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Propagation of G-type seismic waves in heterogeneous layer lying over an initially stressed heterogeneous half-space
Applied Mathematics and Computation 234 (2014) 1–12
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Propagation of G-type seismic waves in heterogeneous layer lying over an initially stressed heterogeneous half-space q S. Kundu ⇑, S. Gupta, S. Manna Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India
a r t i c l e
i n f o
Keywords: G-type seismic wave Heterogeneous layer Initial stress Transform technique Dispersion equation Phase velocity
a b s t r a c t The aim of present paper is to investigate the propagation of G-type seismic waves in a heterogeneous layer overlying a heterogeneous half-space under initial stress. Exponential variations in rigidity and density have been taken in the upper layer. In the lower halfspace both rigidity and density are varying with depth. Dispersion equation has been obtained in closed form. Dispersion equation in case of homogeneous media coincides with the general equation of Love wave. Curves are plotted for different values of inhomogeneity parameters and initial stress parameter. We have seen that the phase velocity decreases with the increase of inhomogeneity parameters. It is observed that initial stress has dominant effect on the propagation of G-type wave. Variation in group velocity has shown for different values of initial stress parameter. We have also drawn surface plots of group velocity with respect to wave number and depth parameter for different values of initial stress parameter. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Seismology is a study of earthquake and seismic waves that through and around the Earth. The propagation of waves in layered media is of central interest to the theoretical Seismologists. A review of this subject has been given by Ewing et al. in a classical monograph [1]. Two basic types of seismic waves generated by an earthquake, one is body waves and another one is surface waves. The body waves propagate within a body of rock. The faster of these body waves is called Primary wave or longitudinal wave and the slower one is called Secondary wave or shear wave. The second general type of wave is called surface wave, because its motion is restricted to near the ground surface. Such waves correspond to ripples of water that travel across a lake. The wave motion is located at the outside surface itself, and as the depth below this surface increases, wave displacement becomes less and less. The study of surface wave is always important to seismologists for understanding the causes and estimation of damage due to earthquakes. Surface waves in earthquakes can be divided into two types, Love waves and Rayleigh waves. Love waves are registered in the horizontal component but Rayleigh waves which are polarized in the vertical plane, are registered both in horizontal and in vertical components. The VI Generation of Love and other types of SH waves was formulated by Sato [2].
q
This work is supported by Indian School of Mines, Dhanbad, under Grant number ISM-JRF/Acad/2012/75.
⇑ Corresponding author.
E-mail addresses: [email protected] (S. Kundu), [email protected] (S. Gupta), [email protected] (S. Manna). http://dx.doi.org/10.1016/j.amc.2014.01.166 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
Abd-Alla et al. [3] discussed about propagation of Love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium. Noyer [4] studied the effect of variations in layer thickness of Love waves. The dispersion curve for Love waves due to irregularity in the thickness of the transversely isotropic crustal layer was observed by Bhattacharya [5]. Chattopadhyay [6] obtained the dispersion equations for Love waves due to irregularity in the thickness of non-homogeneous crustal layer. Wolf [7] investigated the propagation of Love waves in Layers with irregular boundaries. Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity was studied by Ghorai et al. [8]. Singh [9] discussed the Love wave at a layer medium bounded by irregular boundary surfaces. Love waves of long periods (60–300 s) are also called G-waves named after Gutenberg [10]. It takes about 2.5 h for G waves to make a round trip of the Earth. For large earthquakes, surface waves that travel around the Earth more than once are observed. The word ‘initial stress’ is stress which developed a medium before it is being used for study. The Earth is initially stressed medium, due to presence of external loading, slow process of creep and gravitational field, considerable amount of stresses which are called prestresses or initial stresses, remain naturally present in the layers. Many researchers have given their contribution in the study of G-type wave. Notable are Gutenberg [10], Lehmann [11], Mal [12] etc. The propagation of G waves, estimation of earthquake moment, released energy and stress–strain drop from the G wave spectrum was discussed by Aki [13] from the Niigata earthquake of June 16, 1964. Chattopadhyay and Keshri [14] pointed the generation of G-type seismic waves under initial stress in a layer media. Propagation of G-type seismic waves in viscoelastic medium was established by Chattopadhyay et al. [15]. Chattopadhyay and Singh [16] investigated the G-type seismic waves in fiber reinforced media. Some other notable works in this field were done by many researchers like Jeffreys [17], Bhattacharya [18], Haskell [19] etc. Recently extensive and laudable works on the surface waves propagation have been done by many researchers. Gupta et al. [20] discussed the propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space. Torsional surface wave propagation in an initially stressed non-homogeneous layer over a nonhomogeneous half-space was investigated by Gupta et al. [21]. Manna et al. [22] formulated the Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space. SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media was studied by Kundu et al. [23]. In this paper we have obtained the dispersion equation for G-type wave in a heterogeneous layer overlying a heterogeneous half-space under initial stress. We have taken exponential variation in rigidity and density in upper layer that is l ¼ l1 eaz and q ¼ q1 eaz respectively where a is constant having dimension that is inverse of length. Also we have represented the half-space by assuming the variation l ¼ l2 ð1 ecos szÞ and q ¼ q2 ð1 ecos szÞ where e is small positive constant and s is real depth parameter. With this law of variation the equations of motion reduce to Hill’s equation with periodic coefficients which has been solved by the method given by Valeev [24]. Valeev considered a certain class of system of linear differential equations with periodic coefficients which have the property that, by means of Laplace transformation, they may be converted to a system of linear difference equations, which in turn may be solved by the method of infinite determinants. We obtained the Laplace transform of the displacement by considering the terms up to first order. 2. Formulation of the problem Let us consider a heterogeneous medium of thickness H overlying a semi-infinite heterogeneous medium under initial stress. The x-axis is taken as horizontal axis, z-axis as vertically downwards and origin has taken at the interface of layer and half-space. The rigidity and density of upper layer are exponential variable i.e., l ¼ l1 eaz and q ¼ q1 eaz respectively. The variation of rigidity and density for half-space are taken in following manner
where
l ¼ l2 ð1 ecos szÞ;
ð1Þ
q ¼ q2 ð1 ecos szÞ;
ð2Þ
e is small positive constant and s is real depth parameter (see Fig. 1).
3. Solution of the problem We consider the propagation of horizontally polarized surface waves of shear type, propagating along x axis. So, the displacement components are u ¼ 0; w ¼ 0 and v ¼ v ðx; z; tÞ. Therefore, the equation of motion for upper heterogeneous layer is
@ @x
l1 eaz
@v 1 @ @v @2v þ l1 eaz 1 ¼ q1 eaz 21 : @z @x @z @t
ð3Þ
In the lower inhomogeneous medium under initial stress the displacement
@ @x
l2 ð1 ecos szÞ
@v 2 @ þ @z @x
l2 ð1 ecos szÞ
where P is the initial stress parameter.
v 2 ðx; z; tÞ satisfies differential equation
@v 2 P @2v 2 @2v 2 ¼ ½q2 ð1 ecos szÞ 2 : 2 @x2 @z @t
ð4Þ
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
3
Fig. 1. Geometry of the problem.
Since stresses and displacements are continuous at the interface and the upper layer is stress free hence the boundary conditions are
8 > < ðiÞ ðiiÞ > : ðiiiÞ
v1 ¼ v2
at z ¼ 0;
l1 eaz @@zv 1 ¼ l2 ð1 ecos szÞ @@zv 2 at z ¼ 0; @v 1 @z
¼ 0 at z ¼ H:
Now, using the separation of variable we substitute 2
d V1 2
dz
þa 2
where m21 ¼ k
ð5Þ
v 1 ðx; z; tÞ ¼ V 1 ðzÞeikðxctÞ , in (3) we obtain
dV 1 þ m21 V 1 ¼ 0; dz
ð6Þ
qffiffiffiffi c2 1 ; b1 ¼ lq1 and x ¼ kc; k is the wave number and c is the phase velocity. b2 1
1
a
Again, substituting V 1 ðzÞ ¼ VðzÞe2z in (6) we get 2
d V þ n2 V ¼ 0; dz
2
ð7Þ
where n2 ¼ m21 a4 . Therefore, we have a 2
v 1 ðx; z; tÞ ¼ e z ½Acos nz þ Bsin nzeikðxctÞ :
ð8Þ
As upper surface is stress free, hence using boundary condition (iii) of Eq. (5) we get
A B ¼ a ¼ R0 þ cos nH 2n cos nH sin nH
say:
ð9Þ
ha i sin nðz þ HÞ þ cos nðz þ HÞ eikðxctÞ : 2n
ð10Þ
a sin nH 2n
Hence by (8) and (9) we have a 2
v 1 ¼ e z R0
In the lower non-homogeneous half-space the displacement
@ @x
l2 ð1 ecos szÞ
@v 2 @ þ @z @x
l2 ð1 ecos szÞ
v 2 ðx; z; tÞ satisfies differential equation
@v 2 P @2v 2 @2v 2 v ¼ ½ q ð1 e cos szÞ ; 2 2 @x2 @z @t 2
ð11Þ
where the density q2 is assumed to be constant. Now, taking
v 2 ðx; z; tÞ ¼ V 2 ðzÞeikðxctÞ
ð12Þ
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
and using Eq. (11), the equation of motion for lower non-homogeneous medium may be written as
"
2
d V2
þ eisz
2
dz
"
þe
e q2 2 2 e 2 e d2 V 2 esi dV 2 k c V 2 ðzÞ þ k V 2 ðzÞ þ 2 dz 2 l2 2 2 dz2 #
#
2 k V 2 ðzÞ ¼ 0:
e q2 2 2 e 2 e d2 V 2 esi dV 2 q c2 P k c V 2 ðzÞ þ k V 2 ðzÞ 1 þ 2 þ 2 2 dz 2l2 2 l2 2 2 dz l2
isz
ð13Þ
This is the differential equation, which will solved by the method given by Valeev [24]. We apply Laplace transform with respect to z, that is we multiply Eq. (13) by erz and integrate with respect to z from 0 to 1 we get
Z
( ðrþisÞz
e
0
þ
Z
)
e q2 2 2 e 2 e d2 V 2 esi dV 2 k c V 2 ðzÞ þ k V 2 ðzÞ þ dz 2 dz 2 l2 2 2 dz2
1
(
1 ðrisÞz
e
0
)
e q2 2 2 e 2 e d2 V 2 esi dV 2 k c V 2 ðzÞ þ k V 2 ðzÞ dz þ 2 l2 2 2 dz2 2 dz
Z
1
e
rz
0
" 2 d V2
q2 c2 P þ þ 1 2 l l2 2 dz 2
#
2 k V 2 ðzÞ dz ¼ 0: ð14Þ
Using the boundary condition ðiÞ of (5) we get
V 2 ð0Þ ¼ R0
ha i sin nH þ cos nH : 2n
Now, we consider qð0Þ ¼
dV 2 dz
qð0Þ ¼
z¼0
and
ð15Þ
lð0Þ 2 ¼ l2 ð1 eÞ, then ðiiÞ of (5) gives
l1 a2 þ 4n2 sin nH: R ð0Þ 0 4n l2
ð16Þ
Defining the Laplace transform of V 2 ðzÞ as
FðrÞ ¼
Z
1
erz V 2 ðzÞdz:
ð17Þ
0
Using Eqs. (16) and (17) in Eq. (14) we obtain
e q2 2 2 e 2 esi e 2 k c þ k þ ðr þ isÞ ðr þ isÞ Fðr þ isÞ 2 2l 2 2 2
e q2 2 2 e 2 esi e 2 þ k c þ k ðr isÞ ðr isÞ Fðr isÞ þ ðr 2 w2 ÞFðrÞ ¼ rk1 þ k2 ; 2 2 l2 2 2
ð18Þ
where
k1 ¼ ð1 eÞV 2 ð0Þ;
2
k2 ¼ ð1 eÞqð0Þ and w2 ¼ k
P q c2 : 1 2 2l 2 l2
ð19Þ n
To find FðrÞ from Eq. (18), we replace r by r þ ism and then divide throughout by ðismÞ ðm – 0Þ. We obtained the following infinite system of linear algebraic equations in the quantities Fðr þ ismÞ; ðm ¼ 0; 1; 2; . . .Þ
e q2 2 2 e 2 esi e 2 k c þ k þ fr þ isðm þ 1Þg fr þ isðm þ 1Þg Ffr þ isðm þ 1Þg 2 2 l2 2 2 e q2 2 2 e 2 esi e n 2 þ ðismÞ k c þ k fr þ isðm 1Þg fr þ isðm 1Þg Ffr þ isðm 1Þg 2 2 l2 2 2
ðismÞ
n
n
2
n
þ ðismÞ fðr þ ismÞ w2 gFðr þ ismÞ ¼ ðismÞ fðr þ ismÞk1 þ k2 g;
ð20Þ
where r may be considered as a parameter in the coefficients. It should be noted that in order not to consider the special case n m ¼ 0 separately, we include Eq. (18) in Eq. (20) by agreeing to regard ðismÞ ¼ 1 when m ¼ 0. Solving the system of difference Eqs. (20), we obtain FðrÞ as the ratio of two infinite determinants, viz,
FðrÞ ¼
41 ; 42
ð21Þ
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
where
... ... . . . ðisÞn fðr þ isÞ2 w2 g 2 2 ... 2e lq2 k c2 þ 2e k 2 41 ¼ 2 þ e2si ðr þ isÞ 2e ðr þ isÞ ... 0 ... ...
...
... . . . . . . . . . ...
...
n
ðisÞ fðr þ isÞk1 þ k2 g
0 2 2e lq2 k c2 2
ðrk1 þ k2 Þ
þ 2e k
2
e2si ðr isÞ 2e ðr isÞ n
2
2
n
ðisÞ fðr isÞk1 þ k2 g
ðisÞ fðr isÞ w2 g
...
...
and
... ... n 2 2 . . . ðisÞ fðr þ isÞ w g 2 2 ... 2e lq2 k c2 þ 2e k 2 2 42 ¼ þ e2si ðr þ isÞ 2e ðr þ isÞ ... 0 ... ...
... n o 2 2 2e lq2 k c2 þ 2e k e2si r 2e r 2 2
... 0 2 2e lq2 k c2 2
2
2
ðr þ w Þ
þ 2e k
2
e2si ðr isÞ 2e ðr isÞ
n o 2 2 n ðisÞ 2e lq2 k c2 þ 2e k 2
n ðisÞ e2si r þ 2e r2
2
n
2
ðisÞ fðr isÞ w2 g
...
...
. . . ... . . . : . . . ...
The first approximation of Eq. (21) is
FðrÞ ¼
rk1 þ k2 r 2 w2
rð1 eÞB1 r 2 w2
ð1 eÞB2 ; 2 2 lð0Þ 2 ðr w Þ
ð22Þ
where
na o sin nH þ cos nH ; 2n 2
a þ 4n2 sinnH : B2 ¼ R0 l1 4n
B1 ¼ R0
The second approximation of Eq. (21) is
FðrÞ ¼
43 ; 44
ð23Þ
where
ðisÞn fðr þ isÞ2 w2 g 2 2 2e lq2 k c2 þ 2e k 2 43 ¼ þ esi ðr þ isÞ e ðr þ isÞ2 2 2 0
2 q2 2 2 e e ðrk1 þ k2 Þ 2 l k c þ 2k 2 2 e si e 2 ðr isÞ 2 ðr isÞ n n 2 ðisÞ fðr isÞk1 þ k2 g ðisÞ fðr isÞ w2 g
n 2 ðisÞ fðr þ isÞ w2 g 2 2 2e lq2 k c2 þ 2e k 2 44 ¼ þ esi ðr þ isÞ e ðr þ isÞ2 2 2 0
2 q2 2 2 2 2 e e ðr þ w Þ 2 l k c þ 2k 2 : 2 e2si ðr isÞ 2e ðr isÞ n o 2 2 n n 2 ðisÞ fðr isÞ w2 g ðisÞ 2e lq2 k c2 þ 2e k e2si r 2e r 2 2
n
ðisÞ fðr þ isÞk1 þ k2 g
0
and
Neglecting the terms containing 2
n
2e lq2 k c2 þ 2e k e2si r 2e r 2 2 2
2
o
0
e2 and higher powers, we get 2
2
s2n 43 ¼ ðk1 r þ k2 Þfðr þ isÞ w2 gfðr isÞ w2 g þ fðr isÞk1 þ k2 gfðr þ isÞ w2 g
e q2 2 2 e 2 esi e 2 k c k þ ðr isÞ þ ðr isÞ þ fðr þ isÞk1 þ k2 g 2 2 l2 2 2
e q e esi e 2 2 2 2 2 fðr isÞ w2 g k c2 k ðr þ isÞ þ ðr þ isÞ 2 2 l2 2 2 and
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 2
2
s2n 44 ¼ fr2 w2 gfðr isÞ w2 gfðr þ isÞ w2 g Hence Eq. (23) transforms to
FðrÞ ¼
e fV 2 ð0Þðr þ isÞ þ qð0Þg
q2 2 2 2 2 k c k siðr þ isÞ þ ðr þ isÞ 2 ðr 2 w Þfðr þ isÞ w g l2 r2 w
e fV 2 ð0Þðr isÞ þ qð0Þg q2 2 2 2 2 : þ k c k þ siðr isÞ þ ðr isÞ 2 ðr 2 w2 Þfðr isÞ2 w2 g l2 k1 r þ k2 2
þ
2
2
2
ð24Þ
Then V 2 ðzÞ will be given by the inversion formula as
V 2 ðzÞ ¼
1 2p
Z
cþi1
FðrÞerz dr:
ð25Þ
ci1
The residues R1 ; R2 ; R3 at the poles r ¼ w; r ¼ w þ is; r ¼ w is are given respectively by
) ( )
(
wV 2 ð0Þ þ qð0Þ ew2 eA1 qð0Þ wV 2 ð0Þ wz eV 2 ð0Þ s2 þ 2w2 wz wz e e ; ð1 eÞ þ e þ R1 ¼ 2w 2w 2 s2 þ 4w2 s2 þ 4w2 s2 þ 4w2 R2 ¼
ie fwV 2 ð0Þ þ qð0ÞgfA1 þ w2 þ iswg ðisþwÞz e wð2w þ isÞ 4s
ð26Þ
ð27Þ
and
R3 ¼
e fwV 2 ð0Þ þ qð0ÞgfA1 þ w2 þ iswg wð2w isÞ
4s
eðisþwÞz ;
ð28Þ
where A1 ¼ lq2 k c2 k . 2 2
2
Eqs. (26)–(28) show that the conditions for a large amount of energy to be confined near the surface are
qð0Þ þ wV 2 ð0Þ ¼ 0;
ð29Þ
qð0Þ wV 2 ð0Þ ¼ 0;
ð30Þ
and 2w2 þ s2 ¼ 0:
ð31Þ
Now, Eqs. (29) and (30) gives
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 ffi 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 > > ! > > 1 bc 2 þ 2lP < u < = a 2 = u c2 2 l ð1 eÞ 2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi tan kHt 2 1 ¼ 2 ; a 2 a > : 2k ; l1 > b1 > > c2 : ; 1 T þ 2 2k 2k
ð32Þ
b1
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi9 l > > a c2 P > = 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r T¼ ; a 2 > > > > c2 : ; 2 1 2k b
b21 ¼
l1 l and b22 ¼ 2 : q1 q2
1
The Eq. (32) gives us the dispersion equation for propagation of G-type wave in a heterogeneous layer overlying a heterogeneous half space under initial stress. We will consider only the positive sign for further discussion. Now, from Eq. (31) we have
kc ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi l2 P 2 þ s2 : 2k 1 2l2 2q2
ð33Þ
Then the group velocity U is given by
pffiffiffi 2k 1 2lP d 2 ffi: U ¼ ðkcÞ ¼ b2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk 2 2k 1 2lP þ s2 2
ð34Þ
7
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 1.9 1. a/2k=0.0
1.8
2. a/2k=0.1 1.7
3. a/2k=0.2
c/β
1
1.6
4.
a/2k=0.3
1.5 4
1.4
3 1.3
2 1
1.2
1.1 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
kH
Fig. 2. Variation of dimensionless phase velocity
c b1
against dimensionless wave number kH.
4. Particular cases 4.1. Case I When the upper layer is homogeneous i.e., a ¼ 0 then the Eq. (32) transform to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! c2 l ð1 eÞ tan kH 1 ¼ 2 l1 b21
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1 bc 2 þ 2lP 2 2 ffi qffiffiffiffiffiffiffiffiffiffiffiffi : c2 1 b2
ð35Þ
1
This is the dispersion equation for G-type waves in a homogeneous layer overlying a heterogeneous half space with initial stress.
1.8 1. P/2μ =0.0 2
1.7
2. P/2μ2=0.1 1.6
3. P/2μ =0.2 2
c/β1
1.5
4. P/2μ =0.3
1
2
2
1.4
1.3
3 4
1.2
1.1
1 0
0.1
0.2
0.3
0.4 kH
Fig. 3. Variation of dimensionless phase velocity
c b1
0.5
0.6
against dimensionless wave number kH.
0.7
0.8
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 1.8
1.7
1. ε=0.0 2. ε=0.1
1.6
3. ε=0.2
c/β1
1.5
4. ε=0.3
1
1.4
2 3
1.3
4 1.2
1.1
1 0
0.1
0.2
0.3
0.4 kH
Fig. 4. Variation of dimensionless phase velocity
c b1
0.5
0.6
0.7
0.8
against dimensionless wave number kH.
4.2. Case II For the heterogeneous medium lying over a homogeneous half-space, that is for e ¼ 0 under initial stress, Eq. (32) can be reduces to
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 > ! c2 P > 1 þ < u = < 2 2l2 u c2 b2 a 2 l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan kHt 2 1 ¼ 2 r ffi : 2k ; l1 > 2 b1 2 > c : a þ 2 1 2k
b1
a 2k
9 > > = ; > ; T>
ð36Þ
1 1
0.9
2 3 4
0.8 0.7
U/β2
0.6 0.5
1. P/2μ2=0.0
0.4
2. P/2μ =0.1 2
0.3
3. P/2μ2=0.2
0.2
4. P/2μ =0.3 2
0.1 0 0
1
2
3
4
5 k/s
Fig. 5. Variation of dimensionless phase velocity
6
U b2
7
against scaled wave number ks.
8
9
10
9
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 Surface Graph for P/2μ =0.0 2
0.02
U−−−−−>
0.01 0 −0.01 −0.02 2
2
1
1 0
0 −1
−1 −2
s−−−−−>
−2
k−−−−−>
Fig. 6. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:0. 2
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi9 l > > a c2 P > = 2 ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T¼ : a 2 > > > > c2 : ; 1 2k b2 1
Eq. (36) gives the dispersion relation of G-type wave for the case when there is no inhomogeneity in the half-space. 4.3. Case III For finite thickness homogeneous layer lying over a homogeneous half-space in the absence of initial stress, i.e., when a ¼ 0; e ¼ 0 and P ¼ 0 the Eq. (32) reduces to the general dispersion equation of Love waves (Ewing et al. [1])
qffiffiffiffiffiffiffiffiffiffiffiffi2ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! c c2 l2 1 b22 ffiffiffiffiffiffiffiffiffiffiffiffi ffi; q ¼ tan kH 1 l1 c22 1 b21 b
ð37Þ
1
Surface Graph for P/2μ2=0.1
0.02 0.015
U−−−−−>
0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 2 1
2 1
0
0
−1 s−−−−−>
−1 −2
−2
k−−−−−>
Fig. 7. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:1. 2
10
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 Surface Graph for P/2μ =0.2 2
0.02
U−−−−−>
0.01 0 −0.01 −0.02 2 1 0 −1 −2
s−−−−−>
−2
0
−1
1
2
k−−−−−>
Fig. 8. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:2. 2
Surface Graph for P/2μ2=0.3
0.02
U−−−−−>
0.01 0 −0.01 −0.02 2 1 0 −1 s−−−−−>
−2
−2
−1
0
1
2
k−−−−−>
Fig. 9. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:3. 2
which is the usual dispersion equation for Love waves in a homogeneous medium over a homogeneous half-space with b1 < c < b2 where b21 ¼ lq1 and b22 ¼ lq2 . 1
2
5. Numerical calculations, results and discussions For the graphical representation of phase and group velocity of G-type wave in heterogeneous layer overlying a heterogeneous half-space, we have taken lq1 ¼ 0:02; lq2 ¼ 0:2 and ll2 ¼ 0:1 and the results are presented in Figs. 2–6. In Fig. 2–4 we 1
2
1
have computed the values of phase velocity bc1 from Eq. (32) for taking R0 ¼ 1 as a function of kH. In Fig. 2, we have shown the variation in dimensionless phase velocity parameter
a 2k
c b1
against dimensionless wave number kH for different values of inhomogeneity
P
a and fixed values of 2l ¼ 0:2 and e ¼ 0:2. The values of inhomogeneity parameters 2k for curves 1, curve 2, curve 2
3 and curve 4 have been taken as 0.0, 0.1, 0.2 and 0.3 respectively. From this figure we have seen that curves are accumulating initially and gradually takes little far apart from each other. Also we can observe that the phase velocity decreases a slightly with the increases of parameter 2k . Fig. 3 represent the variation in dimensionless phase velocity bc1 against dimensionless wave number kH for fixed values of e ¼ 0:2 and 2ka ¼ 0 and for different values of initial stress parameter 2lP . The values of initial stress parameter for curves 1, 2
curve 2, curve 3 and curve 4 have been taken as 0.0, 0.1, 0.2 and 0.3 respectively. It follows from the Fig. 3 that the phase velocity decreases with the increases of initial stress parameter.
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
11
In Fig. 4, an attempt has been made to study the effect of inhomogeneity parameter e. This Figure shows the variation in dimensionless phase velocity bc1 against dimensionless wave number kH. The curves are plotted for different values of e and a for fixed values of 2lP ¼ 0:2 and 2k ¼ 0. The values of e for curves 1, curve 2, curve 3 and curve 4 have been taken as 0.0, 0.1, 2 0.2 and 0.3 respectively. From this Figure it is clear that phase velocity decreases slightly with increases of inhomogeneity parameter e. By comparative study of these Fig. 2–4 we can conclude that velocity decreases with the increase of inhomogeneity parameters. Variation in dimensionless group velocity bU2 with respect to scaled wave number ks is shown in Fig. 5 for different values of P . The values of initial stress 2lP for curves 1, curve 2, curve 3 and curve 4 have been taken as 0.0, 0.1, 0.2 and 0.3 respec2l2 2 tively. From this figure we have seen that group velocity U approaches b2 asymptotically with increases in scaled wave numk ber s and it is decreases with the increases of initial stress parameter. Fig. 6 represent a set of surface plots for variation of group velocity U with respect to parameter k and s for different values of 2lP . It has been observed from the figure that group velocity depends on initial stress. So, we can say that phase velocity 2 and group velocity is influenced by initial stress, and it depends on wave number, depth parameter s and parameter e. This study may be useful to understand the nature of seismic propagated during earthquake (see Figs. 7–9). 6. Conclusions We have obtained the dispersion equation of dimensionless phase velocity of G-type seismic wave in heterogeneous medium overlying a heterogeneous half-space under initial stress as a function of dimensionless wave number by using the transform technique and Valeev’s method [24]. We have seen that phase velocity is influenced by inhomogeneity parameters and initially stress parameter. We have found the condition for large amount of energy to be confined near the surface and obtained the expression for group velocity. Variation in phase and group velocity has been plotted graphically against wave number and scaled wave number respectively. From the above figures we may conclude with the following remarks: i. Dimensionless phase velocity bc1 of G-type seismic wave increases with decreases of non-dimensional wave number kH. a ii. Phase velocity decreases with the increases of inhomogeneity parameter 2k in the upper layer. iii. Dimensionless phase velocity decreases as the value of inhomogeneity parameter e increases in the lower half-space. iv. Phase velocity of G-type seismic wave decreases when the initial stress parameter increases. v. Also we have seen that group velocity is effected by initial stress and it depends on wave number k, depth parameter s and inhomogeneity parameter. vi. Dimensionless group velocity bU2 increases with the increases of scaled wave number ks up to a certain point. vii. The group velocity decreases with the increases of initial stress parameter. The results of present paper may be useful to understand the cause of damage during deep earthquakes and it may be helpful to understand the nature of long period Love waves. References [1] W.M. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media, Mcgraw Hill Pub, New York, 1957. [2] Y. Sato, Study on surface waves, VI generation of Love and other types of SH waves, Bull. Earthquake Res. Inst. 30 (1952) 101–120. [3] A.M. Abd-Alla, S.M. Ahmed, Propagation of Love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium, Appl. Math. Comput. 106 (2) (1999) 265–275 (11p). [4] J. De Noyer, The effect of variations in layer thickness of Love waves, Bull. Seismol. Soc. Am. 51 (1961) 227–235. [5] J. Bhattacharya, On the dispersion curve for Love waves due to irregularity in the thickness of the transversely isotropic crustal layer, Gerl Beitr Z Geophys. 71 (1962) 324. [6] A. Chattopadhyay, On the dispersion equations for Love waves due to irregularity in the thickness of non homogeneous crustal layer, Acta Geophys. Pol. 23 (1975) 307–315. [7] B. Wolf, Propagation of Love waves in layers with irregular boundaries, Pure Appl. Geophys. 78 (1970) 48–57. [8] A.P. Ghorai, S.K. Samal, N.C. Mahanti, Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity, Appl. Math. Model. 34 (7) (2010) 1873–1883. [9] S.S. Singh, Love wave at a layer medium bounded by irregular boundary surfaces, J. Vib. Control 17 (5) (2010) 789–795. [10] B. Gutenberg, Wave velocities at depths between 50 and 600 kilometers, Bull. Seismol. Soc. Am. 43 (1953) 223–232. [11] I. Lehmann, S-waves and the structure of the upper mantle, Geophys. J.R. Astron. Soc. 3 (1961) 529–538. [12] A.K. Mal, On the generation of G-waves, Gerlands Beitr. Geophys. 72 (1962) 82–88. [13] K. Aki, Generation and propagation of G waves from the Niigata earthquake of June 16, 1964. Part 2. Estimation of earthquake moment, released energy, and stress-strain drop from the G wave spectrum, Bull. Earthquake Res. Inst. 72 (1966) 82–88. [14] A. Chattopadhyay, A. Keshri, Generation of G-type seismic waves under initial stress, Indian J. Pure Appl. Math. 17 (8) (1986) 1042–1055. [15] A. Chattopadhyay, S. Gupta, V.K. Sharma, P. Kumari, Propagation of G type seismic waves in viscoelastic medium, Int. J. Appl. Math. Mech. 6 (9) (2010) 63–75. [16] A. Chattopadhyay, A.K. Singh, G-type seismic waves in reinforced media, Meccanica 47 (2012) 1775–1785. [17] H. Jeffreys, The Earth, fourth ed., Cambridge University Press, Cambridge, 1959. [18] J. Bhattacharya, On short period signals obtained from explosions within the Earth, Geofis Pura e Appl. 55 (1963) 6371. [19] N.A. Haskell, Radiation pattern of surface waves from point sources in multi-layered medium, Bull. Seismol. Soc. Am. 54 (1) (1964) 377–393. [20] S. Gupta, D.K. Majhi, S. Kundu, S.K. Vishwakarma, Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space, Appl. Math. Comput. 218 (2012) 5655–5664.
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[21] S. Gupta, D.K. Majhi, S.K. Vishwakarma, Torsional surface wave propagation in an initially stressed non-homogeneous layer over a non-homogeneous half-space, Appl. Math. Comput. 219 (2012) 3209–3218. [22] S. Manna, S. Kundu, S. Gupta, Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space, J. Vib. Control (2013), http://dx.doi.org/10.1177/1077546313513626. [23] S. Kundu, S. Gupta, S. Manna, SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media, Meccanica (2013), http://dx.doi.org/10.1007/s11012-013-9825-5. [24] K.G. Valeev, On Hill’s method in the theory of linear differential equations with periodic coefficients, J. Appl. Math. A. Mech. 24 (1960) 1493–1505.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Propagation of G-type seismic waves in heterogeneous layer lying over an initially stressed heterogeneous half-space q S. Kundu ⇑, S. Gupta, S. Manna Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India
a r t i c l e
i n f o
Keywords: G-type seismic wave Heterogeneous layer Initial stress Transform technique Dispersion equation Phase velocity
a b s t r a c t The aim of present paper is to investigate the propagation of G-type seismic waves in a heterogeneous layer overlying a heterogeneous half-space under initial stress. Exponential variations in rigidity and density have been taken in the upper layer. In the lower halfspace both rigidity and density are varying with depth. Dispersion equation has been obtained in closed form. Dispersion equation in case of homogeneous media coincides with the general equation of Love wave. Curves are plotted for different values of inhomogeneity parameters and initial stress parameter. We have seen that the phase velocity decreases with the increase of inhomogeneity parameters. It is observed that initial stress has dominant effect on the propagation of G-type wave. Variation in group velocity has shown for different values of initial stress parameter. We have also drawn surface plots of group velocity with respect to wave number and depth parameter for different values of initial stress parameter. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Seismology is a study of earthquake and seismic waves that through and around the Earth. The propagation of waves in layered media is of central interest to the theoretical Seismologists. A review of this subject has been given by Ewing et al. in a classical monograph [1]. Two basic types of seismic waves generated by an earthquake, one is body waves and another one is surface waves. The body waves propagate within a body of rock. The faster of these body waves is called Primary wave or longitudinal wave and the slower one is called Secondary wave or shear wave. The second general type of wave is called surface wave, because its motion is restricted to near the ground surface. Such waves correspond to ripples of water that travel across a lake. The wave motion is located at the outside surface itself, and as the depth below this surface increases, wave displacement becomes less and less. The study of surface wave is always important to seismologists for understanding the causes and estimation of damage due to earthquakes. Surface waves in earthquakes can be divided into two types, Love waves and Rayleigh waves. Love waves are registered in the horizontal component but Rayleigh waves which are polarized in the vertical plane, are registered both in horizontal and in vertical components. The VI Generation of Love and other types of SH waves was formulated by Sato [2].
q
This work is supported by Indian School of Mines, Dhanbad, under Grant number ISM-JRF/Acad/2012/75.
⇑ Corresponding author.
E-mail addresses: [email protected] (S. Kundu), [email protected] (S. Gupta), [email protected] (S. Manna). http://dx.doi.org/10.1016/j.amc.2014.01.166 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
Abd-Alla et al. [3] discussed about propagation of Love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium. Noyer [4] studied the effect of variations in layer thickness of Love waves. The dispersion curve for Love waves due to irregularity in the thickness of the transversely isotropic crustal layer was observed by Bhattacharya [5]. Chattopadhyay [6] obtained the dispersion equations for Love waves due to irregularity in the thickness of non-homogeneous crustal layer. Wolf [7] investigated the propagation of Love waves in Layers with irregular boundaries. Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity was studied by Ghorai et al. [8]. Singh [9] discussed the Love wave at a layer medium bounded by irregular boundary surfaces. Love waves of long periods (60–300 s) are also called G-waves named after Gutenberg [10]. It takes about 2.5 h for G waves to make a round trip of the Earth. For large earthquakes, surface waves that travel around the Earth more than once are observed. The word ‘initial stress’ is stress which developed a medium before it is being used for study. The Earth is initially stressed medium, due to presence of external loading, slow process of creep and gravitational field, considerable amount of stresses which are called prestresses or initial stresses, remain naturally present in the layers. Many researchers have given their contribution in the study of G-type wave. Notable are Gutenberg [10], Lehmann [11], Mal [12] etc. The propagation of G waves, estimation of earthquake moment, released energy and stress–strain drop from the G wave spectrum was discussed by Aki [13] from the Niigata earthquake of June 16, 1964. Chattopadhyay and Keshri [14] pointed the generation of G-type seismic waves under initial stress in a layer media. Propagation of G-type seismic waves in viscoelastic medium was established by Chattopadhyay et al. [15]. Chattopadhyay and Singh [16] investigated the G-type seismic waves in fiber reinforced media. Some other notable works in this field were done by many researchers like Jeffreys [17], Bhattacharya [18], Haskell [19] etc. Recently extensive and laudable works on the surface waves propagation have been done by many researchers. Gupta et al. [20] discussed the propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space. Torsional surface wave propagation in an initially stressed non-homogeneous layer over a nonhomogeneous half-space was investigated by Gupta et al. [21]. Manna et al. [22] formulated the Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space. SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media was studied by Kundu et al. [23]. In this paper we have obtained the dispersion equation for G-type wave in a heterogeneous layer overlying a heterogeneous half-space under initial stress. We have taken exponential variation in rigidity and density in upper layer that is l ¼ l1 eaz and q ¼ q1 eaz respectively where a is constant having dimension that is inverse of length. Also we have represented the half-space by assuming the variation l ¼ l2 ð1 ecos szÞ and q ¼ q2 ð1 ecos szÞ where e is small positive constant and s is real depth parameter. With this law of variation the equations of motion reduce to Hill’s equation with periodic coefficients which has been solved by the method given by Valeev [24]. Valeev considered a certain class of system of linear differential equations with periodic coefficients which have the property that, by means of Laplace transformation, they may be converted to a system of linear difference equations, which in turn may be solved by the method of infinite determinants. We obtained the Laplace transform of the displacement by considering the terms up to first order. 2. Formulation of the problem Let us consider a heterogeneous medium of thickness H overlying a semi-infinite heterogeneous medium under initial stress. The x-axis is taken as horizontal axis, z-axis as vertically downwards and origin has taken at the interface of layer and half-space. The rigidity and density of upper layer are exponential variable i.e., l ¼ l1 eaz and q ¼ q1 eaz respectively. The variation of rigidity and density for half-space are taken in following manner
where
l ¼ l2 ð1 ecos szÞ;
ð1Þ
q ¼ q2 ð1 ecos szÞ;
ð2Þ
e is small positive constant and s is real depth parameter (see Fig. 1).
3. Solution of the problem We consider the propagation of horizontally polarized surface waves of shear type, propagating along x axis. So, the displacement components are u ¼ 0; w ¼ 0 and v ¼ v ðx; z; tÞ. Therefore, the equation of motion for upper heterogeneous layer is
@ @x
l1 eaz
@v 1 @ @v @2v þ l1 eaz 1 ¼ q1 eaz 21 : @z @x @z @t
ð3Þ
In the lower inhomogeneous medium under initial stress the displacement
@ @x
l2 ð1 ecos szÞ
@v 2 @ þ @z @x
l2 ð1 ecos szÞ
where P is the initial stress parameter.
v 2 ðx; z; tÞ satisfies differential equation
@v 2 P @2v 2 @2v 2 ¼ ½q2 ð1 ecos szÞ 2 : 2 @x2 @z @t
ð4Þ
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
3
Fig. 1. Geometry of the problem.
Since stresses and displacements are continuous at the interface and the upper layer is stress free hence the boundary conditions are
8 > < ðiÞ ðiiÞ > : ðiiiÞ
v1 ¼ v2
at z ¼ 0;
l1 eaz @@zv 1 ¼ l2 ð1 ecos szÞ @@zv 2 at z ¼ 0; @v 1 @z
¼ 0 at z ¼ H:
Now, using the separation of variable we substitute 2
d V1 2
dz
þa 2
where m21 ¼ k
ð5Þ
v 1 ðx; z; tÞ ¼ V 1 ðzÞeikðxctÞ , in (3) we obtain
dV 1 þ m21 V 1 ¼ 0; dz
ð6Þ
qffiffiffiffi c2 1 ; b1 ¼ lq1 and x ¼ kc; k is the wave number and c is the phase velocity. b2 1
1
a
Again, substituting V 1 ðzÞ ¼ VðzÞe2z in (6) we get 2
d V þ n2 V ¼ 0; dz
2
ð7Þ
where n2 ¼ m21 a4 . Therefore, we have a 2
v 1 ðx; z; tÞ ¼ e z ½Acos nz þ Bsin nzeikðxctÞ :
ð8Þ
As upper surface is stress free, hence using boundary condition (iii) of Eq. (5) we get
A B ¼ a ¼ R0 þ cos nH 2n cos nH sin nH
say:
ð9Þ
ha i sin nðz þ HÞ þ cos nðz þ HÞ eikðxctÞ : 2n
ð10Þ
a sin nH 2n
Hence by (8) and (9) we have a 2
v 1 ¼ e z R0
In the lower non-homogeneous half-space the displacement
@ @x
l2 ð1 ecos szÞ
@v 2 @ þ @z @x
l2 ð1 ecos szÞ
v 2 ðx; z; tÞ satisfies differential equation
@v 2 P @2v 2 @2v 2 v ¼ ½ q ð1 e cos szÞ ; 2 2 @x2 @z @t 2
ð11Þ
where the density q2 is assumed to be constant. Now, taking
v 2 ðx; z; tÞ ¼ V 2 ðzÞeikðxctÞ
ð12Þ
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S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
and using Eq. (11), the equation of motion for lower non-homogeneous medium may be written as
"
2
d V2
þ eisz
2
dz
"
þe
e q2 2 2 e 2 e d2 V 2 esi dV 2 k c V 2 ðzÞ þ k V 2 ðzÞ þ 2 dz 2 l2 2 2 dz2 #
#
2 k V 2 ðzÞ ¼ 0:
e q2 2 2 e 2 e d2 V 2 esi dV 2 q c2 P k c V 2 ðzÞ þ k V 2 ðzÞ 1 þ 2 þ 2 2 dz 2l2 2 l2 2 2 dz l2
isz
ð13Þ
This is the differential equation, which will solved by the method given by Valeev [24]. We apply Laplace transform with respect to z, that is we multiply Eq. (13) by erz and integrate with respect to z from 0 to 1 we get
Z
( ðrþisÞz
e
0
þ
Z
)
e q2 2 2 e 2 e d2 V 2 esi dV 2 k c V 2 ðzÞ þ k V 2 ðzÞ þ dz 2 dz 2 l2 2 2 dz2
1
(
1 ðrisÞz
e
0
)
e q2 2 2 e 2 e d2 V 2 esi dV 2 k c V 2 ðzÞ þ k V 2 ðzÞ dz þ 2 l2 2 2 dz2 2 dz
Z
1
e
rz
0
" 2 d V2
q2 c2 P þ þ 1 2 l l2 2 dz 2
#
2 k V 2 ðzÞ dz ¼ 0: ð14Þ
Using the boundary condition ðiÞ of (5) we get
V 2 ð0Þ ¼ R0
ha i sin nH þ cos nH : 2n
Now, we consider qð0Þ ¼
dV 2 dz
qð0Þ ¼
z¼0
and
ð15Þ
lð0Þ 2 ¼ l2 ð1 eÞ, then ðiiÞ of (5) gives
l1 a2 þ 4n2 sin nH: R ð0Þ 0 4n l2
ð16Þ
Defining the Laplace transform of V 2 ðzÞ as
FðrÞ ¼
Z
1
erz V 2 ðzÞdz:
ð17Þ
0
Using Eqs. (16) and (17) in Eq. (14) we obtain
e q2 2 2 e 2 esi e 2 k c þ k þ ðr þ isÞ ðr þ isÞ Fðr þ isÞ 2 2l 2 2 2
e q2 2 2 e 2 esi e 2 þ k c þ k ðr isÞ ðr isÞ Fðr isÞ þ ðr 2 w2 ÞFðrÞ ¼ rk1 þ k2 ; 2 2 l2 2 2
ð18Þ
where
k1 ¼ ð1 eÞV 2 ð0Þ;
2
k2 ¼ ð1 eÞqð0Þ and w2 ¼ k
P q c2 : 1 2 2l 2 l2
ð19Þ n
To find FðrÞ from Eq. (18), we replace r by r þ ism and then divide throughout by ðismÞ ðm – 0Þ. We obtained the following infinite system of linear algebraic equations in the quantities Fðr þ ismÞ; ðm ¼ 0; 1; 2; . . .Þ
e q2 2 2 e 2 esi e 2 k c þ k þ fr þ isðm þ 1Þg fr þ isðm þ 1Þg Ffr þ isðm þ 1Þg 2 2 l2 2 2 e q2 2 2 e 2 esi e n 2 þ ðismÞ k c þ k fr þ isðm 1Þg fr þ isðm 1Þg Ffr þ isðm 1Þg 2 2 l2 2 2
ðismÞ
n
n
2
n
þ ðismÞ fðr þ ismÞ w2 gFðr þ ismÞ ¼ ðismÞ fðr þ ismÞk1 þ k2 g;
ð20Þ
where r may be considered as a parameter in the coefficients. It should be noted that in order not to consider the special case n m ¼ 0 separately, we include Eq. (18) in Eq. (20) by agreeing to regard ðismÞ ¼ 1 when m ¼ 0. Solving the system of difference Eqs. (20), we obtain FðrÞ as the ratio of two infinite determinants, viz,
FðrÞ ¼
41 ; 42
ð21Þ
5
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
where
... ... . . . ðisÞn fðr þ isÞ2 w2 g 2 2 ... 2e lq2 k c2 þ 2e k 2 41 ¼ 2 þ e2si ðr þ isÞ 2e ðr þ isÞ ... 0 ... ...
...
... . . . . . . . . . ...
...
n
ðisÞ fðr þ isÞk1 þ k2 g
0 2 2e lq2 k c2 2
ðrk1 þ k2 Þ
þ 2e k
2
e2si ðr isÞ 2e ðr isÞ n
2
2
n
ðisÞ fðr isÞk1 þ k2 g
ðisÞ fðr isÞ w2 g
...
...
and
... ... n 2 2 . . . ðisÞ fðr þ isÞ w g 2 2 ... 2e lq2 k c2 þ 2e k 2 2 42 ¼ þ e2si ðr þ isÞ 2e ðr þ isÞ ... 0 ... ...
... n o 2 2 2e lq2 k c2 þ 2e k e2si r 2e r 2 2
... 0 2 2e lq2 k c2 2
2
2
ðr þ w Þ
þ 2e k
2
e2si ðr isÞ 2e ðr isÞ
n o 2 2 n ðisÞ 2e lq2 k c2 þ 2e k 2
n ðisÞ e2si r þ 2e r2
2
n
2
ðisÞ fðr isÞ w2 g
...
...
. . . ... . . . : . . . ...
The first approximation of Eq. (21) is
FðrÞ ¼
rk1 þ k2 r 2 w2
rð1 eÞB1 r 2 w2
ð1 eÞB2 ; 2 2 lð0Þ 2 ðr w Þ
ð22Þ
where
na o sin nH þ cos nH ; 2n 2
a þ 4n2 sinnH : B2 ¼ R0 l1 4n
B1 ¼ R0
The second approximation of Eq. (21) is
FðrÞ ¼
43 ; 44
ð23Þ
where
ðisÞn fðr þ isÞ2 w2 g 2 2 2e lq2 k c2 þ 2e k 2 43 ¼ þ esi ðr þ isÞ e ðr þ isÞ2 2 2 0
2 q2 2 2 e e ðrk1 þ k2 Þ 2 l k c þ 2k 2 2 e si e 2 ðr isÞ 2 ðr isÞ n n 2 ðisÞ fðr isÞk1 þ k2 g ðisÞ fðr isÞ w2 g
n 2 ðisÞ fðr þ isÞ w2 g 2 2 2e lq2 k c2 þ 2e k 2 44 ¼ þ esi ðr þ isÞ e ðr þ isÞ2 2 2 0
2 q2 2 2 2 2 e e ðr þ w Þ 2 l k c þ 2k 2 : 2 e2si ðr isÞ 2e ðr isÞ n o 2 2 n n 2 ðisÞ fðr isÞ w2 g ðisÞ 2e lq2 k c2 þ 2e k e2si r 2e r 2 2
n
ðisÞ fðr þ isÞk1 þ k2 g
0
and
Neglecting the terms containing 2
n
2e lq2 k c2 þ 2e k e2si r 2e r 2 2 2
2
o
0
e2 and higher powers, we get 2
2
s2n 43 ¼ ðk1 r þ k2 Þfðr þ isÞ w2 gfðr isÞ w2 g þ fðr isÞk1 þ k2 gfðr þ isÞ w2 g
e q2 2 2 e 2 esi e 2 k c k þ ðr isÞ þ ðr isÞ þ fðr þ isÞk1 þ k2 g 2 2 l2 2 2
e q e esi e 2 2 2 2 2 fðr isÞ w2 g k c2 k ðr þ isÞ þ ðr þ isÞ 2 2 l2 2 2 and
6
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 2
2
s2n 44 ¼ fr2 w2 gfðr isÞ w2 gfðr þ isÞ w2 g Hence Eq. (23) transforms to
FðrÞ ¼
e fV 2 ð0Þðr þ isÞ þ qð0Þg
q2 2 2 2 2 k c k siðr þ isÞ þ ðr þ isÞ 2 ðr 2 w Þfðr þ isÞ w g l2 r2 w
e fV 2 ð0Þðr isÞ þ qð0Þg q2 2 2 2 2 : þ k c k þ siðr isÞ þ ðr isÞ 2 ðr 2 w2 Þfðr isÞ2 w2 g l2 k1 r þ k2 2
þ
2
2
2
ð24Þ
Then V 2 ðzÞ will be given by the inversion formula as
V 2 ðzÞ ¼
1 2p
Z
cþi1
FðrÞerz dr:
ð25Þ
ci1
The residues R1 ; R2 ; R3 at the poles r ¼ w; r ¼ w þ is; r ¼ w is are given respectively by
) ( )
(
wV 2 ð0Þ þ qð0Þ ew2 eA1 qð0Þ wV 2 ð0Þ wz eV 2 ð0Þ s2 þ 2w2 wz wz e e ; ð1 eÞ þ e þ R1 ¼ 2w 2w 2 s2 þ 4w2 s2 þ 4w2 s2 þ 4w2 R2 ¼
ie fwV 2 ð0Þ þ qð0ÞgfA1 þ w2 þ iswg ðisþwÞz e wð2w þ isÞ 4s
ð26Þ
ð27Þ
and
R3 ¼
e fwV 2 ð0Þ þ qð0ÞgfA1 þ w2 þ iswg wð2w isÞ
4s
eðisþwÞz ;
ð28Þ
where A1 ¼ lq2 k c2 k . 2 2
2
Eqs. (26)–(28) show that the conditions for a large amount of energy to be confined near the surface are
qð0Þ þ wV 2 ð0Þ ¼ 0;
ð29Þ
qð0Þ wV 2 ð0Þ ¼ 0;
ð30Þ
and 2w2 þ s2 ¼ 0:
ð31Þ
Now, Eqs. (29) and (30) gives
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 ffi 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 > > ! > > 1 bc 2 þ 2lP < u < = a 2 = u c2 2 l ð1 eÞ 2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi tan kHt 2 1 ¼ 2 ; a 2 a > : 2k ; l1 > b1 > > c2 : ; 1 T þ 2 2k 2k
ð32Þ
b1
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi9 l > > a c2 P >
b21 ¼
l1 l and b22 ¼ 2 : q1 q2
1
The Eq. (32) gives us the dispersion equation for propagation of G-type wave in a heterogeneous layer overlying a heterogeneous half space under initial stress. We will consider only the positive sign for further discussion. Now, from Eq. (31) we have
kc ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi l2 P 2 þ s2 : 2k 1 2l2 2q2
ð33Þ
Then the group velocity U is given by
pffiffiffi 2k 1 2lP d 2 ffi: U ¼ ðkcÞ ¼ b2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk 2 2k 1 2lP þ s2 2
ð34Þ
7
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 1.9 1. a/2k=0.0
1.8
2. a/2k=0.1 1.7
3. a/2k=0.2
c/β
1
1.6
4.
a/2k=0.3
1.5 4
1.4
3 1.3
2 1
1.2
1.1 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
kH
Fig. 2. Variation of dimensionless phase velocity
c b1
against dimensionless wave number kH.
4. Particular cases 4.1. Case I When the upper layer is homogeneous i.e., a ¼ 0 then the Eq. (32) transform to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! c2 l ð1 eÞ tan kH 1 ¼ 2 l1 b21
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1 bc 2 þ 2lP 2 2 ffi qffiffiffiffiffiffiffiffiffiffiffiffi : c2 1 b2
ð35Þ
1
This is the dispersion equation for G-type waves in a homogeneous layer overlying a heterogeneous half space with initial stress.
1.8 1. P/2μ =0.0 2
1.7
2. P/2μ2=0.1 1.6
3. P/2μ =0.2 2
c/β1
1.5
4. P/2μ =0.3
1
2
2
1.4
1.3
3 4
1.2
1.1
1 0
0.1
0.2
0.3
0.4 kH
Fig. 3. Variation of dimensionless phase velocity
c b1
0.5
0.6
against dimensionless wave number kH.
0.7
0.8
8
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 1.8
1.7
1. ε=0.0 2. ε=0.1
1.6
3. ε=0.2
c/β1
1.5
4. ε=0.3
1
1.4
2 3
1.3
4 1.2
1.1
1 0
0.1
0.2
0.3
0.4 kH
Fig. 4. Variation of dimensionless phase velocity
c b1
0.5
0.6
0.7
0.8
against dimensionless wave number kH.
4.2. Case II For the heterogeneous medium lying over a homogeneous half-space, that is for e ¼ 0 under initial stress, Eq. (32) can be reduces to
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 > ! c2 P > 1 þ < u = < 2 2l2 u c2 b2 a 2 l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan kHt 2 1 ¼ 2 r ffi : 2k ; l1 > 2 b1 2 > c : a þ 2 1 2k
b1
a 2k
9 > > = ; > ; T>
ð36Þ
1 1
0.9
2 3 4
0.8 0.7
U/β2
0.6 0.5
1. P/2μ2=0.0
0.4
2. P/2μ =0.1 2
0.3
3. P/2μ2=0.2
0.2
4. P/2μ =0.3 2
0.1 0 0
1
2
3
4
5 k/s
Fig. 5. Variation of dimensionless phase velocity
6
U b2
7
against scaled wave number ks.
8
9
10
9
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 Surface Graph for P/2μ =0.0 2
0.02
U−−−−−>
0.01 0 −0.01 −0.02 2
2
1
1 0
0 −1
−1 −2
s−−−−−>
−2
k−−−−−>
Fig. 6. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:0. 2
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi9 l > > a c2 P >
Eq. (36) gives the dispersion relation of G-type wave for the case when there is no inhomogeneity in the half-space. 4.3. Case III For finite thickness homogeneous layer lying over a homogeneous half-space in the absence of initial stress, i.e., when a ¼ 0; e ¼ 0 and P ¼ 0 the Eq. (32) reduces to the general dispersion equation of Love waves (Ewing et al. [1])
qffiffiffiffiffiffiffiffiffiffiffiffi2ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! c c2 l2 1 b22 ffiffiffiffiffiffiffiffiffiffiffiffi ffi; q ¼ tan kH 1 l1 c22 1 b21 b
ð37Þ
1
Surface Graph for P/2μ2=0.1
0.02 0.015
U−−−−−>
0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 2 1
2 1
0
0
−1 s−−−−−>
−1 −2
−2
k−−−−−>
Fig. 7. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:1. 2
10
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12 Surface Graph for P/2μ =0.2 2
0.02
U−−−−−>
0.01 0 −0.01 −0.02 2 1 0 −1 −2
s−−−−−>
−2
0
−1
1
2
k−−−−−>
Fig. 8. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:2. 2
Surface Graph for P/2μ2=0.3
0.02
U−−−−−>
0.01 0 −0.01 −0.02 2 1 0 −1 s−−−−−>
−2
−2
−1
0
1
2
k−−−−−>
Fig. 9. Variation of group velocity U with respect to parameter k and s for 2lP ¼ 0:3. 2
which is the usual dispersion equation for Love waves in a homogeneous medium over a homogeneous half-space with b1 < c < b2 where b21 ¼ lq1 and b22 ¼ lq2 . 1
2
5. Numerical calculations, results and discussions For the graphical representation of phase and group velocity of G-type wave in heterogeneous layer overlying a heterogeneous half-space, we have taken lq1 ¼ 0:02; lq2 ¼ 0:2 and ll2 ¼ 0:1 and the results are presented in Figs. 2–6. In Fig. 2–4 we 1
2
1
have computed the values of phase velocity bc1 from Eq. (32) for taking R0 ¼ 1 as a function of kH. In Fig. 2, we have shown the variation in dimensionless phase velocity parameter
a 2k
c b1
against dimensionless wave number kH for different values of inhomogeneity
P
a and fixed values of 2l ¼ 0:2 and e ¼ 0:2. The values of inhomogeneity parameters 2k for curves 1, curve 2, curve 2
3 and curve 4 have been taken as 0.0, 0.1, 0.2 and 0.3 respectively. From this figure we have seen that curves are accumulating initially and gradually takes little far apart from each other. Also we can observe that the phase velocity decreases a slightly with the increases of parameter 2k . Fig. 3 represent the variation in dimensionless phase velocity bc1 against dimensionless wave number kH for fixed values of e ¼ 0:2 and 2ka ¼ 0 and for different values of initial stress parameter 2lP . The values of initial stress parameter for curves 1, 2
curve 2, curve 3 and curve 4 have been taken as 0.0, 0.1, 0.2 and 0.3 respectively. It follows from the Fig. 3 that the phase velocity decreases with the increases of initial stress parameter.
S. Kundu et al. / Applied Mathematics and Computation 234 (2014) 1–12
11
In Fig. 4, an attempt has been made to study the effect of inhomogeneity parameter e. This Figure shows the variation in dimensionless phase velocity bc1 against dimensionless wave number kH. The curves are plotted for different values of e and a for fixed values of 2lP ¼ 0:2 and 2k ¼ 0. The values of e for curves 1, curve 2, curve 3 and curve 4 have been taken as 0.0, 0.1, 2 0.2 and 0.3 respectively. From this Figure it is clear that phase velocity decreases slightly with increases of inhomogeneity parameter e. By comparative study of these Fig. 2–4 we can conclude that velocity decreases with the increase of inhomogeneity parameters. Variation in dimensionless group velocity bU2 with respect to scaled wave number ks is shown in Fig. 5 for different values of P . The values of initial stress 2lP for curves 1, curve 2, curve 3 and curve 4 have been taken as 0.0, 0.1, 0.2 and 0.3 respec2l2 2 tively. From this figure we have seen that group velocity U approaches b2 asymptotically with increases in scaled wave numk ber s and it is decreases with the increases of initial stress parameter. Fig. 6 represent a set of surface plots for variation of group velocity U with respect to parameter k and s for different values of 2lP . It has been observed from the figure that group velocity depends on initial stress. So, we can say that phase velocity 2 and group velocity is influenced by initial stress, and it depends on wave number, depth parameter s and parameter e. This study may be useful to understand the nature of seismic propagated during earthquake (see Figs. 7–9). 6. Conclusions We have obtained the dispersion equation of dimensionless phase velocity of G-type seismic wave in heterogeneous medium overlying a heterogeneous half-space under initial stress as a function of dimensionless wave number by using the transform technique and Valeev’s method [24]. We have seen that phase velocity is influenced by inhomogeneity parameters and initially stress parameter. We have found the condition for large amount of energy to be confined near the surface and obtained the expression for group velocity. Variation in phase and group velocity has been plotted graphically against wave number and scaled wave number respectively. From the above figures we may conclude with the following remarks: i. Dimensionless phase velocity bc1 of G-type seismic wave increases with decreases of non-dimensional wave number kH. a ii. Phase velocity decreases with the increases of inhomogeneity parameter 2k in the upper layer. iii. Dimensionless phase velocity decreases as the value of inhomogeneity parameter e increases in the lower half-space. iv. Phase velocity of G-type seismic wave decreases when the initial stress parameter increases. v. Also we have seen that group velocity is effected by initial stress and it depends on wave number k, depth parameter s and inhomogeneity parameter. vi. Dimensionless group velocity bU2 increases with the increases of scaled wave number ks up to a certain point. vii. The group velocity decreases with the increases of initial stress parameter. The results of present paper may be useful to understand the cause of damage during deep earthquakes and it may be helpful to understand the nature of long period Love waves. References [1] W.M. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media, Mcgraw Hill Pub, New York, 1957. [2] Y. Sato, Study on surface waves, VI generation of Love and other types of SH waves, Bull. Earthquake Res. Inst. 30 (1952) 101–120. [3] A.M. Abd-Alla, S.M. Ahmed, Propagation of Love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium, Appl. Math. Comput. 106 (2) (1999) 265–275 (11p). [4] J. De Noyer, The effect of variations in layer thickness of Love waves, Bull. Seismol. Soc. Am. 51 (1961) 227–235. [5] J. Bhattacharya, On the dispersion curve for Love waves due to irregularity in the thickness of the transversely isotropic crustal layer, Gerl Beitr Z Geophys. 71 (1962) 324. [6] A. Chattopadhyay, On the dispersion equations for Love waves due to irregularity in the thickness of non homogeneous crustal layer, Acta Geophys. Pol. 23 (1975) 307–315. [7] B. Wolf, Propagation of Love waves in layers with irregular boundaries, Pure Appl. Geophys. 78 (1970) 48–57. [8] A.P. Ghorai, S.K. Samal, N.C. Mahanti, Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity, Appl. Math. Model. 34 (7) (2010) 1873–1883. [9] S.S. Singh, Love wave at a layer medium bounded by irregular boundary surfaces, J. Vib. Control 17 (5) (2010) 789–795. [10] B. Gutenberg, Wave velocities at depths between 50 and 600 kilometers, Bull. Seismol. Soc. Am. 43 (1953) 223–232. [11] I. Lehmann, S-waves and the structure of the upper mantle, Geophys. J.R. Astron. Soc. 3 (1961) 529–538. [12] A.K. Mal, On the generation of G-waves, Gerlands Beitr. Geophys. 72 (1962) 82–88. [13] K. Aki, Generation and propagation of G waves from the Niigata earthquake of June 16, 1964. Part 2. Estimation of earthquake moment, released energy, and stress-strain drop from the G wave spectrum, Bull. Earthquake Res. Inst. 72 (1966) 82–88. [14] A. Chattopadhyay, A. Keshri, Generation of G-type seismic waves under initial stress, Indian J. Pure Appl. Math. 17 (8) (1986) 1042–1055. [15] A. Chattopadhyay, S. Gupta, V.K. Sharma, P. Kumari, Propagation of G type seismic waves in viscoelastic medium, Int. J. Appl. Math. Mech. 6 (9) (2010) 63–75. [16] A. Chattopadhyay, A.K. Singh, G-type seismic waves in reinforced media, Meccanica 47 (2012) 1775–1785. [17] H. Jeffreys, The Earth, fourth ed., Cambridge University Press, Cambridge, 1959. [18] J. Bhattacharya, On short period signals obtained from explosions within the Earth, Geofis Pura e Appl. 55 (1963) 6371. [19] N.A. Haskell, Radiation pattern of surface waves from point sources in multi-layered medium, Bull. Seismol. Soc. Am. 54 (1) (1964) 377–393. [20] S. Gupta, D.K. Majhi, S. Kundu, S.K. Vishwakarma, Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space, Appl. Math. Comput. 218 (2012) 5655–5664.
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[21] S. Gupta, D.K. Majhi, S.K. Vishwakarma, Torsional surface wave propagation in an initially stressed non-homogeneous layer over a non-homogeneous half-space, Appl. Math. Comput. 219 (2012) 3209–3218. [22] S. Manna, S. Kundu, S. Gupta, Love wave propagation in a piezoelectric layer overlying in an inhomogeneous elastic half-space, J. Vib. Control (2013), http://dx.doi.org/10.1177/1077546313513626. [23] S. Kundu, S. Gupta, S. Manna, SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media, Meccanica (2013), http://dx.doi.org/10.1007/s11012-013-9825-5. [24] K.G. Valeev, On Hill’s method in the theory of linear differential equations with periodic coefficients, J. Appl. Math. A. Mech. 24 (1960) 1493–1505.