ON THE PROPAGATION OF LOVE WAVES IN A NONHO. MOGENEOUS INTERNAL STRATUM OF FINITE DEPTH LYING
BETWEEN
TWO SEMIINFINITE MEDIA
ISOTROPIC
b y SUBaAS DUTTA (*)
Summary   The possibility of propagation of Love waves in a ,.onhomogeneous internal stratum of finite depth lying between two semiinfinite isotropic media has been studied in this paper. The density and rigidity of the internal stratum are taken to vary exponentially with depth.
Introduction.   "R. STO~ELEY (1) (1924) studied the problem of propagation of generalised type of Love waves in a homogeneous medium of finite depth lying between two semiinfinite isotropic media. He has shown t h a t the existence of the Lovetype wave is possible, if the wave length is n o t very large or the thickness of the middle layer is n o t too thin. He further showed that, when the distortional wave velocity in the upper semiinfinite medium is less or greater t h a t t h a t of the lower semiinfinite medium, the Lovetype wave can exist in both the cases. The aim of this paper is to study a similar problem in which the i n t e r m e d i a t e layer is nonhomogeneous, the rigidity and density both v a r y i n g exponentially with depth. The result obtained in this case is very similar to t h a t obtained b y R. STONELEY. The possible limits of the roots of the frequeney equation are calculated with suitable values of the rigidity a n d density of different media 9 Let us assume t h a t the m e d i u m (1) is extended from z = ~ to z = 0, the medium (2) is extended from z = 0 to z = H, and the m e d i u m (3) is extended fromz ~I4to z ~oo. The components of displacement (u, v, w) in a plane wave travelling i n the direction x increasing, in a n y m e d i u m m a y be assumed to be the real part of (0, V, 0). 9 exp I l k ( x   c t ) ] , where V is a function of z alone. The equation of motion is
(1)
~
(@) +
a ( ~ ) +  a (~) = p 82~ ~y 8z ~t2
(*) Department of Mathematics, Bangabasi College, C a l c u t t a

9, India.

The nonzero
stressstrain
32
relations

are
i xy = ~Zexy,
(2) i yz
= ,l~eyz.
Let ~1 and 92 be the displacement components in medium (1) and (3) respectively. Substituting (2) in (1) we get the equations of motion in media [1) and (3) as follows : a2~i (3)
~x 2
a231
~23 i
I
az 2
ci 2
+
at ~ ,
~231
~2&3 ~233 I  }~x 2 az 2 c32
(4)
~t ~
,
where
V
Y
#i
#3
Pi
P3
are the velocities in the upper and lower media respectively; ~zl, ~zs are the rigidity and ~z, P3 are the densities of the two media. Let
31 = Fie ~kcx ct)
and
33 =
in m e d i u m (1)
Vze i~(zct)
in m e d i u m
(3),
where Vi and V 8 are functions of z only; then equations (3) and (4) reduce to d2V1
(5)
sl2V1 = O,
dz 2
(6)
s32V ~
dz 2
0,
=
xwhere
sl=k
1
2, Cl2
(7) c~ }I '~
C3 2
The solutions of the equations (5) and (6) are easily obtained as follows g l ~ AeSl z ,
(8)
The equation
of motion in the second medium
[EwI~G
& PRESS, (2)]

33
is
1
~ ~
~2~.  
:
P2
Ot ~
,
where ~2 is the displacement component in the second m e d i u m ; [~2 is the rigidity and ~2 the density of the m a t e r i a l of t h e same m e d i u m and are functions of z only. E q u a t i o n (9) can be w r i t t e n as (10)
 a2~ax ~+ ~~2~ 2az ~ 1
~2
~2
Let ~2=
Oz
~2
~t2
where V2 is a function of z only. So (10) reduces to
Vsei~(z~t),
dz V~ dz ~
(11)
0~2 __ P2 ~92
~z
[
1
d[l.2
d V2
[z2
dz
dz
[ k ~ [1
pz c 2] V 2 = O . [z2
L e t the equation (11) in V2 be t r a n s f o r m e d to equation in V' b y p u t t i n g V' V2 ~ ]/~2
(lla) and there b y we get d~V ~ dz 2
(12)
1
v
22
d2~.2 1 dz 2 4 4 .~ 2
vt(d~t212 .\ ~  z
k2(1
] . 
.
t~2
)V'
~.~ c2
O.
Suppose ~
(13)
~2 = ~0e
mz
and p~ = poe
where m is constant and ~0 and ?0 are the values of the r i g i d i t y and d e n s i t y at the interface and are constants. P u t t i n g (13) in (12) we get
~ v'+k2{(c~~  1 ) ~ "~}
(la)
dz 2
V'= 0
where (14a)
c2 =
 9 ~o
The solution of t h e equation (14) is (15)
V~= Ccosnz ~ Dsinnz,
where rt 2 =
k2~ 2 ,
(15a) I~ e u c e
(16)
 89 v2 = ~0 ( c cos nz + D sin nz) e
2

34
T h e b o u n d a r y conditions are i
V2 :
V1, at
z =
O,
V2 : Va, at z =   H dV 1
(17)
~1~
dV 2
:~2
~
dv~
, at z : O,
dV~
~ 2  ~ zz
:~3~zz
, at z :   H .
From the boundary conditions (17) we get  89 (17a)
A :
89
(17 b)
~0
(17 c)
C~0
, mH
(C cos nH   D sin nil) e"~ : BeSa H
~lAsl
:
V'o
[Dn
m/2
C]
and /
mH
~0 e
(17 d) i
~ [{CnsinnH}DncosnH)~(CcosnHDsinnH}]
= tL3 Bs3 eH'r 9
E l i m i n a t i n g A, B, C, D f r o m t h e a b o v e we get t h e f r e q u e n c y e q u a t i o n Won ~0m    ~V1 2
(18)
~on ~om _
_
_
2
_
tan n H +
1 m
~L3S3
2n
tan nH
mH ~on e
~1sl
On simplification equation (18) reduces to
f (c) = [n2 emH + ( ~# s3} __m emH) ( m_2
"z
~1 t a a) n H 8 1 ~o

n
(19) 
Sa }

s l e raft
Discussion of the roots of the frequency equation.   I n order t h a t n 2 is p o s i t i v e we m u s t h a v e
C2
)
//~2 w h i c h indicates
or
c >c~_.

35
We suppose C2 <
C <
s
<
('3"
F o r a g i v e n v a l u e o f m, let c = c' > ca m a k e s n = 0, t h e n m/2 = k L/c~c2  1 , a n d s 2 are p o s i t i v e . N o w f ( c ) will b e n e g a t i v e i f
sI
T~
V'Isl> T i.
~0
e.
w h i c h is p o s s i b l e w h e n c' is g r e a t e r t h a n c2 b y a s m a l l q u a n t i t y . As C i n c r e a s e s sl, s 3 d i m i n i s h e s a n d t a n n H / n i n c r e a s e s . W h e n c   cI i.e. s 1 ~ 0 t h e n f ( c ) will b e p o s i t i v e i f t a n n H i.e. t a n k ~ H > q a t c  cl, w h e r e ~ 
S3
q ~
at c = cl 2
s~=k
,
~o
1. C22
H e n c e t h e r e will b e a r o o t b e t w e e n c' a n d cl, i f k H is m a d e sufficiently large b y m a k i n g k or H large. I f k H is so s m a l l t h a t t h r o u g h o u t t h e r a n g e c2 < c < q , k ~ H r e m a i n i n g less t h a n ~/2 a n d a t c  q , t a n k ~ H < q, t h e r e will be no r o o t . T h u s i f t h e w a v e l e n g t h is v e r y l o n g or t h e t h i c k n e s s o f t h e i n t e r m e d i a t e l a y e r b e t o o s m a l l n o w a v e m o t i o n o f L o v e t y p e is possible. If c2 < ci < c < ca , t h e n s 1 is i m a g i n a r y . So t h e f r e q u e n c y e q u a t i o n will h a v e n o r o o t . If I f c2 < c < c3 < ci t h e n a r g u i n g as b e f o r e it c a n be s h o w n t h a t t h e r e is a p o s s i b l e r o o t o f t h e f r e q u e n c y equation. A n d i f c2 < ca < c < c1 , t h e n s a b e c o m e s i m a g i n a r y , t h e n u n d e r s u c h a cond i t i o n t h e r e is n o p o s s i b l e r o o t o f t h e f r e q u e n c y e q u a t i o n .
Nunterical results.   F o r t h e n u m e r i c a l e v a l u a t i o n o f t h e r o o t s o f t h e f r e q u e n c y e q u a t i o n w e t a k e t h e v a l u e s o f t h e elastic c o n s t a n t s a n d d e n s i t i e s as f o l l o w s : ~l P0 ~a 72
 3.00  5.00 = 6.47 ~ 2.72
X 1011 d y n e s / c m 2 [ u p p e r l a y e r ' ( I ) ] X 1012 d y n e s / c m 2 (at t h e i n t e r f a c e ) X 1011 d y n e s / c m 2 [lower l a y e r (3)] g m s / c m a ( u p p e r layer) Ro ~ 9.89 g m s / c m a ( a t t h e i n t e r f a c e ) ?3 = 3.40 g m s / c m a (lower layer)
36
cl
~'~1/Pl :
3.321
c2 ~ ~/~0/P0 :
=
2.249

c8 ~ ~ 3 / ~ 3 : 4.362 H = 37.5 K i n . W e a s s u m e k H  ~ 0.75 a n d m H = 1.62. N o w c : c' ~ 3.3097 m a k e s n = 0. T h e n f ( 3 . 3 0 9 7 ) =   .02661 A n d a t c = c1 = 3.321 f ( 3 , 3 2 1 ) ~ ~ .0282. H e n c e t h e r e is a r o o t of t h e f r e q u e n c y e q u a t i o n b e t w e e n 3.3097 a n d 3.321. ~ F i n a l l y I e x p r e s s m y g r a t i t u d e t o D r . B. B. S r ~ , D. Sc., F . N . I . for his h e l p a n d g u i d a n c e a t e v e r y s t a g e of t h e w o r k .
REFERENCES
(1) R. STObTELEY: Elastic Waves at the surface of separation of two Solids. (Transverse waves in an I n t e r n a l s t r a t u m ) , Proc. Roy. Soc. Series A, Vol. 106, p. 424 (1924). (2) W. M, EWIrCG & E. PRESS: Elastic Waves in Layered Media (1957), p. 342. (Received 9th January 1963)