the same as the times taken by the disturbance to travel the portions O1A and AP of the limiting ray. We shall calculate the times tl and t2 from the limiting ray.

R E F L E C T I O N AND D I F F R A C T I O N OF SH WAVES

385

Now, using the relations d s / d t = C, d z / d s - - --sin X and cos X = = C/b on the limiting ray, we obtain 6.

f

0

;

dt =

0

0

Ca/1

f

C2tb 2

Zo

b e -~'z a/1 -- e-2~z

Zo

which yields 1 x/e2VZo-- 1 1 tl -----y~= ~b-,,/~-

1.

Similarly, t2, the travel time from A to P along the limiting ray can be obtained as

1 t2=7~/V--1. Therefore, the time taken by the disturbance to travel the portions of the path 01A and A2P2 of the diffracted ray is given by

1 -- P V - x / ~ - 1 q- x / ~ - -

17.

(3.9)

o)

Now, introducing the time factor e -iot in the expression the displacement in the shadow region, we obtain v(x, z, t)

--

(3.6)

for

2~i7 e-) (zo-~) X #

Vm e i [ p ( V ~ +

×E

V~l)-~,.(cos-~

1/~o+ cos -~ ll~)+k.,x-~!2-mt]

(3.10)

kmC~m(1 - - 1I~2)~ (1 -- 1I~0~)~

Substituting the values of p ( x / ~ 1 + ,/~1), cos -1 1I~ and cos -1 I/t0 from (3.9), (3.8) and (3.7) and also the value of am from (3.2) in the above expression we find: v(x, z, t) =

2~

z~

e T (,o-~) E

~'m7 ei=13

ei[k~x-vmT(x~+xD-a~{t-(h+tD}] / 3r~ k~

k m ( 4 m + 1)~ ~ - @ - J

386

M.L. GHOSH

Finally, using the relation that vm ~ km/y from (3.3) and replacing x -- (xl + x2) by R, we obtain the expression for the displacement as follows v(x, z, O =

c~ 2= e y (zo-z)

ei=/3 ei[k.~R-w{t--(h+h)}]

Z

(3.11)

P

(4re:q- 1)* \ ~ - ]

From (3.4) km ~ V m y ~ - - - p y

1 q-½(4mq-

1)~\-~-]

e i~/a

-~

-- ~° b [1 q- ½-(4m q- 1)-: (--~-)~ 31~/ie it becomes clear that the imaginary part of km introduces an exponential decay term in the amplitude of the diffracted wave. Therefore, the significant contribution will be made by the values of km of which the imaginary parts are small. This causes us to seek for the values of vm of which the imaginary parts are small. Substituting the value of km in (3.11); the displacement in the shadow region takes the form c~

v(x, z, t) =

2= e y (zo-z) E

ei[=/~ + ~(4m+ 1 )~.(3r~/47))~pTR]

X

/ 3= V

P

+

')'iV)

X e -(V~/4)pT(4m+lp(3~Av):R )< e -i°[{t-(tl+t2)}-R/b].

(3.12)

The first exponential factor gives the total phase change as the disturbance travels from the source point (0, Zo) to the point Pc(x, z) along the ray path O1AA2P2, and the second exponential factor shows that exponential decay takes place as the ray travels along the horizontal path AA2. This is in accordance with our expectation from the physical standpoint since from each point of the ray AA2 waves are diffracted into the medium. § 4. Reflected w a v e

In order to study the displacement in the illuminated region, the integral for displacement has to be evaluated by the saddle point method. From (2.6), the expression for the displacement at the

R E F L E C T I O N A N D D I F F R A C T I O N OF SH W A V E S

387

point Pl(x3, z) (cf. Fig. 3) is a

7.c e 2

(zo-z)

4i~ oo

x

H~2)(PS)-- pyH~l)'(p) -- ~/2 H~l)(p) --co

X H~I)(p~0) e ikx~ dk.

(4.1)

At first, we consider the evaluation of the part oo

-- 4i~-

pyH~l),(p) -

o /2

(1)(p)

X

--oo

x H~')(p~) H~')(p~o) eikx' dk

(4.2)

of the above integral by the saddle point method. Multiplication of the quantity within the paranthesis of the integrand of (4.2) by the factor 1/[H~2)(p)]2 annuls its exponential character. By multiplying the quantities within and outside the paranthesis by 1/[H~)(p)12 and [H~)(p)] 2 respectively and writing A(k) for the resulting expression within the paranthesis, the above integral (4.2) can be written in the form xe2

(zo-z)

X

4i7 oo

X f A (k) H~l)(p~) H~2)(p) H~l)(p~o) H~)(p) e~kx3dk.

(4.3)

We are now prepared to transform each of the factors except A (k) into an exponential integral by the aid of the expressions r]--ioo

H~l)(z ) _ =1

f --ri+ioo

ei~°°s:'l+i~(t''-~/~) d#l

388

M.L. GHOSH

and

,f

~-r:-ioo

e -izc°sm-i~(~l-r:/2) d/*l

TC

x--~+ic~

where --argz<~

<~--argz.

Therefore, the integral (4.3) can be written as e 2

A (k) ei~ d/*l d#2 d/*3 d/*4 dk

4i7~a

(4.4)

--oo

where W = P~ cos/.1 + P~0 cos/*2 -- P cos/*a -- P cos/.4 +

-4-V/*l@V/*2--v/*a--v/*4-/kxa. The saddle points are determined b y

~/*1

~/.2

~/.3

~/.4

~k

--0.

This gives Vs -----p~ sin #is = P~0 sin #2s = p sin #as = P sin #4s

(4.5)

and ks X3 _t_ ~,2~_ [/*ls -}- /ZiS - - /*3s - - /*4S] =

Replacing #is, #2s, #3s and #4s by re~2 - and ~/2 - - X 4 respectively, (4.5) becomes ~s bib

cos X1 b e -~z

cos X2 b e-~Z0

cos X1 . .

cos Xg. . .

X1,

0.

"r;/2 - - X 2 ,

cos X3 b

cos X4 b

cos Xa . .

cos X4

(4.6)

~/2 -- X3

or Vs

bp

.

C1

C2

.

C3

C3

(4.7)

where C1, Ce and C3 are respectively the values of the wave velocity at the observation point, source point and at the surface. X2 and Xa are the grazing angles of the incident ray at the source and at the surface; X1 and X4 are the grazing angles of the reflected

REFLECTION AND DIFFRACTION OF SH WAVES

389

ray at the observation point Pl(x3, z) and at the surface. From (4.7), we obtain the relation Xa = X4 which shows that the condition of reflection at the free surface is satisfied. Now, the saddle point method evaluates the integral (4.4) in the form

1 e4(Zo-z)A(ks ) . ~

(C i/4 x / ~ ) 5 elW= . -x/Hs

4iyrcS

(e<~J~)5

e V (=o-=)

--

4iy~s

A (ks)

x/Hs

x

X e in1 e i[(p&c°sv~=-pe°s~==)+(pSe°sm~-~°e°sm~)].

(4.8)

Therefore, introducing the time factor e -iot, the displacement arising out of (4.2) is _ ~ e V (=°-=) A (k=) 4iyr~3

x

x/Hs

X e in~ e i [ ( p ~ ° s i n X ~ - p s i n X S ) + ( p ~ s i n X ~ - p s i n X D - ° ) t l where ~1 = IX3 -~ 2 4 - -

Xl--

X2]

(

~S

(4.9)

~2fS

and Hs is the Hesse's determinant [101 defined by

[ I~#iS#~ ~2Y~ s'

i,k=1,2,3,4,5 and#5=k.

Now, the time required for the disturbance to travel along the incident ray OA1 (Fig. 3) is given by ta

0

dt = -0

C sin X z0

using the relation, cos X

cos X2 - -

C we obtain 1

-

-

C2

cos X i.e.

-

b e-VZ -

cos X2 - -

be-;,Zo

390

M . L . GEOSI-I

which with the help of the relation ~o cos X2 = cos X3 from (4.7) reduces to the form t3 = ~

1

(P~o sin X2 -- p sin X3).

Therefore, o)ta ---- (P~0 sin X2 -- p sin X3).

(4.10)

Similarly, the time t4 required for the disturbance to travel along the reflected ray AlP1 may be shown to satisfy the equation tot4 = (p~ sin X 1

-

-

p sin X4).

(4.11)

Taking help of (4.9), (4.10) and (4.11) and introducing the time factor e -iot, the displacement at the point Pl(x3, z) due to the second term of (4.1) is 1

<~ (e=i/4 J ~ ) 5 e Y (z°-~)A(ks)

4i~a

e ~ e -x'Et-(t~+t~)l.

(4.12)

x/Hs

The factor eis~ gives the total phase shift as the disturbance travels from the point 01(0, z0) to the point Pl(X3, z) along the path OAIP1. The displacement arising out of the first term of (4.1) viz. o~ ( z o - z )

oo

H 2)(p~) H~l)(p~0 ) eik, l dk

4iv --oo

can similarly be evaluated by the saddle point method. It gives the direct wave from the source point (0, z0) to the point Pl(X3, z) without any reflection on the boundary. The path has been shown in Fig. 3 by the ray OA3P1. The evaluation of the above integral by the saddle point method yields,

4i w

x/H

where the factor ein"= ei(~'-m')[vr(~s"/r"v~)l gives the total phase change as the disturbance travels from the point 01, to the point P1 and H'~ is the determinant given by

REFLECTION AND DIFFRACTION OF SH WAVES

p~ cos #is

0

-

0

--cos #~s

391

-Ff s

~-

s

, / a2~

ks

The subscript s denotes the values of the quantities at the saddle point. It can easily be shown that the factor e i[~°~°e°sa%~-~e°s#'lsl is equal to ei~t5 where t5 is the time required by the disturbance to travel from the point O1 to P; along the ray 01AaP1. Therefore the resulting displacement which is a combination of the direct and the reflected wave arriving at Pl(x3, z) is

4~:i7

L +

,/Hi 4H s

+

A(ks) e i•1 e -k°{t (ta+/4)}j.

(4.13)

§ 5. Conclusion

Firstly, evaluating the integral for the displacement by the residue theory, the diffraction field at any point in the shadow zone is deiived and the appearance of an exponential decay term in the expression for the displacement is shown. This exponential decay takes place as the ray travels along the horizontal portion of its path on the boundary. Secondly, the saddle point method of evaluation of the integral yields the displacement at any point in the illuminated region and a continuous phase shift takes place as the ray travels from the source point to the observation point, a phenomenon which is absent in the homogeneous medium. In this paper, the shear wave velocity has a exponentially decreasing character leading to the formation of a shadow zone. If, instead, the shear wave velocity is exponentially increasing with the depth, the displacement at any point near the surface will suffer from the rays successive reflections on the boundary. Due to the reflection of different rays on the boundary, caustics, which are the envelops

392

R E F L E C T I O N A N D D I F F R A C T I O N OF SH W A V E S

of different rays striking the boundary, will also be formed. In a later investigation, the author intends to study the field at points near and away from the caustics in the case when the shear wave velocity has a exponentially increasing character. Received 4 September 1969 In final form 22 J a n u a r y 1970

REFERENCES

[1] DERESIEWICZ,H., Bull. Seis. See. Amer. 52 (1962) 639 [2]

HUDSON, J. A., Geoph. J. 6 (1962) 131.

[3] WILSON,J. T., Bull. Seis. See. Amer. 32 (1942) 297. [4] [5] [6] [7] [8] [9]

MITRA, M., Geofis. Pura Appl. 41 (1958) 86. HOOK, J. F., J. App. Mech. 29 (1962) 293. PEKERIS, C. L., J. Acoust See. Amer. 19 (1946) 295. BENMENA~tE~I, A., Bull. Seis. See. Amer. 50 (1960) 15. VLAAR, N. J., Bull. Seis. See. Amer. 50 (1966) 715. STRIEFE~, W. and R. D. KEPIS, Quart. App. Math. 21 (1964) 2$~ [101 VANDERPOL,B. and H. BREMMER,Phil. Mag. 24 (1937) 141.