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The present paper is concerned with the propagation of plane waves in an isotropic two-temperature generalized thermoelastic solid half-space in context of Green and Naghdi theory of type II (without energy dissipation). The governing equations in x – z plane are solved to show the existence of three coupled plane waves. The reflection of plane waves from a thermally insulated free surface is considered to obtain the relations between the reflection coefficients. A particular example of the half-space is chosen for numerical computations of the speeds and reflection coefficients of plane waves. Effects of two-temperature and rotation parameters on the speeds and the reflection coefficients of plane waves are shown graphically.

Lord and Shulman [

Gurtin and Williams [14,15] suggested the second law of thermodynamics for continuous bodies in which the entropy due to heat conduction was governed by one temperature, that of the heat supply by another temperature. Based on this suggestion, Chen and Gurtin [

In the present paper, we have applied Youssef [

We consider a two-temperature thermoelastic medium, which is rotating uniformly with an angular velocity, where is a unit vector representing the direction of the axis of rotation.The displacement equation of motion in the rotating frame of reference has two additional terms: Centripetal acceleration, due to time-varying motion only and the Corioli's acceleration, where is the dynamic displacement vector. These terms do not appear in non-rotating media. Following Youssef [

(i) The heat conduction equation

(ii) The displacement-strain relation

(iii) The equation of motion

(iv) The constitutive equations

where is a coupling parameter and is the thermal expansion coefficient. and are called Lame’s elastic constants, is the Kronecker delta, is material characterstic constant, T is the mechanical temperature, is the reference temperature, with, is the stress tensor, is the strain tensor, is the mass density, is the specific heat at constant strain, are the components of the displacement vector, is the conductive temperature and satisfies the relation

where is the two-temperature parameter.

We consider a homogeneous and isotropic thermoelastic medium of an infinite extent with Cartesian coordinates system, which is previously at uniform temperature. The origin is taken on the plane surface and the z-axis is taken normally into the medium. The surface is assumed stress-free and thermally insulated. The present study is restricted to the plane strain parallel to x-z plane, with the displacement vector and rotational vector. Now, the Equation (3) has the following two components in x-z plane

The heat conduction Equation (1) is written in x-z plane as

and, the Equation (5) becomes,

The displacement components and are written in terms of potentials and as

Using Equations (9)-(10) in Equations (6)-(8), we obtain

Solutions of Equations (11)-(13) are now sought in the form of harmonic travelling wave

in which is the phase speed, is the wave number and denotes the projection of wave normal onto x-z plane. Making use of Equation (14) into the Equations (11)-(13), we obtain a homogenous system of equations in A, B and C, which admits the non-trivial solution if

where

and

The three roots of the cubic Equation (15) are complex. Using the relation, we obtain three real values of the speeds of three plane waves, namely, waves, respectively.

In absence of rotation parameters, we have and the velocity Equation (15) reduces to

which gives the speeds of P, thermal and SV waves in an isotropic two-temperature thermoelastic medium without energy dissipation.

In absence of rotation and thermal parameters, we have and the Equation (15) reduces to

which gives the speeds of P and SV waves in an isotropic elastic media.

We consider the incidence of wave. The boundary conditions at the stress-free thermally insulated surface are satisfied, if the incident wave gives rise to a reflected waves. The required boundary conditions at free surface are as

(i) Vanishing of the normal stress component

(ii) Vanishing of the tangential stress component

(iii) Vanishing of the normal heat flux component

where

The appropriate displacement and temperature potentials are taken in the following form

where the wave normal to the incident wave makes angle with the positive direction of z-axis and those of reflected waves make angles and, respectively with the same direction, and

where

The ratios of the amplitudes of the reflected waves to the amplitude of incident wave, namely and are the reflection coefficients (amplitude ratios) of reflected wave, respectively. The wave numbers and the angles are connected by the relation

at surface z = 0. In order to satisfy the boundary conditions (18)-(20), the relation (26) is also written as

with the help of the potentials given by Equations (23)- (25) and the Snell’s law Equations (26) and (27), the boundary conditions (18)-(20) results into a system of following three non-homogeneous equations

where are the reflection coefficients of reflected waves, and

To study the effects of two-temperature and rotation parameters on the speeds of propagation and reflection coefficients of plane waves, we consider the following physical constants of aluminium as an isotropic thermoelastic solid half space

Using the relation in Equation (15), the real values of the propagation speeds of waves are computed for the range of two-temperature parameter, when . The speeds of waves are shown graphically versus the two-temperature parameter in

With the help of Equation (28), the reflection coefficients of reflected waves are computed for the incidence of wave. For the range of the angle of incidence of wave, the reflection coefficients of the waves are shown graphically in

For the range of the angle of incidence of P_{1} wave, the reflection coefficients of the waves are shown graphically in

at. The reflection coefficient of wave decreases from its maximum value at to its minimum value zero at. From

Two-dimensional solution of the governing equations of an isotropic two-temperature thermoelastic medium without energy dissipation indicates the existence of three plane waves, namely, waves. The appropriate solutions in the half-space satisfy the required boundary conditions at thermally insulated free surface and the relations between reflection coefficients of reflected waves are obtained for the incidence of wave. The speeds and reflection coefficients of plane waves are computed for a particular material representing the model. From theory and numerical results, it is observed that the speeds and reflection coefficients of plane waves are significantly affected by the two-temperature and rotation parameters.