Analysis of dispersion and absorption characteristics of shear waves in sinusoidally corrugated elastic medium with void pores

This theoretical work reports the dispersion and absorption characteristics of horizontally polarized shear wave (SH-wave) in a corrugated medium with void pores sandwiched between two dissimilar half-spaces. The dispersion and absorption equations have been derived in a closed form using the method of separation of variables. It has been established that there are two different kinds of wavefronts propagating in the proposed media. One of the wavefronts depends on the modulus of rigidity of elastic matrix of the medium and satisfies the dispersion equation of SH-waves. The second wavefront depends on the changes in volume fraction of the pores. Numerical computation of the obtained relations has been performed and the results are depicted graphically. The influence of corrugation, sandiness on the phase velocity and the damped velocity of SH-wave has been studied extensively.


Introduction
In the last few decades, the studies on propagation of shear wave (SH-wave) through isotropic media have been of outstanding attention, as it served to analyse the Earth's interior structure. The seismological research has focused on specific areas of the Earth, such as the crust, mantle, outer/inner core and the interfaces between different strata. It is our great interest to examine the propagation of SH-waves in an elastic isotropic medium with void pores as it plays an indispensable function in material fracture and failure. Originally, Cowin & Nunziato [1] developed a nonlinear theory of elastic solids with voids. Later, they [2] introduced the linear theory of elastic materials as a specialization of the article [1] 2017 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. which is a valuable extension to the classical theory of elasticity. The study interpreted that the linear theory differs quite considerably from the classical linear elasticity as the volume fraction was considered as an independent kinematical variable.
Some remarkable works on the propagation of SH-waves in various media with different geometries have been studied by Dey et al. [3][4][5]. Singh [6] studied the reflection and transmission of couple longitudinal waves at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids. Khan et al. [7] observed the gravitational effect on surface waves in homogeneous fibrereinforced anisotropic general viscoelastic media of higher and fractional order with voids. Rani et al. [8] studied the torsional vibrations of initially stressed composite poroelastic cylinders. Golamhossen [9] studied the propagation of waves in an elastic cylinder with voids. Vishwakarma et al. [10] discussed the influence of the rigid boundary on the Love wave propagation in an elastic layer with void pores.
It has been observed by many articles that the materials of various layers under the surface of the Earth are not always elastic or isotropic. It may be a sandy medium consisting of sandy particles having no moisture or water vapour. The resistance to shear is much less in dry sandy medium because of individual granules' slippage over each other; for that reason the granules of sandy medium are very much responsible for greater shear deflection [11]. According to the classical theory of elasticity, Young's modulus, modulus of rigidity and Poisson's ratio are related by the expression E/μ 2 = 2(1 + ν), where E is Young's modulus, μ 2 is the rigidity of elastic medium and ν is Poisson's ratio. The relation may be rewritten for dry sandy medium as where η is the sandiness parameter. The sandiness parameter η > 1 corresponds to the sandy medium and η = 1 for an elastic medium. The value of Young's modulus E when compared with modulus of rigidity μ 2 is much greater in sandy medium than a solid material. Recently, some research articles on the propagation of surface wave in dry sandy medium have been studied by Kundu et al. [12], Pal & Ghorai [13] and Dey et al. [14]. Generally, the interfaces between two layers of the Earth's interior structure can neither be perfectly flat nor in a regular shape to a certain extent. However, some boundaries/interfaces may be approximated as regular ones, while others may be considered as irregular. The effects of irregular boundary surfaces on the wave propagation have been studied in many previous articles, such as Zhang & Shinozuka [15], Singh [16], Chattopadhyay & Singh [17] and many more. The shape of irregular interfaces may be in different forms, viz. rectangular, triangular, parabolic, corrugated, etc. The propagation and attenuation of waves through such type of corrugated boundaries get affected by these undulations or roughness. The reflection and transmission of SH-waves at a corrugated interface between two laterally and vertically heterogeneous anisotropic elastic solid half-spaces has been investigated by Tomar & Kaur [18]. Later, they [19] have shown the reflection and refraction of SH-waves at a corrugated interface between two laterally and vertically heterogeneous viscoelastic solid half-spaces. Recently, Kundu et al. [20] have shown the effects of periodic corrugation, reinforcement, heterogeneity and initial stress on the propagation of Love waves. Singh et al. [21] studied the influence of corrugated boundary surfaces, reinforcement, hydrostatic stress, heterogeneity and anisotropy on Lovetype wave propagation. Chattaraj & Samal [22] obtained the dispersion of Love-type surface wave in anisotropic porous layer with periodic non-uniform boundary surface.
To the best of the authors' knowledge, there are no analytical results available in the literature for the SH-wave propagation in a sinusoidally corrugated elastic medium with void pores lying between two dissimilar half-spaces. Consequently, this motivates us to examine the influence of corrugated boundary surfaces and sandiness parameter on the propagation of SH-waves in the proposed media.
The objective of this issue is to characterize the influence of corrugated boundary surfaces and sandiness on the propagation of SH-waves in the corrugated elastic medium with void pores sandwiched between two half-spaces. It has been considered that elastic medium with void pores is bounded by two sinusoidally corrugated surfaces z = −H + f 1 (x) and z = f 2 (x). f 1 (x) and f 2 (x) are the continuous periodic functions of x, independent of y (see §2). The dispersion and absorption relations are introduced in closed simple form. It is shown that there is a possibility of propagation of SH-waves in two different wavefronts, one of the wavefronts is associated with the elasticity of the medium and the other wavefront is associated with the changes in void's volume fraction.

Formulation of physical model and its solution
We consider a corrugated elastic medium with void pores sandwiched between an isotropic half-space and a sandy half-space as shown in figure 1. A rectangular Cartesian coordinate system (x, y, z) is taken in such a way that x-axis is taken along the direction of wave propagation and z-axis is pointing vertically downward into the lower half-space. Since the constituent particles of the media are aligned on a line parallel to y-axis and are displacing equally, the field quantities are independent of y-direction, i.e. ∂/∂y ≡ 0. The average thickness of the corrugated layer is assumed to be H. The average thickness of the corrugated layer may be defined as f 1 (x) − H ≤ z ≤ f 2 (x); and the upper and lower half-spaces as −∞ ≤ z ≤ f 1 (x) − H and f 2 (x) ≤ z ≤ ∞, respectively, where the Fourier series expansion of the functions f 1 (x) and f 2 (x) may be given as [16]  In this article, we shall consider the above series expansion is up to first order, hence the boundary surfaces may be expressed as only one cosine term, i.e. f i (x) = a i cos(bx), i = 1, 2, where a i are the corresponding amplitudes of the undulated surface. It will be convenient to describe the undulated nature on the propagation and attenuation of shear waves. Shear-wave condition. Let the displacement components of upper isotropic half-space, elastic medium with void pores and lower half-space be (u 0 , v 0 , w 0 ), (u 1 , v 1 , w 1 ) and (u 2 , v 2 , w 2 ), respectively, so that the condition for the propagation of SH-waves may be given by

Wave propagation in the isotropic half-space
The equations of motion for the propagation of SH-wave in an isotropic elastic solid in the absence of body forces may be written in compound form as Biot [23]  where σ ij are the stress components and ρ 0 is the density of the medium. Equation (2.4) may be written as Applying Hooke's Law for an isotropic medium gives where λ, , δ ij , μ 0 are Lame's constant, cubical dilation, Kronecker delta function and rigidity of the medium, respectively, and e ij = 1 2 (∂u i /∂x j + ∂u j /∂x i ) are components of strain tensor. Now, using the SH-wave condition, the equation of motion get reduced to where β 0 = √ μ 0 /ρ 0 is the SH-wave velocity in the isotropic medium.
Considering the solution of (2.7) as The appropriate expression for the displacement component in the upper isotropic half-space may be obtained as where k is wavenumber, c is wave velocity and t is time.
The solution of upper half-space vanishes as z → −∞. Hence, the following equation: is the displacement component of the isotropic upper half-space.

Wave propagation in elastic medium with void pores
The field equations governing the displacement u(x, t) and the volume fraction φ(x, t) for a homogeneous and isotropic elastic medium in the absence of body forces as given by Cowin & Nuziato [1] are and α, γ , ξ , ω are the parameters of material due to void field;k is equilibrated inertia, ρ is the density of the medium. The above equations of motion are welded by the following constitutive equations: where τ ij are components of stress tensor. Moreover, Again using the SH-wave condition, the non-vanishing equations of motion (2.13) and (2.14) in elastic medium with void pores in the absence of body forces may be written as Let us consider the solution of equations (2.17) and (2.18) as wave velocity in elastic void pores. Therefore, the solution of equations (2.17) and (2.18) may be expressed as

Wave propagation in dry sandy half-space
With the help of equation (2.3) and using the stress-strain relations [23], the components of stress for the concerned medium may be procured as follows: Therefore, the non-vanishing equation of motion for the propagation of SH-waves in the dry sandy medium with rigidity μ 2 /η and density ρ 2 is given by considering the solution of equation (2.24) as where s 2 = k 1 − c 2 /β 2 2 and β 2 (= √ μ 2 /ηρ 2 ) is SH-wave velocity of the sandy medium. The solution of equation (2.26) may be obtained as

Boundary conditions and frequency equations
The required boundary conditions for the concerned problem include the continuity of the stress tensors, displacements and volume fraction at the solid interfaces. Therefore, at the corrugated interfaces some suitable boundary conditions may be mathematically given as follows: Mathematically, wheren is the unit vector normal to the external boundary and Substituting equations (2.12), (2.22a,b) and (2.29) on the above boundary conditions (3.1)-(3.6), following equations may be obtained: where Γ 1 , Γ 2 , T 1 , T 2 , T 3 and T 4 are provided in appendix A. Equation (3.13) implies that (3.14) or The above equations elucidate that the SH-wave propagates through elastic medium with void pores in two wavefronts. Equations (3.14) and (3.15) are the required frequency equations of SH-wave of the first and second kind, respectively, in elastic medium with void pores between two dissimilar half-spaces. It is also noticed that these frequency equations are in complex-valued form and are implicit equations.
Owing to dissipation of the elastic solids, waves are attenuated. If the dissipation coefficient is non-zero, the wavenumber k should be complex, i.e. k = k 1 + ik 2 . Therefore, separating the real and imaginary parts of equation (3.14) yields sin(2Γ 1 ) cos(2Γ 1 ) + cosh(2Γ 2 ) and sinh(2Γ 2 ) cos(2Γ 1 ) + cosh( respectively. With the similar approach, the separation of real and imaginary part of equation (3.15) gives

Case IV
When the amplitude of the corrugated boundary surfaces are ignored (a 1 = a 2 = 0), and sandiness of lower half-space is neglected (η = 1) and attenuation is neglected (δ = 0), (T 2 = 0, and the absorption relation (3.17) vanishes identically. Equation (4.7) coincides with the classical equation [24] of SH-waves sandwiched between two dissimilar isotropic half-spaces. The range of existence of the real root of equation (4.7) is β < c < β 0 or β 2 .

Numerical discussion
To illustrate the results of the dispersion and absorption characteristics, numerical calculations are presented using above-mentioned formulation with the help of Mathematica software. The set of parameter values has been considered from the references [25,26] (table 1). The illustrated results are presented graphically with three cases I, II and III.  Figure 10 represents the dimensionless phase and damped velocity c 2 /c v against dimensionless displacement parameter k 1 R for the second kind of SH-waves. In addition, a comparative study has been made to compare the velocity of SH-wave in classical case with respect to corrugated case in figure 11. In dispersion curves, we note   that the phase velocity reduces with increase of wavenumber. In absorption curves, we observe that a maximum attenuation corresponding to the damped velocity increases initially and it becomes stable in higher range of wavenumber.  In figure 3a, the curves are very much influenced by the parameters x/H and bH over the considered frequency range. Figure 3b illustrates the effect of x/H and bH on damped velocity is much larger for high frequency in comparison with the low-frequency range. It is noted that the phase and damped velocity of SH-waves increase with increasing value of x/H and bH in the absence of lower corrugated boundary. Figure 4a depicts the effect of parameters x/H and bH on dispersion characteristics of SH-wave, which establishes the phase velocity increases with the increase of parameters x/H and bH. Figure 4b shows the parameters x/H and bH affect the absorption characteristics significantly. It is observed that x/H and bH have positive effect on the phase and damped velocity of SH-waves in presence of corrugated boundary interfaces.

Effect of sandiness parameter
The figures 5-7 are associated with the variation of sandiness parameter η on the phase and damped velocity with respect to real wavenumber. To study the effect, the numerical values of sandiness parameter η have been considered as 1.10, 1.11 and 1.12 for cases I, II and III, respectively. We observe that η has a very significant effect on the dispersion and absorption characteristics of SH-waves throughout the considered frequency range in all cases. Figures 5a-7a illustrate, as the value of η increases the dispersion curve shifts downwards; this trend irradiates that the phase velocity decreases with the increase of sandiness parameter η. Figures 5b-7b show the damping nature of SH-waves for different values of η. As the value of η increases, the damped velocity decreases. Hence it can be concluded that the increasing value of η disfavours the dispersion and absorption characteristics of SH-waves in all cases.

Effect of corrugation parameters
The effect of corrugation parameters a 2 b and a 1 b on the SH-waves are exhibited in figures 8 and 9, respectively. Figure 8a  3 have been considered as 0.01, 0.02 and 0.03, respectively. It is noted that the corrugation parameter a 1 b has a cogent effect on the phase velocity of SH-waves over the considered frequency range. Moreover, there is a positive relationship between a 1 b and phase velocity, i.e. phase velocity increases with increase of a 1 b, whereas the damped velocity has an inverse relationship with a 1 b.
The second kind of SH-wave velocities are exhibited in figure 10. The phase and damped velocity c 2 /c v have been plotted from equations (3.18) and (3.19) with respect to displacement parameter k 1 R by taking a specified value of t and dimensionless parameter κ. Figure 10a,b illustrates that the phase velocity decreases with the increase of k 1 R, whereas damped velocity increases. This is evident from the comparative study as shown in figure 11, the dispersion curve associated with the isotropic medium is lying above with respect to corrugated medium. Hence this fact suggests that the SH-waves propagate even faster in isotropic medium than a layered corrugated structure.

Conclusion
The propagation of SH-waves in elastic medium with void pores between two half-spaces is studied in detail. The dispersion and absorption equations for SH-waves are derived separately in closed forms. The sandiness parameter, undulatory parameter, position parameter and corrugation have significant effects on dispersion and absorption characteristics (see the figures). Moreover, the following outcomes of the study can be highlighted: -In elastic solids with void pores, more than one wavefront of SH-waves may exist, which satisfy the requirement for surface waves, i.e. to decay with the depth, whereas in the elastic case there is only ever one wavefront. Both wavefronts of SH-wave are dispersive in nature. -The phase velocity of SH-waves decreases with increase in wavenumber.
-The damped velocity of SH-waves increases gradually with wavenumber and thereafter attenuates in high range of wavenumber. -With the increase of sandiness parameter, the dispersion and absorption curves shift downwards. This trend shows that the phase and damped velocities of SH-waves decrease with the increase of sandiness parameter. It is observed that the sandiness parameter has a cogent effect on dispersion and absorption curves. -The undulatory parameter and position parameter have harmonious effect on the phase velocity while it has inharmonious effect on the damped velocity. The undulatory parameter, position parameter and corrugation have a slight effect on damped velocity in the low range of wavenumber and significant effects in the higher range of wavenumber. -As a particular case, the dispersion relation for the propagation of SH-waves at the common elastic medium between two different homogeneous isotropic half-spaces is validated with the results [24]. It is also noted that the damping part associated with dispersion relation in propagating the waves in an elastic medium sandwiched between two dissimilar isotropic half-spaces vanishes.