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The reflection of plane waves at the free surface of thermally conducting micropolar elastic medium with two temperatures is studied. The theory of thermoelasticity with and without energy dissipation is used to investigate the problem. The expressions for amplitudes ratios of reflected waves at different angles of incident wave are obtained. Dissipation of energy and two-temperature effects on these amplitude ratios with angle of incidence are depicted graphically. Some special and particular cases are also deduced.

The theory of micropolar elasticity was introduced and developed by Eringen [

The linear theory of micropolar thermoelasticity has been developed by extending the theory of micropolar continua to include thermal effect and comprehensive review work on the subject was given by Eringen [

The main difference of thermoelasticity with two temperatures with respect to the classical one is the thermal dependence. Chen et al. [

Youssef [

Kaushal et al. [

Ezzat and Othman [

To study the propagation of thermal waves at finite speed, it may be possible in the foreseeable future to identify an idealized material. Green and Naghdi [

Various investigators have studied the different problems using GN type II and type III theories. Mukhopadhyay and Kumar [

In the present investigation, we study the reflection of plane waves, that is, longitudinal displacement wave (LD wave), thermal wave (T-wave), and transverse wave coupled with microrotational wave (CD-I wave and CD-II wave) at the free surface of thermally conducting micropolar elastic medium with two temperatures with and without energy dissipation. Energy dissipation and two-temperature effects are depicted numerically and depicted graphically on the amplitude ratios for incidence of various plane waves for a particular model.

Following Eringen (1970), Ezzat and Awad [

A homogeneous, isotropic, micropolar, thermoelastic solid half space with two temperatures (medium

The components of displacement and microrotation for two dimensional problem are taken as

The relations between nondimensional displacement components

The following boundary conditions at the free surface

We consider longitudinal displacement wave (LD-wave), thermal wave (T-wave), and coupled transverse and coupled microrotational waves (CD-I wave and CD-II wave) propagating through micropolar thermoelastic with two-temperature solid half space

Geometry of the problem.

In order to solve (

Using (

In view of (

We use the following extension of Snell’s law to satisfy the boundary conditions:

For incident LD-wave:

For incident T-wave:

For incident CD-I wave:

For incident CD-II wave:

By neglecting two-temperature parameters, that is,

If we take

If we take

The following values of relevant parameters for numerical computations are taken.

Following Eringen [

In Figures

Variations of amplitude ratios with angle of incidence for LD-wave.

Variations of amplitude ratios with angle of incidence for T-wave.

Variations of amplitude ratios with angle of incidence for CD-I wave.

Variations of amplitude ratios

Figure

It is evident from Figure

Figure

Figure ^{2}.

Variations of amplitude ratios

Figure

Figure

It is evident from Figure

Figure ^{2}.

Variations of amplitude ratios

Figure

It is depicted from Figure ^{2} and the values for ATS are magnified by multiplying by 10.

It is noticed from Figure

Figure

In the present paper, the expressions for reflection coefficients of various reflected waves have been derived in micropolar thermoelastic solid half space in the context of GN type II and GN type III theories. It is observed that when LD-wave is incident, the values of amplitude ratios for TS remain more than the value for ATS; that is, two-temperature effect increases the magnitude of amplitude ratios. Also when T-wave is incident, the values of amplitude ratios follow oscillatory pattern and the values for KTS and AKTS attain peak value near the grazing incidence. The values of amplitude ratio

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