18
On Love Waves in Heterogeneous Media By TAPAN KUMAR DE 1)
Summary - Love waves in a half space with one homogeneous elastic layer overlying a semiinfinite medium having elastic properties varying with depth has been considered. The frequency equation for small wave lengths has been obtained, considering general variation, and has been shown to involve the first three derivatives of the rigidity of the heterogeneous medium at its interface with the homogeneous layer. Statement of the problem Let us assume that the lower in-homogeneous half-space is characterized by the density 0 and rigidity # while the upper homogeneous layer of thickness H is characterized by density O~ and rigidity pa. Further let x-axis be along the direction of propagation of wave and z-axis vertically downwards with the origin at the interface. The equation of motion for displacements of the form [0, v(x, z, t), O] is
8%
8# 8v
o ~t 2 = ~ V 2 v + ez gz
(1)
and for the upper layer
82vl
o O~-= ~
V2
v~.
The boundary conditions at the interface z = O and on the free surface z = - H
(2) are
(
#1 8 z / ~ = _ ~ = O (v~)~ = o
=
(0=
= o
(3)
1) TAPANKUMARDE, 12 Pannalal Basak Lane, P.O. Liluah, Dist. Howrah, West Bengal, India.
Love Waves in Heterogeneous Media
19
Solution Substituting v = V/,/ti in (1), we get
o
a2V
V2
v+
(14#'2
1 ,,'~
7--2 #)v
If V=e ~('- a)f(z), the above equation becomes
d2f ( 1 #t, dz 2 - \ ~ c 2 + 2 # = I(z) f(z)
1 #,2
~c2c2~) f
4 #2
(4)
say
where if'
1
1 #,2
I(z)=rc 2 + 2 #
4 #2
/,c2 c 2
#
Supposing # and ,o to be slowly varying function of z, we have I - ~/r is a slowly varying bounded function of z. Hence for large ~c, taking w.k.b, approximation, we have
1 Hence the solution of equation can be written as v = ~ /A~ el,C(:,_ ct) I - W4- e
-- 5 1 llz dz o
(7)
the coefficient of ~c2 in I being taken positive for all z. The solution of equation (2) can be at once written as
e~sl= +
vl = ei~(x-C~
B1
e~s~]
where
s,2
r ~ ----
p~
1,
Applying the boundary conditions (3),
A1 e -iK~IH + BI A
1
1-1/4
e i~sIH =
e
0
= A t
q- B 1
z=0
A
.7+~)+I
=i~cst#l(A1-B1).
1/2 e z=O
(8)
20
T.K. De
(Pageoph,
Eliminating A, A1, B1 the period equation becomes
1 [ # (1#'
tan tc sl H -
+ 11' + 11/21 ]
.
(9)
Since the term on the right hand side inside the bracket is always positive, therefore si is positive. Hence from (8) c > ill" (10) Also the present solution is valid under e
1
H-
/~s .
[/~ Ilia]z= o --
(11)
Again I'
-- = kt 1
kt" ~C2 +
2
+ - _ _ + tr 4 #3
t*
1 if'
1 t2
2/~
4/12
1,
O02 0
#
#
Thus the frequency equation (9) involves derivatives of # up to third order. Special cases (i)
/2=]2 2
e'SZ; O=Oz e oz. (DAS GUPTA [112))
In this case the period equation, from (9), is
tan](~g~H--~2I'/~
~c~lSl 2~+ ~2_~
(ii)
#=#2 (l+bz)Z;
(JEFFREYS[2])
0=~2=constant.
The period equation from (9) becomes tantr Sl H (iii)
P=#2 (l+bz)2;
#2
Ix/
0=e2 (l+bz).
~_o02]fl~+
--+b]
1 (o02/fl22)b K2
(MaTUZhWA [3])
The period equation is tan ~csl H =
/~2
~ - - C2/fl20 +
[~1 S1
2) Numbers in brackets refer to References, page 21.
.~_ 4 ~c 1 - ca/fl~
Vol. 70, 1968/I[)
Love Waves in Heterogeneous Media
21
where /~o 2 =/~2. ~2
(iv)
# =/22-t-/~ Z ;
~O~---Q2~- ~z.
The period equation is tan ~cst
1 H
K
s t ~j
where po~
~2
Q2
Acknowledgement [ express my thanks to Dr.S.K. CHAICRAVORTV, Department of Mathematics, University of Burdwan, Burdwan for his kind suggestion and help. REFERENCES [1] S. C. DAS GUPTA,Note on Love waves in a homogeneous crust laid upon heterogeneous medium, I, J. Appl. Phys. 23 (1952), 1276-1277. [2] H. JEFFR~VS,The effect on Love waves o f heterogeneity in the lower layer, M.N.R.A.S., Geophys. Suppl. 2 (1928), 101-111. [3] T. MATUZAWA,Observation o f some recent earthquakes and their time-distance curves, Bull. Earthquake Research Institute (Tokyo) 6 (1929), 225-228. (Received l lth September 1967)
