In this paper, we investigated the inuence of rotating half-space on the propagation of Rayleigh waves in a homogeneous isotropic, generalized thermo-elastic body, subject to the boundary conditions that the surface is traction free. In addition, it is subject to insulating thermal conduction. A general solution is obtained by using Lame’ potential’s and Hankel transform. The dispersion equations has been derived separately for two types of Rayleigh wave propagation properties by solving the equations of motion with appropriate boundary conditions. It is observed that the rotation, frequency and r exert some influence in the homogeneous isotropic medium due to propagation of Rayleigh waves. The frequency equation has been derived of homogeneous properties by solving the equations of motion with appropriate boundary conditions. It has been found that the frequency equation of waves contains a term involving the rotating. Therefore, the phase velocity of Rayleigh waves changes with respect to this rotating. When the rotating vanishes, the derived frequency equation reduces to that obtained in classical generalized thermo-elastic case which includes the relaxation time of heat conduction. In order to illustrate the analytical development, the numerical solution is carried out and computer simulated results in respect of Rayleigh wave velocity and attenuation coefficient are presented graphysically. A comparative and remarkable study has been carried out through various graphs to deliberate the consequences of different parameter on the frequency equation. The obtained results can be very useful in the design and optimization of Rayleigh wave.

There are many previous studies on generalized thermoelastic waves. Abd-Alla [

This paper brings out the analytical study of generalized thermoelastic medium subjected to rotation. The generalized thermoelastic cylinder is assumed under the influence of rotation and the relaxation time. The main aim of the paper is to investigate the effects of involved parameters on the Rayleigh wave velocity and attenuation coefficient of the wave. Numerical computation has been accomplished to manifest the effect of rotation and relaxation time on the Rayleigh wave velocity and attenuation coefficient of the wave for different types of parameters. The numerical results have been obtained and presented graphically.

Let us consider a homogeneous isotropic elastic solid with an infinite circular cylinder under initial temperature

The dynamic equation of motion is given by Sharma et al. [

The heat conduction equation is given by [

Where _{v}_{rr}_{zz}_{rz}

The stress–strain relations are given by Abd-Alla [

where,

The strain components and the rotation are given by Bagri et al. [

Using

By Helmholtz’s theorem [

Where the scalar

From

Substituting from

Where,

Let us consider a homogeneous and isotropic elastic solid with an infinite cylinder of the radius

The thermal boundary condition is

Assume that the temperature and potential functions of solid satisfy

where

Where

The general solution of

We obtain

By putting

where

If

The above roots correspond to the case in which the elastic wave and generalized heat condition equations are not coupled. For small

Then the solution of

which leads to

The solution of

Where

Substituting

The stress components _{rr}_{rz}

Substituting

In this section, we are going to obtain the frequency equation for the boundary conditions which specify that the outer surface of the cylinder is traction free and the thermal boundary conditions are illustrated. Substituting

By eliminating constraints

where

The frequency

The numerical calculation was carried out of the Rayleigh waves velocity and attenuation coefficient. To illustrate the theoretical results obtained in the preceding section, we now present some numerical results. The material chosen for this purpose was carbon steel, the physical data for which are given below [

The governing field equations for linear homogeneous and isotropic thermoelastic materials with rotation are solved to work out appropriate surface wave solutions in an infinite cylinder. The frequency equation for the Rayleigh surface wave is obtained. The numerical results are illustrated graphically against frequency for different values of rotation and relaxation time. Some concluding remarks are given as follows

The rotation and relaxation time significantly influence the variations of the Rayleigh wave.

Analysis of Rayleigh wave developed into a body due to rotation and relaxation time.

The rotation and relaxation time of an infinite cylinder give the same effect in the problem as mentioned above in the results.

The present theoretical results may provide interesting information for experimental scientists, researchers, and seismologists working on this subject.