Propagation of S-wave in a non-homogeneous anisotropic
Applied Mathematics and Computation 217 (2011) 4321–4332
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Propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field A.M. Abd-Alla a,1, S.R. Mahmoud b,⇑, S.M. Abo-Dahab a,2, M.I. Helmy a a b
Mathematics Department, Faculty of Science, Taif University, Saudi Arabia Mathematics Department, Faculty of Education, King Abdul Aziz University, Saudi Arabia
a r t i c l e Keywords: Incompressible Initial stress Anisotropic S-wave Gravity field
i n f o
a b s t r a c t In this paper, propagation of shear waves in a non-homogeneous anisotropic incompressible, gravity field and initially stressed medium is studied. Analytical analysis reveals that the velocity of propagation of the shear waves depends upon the direction of propagation, the anisotropy, gravity field, non-homogeneity of the medium, and the initial stress. The frequency equation that determines the velocity of the shear wave has been obtained. The dispersion equations have been obtained and investigated for different cases. A comparison is made with the results predicted by Abd-Alla et al. [22] in the absence of initial stress and gravity field. The results obtained are discussed and presented graphically. Published by Elsevier Inc.
1. Introduction In recent years, more attention has been given to using the anisotropic material in engineering applications in considerable research activity. The problem of shear waves in an orthotropic elastic medium is very important for the possibility of its extensive application in various branches of science and technology, particularly in optics, earthquake science, acoustics, geophysics and plasma physics. Shear waves propagating over the surface of homogeneous and non-homogeneous elastic half-spaces are a well-known and prominent feature of wave theory. Mahmoud [1] studied the wave propagation in cylindrical poroelastic dry bones and Abd-Alla and Farhan [2] investigated the effect of inhomogeneity on the composite infinite cylinder of orthotropic material. Abd-Alla and Farhan [2] and Abd-Alla and Mahmoud [3] solved the magneto-thermoelastic problem in rotating a non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, and investigated the effect of the rotation on propagation of thermoelastic waves in a non-homogeneous infinite cylinder of isotropic material. Abd-Alla et al. [5] presented Rayleigh wave in a magnetoelastic half-space of orthotropic material under influence of initial stress and gravity field. The shear waves in acoustic anisotropic media have been discussed by Grechka et al. [6]. Many authors such as AbdAlla and Ahmed [7,10] and El-Naggar et al. [11] studied the effect of gravity on the propagation of Rayleigh waves on elastic solid medium. Ahmed [8] studied Rayleigh waves in a thermoelastic granular medium under initial stress. The influence of gravity on the propagation of waves in granular medium is discussed by Ahmed [9]. Das et al. [12] studied the influence of gravity on surface waves in a non-homogeneous isotropic elastic medium. Datta [13] discussed the influence of gravity field on Rayleigh waves propagation in inhomogeneous elastic medium. Bouden and Datta [14] investigated Rayleigh waves in a granular medium over an initially stressed elastic half-space. Influence of gravity on propagation of waves in composite layer has been illustrated by Bhattacharya and Sengupta [15]. ⇑ Corresponding author. Permanent address: Mathematics Department, Faculty of Science, University of Sohag, Egypt. 1 2
E-mail address: [email protected] (S.R. Mahmoud). Permanent address: Mathematics Department, Faculty of Science, University of Sohag, Egypt. E-mail: [email protected] (A.M. Abd-Alla). Permanent address: Mathematics Department, Faculty of Science, SVU, Qena 83523, Egypt. E-mail: [email protected] (S.M. Abo-Dahab).
0096-3003/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2010.10.029
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Acharya and Sengupta [16] discussed the influence of gravity field on the propagation of waves in a thermoelastic layer. De and Sengupta [17], investigated many problems of elastic waves and vibration under the influence of gravity field. Surface waves under the influence of gravity in a homogeneous medium were considered by De [18] and De and Sengupta [19]. Mahmoud [20] studied the effect of the non-homogeneity on wave propagation on orthotropic elastic media. Abd-Alla and Abo-Dahab [21] studied the time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation. In this work, the effect of gravity field on the propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium has been discussed using the wave equations which satisfied by the displacement potentials /. The frequency equations that determines the velocity of shear wave have been obtained. The dispersion equations have been obtained, and investigated for different cases. Numerical results are presented for the variation of velocity of shear waves with respect to depth b and angle h. Numerical results presented graphically. The effect of non-homogeneous, initial stress and gravity field are very pronounced. 2. Formulation of the problem Most materials behave as incompressible media and the velocities of longitudinal waves in them are very high. The varieties of hard rocks present in the earth are also almost incompressible due to factors like external pressure, slow process of creep, difference in temperature, manufacturing processes, nitriding, pointing etc., the medium stays under high stresses. These stresses are regarded as initial stresses. Owing to the variation of elastic properties and the presence of these initial stresses, the medium becomes isotropic as well. We consider an unbounded incompressible anisotropic medium under initial stresses s11 and s22 along the x-, y-directions, respectively. When the medium is slightly disturbed (u, v), the incremental stresses s11, s12 and s22 are developed, and the equations of motion in incremental state become [4]
@s11 @s12 @w @v @2u qg þ P ¼q 2; @y @x @y @x @t
ð1Þ
@s12 @s22 @w @u @2v þ qg ¼q 2; þ P @x @x @x @y @t
ð2Þ
s11 ¼ 2Nexx þ s;
ð3Þ
, and q represents the density of the medium, also, sij are incremental stresses, (u, v) are where, P ¼ S22 S11 ; w ¼ 12 @@xv @u @y incremental deformation, w is the rotational component about the z-axis, and g is the acceleration due to gravity. The incremental stress–strain relations for an incompressible medium may be taken as [4]
s22 ¼ 2Neyy þ s and s12 ¼ 2Qexy ;
s11 þs22 2
where, s ¼ ; eij are incremental strain components, and N and Q are rigidities of medium. The incompressibility condition exx + eyy = 0 is satisfied by
u¼
@u @y
and
v¼
@u : @x
ð4Þ
Substituting from Eqs. (3) and (4) into Eqs. (1) and (2), we get
" # " !# ! @s @3/ @ @2u @2u P @3u @3u @2u @3u ; ¼q g 2 2 2N 2 þ Q 2 þ @x @x @y @y 2 @x2 @y @y3 @x2 @y @x @t @y ! " # ! ! @s @3u @3u @ @2u P @3u @3u @2u @3u þQ 2N þ : þ ¼q g þ @y @y 2 @x3 @x@y2 @x3 @x@y2 @x@y @x@y @t 2 @x
ð5Þ
ð6Þ
Assuming non-homogeneities as
Q ¼ Q 0 ð1 þ ayÞ; N ¼ N0 ð1 þ byÞ;
q ¼ q0 ð1 þ cyÞ;
ð7Þ
where, N0 and Q0 are rigidities, and q0 is the density of the medium at the surface (y = 0). Substituting from (7) into Eqs. (5) and (6) we get
P @4u @4u @3u P @4u Q 0 ð1 þ ayÞ þ ½ 4N ð1 þ byÞ 2Q ð1 þ ayÞ þ ½4N b 2aQ ð1 þ ayÞ þ þ Q 0 0 0 0 0 2 @x4 2 @y4 @x2 @y2 @x2 @y " # " # 3 4 4 2 3 @ u @ u @ u @ u @ u : q0 c g 2 2 þ 2aQ 0 3 ¼ q0 ð1 þ cyÞ þ @y @x @x2 @t 2 @y2 @t2 @t @y
ð8Þ
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3. Solution of problem For propagation of sinusoidal waves in my arbitrary direction, assuming harmonic time variation of Eq. (8) as
/ðx; y; tÞ ¼ Aeikðp1 xþp2 yc1 tÞ ;
ð9Þ
where, p1 and p2 are cosine of angles made by direction of propagation with x- and y-axes, and c1 and k are the velocity of propagation and the wave number, respectively. Using Eq. (9) into Eq. (8) and equating real and imaginary parts separately, we get
( ) 2 c1 1 P 2N0 P gc 2 4 2 2 4 P þ2 p ð1 þ ayÞ ¼ ð1 þ byÞ ð1 þ ayÞ p1 p2 þ 1 þ ay þ p ð1 þ cyÞ 2Q 0 1 2Q 0 2 k2 b2 1 b Q0
ð10Þ
2 a c1 N0 b a 2 p1 þ 2 p2 ; ¼2 2 c 2 b Q0 c c
ð11Þ
and
12
where, b ¼ Qq 0 is the velocity of shear waves in homogeneous isotropic medium. Eq. (10) gives the velocity of propagation 0 of shear wave and Eq. (11) gives the damping. Eq. (10) shows that the velocity cb1 depends much on the anisotropy factor, the initial stress factor and also on the direction of propagation denoted by (p1, p2). 4. Particular cases 4.1. Analysis of Eq. (10) In order to gain more insight information the following cases have been discussed: analysis of Eq. (10) obtained by equating the real part of equation of motion. Case I. In case Q is homogeneous (a ? 0), i.e., rigidity along vertical direction is constant
( ) 2 c1 1 P N0 p gc p41 þ 2 2 p42 2 2 p21 : 1 ¼ ð1 þ byÞ 1 p21 p22 þ 1 þ 1 þ cy 2Q 0 2Q 0 b Q0 k b
ð12Þ
The velocity in the x-direction is (p1 = 1, p2 = 0, c = c11)
( ) 2 c11 1 p gc ; ¼ 1 1 þ cy 2Q 0 b2 k2 b ( ) b2 p gc c211 ¼ 2 2 : 1 2Q 0 k b 1 þ cy
ð13Þ ð14Þ
In case the medium is free from initial stress (p ? 0, c ? 0) and c11 = b. Similarly the velocity of propagation along y-direction (p1 = 0, p2 = 1, c1 = c22) obtained as
c222 ¼
b2 p : 1þ 2Q 0 1 þ cy
ð15Þ
Subtracting Eq. (15) from Eq. (14), we get
c222 c211 b2
! 1 p gc ; ¼ 1 þ cy 2Q 0 k2 b2
ð16Þ
which is a function of initial stress, gravity field and density. It may also be observed that if p = S22 S11 > 0, the effect of initial stresses on the body is compressive along x-direction and tensile along y-direction. The compressive initial stress reduces while tensile stress increases the velocity of shear wave along x-direction. A reverse effect is obtained along y-direction. Case II. In case N is homogeneous (b ? 0), i.e., rigidity along horizontal direction is constant
( ) 2 c1 1 p N0 gc 2 p 4 2 2 p þ2 2 p4 ; 1 þ ay ¼ ð1 þ ayÞ p1 p2 2 2 p1 þ 1 þ ay þ 1 þ cy 2Q 0 1 2Q 0 2 b Q0 k b
ð17Þ
the velocity along x-direction (p1 = 1, p2 = 0, c1 = c11) is given by
c211
( ) b2 p gc ; ¼ 1 þ ay 2Q 0 k2 b2 1 þ cy
ð18Þ
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which depends on the depth y and gravity and the wave is dispersive, the velocity along y-direction is (p1 = 0, p2 = 1, c1 = c22)
c222 ¼
b p ; 1 þ ay þ 1 þ cy 2Q 0
ð19Þ
Fig. 1. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of density parameter ¼ 4; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; c ¼ 0:7 ; 0:8 ; 0:9 . . .. c : a
Fig. 2. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of density parameter ¼ 4; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; c ¼ 0:7 ; 0:8 ; 0:9 . . .. c : a
Fig. 3. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of rigidities parameter : c ¼ 8; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; a ¼ 3 ; 3:5 ; 4 . . .. a
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Fig. 4. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of rigidities parameter ¼ 3 ; 3:5 ; 4 . . .. : c ¼ 8; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; a a
Fig. 5. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of N (anisotropy): ¼ 4; N ¼ 2 ; 2:5 ; 3 . . .. c ¼ 8; P ¼ 0:5; g ¼ 0:3; a
Fig. 6. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of N (anisotropy): ¼ 4; N ¼ 2 ; 2:5 ; 3 . . .. c ¼ 8; P ¼ 0:5; g ¼ 0:3; a
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Fig. 7. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of initial stress ¼ 4; N ¼ 2:5; P ¼ 0:8 ; 0:0 ; 0:8 . . .. ¼ 0:3; a P : c ¼ 8; P ¼ 0:5; g
Fig. 8. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of initial stress ¼ 4; N ¼ 2:5; P ¼ 0:8 ; 0:0 ; 0:8 . . .. ¼ 0:3; a P : c ¼ 8; P ¼ 0:5; g
Fig. 9. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of gravity parameter ¼ 4; N ¼ 2:5; g ¼ 0:1 ; 0:3 ; 0:5 . . .. g : c ¼ 8; P ¼ 0:5; a
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for p > 0, the velocity along y-direction may increase considerably at a distance y from free surface and the wave becomes dispersive. Case III. In case N, Q and c are homogeneous
p N0 p 4 p41 þ 2 2 c21 ¼ b 1 1 p21 p22 þ 1 þ p : 2Q 0 2Q 2 Q0
ð20Þ
In the absence of initial stress the velocity equation becomes
12 c1 N0 2 2 p1 p2 ¼ 14 1 b Q0
ð21Þ
in x-direction and y-direction c1 = b, then the velocity does not depend on an isotropy effects the velocity for isotropic medium N0 = Q0, c1 = b. Case IV. In the absence of initial stress P ? 0, the velocity is obtained as
( ) 2 c1 1 N0 gc 2 4 2 2 4 ð1 þ ayÞp1 þ 2 2 ¼ ð1 þ byÞ ð1 þ ayÞ p1 p2 2 2 p1 þ ð1 þ ayÞp2 1 þ cy b Q0 k b
ð22Þ
in x-direction (p1 = 1, p2 = 0, c1 = c11), Eq. (22) reduces to
( ) 2 c11 1 gc 1 þ ay 2 2 ¼ 1 þ cy b k b
ð23Þ
and along y-direction (p1 = 0, p2 = 1, c = c22), Eq. (22) tends to
2 c22 1 þ ay ¼ : 1 þ cy b
ð24Þ
Case V. In the absence of gravity field the velocity is obtained as
2 c1 1 P N0 P p41 þ 2 2 p4 : 1 þ ay ¼ ð1 þ byÞ ð1 þ ayÞp21 p22 þ 1 þ ay þ 1 þ cy 2Q 0 2Q 0 2 b Q0
ð25Þ
4.2. Analysis of Eq. (11) Analysis of Eq. (11) obtained by equating imaginary parts of equation of motion. In the absence of q and P in Eq. (11) three cases have been analyzed as follows: Case I. In case Q is homogeneous (a ? 0) i.e., rigidity along vertical direction is constant, one may obtain
2 c1 2N0 b 2 p : ¼2 b Q0 c 1
ð26Þ
Fig. 10. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of gravity parameter ¼ 4; N ¼ 2:5; g ¼ 0:1 ; 0:3 ; 0:5 . . .. g : c ¼ 8; P ¼ 0:5; a
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This shows that velocity of shear wave is always damped. The velocity of wave along x-direction (p1 = 1, p2 = 0, c = c11) is obtained as
2 c11 N0 b : ¼2 2 b Q0 c
ð27Þ
0 b , and no damping takes place along y-direction. This shows that actual velocity in x-direction is damped by 2 2N Q0 c Case II. In case N is homogeneous (b ? 0) i.e., rigidity along horizontal direction is constant
2 a a c1 p2 ; ¼ 2 p21 þ 2 c c 2 b
ð28Þ
the velocity of wave along x-direction (p1 = 1, p2 = 0, c = c11) is given by
¼ 4; N ¼ 2:5; (a) P ¼ 0:8, ¼ 0:3; a Fig. 11. Variation in velocities of shear wave with respect to the angle h and different values of density parameter c : g (b) P ¼ 0:0, (c) P ¼ 0:8; c ¼ 0:7 ; 0:8 ; 0:9 . . ..
A.M. Abd-Alla et al. / Applied Mathematics and Computation 217 (2011) 4321–4332
2 a c11 ¼2 : c b
4329
ð29Þ
Existence of negative sign shows that damping does not takes place along x-direction for (b ? 0), the velocity along y-direction is given by
2 a c22 ¼2 c b
ð30Þ
indicating a damping of magnitude 2a takes place along y-direction. c Case III. In case N and Q are homogeneous but density is linearly varying with depth
c1 ¼0 b
ð31Þ
i.e., no damping takes place.
: c ¼ 0:8; g ¼ 0:3; N ¼ 2:5; (a) Fig. 12. Variation in velocities of shear wave with respect to the angle h and different values of rigidities parameter a ¼ 3 ; 3:5 ; 4 . . .. P ¼ 0:8, (b) P ¼ 0:0, (c) P ¼ 0:8; a
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¼ 4; g ¼ 0:3; (a) P ¼ 0:8, Fig. 13. Variation in velocities of shear wave with respect to the angle h and different values of anisotropy parameter N : c ¼ 8; a (b) P ¼ 0:0, (c) P ¼ 0:8; N ¼ 2 ; 2:5 ; 3 . . ..
5. Numerical results and discussions To get numerical information on the velocity of shear wave in the non-homogeneous initially stressed medium we introduce the following non-dimensional parameters:
a ¼ ; a b
¼ by; b
c c ¼ ; b
c1 ¼
c1 ; b
N¼
N0 ; Q0
P¼
P ; 2Q 0
g ¼
gb 2
k b2
Using these parameters in the Eq. (10) we obtain:
c1 2 ¼
1 ð1 þ cbÞ
n
o ð1 þ a 2 p2 þ ½1 þ a þ Pp4 cgp2 : PP4 þ 2½2Nð1 þ bÞ bÞp b b ½1 þ a 1 2 2 1 1
and h, and the results are presented in Figs. 1–14. ; c; N; P; g; b; One may calculate c1 for different values of a
ð32Þ
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¼ 4; N ¼ 2:5; (a) P ¼ 0:8, (b) : c ¼ 8; a Fig. 14. Variation in velocities of shear wave with respect to the angle h and different values of gravity parameter g P ¼ 0:0, (c) P ¼ 0:8; g ¼ 0:1 ; 0:3 ; 0:5 . . ..
Figs. 1, 3, 5, 7 and 9 show the effects of density, rigidities, anisotropy, initial stress, respectively, if the angle h = 30° on Figs. 2, 4, 6, 8 and 10 show the effects of density, rigidities, anisotropy, initial shear wave velocity c with respect to depth b, It is clear that shear wave velocity stress, respectively, if the angle h = 60° on shear wave velocity c1 with respect to depth b. c1 increases with an increase of the depth b. and the From Figs. 1–9, it is obvious that shear wave velocity c1 decreases with an increase of the density parameter a and anisotropy parameter N if h = {30°, 60°}. Also, it apgravity g but increases with an increase of the rigidity parameter a pears that shear wave velocity c1 decreases with an increase of the initial stress if h = 30° but c1 increases with an increase of the initial stress if h = 60°. One can mention that shear wave velocity c1 has small values if h = 30° compared to its values if h = 60°. Figs. 11–14 display the variation in velocities of shear wave with respect to the angle h with the varying values of the density, rigidities, anisotropy, and gravity parameters, respectively, for compression, tensional initial stress, and without initial stress.
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It is seen that shear wave velocity c1 decreases with an increase of angle h and then increases with the high values of h. From Figs. 11 and 14, it is mentioned that shear wave velocity c1 decreases with an increase of the density and gravity parameters. Also, it is obvious that shear wave velocity c1 increases with an increase of rigidities and anisotropy parameters. Finally, from Figs. 11–14, it is concluded that the increasing values of initial stress (compression, without initial stress, and tensional initial stress), the shear wave velocity c1 decreases. 6. Conclusions The anisotropy, gravity field, non-homogeneity of the medium, the initial stress, the direction of propagation and the depth have considerable effect on the velocity of propagation of shear wave and attracts the attention of earth scientists in their work. Numerical computation shows that the presence of initial compressive stress in the medium gravity field, reduces the velocity of propagation while the tensile stress increases it. It is found that the variation in parameters associated with anisotropy and non-homogeneity of the medium directly affects the velocity of the wave. The velocity of propagation also depends on the inclination of the direction of propagation; an increase in the inclination angle decreases the velocity in the beginning, takes a minimum value before increasing. Finally, it is concluded that the increasing values of initial stress (compression, without initial stress, and tensional initial stress), the shear wave velocity c1 decreases. References [1] S.R. Mahmoud, Wave propagation in cylindrical poroelastic dry bones, Journal of the Applied Mathematics & Information Sciences 4 (2) (2010) 209– 226. [2] A.M. Abd-Alla, A.M. Farhan, Effect of the non-homogenity on the composite infinite cylinder of orthotropic material, Physics Letters A 372 (2008) 756– 760. [3] M. Abd-Alla, S.R. Mahmoud, Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, Meccanica 45 (4) (2010) 451–462. [4] M. Abd-Alla, S.R. Mahmoud, Effect of the rotation on propagation of thermoelastic waves in a non-homogeneous infinite cylinder of isotropic material, International Journal of Mathematical Analysis 4 (2010). [5] A.M. Abd-Alla, H.A.H. Hammad, S.M. Abo-Dahab, Rayleigh waves in magnetoelastic half-space of orthotropic material under influence of initial stress and gravity field, Applied Mathematics and Computation 154 (2, 5) (2004) 583–597. [6] V. Grechka, L. Zhang, W.J. Rector, Shear waves in acoustic anisotropic media, Geophysics 69 (2004) 576–582. [7] A.M. Abd-Alla, S.M. Ahmed, Stoneley and Rayleigh waves in a non-homogeneous orthotropic elastic medium under influence of gravity, Applied Mathematics and Computation 135 (1) (2003) 187–200. [8] S.M. Ahmed, Rayleigh waves in a thermoelastic granular medium under initial stress, International Journal of Mathematics and Mathematical Sciences 23 (2000) 627–637. [9] S.M. Ahmed, Influence of gravity on the propagation of waves in granular medium, Applied Mathematics and Computation 101 (1999) 269–280. [10] A.M. Abd-Alla, S.M. Ahmed, Rayleigh waves in an orthotropic thermoelastic medium under gravity and initial stress, Earth, Moon and Planets 75 (1998) 185–197. [11] A.M. El-Naggar, A.M. Abd-Alla, S.M. Ahmed, Rayleigh waves in a magneto-elastic initially stresses conducting medium with the gravity field, Bulletin of the Calcutta Mathematical Society 86 (1994) 51–56. [12] S.C. Das, D.P. Acharya, P.R. Sengupta, Surface waves in an inhomogeneous elastic medium under the influence of gravity, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 37 (5) (1992) 539–552. [13] B.K. Datta, Some observations on interactions of Rayleigh waves in an elastic solid medium with the gravity field, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 31 (4) (1986) 369–374. [14] M. Bouden, S.K. Datta, Rayleigh waves in granular medium over an initially stressed elastic half-space, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 29 (1984) 271–285. [15] P.C. Bhattacharya, P.R. Sengupta, Influence of gravity on propagation of waves in a composite layer, Ranchi University Mathematical Journal 15 (1984) 53–67. [16] P.R. Sengupta, D. Acharya, The influence of gravity on the propagation of waves in a thermoelastic layer, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 24 (1979) 395–406. [17] S.K. De, P.R. Sengupta, Effect of anisotropy on surface wave under the influence of gravity, Acta Geophysica Polonica XXIV (1978) 4–16. [18] S.N. De, Surface waves in the influence of gravity, Gerlands Beitr, Geophysics 85 (1976) 4–15. [19] S.K. De, P.R. Sengupta, Influence of gravity on wave propagation in an elastic layer, Journal of the Acoustical Society of America 55 (1974) 1–5. [20] S.R. Mahmoud, Effect of the non-homogeneity on wave propagation on orthotropic elastic media, International Journal of Contemporary Mathematical Sciences 5 (45) (2010) 2211–2224. [21] A.M. Abd-Alla, S.M. Abo-Dahab, Time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation, Applied Mathematical Modelling 33 (5) (2009) 2388–2402. [22] M. Abd-Alla, S.R. Mahmoud, M.I.R. Helmi, Effect of initial stress and magnetic field on Propagation of shear wave in non homogeneous Anisotropic medium under gravity field, The Open Applied Mathematics Journal 3 (2009) 49–56.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field A.M. Abd-Alla a,1, S.R. Mahmoud b,⇑, S.M. Abo-Dahab a,2, M.I. Helmy a a b
Mathematics Department, Faculty of Science, Taif University, Saudi Arabia Mathematics Department, Faculty of Education, King Abdul Aziz University, Saudi Arabia
a r t i c l e Keywords: Incompressible Initial stress Anisotropic S-wave Gravity field
i n f o
a b s t r a c t In this paper, propagation of shear waves in a non-homogeneous anisotropic incompressible, gravity field and initially stressed medium is studied. Analytical analysis reveals that the velocity of propagation of the shear waves depends upon the direction of propagation, the anisotropy, gravity field, non-homogeneity of the medium, and the initial stress. The frequency equation that determines the velocity of the shear wave has been obtained. The dispersion equations have been obtained and investigated for different cases. A comparison is made with the results predicted by Abd-Alla et al. [22] in the absence of initial stress and gravity field. The results obtained are discussed and presented graphically. Published by Elsevier Inc.
1. Introduction In recent years, more attention has been given to using the anisotropic material in engineering applications in considerable research activity. The problem of shear waves in an orthotropic elastic medium is very important for the possibility of its extensive application in various branches of science and technology, particularly in optics, earthquake science, acoustics, geophysics and plasma physics. Shear waves propagating over the surface of homogeneous and non-homogeneous elastic half-spaces are a well-known and prominent feature of wave theory. Mahmoud [1] studied the wave propagation in cylindrical poroelastic dry bones and Abd-Alla and Farhan [2] investigated the effect of inhomogeneity on the composite infinite cylinder of orthotropic material. Abd-Alla and Farhan [2] and Abd-Alla and Mahmoud [3] solved the magneto-thermoelastic problem in rotating a non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, and investigated the effect of the rotation on propagation of thermoelastic waves in a non-homogeneous infinite cylinder of isotropic material. Abd-Alla et al. [5] presented Rayleigh wave in a magnetoelastic half-space of orthotropic material under influence of initial stress and gravity field. The shear waves in acoustic anisotropic media have been discussed by Grechka et al. [6]. Many authors such as AbdAlla and Ahmed [7,10] and El-Naggar et al. [11] studied the effect of gravity on the propagation of Rayleigh waves on elastic solid medium. Ahmed [8] studied Rayleigh waves in a thermoelastic granular medium under initial stress. The influence of gravity on the propagation of waves in granular medium is discussed by Ahmed [9]. Das et al. [12] studied the influence of gravity on surface waves in a non-homogeneous isotropic elastic medium. Datta [13] discussed the influence of gravity field on Rayleigh waves propagation in inhomogeneous elastic medium. Bouden and Datta [14] investigated Rayleigh waves in a granular medium over an initially stressed elastic half-space. Influence of gravity on propagation of waves in composite layer has been illustrated by Bhattacharya and Sengupta [15]. ⇑ Corresponding author. Permanent address: Mathematics Department, Faculty of Science, University of Sohag, Egypt. 1 2
E-mail address: [email protected] (S.R. Mahmoud). Permanent address: Mathematics Department, Faculty of Science, University of Sohag, Egypt. E-mail: [email protected] (A.M. Abd-Alla). Permanent address: Mathematics Department, Faculty of Science, SVU, Qena 83523, Egypt. E-mail: [email protected] (S.M. Abo-Dahab).
0096-3003/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2010.10.029
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Acharya and Sengupta [16] discussed the influence of gravity field on the propagation of waves in a thermoelastic layer. De and Sengupta [17], investigated many problems of elastic waves and vibration under the influence of gravity field. Surface waves under the influence of gravity in a homogeneous medium were considered by De [18] and De and Sengupta [19]. Mahmoud [20] studied the effect of the non-homogeneity on wave propagation on orthotropic elastic media. Abd-Alla and Abo-Dahab [21] studied the time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation. In this work, the effect of gravity field on the propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium has been discussed using the wave equations which satisfied by the displacement potentials /. The frequency equations that determines the velocity of shear wave have been obtained. The dispersion equations have been obtained, and investigated for different cases. Numerical results are presented for the variation of velocity of shear waves with respect to depth b and angle h. Numerical results presented graphically. The effect of non-homogeneous, initial stress and gravity field are very pronounced. 2. Formulation of the problem Most materials behave as incompressible media and the velocities of longitudinal waves in them are very high. The varieties of hard rocks present in the earth are also almost incompressible due to factors like external pressure, slow process of creep, difference in temperature, manufacturing processes, nitriding, pointing etc., the medium stays under high stresses. These stresses are regarded as initial stresses. Owing to the variation of elastic properties and the presence of these initial stresses, the medium becomes isotropic as well. We consider an unbounded incompressible anisotropic medium under initial stresses s11 and s22 along the x-, y-directions, respectively. When the medium is slightly disturbed (u, v), the incremental stresses s11, s12 and s22 are developed, and the equations of motion in incremental state become [4]
@s11 @s12 @w @v @2u qg þ P ¼q 2; @y @x @y @x @t
ð1Þ
@s12 @s22 @w @u @2v þ qg ¼q 2; þ P @x @x @x @y @t
ð2Þ
s11 ¼ 2Nexx þ s;
ð3Þ
, and q represents the density of the medium, also, sij are incremental stresses, (u, v) are where, P ¼ S22 S11 ; w ¼ 12 @@xv @u @y incremental deformation, w is the rotational component about the z-axis, and g is the acceleration due to gravity. The incremental stress–strain relations for an incompressible medium may be taken as [4]
s22 ¼ 2Neyy þ s and s12 ¼ 2Qexy ;
s11 þs22 2
where, s ¼ ; eij are incremental strain components, and N and Q are rigidities of medium. The incompressibility condition exx + eyy = 0 is satisfied by
u¼
@u @y
and
v¼
@u : @x
ð4Þ
Substituting from Eqs. (3) and (4) into Eqs. (1) and (2), we get
" # " !# ! @s @3/ @ @2u @2u P @3u @3u @2u @3u ; ¼q g 2 2 2N 2 þ Q 2 þ @x @x @y @y 2 @x2 @y @y3 @x2 @y @x @t @y ! " # ! ! @s @3u @3u @ @2u P @3u @3u @2u @3u þQ 2N þ : þ ¼q g þ @y @y 2 @x3 @x@y2 @x3 @x@y2 @x@y @x@y @t 2 @x
ð5Þ
ð6Þ
Assuming non-homogeneities as
Q ¼ Q 0 ð1 þ ayÞ; N ¼ N0 ð1 þ byÞ;
q ¼ q0 ð1 þ cyÞ;
ð7Þ
where, N0 and Q0 are rigidities, and q0 is the density of the medium at the surface (y = 0). Substituting from (7) into Eqs. (5) and (6) we get
P @4u @4u @3u P @4u Q 0 ð1 þ ayÞ þ ½ 4N ð1 þ byÞ 2Q ð1 þ ayÞ þ ½4N b 2aQ ð1 þ ayÞ þ þ Q 0 0 0 0 0 2 @x4 2 @y4 @x2 @y2 @x2 @y " # " # 3 4 4 2 3 @ u @ u @ u @ u @ u : q0 c g 2 2 þ 2aQ 0 3 ¼ q0 ð1 þ cyÞ þ @y @x @x2 @t 2 @y2 @t2 @t @y
ð8Þ
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3. Solution of problem For propagation of sinusoidal waves in my arbitrary direction, assuming harmonic time variation of Eq. (8) as
/ðx; y; tÞ ¼ Aeikðp1 xþp2 yc1 tÞ ;
ð9Þ
where, p1 and p2 are cosine of angles made by direction of propagation with x- and y-axes, and c1 and k are the velocity of propagation and the wave number, respectively. Using Eq. (9) into Eq. (8) and equating real and imaginary parts separately, we get
( ) 2 c1 1 P 2N0 P gc 2 4 2 2 4 P þ2 p ð1 þ ayÞ ¼ ð1 þ byÞ ð1 þ ayÞ p1 p2 þ 1 þ ay þ p ð1 þ cyÞ 2Q 0 1 2Q 0 2 k2 b2 1 b Q0
ð10Þ
2 a c1 N0 b a 2 p1 þ 2 p2 ; ¼2 2 c 2 b Q0 c c
ð11Þ
and
12
where, b ¼ Qq 0 is the velocity of shear waves in homogeneous isotropic medium. Eq. (10) gives the velocity of propagation 0 of shear wave and Eq. (11) gives the damping. Eq. (10) shows that the velocity cb1 depends much on the anisotropy factor, the initial stress factor and also on the direction of propagation denoted by (p1, p2). 4. Particular cases 4.1. Analysis of Eq. (10) In order to gain more insight information the following cases have been discussed: analysis of Eq. (10) obtained by equating the real part of equation of motion. Case I. In case Q is homogeneous (a ? 0), i.e., rigidity along vertical direction is constant
( ) 2 c1 1 P N0 p gc p41 þ 2 2 p42 2 2 p21 : 1 ¼ ð1 þ byÞ 1 p21 p22 þ 1 þ 1 þ cy 2Q 0 2Q 0 b Q0 k b
ð12Þ
The velocity in the x-direction is (p1 = 1, p2 = 0, c = c11)
( ) 2 c11 1 p gc ; ¼ 1 1 þ cy 2Q 0 b2 k2 b ( ) b2 p gc c211 ¼ 2 2 : 1 2Q 0 k b 1 þ cy
ð13Þ ð14Þ
In case the medium is free from initial stress (p ? 0, c ? 0) and c11 = b. Similarly the velocity of propagation along y-direction (p1 = 0, p2 = 1, c1 = c22) obtained as
c222 ¼
b2 p : 1þ 2Q 0 1 þ cy
ð15Þ
Subtracting Eq. (15) from Eq. (14), we get
c222 c211 b2
! 1 p gc ; ¼ 1 þ cy 2Q 0 k2 b2
ð16Þ
which is a function of initial stress, gravity field and density. It may also be observed that if p = S22 S11 > 0, the effect of initial stresses on the body is compressive along x-direction and tensile along y-direction. The compressive initial stress reduces while tensile stress increases the velocity of shear wave along x-direction. A reverse effect is obtained along y-direction. Case II. In case N is homogeneous (b ? 0), i.e., rigidity along horizontal direction is constant
( ) 2 c1 1 p N0 gc 2 p 4 2 2 p þ2 2 p4 ; 1 þ ay ¼ ð1 þ ayÞ p1 p2 2 2 p1 þ 1 þ ay þ 1 þ cy 2Q 0 1 2Q 0 2 b Q0 k b
ð17Þ
the velocity along x-direction (p1 = 1, p2 = 0, c1 = c11) is given by
c211
( ) b2 p gc ; ¼ 1 þ ay 2Q 0 k2 b2 1 þ cy
ð18Þ
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which depends on the depth y and gravity and the wave is dispersive, the velocity along y-direction is (p1 = 0, p2 = 1, c1 = c22)
c222 ¼
b p ; 1 þ ay þ 1 þ cy 2Q 0
ð19Þ
Fig. 1. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of density parameter ¼ 4; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; c ¼ 0:7 ; 0:8 ; 0:9 . . .. c : a
Fig. 2. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of density parameter ¼ 4; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; c ¼ 0:7 ; 0:8 ; 0:9 . . .. c : a
Fig. 3. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of rigidities parameter : c ¼ 8; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; a ¼ 3 ; 3:5 ; 4 . . .. a
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Fig. 4. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of rigidities parameter ¼ 3 ; 3:5 ; 4 . . .. : c ¼ 8; P ¼ 0:5; g ¼ 0:3; N ¼ 2:5; a a
Fig. 5. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of N (anisotropy): ¼ 4; N ¼ 2 ; 2:5 ; 3 . . .. c ¼ 8; P ¼ 0:5; g ¼ 0:3; a
Fig. 6. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of N (anisotropy): ¼ 4; N ¼ 2 ; 2:5 ; 3 . . .. c ¼ 8; P ¼ 0:5; g ¼ 0:3; a
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Fig. 7. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of initial stress ¼ 4; N ¼ 2:5; P ¼ 0:8 ; 0:0 ; 0:8 . . .. ¼ 0:3; a P : c ¼ 8; P ¼ 0:5; g
Fig. 8. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of initial stress ¼ 4; N ¼ 2:5; P ¼ 0:8 ; 0:0 ; 0:8 . . .. ¼ 0:3; a P : c ¼ 8; P ¼ 0:5; g
Fig. 9. Variation in velocities of shear wave in the direction of h = 30° with x-axis at different depth and different values of gravity parameter ¼ 4; N ¼ 2:5; g ¼ 0:1 ; 0:3 ; 0:5 . . .. g : c ¼ 8; P ¼ 0:5; a
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for p > 0, the velocity along y-direction may increase considerably at a distance y from free surface and the wave becomes dispersive. Case III. In case N, Q and c are homogeneous
p N0 p 4 p41 þ 2 2 c21 ¼ b 1 1 p21 p22 þ 1 þ p : 2Q 0 2Q 2 Q0
ð20Þ
In the absence of initial stress the velocity equation becomes
12 c1 N0 2 2 p1 p2 ¼ 14 1 b Q0
ð21Þ
in x-direction and y-direction c1 = b, then the velocity does not depend on an isotropy effects the velocity for isotropic medium N0 = Q0, c1 = b. Case IV. In the absence of initial stress P ? 0, the velocity is obtained as
( ) 2 c1 1 N0 gc 2 4 2 2 4 ð1 þ ayÞp1 þ 2 2 ¼ ð1 þ byÞ ð1 þ ayÞ p1 p2 2 2 p1 þ ð1 þ ayÞp2 1 þ cy b Q0 k b
ð22Þ
in x-direction (p1 = 1, p2 = 0, c1 = c11), Eq. (22) reduces to
( ) 2 c11 1 gc 1 þ ay 2 2 ¼ 1 þ cy b k b
ð23Þ
and along y-direction (p1 = 0, p2 = 1, c = c22), Eq. (22) tends to
2 c22 1 þ ay ¼ : 1 þ cy b
ð24Þ
Case V. In the absence of gravity field the velocity is obtained as
2 c1 1 P N0 P p41 þ 2 2 p4 : 1 þ ay ¼ ð1 þ byÞ ð1 þ ayÞp21 p22 þ 1 þ ay þ 1 þ cy 2Q 0 2Q 0 2 b Q0
ð25Þ
4.2. Analysis of Eq. (11) Analysis of Eq. (11) obtained by equating imaginary parts of equation of motion. In the absence of q and P in Eq. (11) three cases have been analyzed as follows: Case I. In case Q is homogeneous (a ? 0) i.e., rigidity along vertical direction is constant, one may obtain
2 c1 2N0 b 2 p : ¼2 b Q0 c 1
ð26Þ
Fig. 10. Variation in velocities of shear wave in the direction of h = 60° with x-axis at different depth and different values of gravity parameter ¼ 4; N ¼ 2:5; g ¼ 0:1 ; 0:3 ; 0:5 . . .. g : c ¼ 8; P ¼ 0:5; a
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This shows that velocity of shear wave is always damped. The velocity of wave along x-direction (p1 = 1, p2 = 0, c = c11) is obtained as
2 c11 N0 b : ¼2 2 b Q0 c
ð27Þ
0 b , and no damping takes place along y-direction. This shows that actual velocity in x-direction is damped by 2 2N Q0 c Case II. In case N is homogeneous (b ? 0) i.e., rigidity along horizontal direction is constant
2 a a c1 p2 ; ¼ 2 p21 þ 2 c c 2 b
ð28Þ
the velocity of wave along x-direction (p1 = 1, p2 = 0, c = c11) is given by
¼ 4; N ¼ 2:5; (a) P ¼ 0:8, ¼ 0:3; a Fig. 11. Variation in velocities of shear wave with respect to the angle h and different values of density parameter c : g (b) P ¼ 0:0, (c) P ¼ 0:8; c ¼ 0:7 ; 0:8 ; 0:9 . . ..
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2 a c11 ¼2 : c b
4329
ð29Þ
Existence of negative sign shows that damping does not takes place along x-direction for (b ? 0), the velocity along y-direction is given by
2 a c22 ¼2 c b
ð30Þ
indicating a damping of magnitude 2a takes place along y-direction. c Case III. In case N and Q are homogeneous but density is linearly varying with depth
c1 ¼0 b
ð31Þ
i.e., no damping takes place.
: c ¼ 0:8; g ¼ 0:3; N ¼ 2:5; (a) Fig. 12. Variation in velocities of shear wave with respect to the angle h and different values of rigidities parameter a ¼ 3 ; 3:5 ; 4 . . .. P ¼ 0:8, (b) P ¼ 0:0, (c) P ¼ 0:8; a
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¼ 4; g ¼ 0:3; (a) P ¼ 0:8, Fig. 13. Variation in velocities of shear wave with respect to the angle h and different values of anisotropy parameter N : c ¼ 8; a (b) P ¼ 0:0, (c) P ¼ 0:8; N ¼ 2 ; 2:5 ; 3 . . ..
5. Numerical results and discussions To get numerical information on the velocity of shear wave in the non-homogeneous initially stressed medium we introduce the following non-dimensional parameters:
a ¼ ; a b
¼ by; b
c c ¼ ; b
c1 ¼
c1 ; b
N¼
N0 ; Q0
P¼
P ; 2Q 0
g ¼
gb 2
k b2
Using these parameters in the Eq. (10) we obtain:
c1 2 ¼
1 ð1 þ cbÞ
n
o ð1 þ a 2 p2 þ ½1 þ a þ Pp4 cgp2 : PP4 þ 2½2Nð1 þ bÞ bÞp b b ½1 þ a 1 2 2 1 1
and h, and the results are presented in Figs. 1–14. ; c; N; P; g; b; One may calculate c1 for different values of a
ð32Þ
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¼ 4; N ¼ 2:5; (a) P ¼ 0:8, (b) : c ¼ 8; a Fig. 14. Variation in velocities of shear wave with respect to the angle h and different values of gravity parameter g P ¼ 0:0, (c) P ¼ 0:8; g ¼ 0:1 ; 0:3 ; 0:5 . . ..
Figs. 1, 3, 5, 7 and 9 show the effects of density, rigidities, anisotropy, initial stress, respectively, if the angle h = 30° on Figs. 2, 4, 6, 8 and 10 show the effects of density, rigidities, anisotropy, initial shear wave velocity c with respect to depth b, It is clear that shear wave velocity stress, respectively, if the angle h = 60° on shear wave velocity c1 with respect to depth b. c1 increases with an increase of the depth b. and the From Figs. 1–9, it is obvious that shear wave velocity c1 decreases with an increase of the density parameter a and anisotropy parameter N if h = {30°, 60°}. Also, it apgravity g but increases with an increase of the rigidity parameter a pears that shear wave velocity c1 decreases with an increase of the initial stress if h = 30° but c1 increases with an increase of the initial stress if h = 60°. One can mention that shear wave velocity c1 has small values if h = 30° compared to its values if h = 60°. Figs. 11–14 display the variation in velocities of shear wave with respect to the angle h with the varying values of the density, rigidities, anisotropy, and gravity parameters, respectively, for compression, tensional initial stress, and without initial stress.
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It is seen that shear wave velocity c1 decreases with an increase of angle h and then increases with the high values of h. From Figs. 11 and 14, it is mentioned that shear wave velocity c1 decreases with an increase of the density and gravity parameters. Also, it is obvious that shear wave velocity c1 increases with an increase of rigidities and anisotropy parameters. Finally, from Figs. 11–14, it is concluded that the increasing values of initial stress (compression, without initial stress, and tensional initial stress), the shear wave velocity c1 decreases. 6. Conclusions The anisotropy, gravity field, non-homogeneity of the medium, the initial stress, the direction of propagation and the depth have considerable effect on the velocity of propagation of shear wave and attracts the attention of earth scientists in their work. Numerical computation shows that the presence of initial compressive stress in the medium gravity field, reduces the velocity of propagation while the tensile stress increases it. It is found that the variation in parameters associated with anisotropy and non-homogeneity of the medium directly affects the velocity of the wave. The velocity of propagation also depends on the inclination of the direction of propagation; an increase in the inclination angle decreases the velocity in the beginning, takes a minimum value before increasing. Finally, it is concluded that the increasing values of initial stress (compression, without initial stress, and tensional initial stress), the shear wave velocity c1 decreases. References [1] S.R. Mahmoud, Wave propagation in cylindrical poroelastic dry bones, Journal of the Applied Mathematics & Information Sciences 4 (2) (2010) 209– 226. [2] A.M. Abd-Alla, A.M. Farhan, Effect of the non-homogenity on the composite infinite cylinder of orthotropic material, Physics Letters A 372 (2008) 756– 760. [3] M. Abd-Alla, S.R. Mahmoud, Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, Meccanica 45 (4) (2010) 451–462. [4] M. Abd-Alla, S.R. Mahmoud, Effect of the rotation on propagation of thermoelastic waves in a non-homogeneous infinite cylinder of isotropic material, International Journal of Mathematical Analysis 4 (2010). [5] A.M. Abd-Alla, H.A.H. Hammad, S.M. Abo-Dahab, Rayleigh waves in magnetoelastic half-space of orthotropic material under influence of initial stress and gravity field, Applied Mathematics and Computation 154 (2, 5) (2004) 583–597. [6] V. Grechka, L. Zhang, W.J. Rector, Shear waves in acoustic anisotropic media, Geophysics 69 (2004) 576–582. [7] A.M. Abd-Alla, S.M. Ahmed, Stoneley and Rayleigh waves in a non-homogeneous orthotropic elastic medium under influence of gravity, Applied Mathematics and Computation 135 (1) (2003) 187–200. [8] S.M. Ahmed, Rayleigh waves in a thermoelastic granular medium under initial stress, International Journal of Mathematics and Mathematical Sciences 23 (2000) 627–637. [9] S.M. Ahmed, Influence of gravity on the propagation of waves in granular medium, Applied Mathematics and Computation 101 (1999) 269–280. [10] A.M. Abd-Alla, S.M. Ahmed, Rayleigh waves in an orthotropic thermoelastic medium under gravity and initial stress, Earth, Moon and Planets 75 (1998) 185–197. [11] A.M. El-Naggar, A.M. Abd-Alla, S.M. Ahmed, Rayleigh waves in a magneto-elastic initially stresses conducting medium with the gravity field, Bulletin of the Calcutta Mathematical Society 86 (1994) 51–56. [12] S.C. Das, D.P. Acharya, P.R. Sengupta, Surface waves in an inhomogeneous elastic medium under the influence of gravity, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 37 (5) (1992) 539–552. [13] B.K. Datta, Some observations on interactions of Rayleigh waves in an elastic solid medium with the gravity field, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 31 (4) (1986) 369–374. [14] M. Bouden, S.K. Datta, Rayleigh waves in granular medium over an initially stressed elastic half-space, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 29 (1984) 271–285. [15] P.C. Bhattacharya, P.R. Sengupta, Influence of gravity on propagation of waves in a composite layer, Ranchi University Mathematical Journal 15 (1984) 53–67. [16] P.R. Sengupta, D. Acharya, The influence of gravity on the propagation of waves in a thermoelastic layer, Revue Roumaine des Sciences Techniques – Série Mécanique Apliuquée 24 (1979) 395–406. [17] S.K. De, P.R. Sengupta, Effect of anisotropy on surface wave under the influence of gravity, Acta Geophysica Polonica XXIV (1978) 4–16. [18] S.N. De, Surface waves in the influence of gravity, Gerlands Beitr, Geophysics 85 (1976) 4–15. [19] S.K. De, P.R. Sengupta, Influence of gravity on wave propagation in an elastic layer, Journal of the Acoustical Society of America 55 (1974) 1–5. [20] S.R. Mahmoud, Effect of the non-homogeneity on wave propagation on orthotropic elastic media, International Journal of Contemporary Mathematical Sciences 5 (45) (2010) 2211–2224. [21] A.M. Abd-Alla, S.M. Abo-Dahab, Time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation, Applied Mathematical Modelling 33 (5) (2009) 2388–2402. [22] M. Abd-Alla, S.R. Mahmoud, M.I.R. Helmi, Effect of initial stress and magnetic field on Propagation of shear wave in non homogeneous Anisotropic medium under gravity field, The Open Applied Mathematics Journal 3 (2009) 49–56.
