ON THE PROPAGATION OF LOVE WAVES IN A NON-HO. MOGENEOUS INTERNAL STRATUM OF FINITE DEPTH LYING
BETWEEN
TWO SEMI-INFINITE MEDIA
ISOTROPIC
b y SUBaAS DUTTA (*)
Summary - - The possibility of propagation of Love waves in a ,.on-homogeneous internal stratum of finite depth lying between two semi-infinite 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 semi-infinite isotropic media. He has shown t h a t the existence of the Love-type 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 semi-infinite medium is less or greater t h a t t h a t of the lower semi-infinite medium, the Love-type 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 non-homogeneous, 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 non-zero
stress-strain
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~(z-ct)
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 ~ Ae-Sl 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
2-2
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)--~-(CcosnH--DsinnH}]
= tL3 Bs3 e-H'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 e-mH + ( ~# s3-}- __m e-mH) ( 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)
