Propagation of Love-type wave in an initially stressed porous medium over a semi-infinite orthotropic medium with the irregular interface has been studied. The method of separation of variables has been adopted to get the dispersion relation of Love-type wave. The irregularity is assumed to be rectangular at the interface of the layer and half-space. Finally, the dispersion relation of Love wave has been obtained in classical form. The presence of porosity, irregularity, and initial stress in the dispersion equation approves the significant effect of these parameters in the propagation of Love-type waves in porous medium bounded below by an orthotropic half-space. The scientific effect of porosity, irregularity, and initial stress in the phase velocity of the Love-type wave propagation has been studied and shown graphically.

The Earth contains fluid-saturated porous rocks on or below its surface in the form of sandstone and other sediments permeated by groundwater or oil; the diffusion of fluid and readjustment of fluid pressure have been acting as a triggering mechanism for earthquakes. So, the study of wave propagation in a porous medium has gained prime interest. The propagation of Love-type wave in porous media with irregular boundary surfaces is important leading to better understanding and prediction of behaviour of seismic wave at mountain roots, continental margins, and so forth. Love-type wave propagation in layered media has long been a research subject because of its practical importance in exploration of oil, geophysics, earthquake engineering, and underground water. The current work is concerned with the propagation of Love-type waves in initially stressed porous layer overlying semi-infinite orthotropic medium with irregular interface. It has been noticed that the presence of porosity, irregularity, and initial stress in the dispersion equation approves the significant effect of these parameters in the propagation of Love-type waves.

The intended applications of this theory may be found in the field of geophysics and the manufactured porous solids. Various problems of waves and vibrations based on these theories of elasticity have been attempted by the researchers and have appeared in the open literature. Following Biot ([

In this paper, we use the porous medium (layer) over an orthotropic half-space with the effect of irregular interface in the propagation of Love-type waves. The main attention is paid to the influence of irregularity of interface, porosity, and initial stress on the propagation of Love-type waves in porous-orthotropic medium (Figure

Geometry of the problem.

We have considered a model consisting of initially stressed porous layer of finite thickness

Neglecting the viscosity, in the absence of body forces, the dynamical equations of motion for initially stressed anisotropic porous medium can be written as Biot [

For the propagation of Love waves along the

Therefore, (

In the anisotropic porous medium, the shear wave velocity along the

Thus, one gets the following:

Assume the solution of (

Therefore, the solution of (

The equations of motion for the orthotropic medium under initial stress in the absence of body forces are

The stress-strain relations in the orthotropic medium are

Again, using the Love waves conditions

For wave propagation along

The upper surface of the porous layer is stress-free; that is,

Now, applying the boundary conditions, we have

In case the porous layer has no irregularity, that is,

For the nonporous homogeneous layer

When the semi-infinite medium is initially stress-free and homogeneous with rigidity

Based on dispersion (

To study the effect of porosity, initial stress, and irregularity, we represent the numerical data from Gubbins [

For the orthotropic half-space,

For the anisotropic porous layer,

In all the figures, curves have been plotted as phase velocity

Figure

Variation of phase velocity

Figure

Variation of phase velocity

Figure

Variation of phase velocity

Figure

Variation of phase velocity

Figure

Variation of phase velocity

The study of seismic waves gives important information about the layered Earth structure and has been used to determine the epicenter of the earthquake. Seismologists are able to learn about the Earth’s internal structure by measuring the arrival of seismic waves at stations around the world because these waves travel at different speeds through different materials. Knowing how fast these waves travel through the Earth, seismologists can calculate the time when the earthquake occurred and its location by comparing the times when shaking was recorded at several stations. If a wave arrives late, it passed through a hot, soft part of the Earth.

Propagation of Love-type waves in an initially stressed anisotropic porous layer over an initially stressed orthotropic medium with rectangular irregularity has been discussed. The method of separation of variables has been adopted to solve the equation of motion, separately, for different media using suitable boundary condition at the interface of anisotropic porous layer and orthotropic half-space with irregular interface. The dispersion relation of Love-type wave has been obtained and coincides with the classical dispersion relation of Love wave in particular cases. The presence of porosity, irregularity, and initial stress in the dispersion equation approves the significant effect of these parameters on the propagation of Love-type wave in porous medium bounded below by an orthotropic half-space. It has been observed that the maximum changes happen in phase velocity between

The height

It is observed that the porosity also has a dominant role in the propagation of Love-type wave. When the porosity of the porous layer increases, the phase velocity of the Love wave also increases in such a structure.

The phase velocity increases with increases in initial stress

The height of irregularity has the impact on the phase velocity of Love-type wave in the absence of initial stress

It is observed that the presence of porosity, initial stress, and irregularity affected the phase velocity of Love-type wave and has much dominance at large values of wave number. The initial stress in the porous medium increases the phase velocity of Love-type wave, whereas the phase velocity decreases in orthotropic medium due to initial stress. The phase velocity of Love-type wave also decreases with the depth of irregularity in an orthotropic medium.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors convey their sincere thanks to the Indian School of Mines, Dhanbad, India, for providing them with the best facilities.