37
Propagation of Love Waves in a Non-homogeneous Stratum of Finite Depth Sandwitched Between Two Semi-infinite Isotropic Media By NIRMAL KUMAR SINHA 1)
S u m m a r y - The aim of this paper is to study a problem in which the intermediate layer is nonhomogeneous, the rigidity varying exponentially with depth i.e. }~2 : Q2 vg e~Pz, the density being constant, velocity varies also exponentially with depth according to the law /7 = /l~2/o--~ = v0 eP< The variability of K H with the change of phase velocity is shown graphically.
InW oduclio~, STONELEY [112) studied the problem of propagation of generalized type of Love waves in homogeneous medium of finite depth sandwitched between two semi-infinite isotropic media. He showed that the existence of Love type wave is possible, if the wave length is not very large or the thickness of the middle layer is not too thin. He also showed that when the distortional wave velocityin the upper semi-infinite medium is less or greater than that of the lower semMnfinite medium, the Love type wave can exist in both the cases. DUTTA [21 solved the problem of propagation of Love waves in a non-homogeneous internal stratum of finite depth lying between two semidnfinite isotropic media in three different cases. In one paper variation considered b y him are (I)
z
#2=/zocosh 2~-,
~ 2 = ~ o c o s h 2~.
where #0 and ~Ooare constants.
where #o, ~0 are constants.
i) Tangrakhali Bankim Sardar College, P.O. Tangrakhali, 94-Parganas, India. 2) N u m b e r s in brackets refer to References, page 42,
38
N.K. Sinha
(Pageoph,
a n d in a n o t h e r p a p e r t h e v a r i a t i o n s are (I)
~=#oe
'~'~ a n d
~ o ~ = 9 o e ....
where m is c o n s t a n t a n d #o, ~0 are t h e vatues of the r i g i d i t y a n d d e n s i t y at t h e i n t e r face a n d are constants. /3.~= |l/i;; .~=
V'~-
constant.
-
I n all these p a p e r s t h o u g h he considered t h e r i g i d i t y a n d d e n s i t y variable, velocity is t a k e n constant. H e showed t h a t t h e L o v e t y p e w a v e is possible if t h e p r o d u c t of the w a v e n u m b e r a n d t h e d e p t h of the i n t e r n a l s t r a t u m is sufficiently large. R e c e n t l y PAUL [3] s t u d i e d a similar p r o b l e m of p r o p a g a t i o n of L o v e waves in a f l u i d - s a t u r a t e d porous l a y e r lying b e t w e e n two elastic halfspaces. I t is f o u n d t h a t in t h e case of small p o r o s i t y factor, the w a v e l e n g t h of L o v e w a v e p r o p a g a t e d in a fluid s a t u r a t e d porous l a y e r increases or decreases in c o m p a r i s o n w i t h t h e case of elastic i n t e r m e d i a t e l a y e r according as t h e d e n s i t y of the solid is g r e a t e r or less t h a n t h a t of t h e fluid filling t h e pores. I n this p a p e r d e n s i t y is considered to be constant, t h e r i g i d i t y being v a r i a b l e t h e v e l o c i t y varies w i t h depth. T h e result o b t a i n e d in this p a p e r shows t h a t for c < c, < Ca L o v e t y p e of w a v e is possible a n d t h e v a r i a t i o n of the phase v e l o c i t y w i t h w a v e n u m b e r is shown graphically.
oz=VgC
v~176
i
p = O K~/~
e ___,,_
Mathemalical derivatio~
o -- constant f~ = variable
Ii z
-H
III /5
--c~
L e t us assume t h a t the m e d i u m I is e x t e n d e d from z -= 0 to z = o% t h e m e d i u m I I is e x t e n d e d from z = 0 to z = - - H a n d t h e m e d i u m I I I is e x t e n d e d from z --- - - H to Z ~
--OO.
The c o m p o n e n t of d i s p l a c e m e n t (u, v, w) in a plane w a v e t r a v e l l i n g in t h e direction x increasing in a n y m e d i u m m a y be a s s u m e d to be t h e real p a r t of (0, V, 0) e~(*-~'~/, where V is a function of z only. F o r L o v e w a v e we consider u = z~ - - 0.
Vol. 65, 1966/111)
39
Love Waves in a Non-homogeneous Stratum
The equation of motion is [4] d2v
o
0t 2
--
@ xy 0x
4-
dp y z &
'
(1)
and for Love wave we assume t h a t all displacements are independent of y coordinate. The non-zero stress-strain relations are 6v
P~Y
(Ref.:
EWlRG, JARDETZKY
# ~-x
and
and
6v
Pu~
# & .
(2)
PRESS [4]).
Let vj(] = 1, 3) be the displacement components in m e d i u m I and I I I respectively. P u t t i n g (2) in (1) we have ~
\W]
+ 6zz \ & ]
#a
6t~
( ] = 1,3)
(3)
where (#~-, ~j), (j = 1, 3) are the constant rigidity and density of m e d i u m I and I I I respectively. Let vj - - V j e ik(~-~') , (?" = 1, 3) (4) where Vj(]
1, 3) are functions of z only, the equation (3) reduces to d~V~ dz 2
d 2 V3 dz 2
s~ V~ = 0
(5)
2 V8 = 0
(6)
$3
where sl--k
1--
c~J
and
s3=k
1 - - c~J
'
(7)
The solutions of the equations (5) and (6) suitable for the problem are V~ = C e ....
and
Va=De
"~"z
(8)
where C and D are constants. The equation of motion in the second m e d i u m is d
~v2 ~
d
dx (#2 Ox] + 6~-
(#2 Or2 ~
62v2
&-z] = ~2 dt~ ,
(9)
where v~ is the displacement c o m p o n e n t in the second medium, #~ the rigidity of the m a t e r i a l of the second m e d i u m being function of z only and ~o2is constant. E q u a t i o n (9) can be written as 62v2 6~v2 dt~.z ,u2 ~Tx2- 4- #2 ~z~ 4- &
P u t t i n g vz = Vz
e ik(x-ct)
&2 __ ~ &
62v2 6t2
O0 )
in (10)
d~V2 1 dz2 + - - -r-
d~*2 dz
dV~ k~ ( 1 ~ -dz - --
o2c~) V2 0 ~Its = "
(11)
40
N.K. Sinha
(Pageoph,
A g a i n p u t t i n g V= = Z//,72 in the a b o v e equation, we h a v e cl2Z dz~-
Z ct~#2 Z2 ( d # ~ 2 _ k 2 ( 1 _ 2 ~2 dz~ + 4 ~ ~ dz ]
~2c" t -g2 / z = O "
(12)
N o w p u t t i n g ,u.~ = O~ v~ e 2p* for the second m e d i u m t h e a b o v e e q u a t i o n reduces to d2)r
e~-
_
p2
Z -- k 2
(
1--
--
~o2 e
~l'z
)Z=
O.
(13)
P u t t i n g 7 = e-2P~ in (13) we get, [
dY2 + -7 ~
~2
kc
+
p2__k2 4 y~
Z = 0.
(14)
W r i t i n g (k c/2 vo p) y = ~] (14) becomes d~z _r_ 1 d Z {1 d~ 2 r/ d~- + ~7
m2 } 4~12
Z=0
where P~ + k 2
m 2 --
p2
Hence t h e solution is Z = A J,,(2 ~,'~) + B Y ~ ( 2 1/~). Hence A J~(a e-p~) + B Y,~(a e P~)
V2
1/7~~o,P'
/ g ~ o ,p~"
(15)
where a
:
k c- -
.
t~Op
The b o u n d a r y conditions are
ftl
V2 = V1
at
z = 0
V2
V3
at
z
-- H
d V2 ,u2 dT-z
at
z
0
at
z
--H.
d gl dz d V2
~2
clz
06)
d Va =,uaT
F r o m t h e b o u n d a r y conditions, we have, A Jm(a)
4- B
Ym(a)
-=
C
,
~/~o A J,,(a ePH) .- B Ym(a ePH) __ D e ~3H l / ~ ~'o~+H "2
(16 a)
(16b)
tl
- - 2 02 v0 # , .st C = ,u2E{ - - A a p ( J m - l ( a ) - J,,,+,(a)) (16c) -
p a B(r,,_,(a)
- - Ym+~(a)) } V'~ vo - - 2 ~/e~ vo p { A J,,(a) -}- B Y , , ( a ) } ! ,
Vol. 65, 1966/111)
Love W a v e s in a N o n - h o m o g e n e o u s S t r a t u m
41
and
--t3ap#;{Y=_,(ae __
/--
pH) -- Ym+l(aePH)}]V~voe -pH]
(16d)
!
Where ~.~ = ~ ~ , - ~ ana ~ ' = e, ~ . From (16a), (16b), (16c) and (16d) eliminating equation,
A , B , C, D w e g e t t h e f r e q u e n c y
p 12~ a J m - l ( a epH) + (2 #a sa + 2 p 12~) J m ( a epH) -- p 12~ a Jm+t(a el)H) ,u~p a.]'m-~(a) + (2 12'~p -- 2 121 sl) Jrn(a) - #$. p a J~n+i(c~)
(17)
p t,~ aa Ym-Z(oa e r + (2 123 s3 + 2 p 12~) Y m ( a epH) -- p 12~ a Ym+z(a ePH) lt~ p a Y m - l ( a ) + (2 12~ p -- 2 12I sl) Ym(a) -- 12'~p a Yra+l(a)
F o r t h e n u m e r i c a l calculations we t a k e
kip
= 1/3 so m = 2. E q u a t i o n (17) becomes
p 12~ al J l ( a l r) q- (2 12a sa + 2 p 12~) Jz (al f) - - p 1012g~l J3(al ~) 12'~p a l J l ( a l ) + (2 12~p - 2 #1 sj) J2(al) - 12'~p a l J a ( a l ) _
where
_
(18)
p #~ al Y l ( a l ~') + (2 12a sa + 2 p r Y z ( a l r) - p 126 al Y3(al r) 122 P g l Y I ( g l ) @ (2 12~ p - - 2 ,~1 81) ]('2(gl) - - 126 p a,1 Y3(al)
al -~ 1/a C/Vo a n d r = ekills~5.
Now
14 = ~o2 v~ e -~p~ --/~' e -2p~ = P u t t i n g this v a l u e of r
14_ 1/'2 "
in (18) we get
2S3 23
~,3@_2pY__4p_ug~_pf/2}Jl(al~,)__y{a,lS3 12~/3@a,lpl/--2f16~l}Jo(al $/)
__ {2 sa
-j;
}
12a r a + 2 p r -- 4 p + a~ p r 2
{
# L1 2Sl 12~ + 2 p
Yx(ax r) -- r
}
p a l 2 Yl(al) -
al sa - -
12;
{
sial
r a + al p r -- 2 p a l
121 12~ + a i p
/
}
Yo(az r)
Y0(al)
The n u m e r i c a l calculations are p e r f o r m e d b y the help of 'The T a b l e of F u n c t i o n s ' ~5] a n d we t a k e t h e following p a r t i c u l a r values vo = 3.00, #1 = 2.997, #a = 6.469, cl = 3.363, ca = 4.362 a n d #~' = 2.385. Table 1
al =
/5 c V0
kH
r = e --,_~ V3
1.9
1.7
1.6
1.4
1.3
1.2
1.1
1.00
0.9
0.8
0.7
0.6
0.5
0
2.18
2.77
3.015
3.55
3.85
4.2
4.61
5.105
5.68
6.4
7.12
8.55
10.278
oo
42
N.K. Sinha A ck~,owledg e m e n t
F i n a l l y I express m y g r a t i t u d e to Dr. SuBI~AS DUTTA M. Sc. Ph. D. of B a n g a b a s h i College, C a l c u t t a for his help a n d guidance at every stage of this work. REFERENCES ~1] R. STONELEY, Proc. Roy. Soc. Series A 106 (1924), 424. ~2~ S. DUTTA(i), Geophysics X X V I I I , No. 2 (1963), 156. S. DIJTTA(ii), Geofisica pura e applicata, Milano 55 (1963),31. C3] M. K. PAUL, Bull. Seism. Soe. Alner. 54, No. 6 (1964), 1767. 74] EwI~G, JARDETZKYand PRESS, Elastic waves in layered media (1957). ~5] JAI-INKEand ]:~MDE, The table of fu~ctions with formulate and curves (Dover Publication, 1945), (Received 26th July 1966)
