The two-phase medium made up of a solid part (skeleton) and a liquid part (pore space) is referred to as porous material. The study of wave propagation in porous media is significant to many different branches of science and engineering. The theory behind the phenomena has received substantial research in a variety of fields, including soil mechanics, earth science, acoustics, geotechnical engineering, ocean engineering, geophysics, and many others. Primary characteristic of porous media is their enormous solid-liquid contact area, which leads to novel diffusion and transport phenomena in the fluid in relation to the micro-geometry of the pore space. Wave propagation through porous rocks is a topic of interest in geophysics because it can reveal details about the

In the present paper, Love waves propagating on poroelastic layer between transversely isotropic and an inhomogeneous half-spaces are observed. Few particular cases are observed and discussed. Dispersion equations have been developed and discussed for all the cases under suitable conditions. The phase velocity of plane waves is obtained for a particular model. The effect of initial stress of the half-spaces and inhomogeneity parameters of the lower half-space on the propagation of Love waves has been demonstrated through numerical computation of the dispersion equation. It is found that increasing the initial stress in the lower half of space causes a decrease in phase velocity while having little to no impact on the phase velocity in the upper half-space scenario.

A poroelastic layer with thickness h is considered. It is bounded between transversely poroelastic half-space (upper half-space) and an in-homogeneous elastic half-space (lower half-space) where both are initially stressed. The origin is located at the interface of lower half-space and poroelastic layer. The positive z-axis is taken towards the interior of the lower half space and wave propagates in x-axis direction.

The solutions of equations of motion in the three solids are derived.

In the absence of body forces, the dynamic equations of motion

where

The mass coefficients

Stress-strain relations in a transversely isotropic poroelastic half space under initial

Strain components are expressed in terms of displacements

For propagation of love waves displacements along x and z-axes vanishes thus we have

Combining the equations (2)-(4) with the above displacements, equation (1) reduces to

Assuming harmonic wave solution in the form

where k is a wavenumber, equation (5) yields

Using the solution of the above equations, the displacement

where

In an inhomogeneous elastic solid with initial stress

Where

For love waves

The non-zero stress-strain relations are

In-homogeneous of rigidity and density parameters of the half-space are taken as

Substituting

Using these stresses, the equation of motion (9) reduces to

Assuming wave solution in the form

and substituting it in equation (11), we get

where

Taking g(z) =

where

Now defining the variables

equation (13) can be written as

where

The solution of equation (14) is

where

Now, the displacement component

Considering linear terms of the Whittaker’s functions, equation (16) can be written as

In the absence of body forces, the governing equations of an isotropic poroelastic solid which is homogeneous are

where _{ij} and the fluid pressure s of the solid are given by

The displacement components of the solid

We suppose that the solid and liquid propagation modes

where t represents time, w the circular frequency, k the wave number and i the complex unity.

Using the equation (21) into (20) yields

Solving the two equations of equation (22) for

where

On simplification, equation (23) gives

where E_{1} and E_{2} are constants.

When _{y} can be obtained as

The current problem’s geometry results in the following boundary conditions:

Equations represented in boundary conditions (26) are

where

Equations in (27) constitute a system of four linear equations in four arbitrary constants

Case (1) If

where

Case (1.1) If

Case (1.2) If

Case (1.3) If

Case (2) Upper half space is isotropic poroelastic: If

Case (2.1) Upper half space is isotropic elastic: In addition to

where

Case (3) Layer is isotropic elastic: If

where

Case (3.1) Upper half space is isotropic elastic and Layer is isotropic elastic: If

where

The impact of initial stress of the both lower and upper half-spaces and inhomogeneous parameter of lower half-space has been discussed. Few authors

Neglecting poroelastic constants in the poroelastic layer reduces it into elastic layer and the same is discussed as a particular case. Similarly, frequency equations are obtained by reducing upper transversely isotropic poroelastic half-space into poroelastic and elastic half-spaces as other particular cases. Non-dimensional phase velocity against non-dimensional wave number for various cases is calculated, plotted and discussed.

Fixed value

Phase velocity against wave number in the case of poroelastic layer for fixed initial stress parameter

The non-dimensional in-homogeneous parameters are taken as

of upper half-space and various values of initial stress P_{3 } of lower half-space when lower half-space is homogeneous

Phase velocity is calculated when upper half-space is poroelastic and results are depicted in figure 6. Phase velocity displays a similar pattern of behaviour to that of transversely isotropic poroelastic upper half-space shown in the

The propagation of love waves in poroelastic layer bound to inhomogeneous isotropic half-space and transversely isotropic poroelastic half-space is discussed. Frequency equations of various other particular cases are obtained. The effect of initial stress and in-homogeneity is observed. The results obtained in particular cases are coincided with the results of earlier study. This work can be extended further by discussing effect of gravity, imperfect bonding between the solids.

As the wavenumber increases, phase velocity decreases.

As the inhomogeneous rigidity parameter increases, so does the phase velocity.

Phase velocity drops as density inhomogeneous parameter

Lower inhomogeneous elastic half-space with higher initial stress values generate lower phase velocities.

Inhomogeneous lower half-space has a lower phase velocity than homogeneous lower half-space.

The phase velocity is decreased as the initial stress of the bottom half-space increases.

The phase velocity is not significantly affected by the upper half space's initial stress.

When the upper half-space is poroelastic, phase velocity is higher than it is when the upper half-space is transversely isotropic.