1 Introduction

The Earth contains liquid saturated porous material in the form of sediments found in the ocean beds comprising of sandstone and limestone saturated with water. Porous solids saturated with groundwater and oil are also present inside the Earth's crust. The area concerning liquid saturated porous media has drawn the interest of many researchers working in various fields such as seismology, fluid mechanics, and earthquake engineering, due to its varying and dynamic behavior. Moreover, the study of porous media finds its application in the area of applied sciences and engineering including filtration, acoustics, soil and rock mechanics, construction engineering, hydrogeology, biophysics and material science. The theory of consolidation and settlement of fluid saturated soil and the theory of flow of compressible fluid in a porous material have already been established by Biot^[1]. The study of the problem of deformation in such an elastic porous material aids to determine the flow pattern of the fluid or the developed stresses, which finds application in the field of civil engineering and petroleum geology. Biot^[2] discussed the theories of consolidation and elastic-wave propagation in fluid-saturated porous media and found that two dilatational waves along with one shear wave are possible in such solids. Samal and Chattaraj^[3] obtained the dispersion relationship in the form of a ninth-order determinant, for the propagation of the surface wave in the fibre-reinforced anisotropic elastic layer between a liquid saturated porous half-space and a uniform liquid layer. Sharma et al.^[4] studied the surface wave propagation in a liquid saturated porous layer overlying a homogeneous transversely isotropic half-space and lying under a uniform liquid layer. Sharma et al.^[5] also discussed the dispersion of the surface wave in the transversely isotropic elastic layer overlying a liquid saturated porous solid half-space and lying under a uniform liquid layer. Son and Kang^[6] investigated the influence of anisotropy and porosity on the propagation of shear waves in a transversely isotropic poroelastic layer sandwiched between two elastic layers.

The property of surface waves to propagate through sufficiently smooth curved surfaces, almost unhindered, finds its application in detection of surface cracks, surface hardness, surface crystal structure, thickness of coatings and residual stress distribution. This fact allows surface waves to play an important role in seismology, electro-acoustics and other fields. In most of the real world situations, the surfaces of the layers as well as the boundaries distinguishing the different layers of the Earth's interior are not perfectly plane, but are of undulated nature. Irregularities, such as mountains, basins, mountain roots and salt/ore deposits do affect the energy partition between the reflected and transmitted waves. Therefore, to study the propagation of waves in elastic media, it is necessary to take into account the amplitude of corrugation at the interfaces for better understanding and predicting the seismic wave behavior at continental margins and mountain roots. Many papers have been published illustrating the dispersion of surface waves through various media with irregular boundary surfaces. In addition to this, heterogeneity is a trivial characteristic of a medium. Heterogeneity in a medium is concerned with the variation in its properties throughout the volume. A book by Meissner^[7] enlightened the curve which showed the variation of density against the depth. The figure clearly portrays that there is a large increase in the density with an increase in the depth. Therefore, the elastic constants (properties) of a layer may not be fixed throughout the medium, but may be considered as functions of the depth, i.e., linear, quadratic, and exponential functions.

Tomar and Kaur^[8] examined the propagation of surface waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space. Kumar et al.^[9] studied the reflection/refraction of shear horizontal (SH) waves at a corrugated interface between two different anisotropic and vertically heterogeneous elastic solid half-spaces. Dhua et al.^[10] discussed the propagation of the torsional wave in a composite layer lying over an anisotropic heterogeneous half-space under the initial stress. Gupta et al.^[11] studied the effect of irregularity on the propagation of torsional surface waves in an initially stressed anisotropic poroelastic layer. The propagation of the Love-type wave in a corrugated isotropic layer over a homogeneous isotropic half-space was investigated by Singh^[12]. Sharma^[13] studied the surface wave propagation in the cracked poroelastic half-space lying under a uniform layer of fluid. The frequency equation due to the surface wave propagation in an anisotropic elastic layer sandwiched between a uniform layer of liquid and heterogeneous solid elastic half-space was established by Saini and Tomar^[14]. Recently, Singh et al.^[15-16] and Chattopadhyay et al.^[17] published papers on the dispersion of the Love-type wave through different media containing irregular boundary surfaces. Although there are some papers available in the literature which have considered the propagation of the Rayleigh-type wave in a fluid layer, but till date no attempt has been made to study the effect of corrugation on the propagation of the Rayleigh-type wave in a fluid layer overlying a corrugated substrate.

The objective of the present paper is to investigate the propagation of the Rayleigh-type wave in a fluid layer lying over a corrugated substrate. Two cases are considered, i.e., the first one is of Rayleigh-type wave propagation in a fluid layer overlying a liquid saturated poroelastic corrugated substrate, which can be considered a realistic model for the ocean bottom (Case Ⅰ), and the second one deals with Rayleigh-type wave propagation in a fluid layer lying over a quadratically heterogeneous isotropic elastic corrugated substrate (Case Ⅱ). The dispersion relationship is obtained for Case Ⅰ as well as for Case Ⅱ. The effects of corrugation, porosity and heterogeneity on the dispersion of the Rayleigh-type wave are illustrated through numerical computation and graphical illustration. In the context of the problem, the effects of presence and absence of both heterogeneity and poroelasticity are studied in a comparative manner. As a special case of the problem, the expressions of the phase velocity of the Rayleigh-type wave for both cases, neglecting corrugation at the common interface, are deduced in the closed form.

2 Formulation of problem and governing equations

The propagation of the Rayleigh-type wave in a layer of the average finite thickness H, composed of uniform homogeneous fluid overlying an elastic substrate is considered. The interface of the layer and substrate is assumed to be corrugated. ρ₁ and λ₁ are the density and the Bulk modulus associated with the upper fluid layer, respectively. The elastic substrate is considered to be fluid-saturated poroelastic in Case Ⅰ and quadratically heterogeneous isotropic elastic in Case Ⅱ. The models for both cases are consolidated in Fig. 1.

Consider a two-dimensional problem in the xz-plane in such a way that the x-axis is the direction of wave propagation, the z-axis is normal to the direction of layering, and the origin O is at the interface of the layer and substrate. The corrugated interface of the layer and substrate is defined as z=f(x), where f(x) is a periodic function of x and independent of y. The trigonometric Fourier series of f(x), with the aid of a suitable origin of coordinates, may be represented as^[18]

(1)

where f_l and f_-l are the Fourier expansion coefficients, and l is a series expansion order.

Now, define the constants a, R_l, and I_l as

and

where R_l and I_l are the cosine and sine Fourier coefficients, respectively. For the purpose of numerical computation, the interface of the layer and substrate is expressed with the aid of cosine terms, i.e., f=a cos (bx), where a, b, and 2π/b are the amplitude, the wave number, and the wave length of corrugation, respectively.

Let u_i, v_i, and w_i (i=1, 2, and 3) be the displacement components along the x-, y-, and z-directions, respectively. For the propagation of the Rayleigh-type wave, it is assumed that

(2)

such that u_i and w_i are independent of y. Indices 1, 2, and 3 stand for the fluid layer, the fluid saturated poroelastic substrate (Case Ⅰ), and the quadratically heterogeneous isotropic elastic substrate (Case Ⅱ), respectively. The first and second partial derivatives with respect to time are represented as ∂_t and ∂_tt, respectively. Moreover, d_x and d_xx stand for and , respectively.

3 Dynamics of upper fluid layer

Let φ be the displacement potential of the fluid layer and p⁽¹⁾ be the fluid pressure. The displacement components u₁, w₁, and the fluid pressure, in terms of the displacement potential, may be expressed as

(3)

where τ _zz⁽¹⁾ is the normal stress component in the fluid.

Therefore, with the aid of Eqs. (2) and (3), the equation of motion for the propagation of the Rayleigh-type wave in the fluid layer is obtained as

(4)

where c₁=

Assume the solution to Eq. (4) in the form as follows:

(5)

where c is the phase velocity, and k is the wave number.

Substitution of Eq. (5) into Eq. (4) yields

(6)

which gives

(7)

where and C₁ and C₂ are arbitrary constants.

Therefore, using Eqs. (3) and (7), the expressions of displacement components and liquid pressure associated with the upper fluid layer are given by

(8)

(9)

4 Dynamics of fluid-saturated poroelastic substrate

The components of stresses for the fluid-saturated poroelastic substrate are given by

(10)

(11)

where τ _ij⁽²⁾ (i, j=x, y, and z) and σ are components of stresses corresponding to the solid and fluid parts of the porous aggregate, respectively. B₁, B₂, B₃, and B₄ are the elastic constants of the porous aggregate, and δ_ij is the Kronecker delta. u_S and u_F are displacements in the solid and fluid parts of the porous aggregate, respectively. The strain components in Eq. (10) may be written as

Let ρ₁₁, ρ₁₂, and ρ₂₂ be the mass coefficients such that the inertia of the solid and fluid phases is related to the mass density of solid (ρ_S) and fluid (ρ_F) as

where ρ_S and ρ_F are the masses of solid and fluid per unit volume of aggregate, respectively, and d is the porosity of aggregate. Also, the following relationships must be satisfied by the mass coefficients^[1]:

where B₅ =B₁ +2B₃.

Specifically, B₂ is a measure of coupling between the volume change of solid and fluid, and B₄ is the pressure that must be exerted on fluid to force a given volume of it into the aggregate while the total volume remains to be constant.

Now, the equations of motion for the lower fluid-saturated porous substrate may be written as

(12)

(13)

Let the Helmholtz resolutions of the two displacement vectors u_S and u_F, with φ and ψ as the scalar potentials and M and G as the vector potentials corresponding to solid and fluid parts, respectively, be the following form:

(14)

(15)

With Eqs. (14) and (15), Eqs. (12) and (13) lead to the following set of four equations:

(16)

(17)

(18)

(19)

Assume harmonic wave solutions to Eqs. (16) and (18) as

(20)

Thus, with the aid of Eq. (20), Eqs. (16) and (18) give, respectively,

(E21)

Elimination of d_zzψ from Eq. (21) yields

(22)

where B₆ =B₅ B₄ -B₂².

Equation (22) further leads to

(23)

In view of Eqs. (22) and (23), Eq. (21) reduces to

(24)

where

Since B₆, B₇, and B₈ are non-negative, the solution to Eq. (24) may be written as

(25)

where

(26)

and C₃, C₄, C₅, and C₆ are arbitrary constants.

Thus, the following equation can be obtained:

(27)

where

which gives

(28)

The solutions to Eq. (28) correspond to two dilatational waves, i.e., a fast primary wave, corresponding to φ₁ and propagating with the phase velocity α₁, and a slow primary wave, corresponding to φ₂ and propagating with the phase velocity α₂.

Substituting Eq. (27) into Eq. (22) yields

(29)

where

Moreover, Eqs. (17) and (19) together give

(30)

where α₃^-2 =B₈ B₃^-1ρ₂₂^-1.

For the propagation of the Rayleigh-type wave in the xz-plane, the displacement components u=(u₂, 0, w₂) and U=(U₂, 0, W₂) in the solid and fluid, respectively, may be written as

(31)

where ψ₁ =(-M)_y, which represents the component of M along the y-axis. Further,

(32)

where C₇ and C₈ are arbitrary constants.

The displacement components vanish as z tends to infinity. Thus, the potentials φ₁, φ₂, and ψ₁ are given by

(33)

(34)

(35)

The expressions of the displacement components for the fluid-saturated poroelastic substrate may be obtained from Eq. (31) with the aid of Eqs. (33) -(35).

5 Dynamics of quadratically heterogeneous isotropic elastic substrate

It is assumed that a medium with heterogeneity varying as a quadratic function of the depth constitutes the substrate. If λ₃ and μ₃ are the elastic parameters of the substrate, then they may be written as

where λ₃ and μ₃ are values of λ₃ and μ₃ for the case that the substrate is homogeneous, i.e., the heterogeneity parameter γ =0. The longitudinal and shear wave velocities are given by

respectively, where

and ρ₃ is the density of the medium.

The equation of motion in the xz-plane is given by

(36)

where i_z represents the unit vector along the z-axis, and

φ₃ and ψ₃ are scalar and vector potentials, respectively. Moreover, the displacement vector S is defined as

Now, the wave equations for compressional and shear wave velocities with harmonic time dependence are given by

(37)

(38)

where ω =kc. Assume the solutions to Eqs. (37) and (38) as

(39)

where

Therefore, the expressions for displacement components for the quadratically heterogeneous substrate are

(40)

For λ₃ =μ₃, the stress components take the forms of

(41)

(42)

(43)

The expressions of the displacement components for the quadratically heterogeneous isotropic elastic substrate may be obtained by substituting Eq. (39) into Eq. (40).

6 Boundary conditions and dispersion relationship 6.1 Case Ⅰ

When the layer is composed of fluid, the substrate is composed of the fluid-saturated poroelastic medium, and the relevant boundary conditions at the uppermost free surface and at the common interface of layer and substrate are as follows:

where and are the first-order partial derivatives of w₁, w₂, and W₂ with respect to t, respectively. Moreover, f' is the first-order partial derivative of f with respect to x.

Using the expressions of stresses and displacements for the fluid layer and the fluid saturated poroelastic substrate in the above boundary conditions, the following system of equations may be obtained:

(44)

where l_i (i=1, 2, …, 15) are provided in Appendix A.

Elimination of arbitrary constants from Eq. (44) leads to

(45)

which is the dispersion relationship for the propagation of the Rayleigh-type wave in a fluid layer lying over a fluid-saturated poroelastic substrate.

As d represents the porosity in the substrate, it may be noted that for the cases,

(ⅰ)d→ 0, the substrate becomes non-porous which gives ρ_S → 1 and ρ_F → 0, and hence ρ_S constitutes the whole mass density of the substrate.

(ⅱ)d→ 1, the substrate becomes fluid which gives ρ_S → 0 and ρ_F → 1, and hence ρ_F constitutes the whole mass density of the substrate.

(ⅲ)0 < d < 1, the substrate is porous, and the mass density of the substrate is a total of ρ_S and ρ_F, which are the masses of solid and fluid per unit volume of the aggregate, respectively.

6.2 Case Ⅱ

When the fluid layer is lying over the quadratically heterogeneous isotropic elastic substrate, the relevant boundary conditions at the uppermost free surface and at the common interface of layer and substrate may be stated as

Using the expressions of stresses and displacements for the liquid layer and heterogeneous substrate in the above boundary conditions, the following set of equations may be obtained:

(46)

where t_i (i=1, 2, …, 9) are provided in Appendix B.

Eliminating the arbitrary constants from Eq. (46), the dispersion relationship for the propagation of the Rayleigh-type wave in a fluid layer lying over a quadratically heterogeneous isotropic elastic substrate is obtained as

(47)

7 Special cases

In view of the representation,

the periodic function f(x) leads to a simple periodic interface given by one cosine term only. Therefore, the common interface of the layer and substrate may be expressed as f=a cos (bx), where the wave length of corrugation is 2π /b.

(ⅰ) At f=a cos (bx), the dispersion relationships (45) and (47) reduce to

(48)

and

(49)

respectively. l_i (i=1, 2, …, 15) and t_j (j=1, 2, …, 9) are provided in Appendix A and Appendix B, respectively.

(ⅱ) At f=0, the dispersion relationships (45) and (47) take the form, respectively,

(50)

and

(51)

(i=1, 2, …, 15) and (j=1, 2, …, 9) are provided in Appendix A and Appendix B, respectively.

Equations (50) and (51) are the dispersion relationships for the propagation of Rayleigh-type wave in a fluid layer lying over a fluid saturated poroelastic substrate and a quadratically heterogeneous isotropic elastic substrate, respectively, where the interface of the layer and substrate is without corrugation (or planar).

8 Numerical results and discussion

The following data are considered with a view to perform numerical calculations of the obtained dispersion relationship in Eq. (48), which corresponds to the propagation of the Rayleigh-type wave in a fluid layer lying over a fluid saturated poroelastic substrate^[5]:

In addition to the above values of elastic constants for the fluid layer, the following data are considered for the quadratically heterogeneous isotropic elastic substrate^[19] so as to perform the numerical calculations of the obtained dispersion relationship in Eq. (49) :

Unless otherwise stated, the values of parameters are considered as follows:

Figures 2-9 illustrate the effects of affecting parameters on the dispersion curve for both cases, i.e., Case Ⅰ and Case Ⅱ. In Figs. 2-8, except Fig. 5, the phase velocity (c/c₁) is plotted against the wave number (kH), whereas in Figs. 5 and 9, the phase velocity (c/c₁) is plotted against the position parameter (x/H).

8.1 Case Ⅰ

Figures 2, 3, and 4 show the effects of different affecting parameters on the dispersion curve for the propagation of the Rayleigh-type wave in a fluid layer lying over a fluid saturated poroelastic substrate. All these figures indicate that the phase velocity of the Rayleigh-type wave decreases with the increase in the magnitude of the wave number.

Figure 2 illuminates the effect of porosity (d), associated with the lower fluid saturated poroelastic substrate, on the dispersion curve. Curve 1 in the figure corresponds to the case that the substrate is non-porous, i.e., d=0, curves 2, 3, and 4 indicate the case that the substrate is porous, i.e., 0 < d < 1, and curve 5 corresponds to the case that the substrate becomes fluid, i.e., d=1. It is observed that the dispersion curve shifts downward with the increase in the porosity of the substrate, i.e., the phase velocity of the Rayleigh-type wave decreases with the increase in the porosity of the substrate. It can be easily marked that the dispersion curve is more sensitive to the lower magnitude of porosity compared with the higher magnitude of porosity. Moreover, a remarkable difference in the upper bounds of the phase velocity of the Rayleigh-type wave is marked at the low frequency region for the lower magnitude of porosity, whereas the difference is negligible for the higher magnitude of porosity. The difference in the upper bounds of phase velocity corresponding to the highest and the lowest values of porosity is obtained to be 0.2 approximately.

Figure 3 reveals the effect of corrugation parameter (ab) associated with the common interface of the layer and substrate on the phase velocity of the Rayleigh-type wave. The magnitudes of the initial flatness parameter (a/H) and the undulation parameter (bH) together elucidate the effect of corrugation parameter on the dispersion curve. Therefore, the magnitudes of corrugation under consideration are ab=0.008, 0.018, 0.030, and 0.440 corresponding to curves 1, 2, 3, and 4, respectively. It is found that the phase velocity of Rayleigh-type wave increases with the increase in the corrugation parameter at the interface. The effect of corrugation is most significant at the lower frequency region in comparison with the higher frequency region. More precisely, it is marked that a small increment in the magnitude of initial flatness parameter and undulation parameter is responsible for a significant change in the phase velocity of the Rayleigh-type wave in the low frequency range. The difference in the upper bounds of phase velocity corresponding to the highest and the lowest value of corrugation is obtained to be 0.85 approximately. Further, the difference in the upper bounds of the phase velocity corresponding to different values of corrugation parameter is considerable. However, it is negligible for the lower bounds of the phase velocity.

Figure 4 elucidates the effect of the position parameter (x /H) and the undulation parameter (bH) on the phase velocity of the Rayleigh-type wave. The phase velocity of the Rayleigh-type wave seems to decrease with the increase in the magnitudes of position parameter and undulation parameter. Moreover, like Fig. 3, the influence of position and undulation parameters on the dispersion curve is marked considerable in the low frequency region. However, it is insignificant in the high frequency region. The difference in the upper bounds of the phase velocity is found to be 0.75 approximately corresponding to the highest and lowest values of parameters under consideration.

Figure 5 establishes the variation of phase velocity against the position parameter for different values of corrugation parameter marked through the variation of initial flatness parameter and undulation parameter. It is observed that the phase velocity increases with the increase in the corrugation parameter at the lower value of position parameter. However, as the magnitude of position parameter reaches 0.7 (approximately), the scenario alters, i.e., after the point of inversion, the phase velocity decreases with the increase in the corrugation parameter at the common interface.

8.2 Case Ⅱ

Figures 6, 7, and 8 deal with the effects of heterogeneity (γH), corrugation parameter, and position parameter, respectively, on the dispersion curve. Curve 1 in Fig. 6 depicts the case that the substrate becomes homogeneous, i.e., γH=0. Meticulous examination of Fig. 6 establishes that the phase velocity of the Rayleigh-type wave decreases with the increase in heterogeneity of the substrate. Figure 7 reveals that the corrugation parameter at the common interface of layer and substrate disfavors the phase velocity of the Rayleigh-type wave. Figure 8 implicates that, with the increase in the magnitude of position parameter along with the undulation parameter, the phase velocity of the Rayleigh-type wave increases. Minute observations of all the figures suggest that unlike Figs. 2, 3, and 4, the upper bounds of phase velocity corresponding to each curve in each of Figs. 6, 7, and 8, are almost equal at the lower frequency range.

Figure 9 irradiates the variation of phase velocity against the position parameter for different values of corrugation parameter, which is illustrated by the combined effect of initial flatness parameter and undulation parameter. Therefore, the magnitudes of corrugation parameter are ab=0.008, 0.018, 0.030 and 0.440 corresponding to curves 1, 2, 3, and 4, respectively. The figure concludes that the phase velocity of the Rayleigh-type wave decreases with the increase in the corrugation parameter at the interface of the layer and substrate.

9 Conclusions

The present paper investigates the propagation of the Rayleigh-type wave in a fluid layer lying over a fluid saturated poroelastic substrate in Case Ⅰ and the propagation of the Rayleigh-type wave in a fluid layer lying over a quadratically heterogeneous isotropic elastic substrate in Case Ⅱ. Studies are performed analytically, and numerical computations are also carried out to unravel the hidden facts. The outcomes of the present study are summarized as follows:

(ⅰ) The phase velocity of the Rayleigh-type wave decreases with the increase in the magnitude of the wave number irrespective of the fact that the corrugated substrate is fluid saturated poroelastic or heterogeneous isotropic elastic.

(ⅱ) The increasing magnitude of porosity of the fluid saturated poroelastic substrate has a decreasing effect on the phase velocity of the Rayleigh-type wave. More precisely, the presence of poroelasticity in the substrate is responsible for lowering the phase velocity, while the absence of poroelasticity (when the substrate is simply elastic) comparatively supports more to the phase velocity of the Rayleigh-type wave.

(ⅲ) The phase velocity of the Rayleigh-type wave decreases with the increase in the heterogeneity of the substrate. This leads to the conclusion that a homogeneous substrate (without heterogeneity) supports more to the phase velocity compared with a heterogeneous substrate.

(ⅳ) The influence of corrugated common interface of the layer and substrate on the phase velocity of the Rayleigh-type wave is not found similar for the considered two cases. Comparative studies elucidate that, with the increase in the magnitude of initial flatness parameter along with the undulation parameter of the corrugated interface, the phase velocity increases for the case that the substrate is composed of fluid saturated poroelastic material, while it decreases for the case that the substrate is composed of heterogeneous isotropic elastic material.

(ⅴ) With the increase in the magnitudes of position parameter and undulation parameter of the corrugated common interface, the phase velocity of the Rayleigh-type wave decreases when the substrate is composed of fluid saturated poroelastic material, while it increases when the substrate is composed of heterogeneous isotropic elastic material.

(ⅵ) Considerable variations are found in the upper bounds of phase velocity in the low frequency region for the case that the substrate is fluid saturated poroelastic. However, when the substrate is simply heterogeneous isotropic elastic, these variations are negligible.

Appendix A Appendix B

where m_f and n_f are the values of m and n at z=f, respectively, and and are the first-and second-order derivatives of m and n at z=f, respectively.