# Shear Wave Propagation in Multilayered Medium including an Irregular Fluid Saturated Porous Stratum with Rigid Boundary

Authors: Ravinder Kumar; Dinesh Kumar Madan; Jitander Singh Sikka
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

The present investigation is concerned with the study of propagation of shear waves in an anisotropic fluid saturated porous layer over a semi-infinite homogeneous elastic half-space lying under an elastic homogeneous layer with irregularity present at the interface with rigid boundary. The rectangular irregularity has been taken in the half-space. The dispersion equation for shear waves is derived by using the perturbation technique followed by Fourier transformation. Numerically, the effect of irregularity present is analysed. It is seen that the phase velocity is significantly influenced by the wave number and the depth of the irregularity. The variations of dimensionless phase velocity against dimensionless wave number are shown graphically for the different size of rectangular irregularities with the help of MATLAB.


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## Body

## 1. Introduction

The earth has a layered structure, and this exerts a significant influence on the propagation of elastic waves. Inside the Earth, a very hard layer (also known as “rigid”) is present. Since the composition of the Earth is heterogeneous including a very hard layer, the medium porosity and the rigid interface play significant roles in the propagation of the seismic waves. The propagation of elastic waves in homogeneous layer is of considerable importance in earthquake engineering and seismology. The study of wave propagation in elastic medium with different boundaries is of great importance to seismologists as well as to geophysicists to understand and predict the seismic behavior at different margins of earth. The propagation of shear waves has been studied by many authors assuming different forms of irregularities at the interface. Bhattacharya [1] discussed the dispersion curves for Love wave propagation in a transversely isotropic crustal layer with an irregularity in the interface. Jones [2] discussed wave propagation in a two-layered medium. Chattopadhyay [3] studied the effect of irregularities and nonhomogeneities in the crustal layer on the propagation of Love waves. Biot [4] and Deresiewcz [5] investigated the propagation of surface waves in liquid filled porous solids. Chattopadhyay et al. [6] studied the propagation of SH guided wave in an internal stratum with parabolic irregularity in the lower interface. Chattopadhyay and De [7] studied the dispersion equation for Love waves in a nondissipative liquid filled with porous solid underlain by an isotropic and homogeneous half-space. They derived the dispersion equation by applying the perturbation method, and the phase velocity curve was obtained for different irregularities by using the parameters of the porous medium which were suggested by Biot [8]. Kończak [9] derived dispersion equation for shear waves in a multilayered medium including a fluid saturated porous stratum. The influence of irregularity and rigidity on the propagation of torsional waves has been discussed by Gupta et al. [10]. Love wave propagation in a porous rigid layer lying over an initially stressed half-space is discussed by Kundu et al. [11]. Madan et al. [12] also discussed the propagation of Love waves in an irregular fluid saturated porous anisotropic layer with rigid boundary. For the elastic and viscoelastic waves, a long list of references is available in the monographs of Biot [13], Lamb [14], Miklowitz [15], and Keolsky [16].In this paper we have discussed the propagation of shear waves in a transversely isotropic fluid saturated porous layer resting on a homogeneous elastic half-space, lying under an elastic isotropic and homogeneous layer with rigid boundary with irregularity at the interface. The irregularity is in the form of rectangle. The dispersion curves are depicted by means of graphs for different size of irregularity and different values of common wave velocity. The influence of depth of irregularity on phase velocity and some special cases have been studied.


## 2. Formulation of the Problem

A transversely isotropic fluid saturated porous layer of thicknessH2 resting on a homogeneous elastic half-space and lying under an elastic isotropic and homogeneous layer of thickness H1 has been considered. The Cartesian coordinate system (x, y, z) is chosen with z-axes taken vertically downward in the half-space and x-axes is chosen parallel to the layer in the direction of propagation of the disturbance. We assume the irregularity in the form of a rectangle with length s and depth H′. The origin is placed at the middle point of the interface irregularity. The source of the disturbance is placed on positive z-axes at a distance d(d>H′) from the origin. Therefore, the upper layer describes the medium M1: -(H1+H2)≤z≤H2, the intermediate layer describes the medium M2: -H2≤z≤0,  and the homogeneous elastic half-space describes the medium M3: 0≤z<∞. The geometry of the problem is shown in Figure 1.Figure 1
Geometry of the problem.The interface between the layer and half-space is defined as(1)z=εhx,
where
(2)h(x)=0;x≤-s2,x≥s2f(x);-s2≤x≤s2,
where ε=H′/s and ε≪1.


## 3. Basic Equations

The basic equations for the medium considered are as follows.


### 3.1. MediumM1

The equations of motion, without body force [17], are given by
(3)σij.j(1)=ρ(1)u¨i(1),
where σij(1) are the components of stress tensor, ui(1)  are the components of displacement vector, and ρ(1)  is the density. The comma denotes differentiation with respect to position and dot denotes differentiation with respect to time.The constitutive relations are given by(4)σij(1)=λ(1)ekk(1)δij+2μ(1)eij(1),
where λ(1)  and μ(1) are Lame’s elastic coefficients and δij is the Kronecker delta and
(5)2eij(1)=(ui,j(1)+uj,i(1)),ekk(1)=uk,k(1)=e(1).


### 3.2. MediumM2

The equations of motion for the intermediate fluid saturated porous layer in the absence of body forces are [18]
(6)σij,j(2)=ρ11u¨i(2)+ρ12U¨i(2)-bijU˙j2-u˙j2,σ,i(2)=ρ12u¨i(2)+ρ22U¨i(2)+bijU˙j2-u˙j2,
where σij(2) are the components of stress tensor in the solid skeleton, σ(2)=-fp is the reduced pressure of the fluid (p is the pressure in the fluid, and f is the porosity of the medium), and ui(2)  are the components of the displacement vector of the solid skeleton and Ui(2)  are those of fluid.The stress-strain relations for the transverse-isotropic fluid saturated porous layer are(7)σ11(2)σ22(2)σ33(2)σ23(2)σ31(2)σ12(2)σ(2)=2C1+C2C2C3000C6C22C1+C2C3000C6C3C32C4000C70002C500000002C500000002C10C6C6C7000C8e112e222e332e232e312e122e2,
where
(8)2eij(2)=ui,j2+uj,i2,e(2)=div⁡U(2)≡Uj,j(2),ekk=div⁡u(2)≡uk,k2,
and C1, C2, C3, C4, C5, C6, C7, and C8 are the material constants.


### 3.3. MediumM3

For the lower homogeneous half-space the basic equations of motion, without body force, are [17]
(9)σij,j(3)=ρ(3)u¨i(3),
where σij,j(3)  are the components of stress tensor, ui(3)  are the components of displacement vector, and ρ(3)  is the density.The constitutive relations are given by(10)σij(3)=λ(3)ekk(3)δij+2μ(3)eij(3),2eij(3)=(ui,j(3)+uj,i(3)),ekk(3)=uk,k(3)=e(3),
where λ(3)  and μ(3)  are Lame’s elastic coefficients and are functions of x, y, and z.In this paper, attention is confined to shear waves propagating in thexy-plane. The displacements are parallel to y direction and are independent of the y coordinate. Thus,
(11)u(1)≡w(1)≡0,v(1)≡v(1)(x,z,t),u(2)≡w(2)≡0,v(2)≡v(2)(x,z,t),U(2)≡W(2)≡0,V(2)≡V(2)(x,z,t),u(3)≡w(3)≡0,v(3)≡v(3)(x,z,t),
and the equations of motion (3), (6), and (9) with the help of (4), (5), (7), (8), and (10), respectively, reduce to the form
(12)∂2∂x2+∂2∂z2v(1)=1β12∂2v(1)∂t2,(13)-ρ11∂t2+b11∂t-ρ12∂t2-b11∂t2ρ22∂t2+b11∂tC1∂2∂x2+C5∂2∂z2-ρ11∂t2+b11∂t-ρ12∂t2-b11∂t2ρ22∂t2+b11∂t×(v2,V2)=0,(14)∂2∂x2+∂2∂z2v(3)=1β32∂2v(3)∂t2.
The appropriate boundary conditions for the considered problem are as follows.(i)
At the rigid surfacez=-(H1+H2), the displacement component vanishes; that is,
(15)v(1)(x,z=-(H1+H2),t)=0.(ii)
At the interfacez=-H2, the displacements are continuous; that is,
(16)v(1)(x,z=-H2,t)=v(2)(x,z=-H2,t).(iii)
At the interfacez=-H2, the shear stress components are continuous; that is,
(17)σ321x,z=-H2,t=σ322x,z=-H2,t.(iv)
At the interfacez=εh(x), the displacements are continuous; that is,
(18)v2x,z=εhx,t=v3x,z=εhx,t.(v)
The stresses are continuous at the interfacez=εh(x); that is,
(19)C5∂v(2)∂z-C1εh′x∂v2∂x=μ∂v3∂z-εh′x∂v3∂x,h′x=dhxdx.Thus (12)–(14) with the previous boundary conditions are the governing equations of the problem considered.


## 3.1. MediumM1

The equations of motion, without body force [17], are given by
(3)σij.j(1)=ρ(1)u¨i(1),
where σij(1) are the components of stress tensor, ui(1)  are the components of displacement vector, and ρ(1)  is the density. The comma denotes differentiation with respect to position and dot denotes differentiation with respect to time.The constitutive relations are given by(4)σij(1)=λ(1)ekk(1)δij+2μ(1)eij(1),
where λ(1)  and μ(1) are Lame’s elastic coefficients and δij is the Kronecker delta and
(5)2eij(1)=(ui,j(1)+uj,i(1)),ekk(1)=uk,k(1)=e(1).


## 3.2. MediumM2

The equations of motion for the intermediate fluid saturated porous layer in the absence of body forces are [18]
(6)σij,j(2)=ρ11u¨i(2)+ρ12U¨i(2)-bijU˙j2-u˙j2,σ,i(2)=ρ12u¨i(2)+ρ22U¨i(2)+bijU˙j2-u˙j2,
where σij(2) are the components of stress tensor in the solid skeleton, σ(2)=-fp is the reduced pressure of the fluid (p is the pressure in the fluid, and f is the porosity of the medium), and ui(2)  are the components of the displacement vector of the solid skeleton and Ui(2)  are those of fluid.The stress-strain relations for the transverse-isotropic fluid saturated porous layer are(7)σ11(2)σ22(2)σ33(2)σ23(2)σ31(2)σ12(2)σ(2)=2C1+C2C2C3000C6C22C1+C2C3000C6C3C32C4000C70002C500000002C500000002C10C6C6C7000C8e112e222e332e232e312e122e2,
where
(8)2eij(2)=ui,j2+uj,i2,e(2)=div⁡U(2)≡Uj,j(2),ekk=div⁡u(2)≡uk,k2,
and C1, C2, C3, C4, C5, C6, C7, and C8 are the material constants.


## 3.3. MediumM3

For the lower homogeneous half-space the basic equations of motion, without body force, are [17]
(9)σij,j(3)=ρ(3)u¨i(3),
where σij,j(3)  are the components of stress tensor, ui(3)  are the components of displacement vector, and ρ(3)  is the density.The constitutive relations are given by(10)σij(3)=λ(3)ekk(3)δij+2μ(3)eij(3),2eij(3)=(ui,j(3)+uj,i(3)),ekk(3)=uk,k(3)=e(3),
where λ(3)  and μ(3)  are Lame’s elastic coefficients and are functions of x, y, and z.In this paper, attention is confined to shear waves propagating in thexy-plane. The displacements are parallel to y direction and are independent of the y coordinate. Thus,
(11)u(1)≡w(1)≡0,v(1)≡v(1)(x,z,t),u(2)≡w(2)≡0,v(2)≡v(2)(x,z,t),U(2)≡W(2)≡0,V(2)≡V(2)(x,z,t),u(3)≡w(3)≡0,v(3)≡v(3)(x,z,t),
and the equations of motion (3), (6), and (9) with the help of (4), (5), (7), (8), and (10), respectively, reduce to the form
(12)∂2∂x2+∂2∂z2v(1)=1β12∂2v(1)∂t2,(13)-ρ11∂t2+b11∂t-ρ12∂t2-b11∂t2ρ22∂t2+b11∂tC1∂2∂x2+C5∂2∂z2-ρ11∂t2+b11∂t-ρ12∂t2-b11∂t2ρ22∂t2+b11∂t×(v2,V2)=0,(14)∂2∂x2+∂2∂z2v(3)=1β32∂2v(3)∂t2.
The appropriate boundary conditions for the considered problem are as follows.(i)
At the rigid surfacez=-(H1+H2), the displacement component vanishes; that is,
(15)v(1)(x,z=-(H1+H2),t)=0.(ii)
At the interfacez=-H2, the displacements are continuous; that is,
(16)v(1)(x,z=-H2,t)=v(2)(x,z=-H2,t).(iii)
At the interfacez=-H2, the shear stress components are continuous; that is,
(17)σ321x,z=-H2,t=σ322x,z=-H2,t.(iv)
At the interfacez=εh(x), the displacements are continuous; that is,
(18)v2x,z=εhx,t=v3x,z=εhx,t.(v)
The stresses are continuous at the interfacez=εh(x); that is,
(19)C5∂v(2)∂z-C1εh′x∂v2∂x=μ∂v3∂z-εh′x∂v3∂x,h′x=dhxdx.Thus (12)–(14) with the previous boundary conditions are the governing equations of the problem considered.


## 4. Solution of the Problem

For waves changing harmonically with timet and propagating in x-direction, we obtain
(20)v(1)(z,x,t)=v0(1)(z,x)exp⁡(iωt),v(2)(z,x,t)=v0(2)(z,x)exp⁡(iωt),V(2)(z,x,t)=V0(2)(z,x)exp⁡(iωt),v(3)(z,x,t)=v0(3)(z,x)exp⁡(iωt),
where ω is the angular frequency.Thus equations of motion (12)–(14) take the form of
(21)∂2v0(1)∂x2+∂2v0(1)∂z2+ω2β12v0(1)=0,C1∂2∂x2+C5∂2∂z2+ξ12(v0(2),V0(2))=0,∂2v0(3)∂x2+∂2v0(3)∂z2+ω2β32v0(3)=0,
where ξ12=α1+iα2, α1 and α2 are given in Appendix section.Define Fourier transformationv-0(1)(z,η) of v0(1)(z,η) as
(22)v-0(1)(z,η)=∫-∞∞v01z,xeiηxdx.
And inverse Fourier transformation is given by
(23)v01(z,x)=12π∫-∞∞v-0(1)(z,η)e-iηxdη,
and so forth.The Fourier transformation of (21) then is
(24)∂2v-0(1)∂z2+χ12v-0(1)=0,∂2v-0(2)∂z2+χ22v-0(2)=0,∂2V-0(2)∂z2+χ22V-0(2)=0,∂2v-0(3)∂z2-χ32v-03=0,
where
(25)χ12=ω2β12-η2,χ22=C1C5ξ12C1-η2,χ32=η2-ω2β32.
The solutions of (24) are
(26)v-0(1)=Acos⁡χ1z+Bsinχ1z,v-0(2)=Ccos⁡χ2z+Dsinχ2z,V-0(2)=C-cos⁡χ2z+D-sinχ2z,v-0(3)=Eexp⁡(-χ3z),
where A, B, A-, B-, and D are functions of η.Thus, by inverse Fourier transformation, we obtain(27)v0(1)(z,x)=12π∫-∞∞(Acos⁡χ1z+Bsinχ1z)e-iηxdη,v0(2)(z,x)=12π∫-∞∞(Ccos⁡χ2z+Dsinχ2z)e-iηxdη,(28)V0(2)(z,x)=12π∫-∞∞(C-cos⁡χ2z+D-sinχ2z)e-iηxdη,(29)v0(3)(z,x)=12π∫-∞∞(Ee-χ3z+2χ3eχ3ze-χ3d)e-iηxdη,
where the second term in the integrand of v0(3)(z,x) is introduced due to the source in the lower half-space [19].The relations between the constantsC-, D- and C, D are provided by (13).We set the following approximations due to small value ofε:
(30)A≅A0+A1ε,B≅B0+B1ε,C≅C0+C1ε,D≅D0+D1ε,E≅E0+E1ε.
Since the boundary is not uniform, the terms A, B, C, D, and E in (30) are also functions ofε. Expanding these terms in ascending powers of ε and keeping in view that ε is small and so retaining the terms up to the first order of  ε, A, B, C, D, and E can be approximated as in (30). In physical situations, when the depth H′ of the irregular boundary is too small with respect to the length of the boundary s, the above assumptions are justified. Further, for small ε,
(31)e±αεh≅1±αεh,cos⁡χ1εh≅1,sinχ1εh≅χ1εh,
where α is any quantity.Define Fourier transformation ofh(x) as
(32)h-(λ)=∫-∞∞h(x)eiλxdx,
and the inverse Fourier transformation is
(33)hx=12π∫-∞∞h-λe-iλxdλ.
Therefore,
(34)h′x=-i2π∫-∞∞λh-λe-iλxdλ.
Now, by using boundary conditions (15)–(19) along with (27) and (29)-(30) we obtain a system of ten equations (after equating the absolute term (terms not containing ε) and the coefficients of ε):
(35)A0cos⁡(H1+H2)χ1-B0sin(H1+H2)χ1=0,A0cos⁡H2χ1-B0sinH2χ1-C0cos⁡H2χ2+D0sinH2χ2=0,μχ1A0sinH2χ1+B0cos⁡H2χ1-C5χ2C0sinH2χ2+D0cos⁡H2χ2=0,χ3C0-E0=2e-χ3d,μχ3E0+C5χ2D0=2μe-χ3d,A1cos⁡(H1+H2)χ1-B1sin(H1+H2)χ1=0,A1cos⁡H2χ1-B1sinH2χ1-C1cos⁡H2χ2+D1sinH2χ2=0,μχ1A1sinH2χ1+B1cos⁡H2χ1-C5χ2C1sinH2χ2+D1cos⁡H2χ2=0,E1-C1=R1k,μχ3E1+C5χ2D1=R2k,
where R1(k) and R2(k) are given in Appendix section.Solve the system of equations forA0, B0, C0, D0, E0, A1, B1, C1, D1, and E1 and the corresponding values are given in Appendix section.The displacement in the anisotropic layer is(36)v0(2)=12π∫-∞∞4μe-χ3dEk1+εR2-μχ3R1eχ3d4μ×B2+B3e-ikxdk,
where B2 and B3 are given in Appendix section.Now from (1), (2), and (32), we have
(37)h-(λ)=2sλsinλs2.
Using values of R1(k) and R2(k) as given in Appendix section, we obtain
(38)R2-μχ3R1=sπ∫-∞∞ϕk-λ+ϕk+λ1λsinλs2dλ,
where ϕ(k-λ) is given in Appendix section.Using asymptotic formula of Willis [19] and Tranter [20] and neglecting the terms containing 2/s and highest powers of 2/s for large s, we obtain
(39)∫-∞∞[ϕ(k-λ)+ϕ(k+λ)]1λsinλs2dλ≅π2·2ϕ(k)=πϕ(k).
Now using (38) and (39), we obtain
(40)R2-μχ3R1=sϕk=H′εϕk.
Therefore the displacement in the anisotropic layer is
(41)v0(2)=12π∫-∞∞4μe-χ3dE(k)1-H′ψ(k)eχ3dB2+B3e-ikxdk,
where ψk=ϕk/4μ.The value of this integral depends entirely on the contribution of the poles of the integrand. The poles are located at the roots of the equation:(42a)E(k)1-H′ψkeχ3d=0.
This equation is the dispersion equation for SH waves.Ifc is the common wave velocity of wave propagating along the surface, then we can set in (42a)  ω=ck (ω is the circular frequency and kis the wave number), χ1=P1k, χ2=P2k, and χ3=P3k where
(42b)P1=c2β12-1,P2=1C5c2cG2·F(ω)-C1+i1C5·c2cG2·R(ω),P3=1-c2β32.Solving (42a), we obtain
(43)μP1C5P2hhhhhhhh-H′P2k(C5P2tanP2kH2-μP3)1+tanP1kH2tanP1kH1+H2hhhhh×1-P3kH′C5P2+μP3tanP2kH2hhhhhhhi-H′P2k(C5P2tanP2kH2-μP3)=hhhhhhhhi+H′P2k(C5P2-μP3tanP2kH2)tanP1k(H1+H2)-tanP1kH2hhhhhh×+H′P2k(C5P2-μP3tanP2kH2)(1-P3kH′)C5P2tanP2kH2-μP3hhhhhhhhi+H′P2kC5P2-μP3tanP2kH2.
Since the quantityP22 is complex, so we have
(44)P2=k1+ik2,
where k1and k2 are the real and imaginary parts of P2 and are given in Appendix section.Equation (43) is complex and its real part gives the dispersion equation for shear waves.


## 5. Numerical Results and Discussions

In order to investigate the effect of irregularity present in the transversely isotropic fluid saturated porous layer and to compare the results numerically between the phase velocity and the wave number, we will use the values of elastic constants given by Ding et al. [21] for medium M2 and by Kon´czak [9] for media M1 andM3. And by using MATLAB, we obtain the following graph for different values of common wave velocity c for two special cases.Case I. When H2=0 which is the wave propagation in elastic homogeneous layer lying over a homogeneous half-space, the variations of the dimensionless phase velocity (c/cG) against the dimensionless wave number (kH) in an elastic isotropic homogeneous layer over a homogeneous elastic half-space for different values of H′/H(0,0.15,0.30,0.45) and c=0.25, c=0.5, and  c=0.75 are shown in Figures 2, 3, and 4.Figure 2Figure 3Figure 4Case II. When H1=0 which is the wave propagation in a transversely isotropic fluid saturated porous layer lying over a homogeneous half-space, the variations of the dimensionless phase velocity (c/cG) against the dimensionless wave number (kH) in a transversely isotropic fluid saturated porous layer over a homogeneous elastic half-space for different values of H′/H  (0, 0.15, 0.30, 0.45) and c=0.25, c=0.5, and c=0.75 are shown in Figures 5, 6, and 7.Figure 5Figure 6Figure 7The dimensionless phase velocity (c/cG) is plotted against the dimensionless wave number (kH) in Figures 2, 3, 4, 5, 6, and 7. We conclude that the multilayered medium with irregularity and rigid boundary have significant effect on the propagation of shear waves, and the phase velocity in a layer with irregularity is affected by not only the shape of irregularity but also the wave number, the ratio of the depth of the irregularity to layer width and layer structure.



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