47
Propagation of Love Type Waves in a Non-Homogeneous Layer Lying Over a Vertically Semi-Infinite Homogeneous Isotropic Medium By NIRMAL KUMAR SINHA 1) Summary - The aim of this paper is to study the propagation of Love-type waves in a homogeneous half-space overlain by a heterogeneous crust with various types of heterogenety. Frequency equations are obtained in each of the cases and numerical calculations are done in some cases and the results thus obtained are compared with those done by JEFFREYS and MITRA [3]2).
Introduction Many authors have studied in different times the propagation of Love waves in layered half-space. Most of those have considered inhomogeneity in the lower semiinfinite medium. DUTTA [2] studied a case in which the upper layer was taken to be heterogeneous, the rigidity and density were chosen as # = #0/(1 + ~ z), ~ = ~o/(1 + e z). MITRA [3] also studied a similar problem where the rigidity and density were chosen as p = #o e2W, ~ = Co eZvz. AVTAR [10] in a recent paper considered a non-homogeneous half-space with double homogeneous surface layers, where he considered several types of variation in rigidity and density and obtained the solutions of corresponding differential equations. In this paper the model is chosen as that of MITRA and DUTTA. Several types of variations in rigidity, density and shear wave velocity of the upper cruster layer have been studied. Frequency equations are obtained in each case and numerical results are obtained in few of the cases considered. Problems are classified into three different groups (1) when the rigidity #, the density 0 and the distortional velocity fi of the layer are variable, (2) when # and 0 are variable and fi is constant. (3) when # variable and Q constant. In the first group following cases are considered: /#o(1 + 2 z) 2 1) # = # o ( l + 2 z )
2,
~=Oo(l+6z)
fl=~/~o(1-+--6-Z) '
MEtSSNER [4] and DASGUPTA [7] used this type of variation. 1) Department of Mathematics, Bankim Sardar College, P.O. Tangrakhali, Dist. 24-Parganas, West Bengal, India. 2) Numbers in brackets refer to References, page 59.
48
N.K. Sinha 2)
# = #o e
,
(Pageoph,
0 = 0 e"~
#o
e(m-n)/2z
m-fin.
# 0 e vz
3) # = # o e ~ ,
O-vg(l+itz)
fl=v~
4)
O=0o(l+itz)
fl=~/Oo(l~itz)
#=#o(a-b'z)
/.o 1
#o, Vo constants. b' r b', 2 constants.
In the second group the following cases are studied: 1) # = # o ( l + 2 z ) 2 0 = 0 o ( 1 + 2 z ) 2 #o, 00 and fl being constants. 2) # = ~o cosh%Z ~) o = 6o cosh%Z ~) /~o, 0o, it,/3 being constants. #0, 00, 2, fl being constants. 3) # = # 0 ( l + i t z ) 0=0o(l+itz) In the third group the following cases are discussed: 1) P = #o e~Z 2) # = # o ( l + 2 z ) z 3) # = #o(1 - b' z)
0 = constant. O=constant. 0 = constant.
SATO [83 used this variation.
M a t h e m a t i c a l Derivation
We choose the origin of coordinates at the interface and the Z-axis vertically downwards into the homogeneous isotropic medium which extends from Z = 0 to Z = oo and that the layer extends from Z = 0 to Z = - H. The displacement component Z= --g
I Z=0
I Sz=oo corresponding to a Love waves travelling in the direction of x-increasing in any medium may be assumed to be the real part of(0, V, 0) e iK(x-ct), where V i s a function of Z alone. The equation of motion is
~x (Yx) + ~y (Y,) + 3z (r~) = 0 5 + ~ where
Yx, Yy, Y~
(1)
are the stress components across a plane y=constant, and ~ is the
Vol. 73, 1969/II)
Love Type Waves in a Non-Homogeneous Layer
49
density of the material of the medium. The non-zero stress-stain relations are X r = p exy Yz = # eyz
(2)
where # is the modulus of rigidity of the material and exy and ev~ are the strain components. Let v i be the displacement components in the lower medium. Substituting (2) in (1) we get the equation of motion for the lower medium.
62111 62/)1 5X2 "~- 5Z2
~01 (~2111 #1 5 + 2"
(3)
Where ~1 and #1 are the constant density and rigidity. Let 111= V~e ~(~- ~o where V1 is a function of z only. Substitution of vl in (3) yields,
d 2 v~ _ / / 2 V1 = 0
(4)
dz 2
where
/'/1 =
/~
1
c2~89 -- C~J
(5)
The solution of equation (4) suitable for the medium is V1 =
A
(6)
e -"~z.
The equation of motion [1] in the non-homogeneous layer is d2V l dtt d V dz z + # dz dz
K2(
~ 1-
) c 2 V = O.
(7)
Substituting V ' / x / ~ in (7) we get daV ' dz e
V' d2l~ + _ _ 2 # dz 2 4
\~/
- K2
1 -
- c2 ~
V'
=
o.
(8)
The boundary conditions are
v=~ dV # dz
0
dV dVl # d z =it1 -dz -
at
z=0
at
z =
at
z = 0
-
H
(9)
Group I Case 1 - when
tt=#o(l+2z)
2 O=Qo(l+fz)
(10)
2 and 5 being constants 2 # 6. The same variations where 2 = ~ were used by MEISSNER [4] in his problem of heterogeneous half-space.
4 PAGEOPH73 (1969/II)
50
N.K. Sinha
(Pageoph,
Substituting the values of # and 0 as in (10) in equation (8) we get
d2V~
--
K 2
1 - c2(1 +
where
az) 1 v' = 0
(11)
c2(1 + 2 z)2J
dZ 2
c~2 _/~o ~Oo Now putting 1 - c2 Kc5 C2O'2)~2'
2K(l+2z), = ---
m 2=-
11 1 + 4 cZ(6 - 2) Kz-] 1/2
2
C2o;d
j
(12)
We transform the equation in the form d 2V' I 1 l d4~ +.-4+~+
88 ~2
(13)
j
Its solution is obtained in Whittaker function as v' = E, ~,,m~(r
or
+ F1 W(-,,m)(- 4)
1
V - x/gTo(1 + 2 z) [El W(t,m)(4) + F1 W(-,,m)(- 4)]
(14)
Applying the soundary conditions (9) we get the frequency equation f(c) =
•
(;~ ~o - ~, nl) w(,,m)(~) - ~o az w . , ~ . ) ( r
z o
(1 - ~ / 4 ) 5~ ~ ) - , , m ) ( - - 4) -- X W~_,.~)(-- 4) ~ = - , ,
(15)
•
2 W(,,m)(4) -- (1 -- 2 H) 6z W(z,m)(4)
Z=-H
= 0.
In particular when 2= 5 in (10) the rigidity and density becomes /~=/~o(l+6z), The frequency equation becomes
f(c)
=-
(2/~ o - / q
(16)
p=~o(l+6z).
hi) W(i ' 1/2)(~1) -- ~0 L W(l, 1/2)(~1) Z=0
--/(1 - 2 H) 3z O W(-l, 1]2)(--~1) -~L W(-l, 1/2)(--~1)/_ Z=-H (17) -
(~1 < - .~ ~o) w(_,, 1 / 2 ) ( - ~ ) + ~o ~z w~_,, . ~ ) ( -
•
~ W(l,
1/2)(41) -- (1 -- /~ H) L W(I' 1/2)(r
Z= -H
= 0
r
~=o
Vol. 73, 1969/1I)
51
Love Type Waves in a Non-Homogeneous Layer
where 2K ~ (l+6z),
r
c2 K l=vzo.2~ 6
(18)
Case H - when # = # o e'~Z
(19)
~ = Oo e "z
where mr These values of # and Q reduces equation (7) to d2V d V K 2 1 - - - e (m-")~ V = 0 dz 2 + m dz v~
(20)
The solution of (20) is (21)
V = e=[E2 4 ( f l e~) + F2 yp(fl e~Z)] where o: =
m 2'
v
n-m 2
x/eK2 + m2 P
2[cK'~ fl
l'l -- m
(22)
l'l -- m
Applying the boundary conditions (9) we get f ( c ) "= {(l~~ ~ - v P #~ + #l rll) JP(fl) -I- l~~ V fl JP- l(fl)} x {(c~- v p) yp(fl e -~n) + fl v e -vn yp_ l(fl e-~n)}
I (23)
- {(~o ~ - v p ~o + ~1 nl) yp(/~) + ~o v/~ yp_ 1(/~)} x {(c~ - v p) Yp(fl e-~n) + flv e-~n jp_ l(fl e-~n)} = 0.
/
I
For numerical calculation we put n=2rn
c 2 3 #1 . . . . c~ 4' Po
20 9
and
p=2
and get the table I, Table 1
V'3c/vo
2.00
1.5
1.00
0.5
eI~U/43 lies between
0.4 and .41
.37 and .38
.33 and .35
.21 and .22
Mitra studied a problem on the propagation of Love type waves in a non-homogeneous layer with rigidity /~=#o e2~Z, 0 = 0 o e2"Z lying over a homogeneous semi-infinite media. Putting rn = n we get that particular case, from the above problem. Case I I I - when -- #o e= f12 = Vo2(1+ 2 z) (24) #o, vo2, 22, v being constants. : Substituting (24) in equation (8) we get ~az
+ [vo~(1 + a z) -
K2 +
v ' = 0.
(25)
52
N.K. Sinha
(Pageoph,
t~c /.7
ro 1~' 1.3 /2 0.9 0.7~ ~5
Table 4
Figure l
lz = poe ~z Q= const.
The solution of (25) is obtained in Whittaker function. e - (v/2)z
V-
(26)
[E3 W(t, 1/2)(t) q- F3 W(-I, 1/2)(- t)]
4s
where 1
t=2(gK
K 2C2
2+v2) ~/2(1+2z)
and
(27)
l - 2 v o 2 ( 4 K 2 + v 2 ) 1/2"
Applying the boundary conditions (9) we get the frequency equation.
v
=--
•
)
#, n 1 W(t, 1 / 2 ) ( t ) - ( 4
K2 + vZ)l/Z l~oW('t,x/z)(t)
Z=0
E
- ~ w~_,. ~/2~(- t) + (4 K 2 + v2) ~/2
-
p, nl
1
w~'_,. 1/2~(-
1
0 z=-. (28)
W(-l, , / 2 ) ( - t) - (4 K 2 + v2) '/2/~0 W~'_,, 1 / 2 ) ( - t) Z=0
x
iv
- ~ W(,, 1/2)(0 + (4 K 2 + v2) '/2 W(,, 1/2)(t
= 0 Z= -H
where
w~',. ~/2~(t) = ~t w,. 1/2>(t). Case I V -
when #=#o(1-b'z)
where/~o, Po, b' 2 are contants.
0=0o(l+2z)
(29)
Vol. 73, 1969/II)
Love Type Wavesin a Non-HomogeneousLayer
53
Substituting relations (29) in equation (8) we get
d2V' ~dz+
{~_( b'Z ~cZc2(l + 2z) 1 - - b ' z ) 2 + Vo2(1-b'z)
} K2 V'=0.
(30)
The solution of (30) is in Whittaker function V=~(1
- b' Z) 1/2 [E 4 W(_r,o)(t) + F4 W(r,o)(- t)]
(31)
where l, = _
Kc2(b ' + 2) b'+c22 2 b '2 Vo b'
(32)
Applying the boundary conditions (9) we get the frequency equation f ( c ) - I ( 2 b' +#~n~)W(r,o)(t)+2l~oK
X ] ~ W(-r, o ) ( - t ) + ( 1 + b' H)'2 K
/ (33)
x
I + ~vg b dz
z =o
x 1 ~ W(r, o)(t) + (I + b' H)'2 K + ~o~ b--;dz w(,, o)(t) ~ = - . = o
•
where
2
•/
KZ+
t=
K2 c2 ~. v~b~ (1 - b' z).
-
(34)
b'
Group H When the rigidity and density variable but the distortional velocity is constant.
54
N.K. Sinha
(Pageoph,
Case I - when #=#o(l+2z)
2
z
0=0o(1+2z)
(35)
Substituting the condition (34) in equation (8) we get - - - K
dz z
1-
2
V'=0.
(36)
The solution is V = Es l o g n z + F s sinn z
(37)
~/#o(1 + 2 z) where
(38) Applying the b o u n d a r y condition (9) we have the frequency equation.
n i l { nl H#~ ( 1 - 2 H) + 22
(39)
f(c) -~ tan n H n2 H2(1-2
H)+2H(2
H-#o#1 na H )
for numerical calculation we take
cz
3
c1
4
and
#1 #o
-
20 9
(for 2 H = 0).
(40)
Table 2 e/e2
1.00
KH obtained by JEFFERY oO KH obtained here co
1.05
1.1
1.15
1.2
1.25
1.3
1,333...
4.188 4.195
2.664 2.665
1.943 1.944
1.463 1.464
1.067 1.068
.646 .646
0 0
1,3 cz E /.1
Table 2 Figure 2 /t=/t0(l+2z) 2 0=00(l+2z) z 2H=0
Vol. 73, 1969/II)
Love Type Waves in a Non-Homogeneous Layer
55
Table 3 (for 2 H = 1/160)
c/c2
1.00
KHobtained by Jeffrey ov KH obtained in this case --
1.05
1.1
1.15
1.20
1.25
4.192 4.205
2 . 6 7 0 1.953 1.483 1.110 2.6 1.949 1.47 1.073
1.30
1.333...
-.649
-0
I3 g
7.2 1.I
l,O Table 3 r =/zo(1 + 2 z) 2
Figure 3 O = @o(1 -}- 2 z)2
L H = 1/160
From table 2 and 3 it is clear that the value of K H corresponding to c/c2 as obtained by me almost tally with those obtained by JEFFREYSwhen he worked considering the homogeneous layer over a heterogeneous half-space in which #=po(1 + 2 Z ) 2 and 0 = constant. Case H - when (41) # = #o cosh2( 2 z) 0 = 0o cosh2( 2 z) substituting in equation (8) 22
d2V ' - -
dz 2
+
m 2 V' = 0
where
K2
(42)
V = E6 COSm z + F6 sin m z
(43)
,J o cosh 2 z Applying the boundary conditions (9) we get the frequency equation
f(c) = cosh2 H ( - / q
n 1 costa H + m Po sinm H)
+sinh2H(l~o)~CosmH+2#l
nlm sinm H ) = 0 .
} (44)
Case I I I - when /l=/zo(l+2z)
0=0o(1+2z)-
(45)
56
N.K. Sinha
(Pageoph,
Substituting in (8) we get
dz 2
+ K2
- 1
+ 4 K2(1 + 2 z) 2
V'
0
(46)
where C2 _ rio.
The solution of (46) is 1
V=
x/tXo {A1 Jo(n
~) + B1 yo(n ~)}
(47)
where
~=~x/~
l :=0+~).
(48)
Applying to the boundary conditions (9) the frequency equation is obtained as f(c) = Jl {n(1 - 2 H)} {2 pl n 1 yo(n) -- Po 2 n yl(n)} + Y, {n(1 - 2 H)} {2/q H1 a0(g/) -- ]/0 )L rl a l ( n ) } = 0.
1
(49)
Group III When # is variable but ~ is constant. Case I - when P = #o eVZ. Now the equation (7) becomes
C2e_VZ"~ V = 0 .
d2V dV_ K2(I_ d~ ~ + ~ dz \
~
(50)
/
The solution of (50) is e_(~/2)[A 2 - ~ [ ' 2
K c -(,/2)~
a'i('=+4K=)i't v Co e
(2 K c
)
B2
(52)
Applying the boundary condition (9) we get f ( c ) = M 1 N 2 - M2 N1 = 0
(52)
where M, =
(2
-p)-~l.,
} J.(r) + r~o 2J.-l(r) ,
M2 = {(2 -- p) J.(r # . 2 ) + e~./2 r J._,(r e"/~")}
N2 = {(2 - p) yp(r e~H/2) + e~n/2 r y,_ l(r e~H/2)} where r-
2Kc V C0
and
pZ =
v2+4K 2 V2
(53)
Vol. 73, 1969/II)
Love Type Waves in a Non-Homogeneous Layer
57
F o r numerical calculation we take K
x/3
v
Izl
p=2,
-
2 '
/*o
20
-
c~ 4 --=-. co 3
and
9
(54)
The values ofx/3(C/Co) corresponding to the values of eKn/'/3 is then obtained as in the table (4) and they are plotted in a curve as in Fig. (1). Table 4 ~/3e/e0
1.9
1.7
1.4
1.2
1.00
0.8
0.6
0:5
0
eKH/~/3
1.87
2.08
2.53
2.95
3.53
4.41
5.87
7.04
oo
Case H -
when /* =/*o(1 - b' z),
(55)
0 = constant.
Substituting (55) in equation o f m o t i o n (8) we get
d2V' ~dz+
{
b'2 4(1-b'z)
c2 K 2 z +vo2(1-b'z)
} K 2 V'=0
where
vz - # ~
(56)
The solution of (56) is
V=
x/#o(1
- b' z) llz EP1 w~,,o)(t) + Q1 W~_~,o)(- t)]
(57)
where t-
cZK 2Vo2 b '
2K and
b' z)
t=_b,(1-
.
(58)
SATO [8] worked with this variable in a double h o m o g e n e o u s layer over a heterogeneous half-space. Case III- when tt = #o(1 + 2 z) 2 , 0 = constant (60) we get the equation o f m o t i o n
d2V ldVd# dz z+ # dz dz
K2 (
1-
~0) # c2
V=0.
(61)
JEFFREYS solved this case by an asymptotic solution. But using Bessels function o f imaginary a r g u m e n t we get the solution as K t)] V= tl/2[A4Jv( 2 t) + B4 YP(-2
(62,
where t=i(l+2z)
and
p=
x/)~2 Vo z - 4 K z c2 2 2 Vo
(63)
58
N.K. Sinha
(Pageoph,
Comparison study of graph obtained from different cases are plotted here.
L3 0
L2 l.l
Figure 4 Jeffreys's case l z = p ' ( l + 2 z ) ~ 0--const. ,~H=0
! 1.3 C
12
1.O
Figure 5 Mitra's case ,u/l~o = ~/0o = e T M
Satisfying
the boundary condition (9) we obtain the frequency equation as R1 $ 2 - $ 1
R2=O.
where K
Po
+
K
Y
K
J
K
Y
K
Vol. 73, 1969/II)
Love Type Waves in a Non-Homogeneous Layer
59
A c k n o w le d g e m en t
F i n a l l y I express m y g r a t i t u d e to Dr. SUBHAS DUTTA o f B a n g a b a s i College C a l c u t t a for his helpful guide a n d e n c o u r a g e m e n t at every stage o f this work.
REFERENCES [1] M. EW~NG, W. JAgDETZKYand F. PRESS, Elastic Waves in Layered Media (McGraw-Hill, New York 1957). [2] S. DUTTA, Geophys. J. Roy. Astron. Soc. 8 (1963), 231. [3] M. MITRA, Bull. Seism. Soc. Amer. 48 (1958), 399-402. [4] E. MEISSNEg,Proc. Second Intern. Congr. Appl. Mech. (1926), 3-11. [5] H. JEFFREYS,Monthly Notices Roy. Astron. Soc., Geophys. Suppl. 2 (1928), 101-111. [6] J. T. WILSON,Bull. Seism. Soc. Amer. 32 (1942), 297-304. [7] (a) S. C. DASGUPTA,J. Appl. Phys. 23 (1952), 1276-1277. (b) S. C. DASGtreTA,Indian J. Theor. Phys. 1 (1953). [8] Y. SATO,Bull. Earthquake Research Inst. 30 (1952). [9] T. MATOZAWA,Bull. Earthquake Research Inst. 6 (1929). [10] P. AVTAR,Pure and Applied Geophys. 66 (1967), 40. [11] F.T. WHITTAKERand G. N. WATSON,A Course of Modern Analysis (Cambridge University Press, 1927). (Received 20th April 1968)
