A Comparative Study of magneto-thermo-elastic wave propagation in a finitely conducting medium under thermoelasticity of type I, II, III

TL;DR

This study compares magneto-thermo-elastic wave propagation in a finitely conducting medium using Green-Naghdi theories of types I, II, and III, revealing distinct wave behaviors and effects of magnetic fields.

Contribution

It introduces a comprehensive analysis of wave propagation under different Green-Naghdi models, highlighting their contrasting effects on wave attenuation, dispersion, and thermal wave velocity.

Findings

GN-II model predicts un-attenuated, nondispersive waves.

Thermal wave velocity varies with frequency across models.

Magnetic field influences wave characteristics significantly.

Abstract

The present work is concerned with the propagation of electro-magneto-thermoelastic plane waves of assigned frequency in a homogeneous isotropic and finitely conducting elastic medium permeated by a primary uniform external magnetic field. We formulate our problem under the theory of Green and Naghdi of type-III (GN-III) to account for the interactions between the elastic, thermal as well as magnetic fields. A general dispersion relation for coupled waves is deduced to ascertain the nature of waves propagating through the medium. Perturbation technique has been employed to obtain the solution of dispersion relation for small thermo-elastic coupling parameter and identify three different types of waves. We specially analyze the nature of important wave components like, phase velocity, specific loss and penetration depth of all three modes of waves. We attempt to compute these wave components numerically to observe their variations with frequency. The effect of presence of magnetic field is analyzed. Comparative results under theories of type GN-I, II and III have been presented numerically in which we have found that the coupled thermoelastic waves are un-attenuated and nondispersive in case of Green-Naghdi-II model which is completely in contrast with the theories of type-I and type-III. Furthermore, the thermal mode wave is observed to propagate with finite phase velocity in case of GN-II model, whereas the phase velocity of thermal mode wave is found to be an increasing function of frequency in other two cases. We achieve significant variations among the results predicted by all three theories.

Equations

$$\mu {\nabla }^2\vec{u}+(\lambda +\mu )\vec{\nabla }(\vec{\nabla }.\vec{u})+\vec{J}\times \vec{B}-\nu \vec{\nabla }\theta =\rho \ddot{\vec{u}}$$

$$K^{\ast }{\nabla }^2\theta +K{\nabla }^2\dot{\theta }=\rho C_v\ddot{\theta }+\nu T_0\ddot{u_{i,i}}$$

$$\vec{\nabla }\times \vec{H}=\vec{J}$$

$$\vec{\nabla }\times \vec{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$$

$$\overrightarrow{\nabla .}\vec{B}=0$$

$$\vec{J}=\sigma [\vec{E}+\frac{\partial u}{\partial t}\times \vec{B}]$$

$$div\overrightarrow{b}=0$$

$${\mu }_e\overrightarrow{J}=\vec{\nabla }\times \overrightarrow{b}.$$

$$\vec{J}=[0,-\frac{ik{b}_z}{{\mu }_e},\frac{ik{b}_y}{{\mu }_e}]$$

$$\vec{J}\times \overrightarrow{B_0}=[-\frac{ik(b_z{B}_3+b_y{B}_2)}{{\mu }_e},\frac{ik{b}_y{B}_1}{{\mu }_e},\frac{ik{b}_z{B}_1}{{\mu }_e}]$$

$$\theta =\alpha u_0$$

$$\alpha =\frac{i\nu {\theta }_{0}k{\omega }^2}{K^{\ast }{k}^2-iK{k}^2\omega -\rho C_v{\omega }^2}$$

$$\sigma [E_x-i\omega (q{B}_3-r{B}_2)]=0$$

$$\sigma [\frac{\omega b_z}{k}-i\omega (r{B}_1-p{B}_3)]=-\frac{ik{b}_z}{{\mu }_e}$$

$$\sigma [-\frac{\omega b_y}{k}-i\omega (p{B}_2-q{B}_1)]=\frac{ik{b}_y}{{\mu }_e}$$

$$u_0(-\rho {\omega }^2+(\lambda +2\mu )k^2+i\nu \alpha k)+\frac{ik}{{\mu }_e}(b_3{B}_3+b_2{B}_2)=0$$

$$v_0(-\rho {\omega }^2+\mu k^2)-\frac{ik}{{\mu }_e}(b_2{B}_1)=0$$

$$w_0(-\rho {\omega }^2+\mu k^2)-\frac{ik}{{\mu }_e}(b_3{B}_1)=0$$

$$u_0\sigma (i\omega B_3)+w_0(-\sigma i\omega B_1)+b_3[\frac{ik}{{\mu }_e}-\frac{\sigma \omega }{k}]=0$$

$$u_0\sigma (-i\omega B_2)+v_0(\sigma i\omega B_1)-b_2[\frac{ik}{{\mu }_e}+\frac{\sigma \omega }{k}]=0$$

$$u_0[-\rho {\omega }^2+(\lambda +2\mu )k^2+i\nu \alpha k]+\frac{ik{B}_2{b}_2}{{\mu }_e}=0$$

$$v_0(-\rho {\omega }^2+\mu k^2)-\frac{ik{B}_1{b}_2}{{\mu }_e}=0$$

$$u_0(\sigma i\omega B_2)-v_0\sigma (i\omega B_1)+b_2(\frac{ik}{{\mu }_e}+\frac{\sigma \omega }{k})=0$$

$$\left|\begin{array}[]{ccc}-\rho\omega^{2}+(\lambda+2\mu)k^{2}+i\nu\alpha k&0&\frac{ikB_{2}}{\mu_{e}}\\ \\ 0&\mu k^{2}-\rho\omega^{2}&\frac{-ikB_{1}}{\mu_{e}}\\ \\ i\sigma\omega B_{2}&0&(\frac{ik}{\mu_{e}}+\frac{\sigma\omega}{k})\end{array}\right|=0$$

$$(s^2{\xi }^2-{\chi }^2)[({\xi }^2-{\chi }^2)({\xi }^2{k}_1-{\chi }^2)-i{k}_2{\xi }^2\chi ({\xi }^2-{\chi }^2)-{ϵ}_{\theta }{\xi }^2{\chi }^2](\chi +i{\xi }^2{ϵ}_H)$$

$$+R_H\chi {\xi }^2[({\xi }^2{k}_1-{\chi }^2)-i{k}_2{\xi }^2\chi ]=0$$

$$[({\xi }^2-{\chi }^2)({\xi }^2{k}_1-{\chi }^2)-i{k}_2{\xi }^2\chi ({\xi }^2-{\chi }^2)-{ϵ}_{\theta }{\xi }^2{\chi }^2](\chi +i{\xi }^2{ϵ}_H)+R_H\chi {\xi }^2[({\xi }^2{k}_1-{\chi }^2)-i{k}_2{\xi }^2\chi ]=0$$

$${\xi }^2=A{\chi }^2$$

$$[({\xi }^2-{\chi }^2)(\chi +i{\xi }^2{ϵ}_H)+R_H\chi {\xi }^2]=0$$

$${\alpha }_{1,2}^2=\frac{i{\chi }^2{ϵ}_H-\chi (1+R_H)\pm {\left\{{\chi }^2(1+R_H{)}^2-{\chi }^4{ϵ}_H^2-2{\chi }^3(1+R_H){ϵ}_{H}i+4i{ϵ}_H{\chi }^3\right\}}^{\frac{1}{2}}}{2i{ϵ}_H}$$

$${\alpha }_{1,2}^2=\frac{i{\chi }^2{ϵ}_H-\chi (1+R_H)\pm (a+ib)}{2i{ϵ}_H}$$

Full text

A Comparative Study of magneto-thermo-elastic wave propagation

in a finitely conducting medium under thermoelasticity of type I, II, III

(*Dr.Rakhi Tiwari, Prof. J. C. Mishra)

*Department of Mathematics, Babasaheb Bhimrao Ambedakar Bihar University

Department of Mathematics, Indian Institute of Engineering, Science and Technology

*E-mail: rakhibhu2117@gmail.com

Abstract: The present work is concerned with the propagation of electro-magneto-thermoelastic plane waves of assigned frequency in a homogeneous isotropic and finitely conducting elastic medium permeated by a primary uniform external magnetic field. We formulate our problem under the theory of Green and Naghdi of type-III (GN-III) to account for the interactions between the elastic, thermal as well as magnetic fields. A general dispersion relation for coupled waves is deduced to ascertain the nature of waves propagating through the medium. Perturbation technique has been employed to obtain the solution of dispersion relation for small thermo-elastic coupling parameter and identify three different types of waves. We specially analyze the nature of important wave components like, phase velocity, specific loss and penetration depth of all three modes of waves. We attempt to compute these wave components numerically to observe their variations with frequency. The effect of presence of magnetic field is analyzed. Comparative results under theories of type GN-I, II and III have been presented numerically in which we have found that the coupled thermoelastic waves are un-attenuated and non-dispersive in case of Green-Naghdi-II model which is completely in contrast with the theories of type-I and type-III. Furthermore, the thermal mode wave is observed to propagate with finite phase velocity in case of GN-II model, whereas the phase velocity of thermal mode wave is found to be an increasing function of frequency in other two cases. We achieve significant variations among the results predicted by all three theories.

Keywords: Generalized electro-magneto-thermoelasticity; Green and Naghdi theory; Thermoelasticity of types I, II and III; Plane waves; Dispersion relation; Phase velocity; Specific loss; Penetration depth.

**1. Introduction **

The development of active materials have drawn the serious attention of researchers towards the interactions between magnetic, thermal and mechanical fields in a thermo-elastic solid in presence of magnetic field. The magnetic field originating inside nuclear reactors and the extremely high temperature affect their design and operations [1]. This terminology is acknowledged as the theory of magneto-thermo-elasticity. This topic has several applications in various fields like, geophysics, optics, acoustics, plasma physics, damping of acoustic wave in magnetic fields and other related topics involving sensing and actuation. The materials with electric-elastic coupling are used as ultrasonic transducers and micro-actuators, while materials with thermal and electric coupling have been used for thermal imaging devices. The concept of elasto-magnetic coupling effects in materials have applications in health monitoring of civil structures. Basically, the coupled theory of electro-magneto-thermoelasticity is the combination of two different disciplines namely theory of electro-magnetism and theory of thermoelasticity. A systematic presentation of governing equations along with the uniqueness and reciprocity theorem for linear thermo-electro-magneto-elasticity can be found in the article by Li [2].

The study of plane wave propagation in thermo-elastic material in presence of external magnetic field has been the topic of high interest during last few decades. Paria[3], Wilson [4] and Purushotama[5] have used the classical theory of thermoelasticity (Biot[6]) along with the electromagnetic theory to characterize harmonically time dependent plane waves of assigned frequency in a homogeneous, isotropic and unbounded solid. But unfortunately, the classical theory of thermoelasticity by Biot [6] is unrealistic from a physical point of view, particularly in problems like those concerned with sudden heat inputs, because it exhibits infinite speed of propagation of thermal waves. Hence, to account for demerits of this theory, some theories have been developed as extended theory of classical thermoelasticity. The theories given by Lord and Shulman [7] and Green and Lindsay [8] are two theories that have been extensively employed by researchers as generalized theories of thermoelasticity. Subsequently, Green and Naghdi [9,10,11] developed three theories of thermoelasticity which we refer to in the present work as Green Naghdi -I, II and III (GN-I, GN-II and GN-III) theory of thermoelasticty. The linearized version of GN-I theory is identical to the classical theory of thermoelasticity. GN-II theory accounts for finite speed of thermal wave, but with no energy dissipation. This implies that undamped waves of second sound effect is identified. However GN-III theory does not follow GN-I and GN-II but shows GN-I & GN-II theories as its special cases.

Nayfeh and Nemat-Nesser [12] and later on, Agarwal [13] reported a detailed study on electro-magneto-thermoelastic plane waves in solids in the context of generalized thermoelasticity theory with the effects of thermal relaxation parameters. Roychoudhuri [14,15] and Roychoudhuri and Debnath [16,17] studied propagation of magneto-thermoelastic plane waves in rotating thermoelastic media permeated by a primary uniform magnetic field using generalized heat conduction equation of Lord and Shulman [7]. Chandrasekharaiah [18] investigated the harmonic plane wave propagation in an unbounded medium by employing GN-II theory of thermoelasticity. Puri and Jordan [19] and later on, Kothari and Mukhopadhyay [20] used GN-III theory to explain the propagation of thermoelastic plane waves. Roychoudhuri and Banerjee (Chattopadhyay) [21] discussed magnetoelastic plane waves in rotating media under thermoelasticity of type-II model. Das and Kanoria [22] also worked on magneto-thermo-elastic waves in a medium with infinite conductivity in the context of GN-III theory. Abbas and Alzahrani [23] studied Green-Naghdi Model in a 2D problem of a mode I crack in an Isotropic Thermoelastic Plate. Moreover Ezzat, El-Karamany and El-Bary [24] presented Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer. Tiwari and Mukhopadhyay [25] studied electromagnetic wave propagation in GN-II model of thermoelasticity.

At present, our motive of this work is to investigate the propagation of magneto-thermo-elastic plane waves of assigned frequency in an infinite, homogeneous, isotropic, thermally and electrically conducting solid in the context of GN-I, GN-II and GN-III theory of thermoelasticity. We have taken the media having finite conductivity permeated by a primary uniform external magnetic field. The basic governing equations are derived and a more general dispersion relation is obtained in GN-III theory of thermoelasticity. ’Perturbation technique’ has been enforced to solve the problem analytically. Three various types of plane waves are identified by solving the general dispersion relation analytically and various physical characterizations of plane waves are deduced. The problem is illustrated with numerical results of various wave characterizations and the nature of waves of different modes are critically analyzed. Since GN-I and GN-II an be obtained as a special cases of GN-III model of thermoelasticity. Therefore, the nature of electro-magnetic harmonic plane waves have been characterized under the theories of Green and Naghdi of types I, II and III (GN-I, GN-II and GN-III) using numerical results and conclusions are explained to highlight the specific features of the present investigation.

**2. Problem formulation and basic governing equations **

For our present study, an infinite, homogeneous, isotropic, thermally and electrically conducting solid permeated by a primary magnetic field $\vec{B_{0}}=(B_{1},B_{2},B_{3})$ is considered. The media is characterized by the density $\rho$ and Lame’s elastic constants $\lambda$ and $\mu$.

Using a fixed rectangular cartesian coordinate system (x,y,z) and employing the thermoelasticity theory of Green and Naghdi [[9]-[11]], the equations of motion and the equation of heat conduction in the presence of magnetic field in the absence of external body force (mechanical) and heat sources can be represented in the following manner:

Equation of motion:

[TABLE]

Equation of heat conduction [9]:

[TABLE]

where $\vec{J}$ is the current density vector and $\vec{J}\times\vec{B}$ is the electromagnetic body force (Lorentz force).

Here, $\vec{u}$ is the displacement field, $\theta$ is the temperature above reference temperature $T_{0}$. Total magnetic field $\vec{B}=\vec{B_{0}}+\vec{b}$ is assumed to be small so that the products with $\vec{b}$ and $\dot{\vec{u}}$ and their derivatives can be neglected for the linearization of the field equations. $\vec{b}=(b_{x},b_{y},b_{z})$ is the perturbed magnetic field. $\lambda,\mu$ are Lame’s constants and $\nu=(3\lambda+2\mu)\alpha_{t}$, where $\alpha_{t}$ is the coefficient of thermal expansion of the solid. Dots denote the derivatives with respect to the time t. $C_{v}$ is the specific heat of the solid at constant volume, $K^{*}$ is the thermal conductivity rate and $K$ is the thermal conductivity of the medium.

Notice that if we put $K*=0$ in equation (2) i.e. the thermal conductivity rate is absent, then the equation is acknowledged by the heat conduction equation for GN-I theory of thermoelasticity and if we substitute $K=0$ in equation (2) i.e. the thermal conductivity rate is absent, we obtain the heat conduction equation of GN-II theory of thermoelasticity.

Due to the presence of magnetic field inside the medium, equation of motion needs to be supplemented by generalized Ohm’s law in a continuous medium with Max well’s electromagnetic field equations.

Max well’s equations (where the displacement current and charge density are neglected) are given by

[TABLE]

[TABLE]

where $\vec{B}=\mu_{e}\vec{H}$ and $\mu_{e}$ is the magnetic permeability.

[TABLE]

Generalized Ohm’s law is given by

[TABLE]

where $\sigma$ is the electrical conductivity, ${\frac{\partial u}{\partial t}}$ is the particle velocity of the medium. Here the small effect of temperature gradient on $\vec{J}$ is neglected.

**3. Dispersion relation and its analytical solution **

We are assuming that plane waves are propagating towards x-direction. Due to this all the field quantities are proportional to $e^{i(kx-\omega t)}$, where k is the wave number and $\omega$ is the angular frequency of plane waves. Here, we have assumed that $\omega$ is real and k may be complex quantity such that $Im(k)\leq 0$ for waves to be physically realistic. In the context of above consideration, we can write all the quantities in the following manner:

$\vec{u}=(u,v,w)=(u_{0},v_{0},w_{0})e^{i(kx-\omega t)}$,$\ \theta=\theta_{0}e^{i(kx-\omega t)}$;

$\vec{E}=(E_{x},E_{y},E_{z}),\vec{\ J}=(j_{1},j_{2},j_{3})e^{i(kx-\omega t)}$;

$\vec{b}=(b_{x},b_{y},b_{z})=(b_{1},b_{2},b_{3})e^{i(kx-\omega t)}$

where $u_{0}$, $v_{0}$, $w_{0}$; $j_{1}$, $j_{2}$, $j_{3}$, $b_{1}$, $b_{2}$, $b_{3}$, $\theta_{0}$ are constants.

  With the help of Max well’s relations, we achieve the following relation:

[TABLE]

Above relation implies that $b_{x}=0.$ Using equation (3) we obtain

[TABLE]

which is in agreement with the following value of $\vec{J}$

[TABLE]

and

[TABLE]

Thus the term $\overrightarrow{J}\times\overrightarrow{B}$ can be replaced by the term $\overrightarrow{J}\times\overrightarrow{B_{0}}$.

  Substituting values of the quantities $\vec{u}$ and $\theta$ in the equation of heat conduction , we achieve the following relation:

[TABLE]

where

[TABLE]

We obtain $\vec{E}=(E_{x},E_{y},E_{z})=(E_{x},{\frac{\omega b_{z}}{k}},-{\frac{\omega b_{y}}{k}})$

  Now, making use of the field quantities $\vec{u}$, $\vec{J}$ and $\vec{E}$ and comparing both sides of the modified Ohm’s law (in which $\vec{B}$ is replaced by $\vec{B_{0}}$), we achieve the following three relations:

[TABLE]

[TABLE]

[TABLE]

Further, substituting the values of field quantities $\overrightarrow{u}$,$\overrightarrow{J}$,$\vec{B_{0}}$,$\vec{\theta}$ in the equation of motion, we achieve the following relations:

[TABLE]

[TABLE]

[TABLE]

Equations (14) and (15) can be rewritten as:

[TABLE]

[TABLE]

Equations [(16)-(20)] constitute a system of five equations with five unknowns namely $u_{0}$, $v_{0}$, $w_{0}$, $b_{2}$, $b_{3}$.

We can make further assumptions: we have taken that $\vec{b}$ is directed along y-axis and we consider $w_{0}=0$ provided that $(\mu k^{2}-\rho\omega^{2}\neq 0)$ so that $b_{3}=0$ (equation (18)). Hence, applied and perturbed magnetic field are taken as $(\overrightarrow{B_{1}},\overrightarrow{B_{2}},0)$ and $(0,\overrightarrow{b_{2}},0)$, respectively. On applying the above assumptions in the equations [(16)-(20)], all equations reduce into three following homogeneous equations having three unknowns $u_{0}$, $v_{0}$ and $b_{2}$ as

[TABLE]

[TABLE]

[TABLE]

In order to find the solution for the system of equations (21), (22) and (23) for u_{0},$$v_{0}and $b_{2}$ equations, we can write

[TABLE]

Further, we assume that the initial magnetic field is directed towards y axis i.e. $\vec{B_{0}}=(0,B_{2},0)$. Therefore $B_{1}=0$ in the determinent. For simplifying the system for achieving the solution, we introduce the following non- dimensional quantities in above:

$\chi=\frac{\omega}{\omega^{*}}$, \xi=\frac{kc_{1}}{\omega^{*}},\$$\epsilon_{H}=\frac{\omega^{*}\nu_{H}}{{c_{1}^{2}}}

$\epsilon_{\theta}=\frac{T_{0}\nu^{2}}{\rho^{2}C_{v}{c_{1}^{2}}},\ \nu_{H}=\frac{1}{\mu_{e}\sigma},$

$k_{1}=\frac{K^{*}}{\rho C_{v}c_{1}^{2}},\ k_{2}.\frac{K\omega^{*}}{\rho C_{v}c_{1}^{2}}$

$c_{1}=\sqrt{\frac{(\lambda+2\mu)}{\rho}}$, $c_{2}=\sqrt{\frac{\mu}{\rho}}$

where $c_{1}$ is the longitudinal elastic wave velocity and $c_{2}$ is the transverse elastic wave velocity. We further assume that

$s=\frac{c_{2}}{c_{1}}$, $R_{H}=\frac{B_{2}^{2}}{\rho c_{1}^{2}\mu_{e}}$.

where $R_{H}$ is the magnetic pressure number. Substituting the value of $\alpha$ from eqn. (12) and employing all dimensionless quantities in the expansion of the above determinant, we find the following dispersion relation:

[TABLE]

The first part $(s^{2}\xi^{2}-\chi^{2})=0$ in (24) corresponds to a transverse elastic wave which is clearly found to be uncoupled by thermal and magnetic field. Hence, we take the second part of (24) as

[TABLE]

Equation (25) is clearly identified as the dispersion relation for coupled thermal-dilatational-electrical waves propagating in the medium in the present context. With the help of this relation, we will characterize the behaviour of plane waves propagating in the present context. We will specially concentrate on the analysis of the important wave characterizations like, phase velocity, specific loss and penetration depth. For solving equation (25), we attempt to find the perturbation solution of the dispersion equation for small values of thermoelastic coupling constant$\epsilon_{\theta}$. Therefore, substituting $\epsilon_{\theta}$= 0 in the dispersion relation, we achieve the following solutions:

[TABLE]

where $A=\frac{k_{1}+ik_{2}\chi}{k_{1}^{2}+k_{2}^{2}\chi^{2}}=a_{1}+ib_{1}$

[TABLE]

Equation (27) being an equation of degree 4, we consider the roots of above equation (27) as $\pm\alpha_{1}$and $\pm\alpha_{2}$.

where

[TABLE]

Which implies:

[TABLE]

Therefore

[TABLE]

[TABLE]

where

$a=\sqrt{\frac{\chi^{2}(1+R_{H})^{2}-\chi^{4}\epsilon_{H}^{2}+\chi^{2}[\{(1+R_{H})^{2}+\chi^{2}\epsilon_{H}^{2}-4\chi\epsilon_{H}\sqrt{R_{H}}\}\{(1+R_{H})^{2}+\chi^{2}\epsilon_{H}^{2}+4\chi\epsilon_{H}\sqrt{R_{H}}\}]^{\frac{1}{2}}}{2}}$

$b=\sqrt{\frac{-\chi^{2}(1+R_{H})^{2}+\chi^{4}\epsilon_{H}^{2}+\chi^{2}[\{(1+R_{H})^{2}+\chi^{2}\epsilon_{H}^{2}-4\chi\epsilon_{H}\sqrt{R_{H}}\}\{(1+R_{H})^{2}+\chi^{2}\epsilon_{H}^{2}+4\chi\epsilon_{H}\sqrt{R_{H}}\}]^{\frac{1}{2}}}{2}}$

As we are attempting to find the perturbation solution for small values of thermoelastic coupling constant $\epsilon_{\theta}.$ Therefore $\xi^{2}$ can be written in the following form-

[TABLE]

[TABLE]

[TABLE]

Substituting equations (32), (33) and (34) in equation (25) and comparing the lowest power of $\epsilon_{\theta}$ on both sides of equation and neglecting the terms of $O(\epsilon_{\theta}^{2})$, we obtain the following solution:

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

Clearly, the above expressions for $\xi_{i},i=1,2,3$ (with $Im(\xi_{i}\leq 0)$ correspond to three modes of plane waves propagating inside the medium.

**4. Analytical expressions of various components of magneto-thermo-elastic plane wave **  In order to calculate several components of waves like, phase velocity, specific loss and penetration depth we use the following formulae:

Phase velocity:

[TABLE]

Specific loss:

[TABLE]

Penetration depth:

[TABLE]

From the solutions obtained as above, we can clearly observe that there are three modes of waves which are dissimilar to each other. We denote the first wave as modified (quasi-magneto) elastic (dilatational) wave, the second one as modified (quasi-magneto) thermal wave and the third one as (quasi-magneto) electrical wave. It is observed that both the elastic and thermal mode wave are influenced by the thermoelastic coupling constants $\epsilon_{\theta}$, magneto-elastic coupling constant $\epsilon_{H}$, as well as magnetic pressure number $R_{H}$. We will specially concentrate on these two modes. Since it is very difficult to characterize the nature of waves from analytical solutions obtained above. Therefore, the various components of magneto-thermoelastic plane wave like, phase velocity, specific loss and penetration depth for all three kinds of waves regarding GN-I, GN-II and GN-III theory of thermoelasticity by numerical results.

5. Numerical results: **Analysis of various features of plane wave **

From the analytical results obtained above, we conclude that three various modes of waves have been extracted from the coupled dispersion relation which are named as modified (quasi-magneto) elastic (dilatational) wave, modified (quasi-magneto) thermal wave and (quasi-magneto) electrical wave. In order to illustrate the analytical solution and to have a critical analysis of the nature of waves with the variation of frequency, we will make an attempt to show the variations of various wave components of the identified waves in three different models of thermolelasticity with the help of numerical results. We will also assess the limiting behaviour of wave components.

We make an attempt to represent the plane wave characterizations numerically with the help of computational work by using programming on Mathematica. Copper material has been chosen for the purpose of numerical evaluations. The physical data for our problem are taken as

$\epsilon_{T}=0.0168$

$\lambda=7.76\times 10^{10}Nm^{-2}$

$\mu=3.86\times 10^{10}Nm^{-2}$

$\mu_{e}=4*10^{-7}NA^{-2}$

$\sigma=5.7*10^{7}Sm^{-1}$

$\omega^{*}=1.72*10^{11}sec^{-1}$

$\rho=8954Kgm^{-3}$

  We assume: $k_{1}=1$and $k_{2}=1$.

We will specially analyze the physical parameters of waves like, phase velocity, specific loss and penetration depth. Using the formulae given by equations (39)-(41), we compute these components of waves of different modes and display our results in various Figures. Our analytical work is devoted to GN-III theory of magneto thermoelasticity for a finitely conducting medium. Furthermore, in order to make a comparison between GN-I, GN-II & GN-III models, we carry out computational work for these two special cases too (GN-I and GN-II). Due to this reason, we have considered the variations in the propagation of waves for GN-I and GN-III model of thermoelasticity in each figure. Since we are obtaining a very different nature of waves in GN-II theory of thermoelasticity as compare to GN-I and GN-II theories of thermoelasticity; therefore, nature of waves in GN-II model of thermoelasticity has been represented in seperate plots. We have taken the values of the non dimensional thermal conductivity rate i.e. $k_{1}=1,$ $k_{2}=1$ in GN-III model. $k_{1}=0$ represents the case of GN-I model. $k_{2}=0$ represents the case of GN-II model. In each Figure, the thin solid line is used for $k_{1}=0$(GN-I), thick dashed line is for $k_{1}=1$ and $k_{2}=1$ (GN-III), We find that the trend of all the wave components of the third mode wave, i.e., electric mode wave are almost similar in the contexts of GN-I, GN-II and GN-III model. However, a prominent difference in the results under GN-I model, GN-II model and GN-III model is indicated for the quasi-magneto elastic wave and quasi-magneto thermal mode wave. We highlight several specific features arisen out of the numerical results in the following sections:

5.1. Analysis of phase velocity:

Using the formula given by (39), we compute the phase velocity of all three modes of waves. Figs.1(a) and 1(b) display the variation of phase velocity of modified quasi-magneto dilatational wave and phase velocity of modified quasi-magneto thermal wave, respectively in GN-I and GN-III model of thermoelasticity. Fig. 1(c) represents the variation of phase velocities of quasi-magneto dilatational wave and quasi-magneto thermal wavein GN-II model of thermoelasticity. We observe from Fig.1(a) that the starting from a intial value, phase velocity of quasi-magneto dilatational wave propagates in increasing mode and after giving a extreme value it becomes constant very soon. In GN-I model, after achieving a local maximum value, wave goes down and becomes constant. But in GN-III model, wave is achieving the local maximum value and at local maximum value it becomes constant. We obtain a slight difference between the extreme values as well as constant values in phase velocity in both models. Surprisingly in all cases each figure goes towards a constant limiting value which is nearer to 1.

Fig. 1(b) describe the variations of phase velocity of quasi-magneto thermal wave in GN-I and GN-III models of thermoelasticity. In the present context, waves are propagating from a constant limiting value greater than 0 when frequency tends to zero value and then starts increasing with respect to frequency and goes towards infinity. Here, we are obtaining the similar nature of wave under GN-I and GN-III models of thermoelasticity.

Here, we are achieving a significant result that a prominent difference in the phase velocity of quasi-magneto dilatational wave in GN-I and GN-III models; however mode of propagation for quasi-magneto dilatational wave is similar in nature ion GN-I and GN-III models of thermoelasticity i.e. quasi-magneto thermal mode wave exhibits a remarkable difference between two models (GN-I and GN-III) as compare to quasi-magneto dilatational wave.

As we have already stated above that when $k_{2}=0$, we obtain the equations for Green and Naghdi-II (GN-II) model of magneto-thermoelasticity. Fig. 1(c) exhibits the behaviour of phase velocities of quasi-magneto dilatational wave and quasi-magneto thermal wave in GN-II model of thermoelasticity. Here we obtain that phase velocity of quasi-magneto dilatational wave as well as of quasi-magneto thermal mode wave show almost similar trend of variation and have constant limiting value. Phase-velocity of thermal wave is greater that the phase velocity of elastic mode wave and both of them reaches to constant value as we increase the frequency. In this case, we achieve completely different nature of quasi-magneto dilational wave and quasi-magneto thermal wave in comparison to the results predicted by two models GN-I and GN-III. We have already identified that the speed of modified (quasi magneto) thermal wave increases with the increases of frequency in cases of GN-I and GN-III models and goes towards infinity, but it is constant in GN-II model and this is the special feature of GN-II model of magneto-thermoelasticity that the speed of magneto-thermal wave is always finite in this case.

In the present case, wave is unaffected with the values of $k_{1}$and $k_{2}$ and the predictions of GN-I, GN-II and GN-III model is almost similar in nature. Therefore, Fig. 4(a) represents the behaviour of phase velocity of quasi-magneto electrical wave in all models . Here, phase velocity of waves increases rapidly with respect to frequency.

5.2. Analysis of specific loss:

Specific loss is defined as the loss of energy per stress cycle as defined by formula (40). Figures 2(a) and 2(b) exhibit the variations of specific loss of quasi-magneto dilatational mode wave and quasi-magneto thermal wave, respectively. In quasi-magneto dilatational wave, initially specific loss is 0 but it starts increasing with respect to frequency and giving a maximum value of specific loss it starts decreasing and goes towards 0 value. This maximum value is different for GN-I and GN-III however the difference is not prominent.

Nature of quasi-magneto thermal wave is completely different from quasi-magneto dilatational wave. It starts from a initial value and after achieving a maximum value which is nearer to 1; it becomes constant. Here we are observing that energy loss is greater in quasi-magneto dilatational wave as compare to quasi-magneto dilatational wave. There is one significant point is that in quasi-magneto dilatational wave, maximum value of specific loss is greater in GN-I as compare to GN-III which implies that GN-III model exhibits better results as compare to GN-I model.

Fig.2(c) represents the combined study of specific loss for Quasi-magneto-dilatational wave and Quasi-Magneto-thermal wave in GN-II model of thermoelasticity. Here the picture is complely changed as compare to GN-I and GN-III model of thermoelasticity. Here, the value of loss is very less nearer to 0. As we have studied that there is no specific loss in GN-II model of thermoelasticity without magnetis field But here we see that magnetic field is effective which means that in presence of magnetic field, the ideal nature of GN-II variates and we are finding wave-energy-loss. Nature of loss for both type of waves are almost similar.

Fig.4(b) expresses the behaviour of specific loss for quasi-magneto-electrical wave. Here we are achieving a constant value of loss and the value is appromately 1. Furthermore, we obtain the result that there is no variation in the behaviour of quasi-magneto electrical wave for GN-I, GN-II and GN-III model of thermoelasticity i.e. we obtain the same result in all three models.

5.3. Analysis of penetration depth:

The behaviour of penetration depth of quasi-magneto dilatational wave and quasi-magneto thermal wave can be seen from Figs. 3(a,b), respectively which show that the penetration depth of quasi-magneto dilatational wave decreases from infinite value as the frequency increases and reaches to constant limiting value nearer to119. But we observe significant differences among the plots of this field for different GN-I and GN-III. We see that in GN-III model, penetration depth of quasi-magneto dilatational wave decreases from infinite value and after giving a local minimum it starts increases and become constant at the limiting value nearer to 119 However in GN-I model, penetration depth decreases from infinite and becomes constant that in GN-III, is more prominent for lower frequency values, although in all cases, it finally reaches constant value nearer to 119. Here, there is no significicant difference between the constant limiting values under two models (GN-I and GN-III).

For quasi-magneto thermal wave, penetration depth decreases ferom infinite and goes down to a very low constant value nearer to 0. THere is no significant difference between two models GN-I and GN-III.

Fig.3(c) displays the penetration depth of Quasi-magneto-dilatational wave and Quasi-Magneto-thermal wave in GN-II model. Here we are getting completely different results as compare to GN-I and GN-III. In the present context, penetration depth decreases from infinite and becoms constant at a very high value nearer to in power of $10^{8}$which is very large as compare to the constant limiting value in case of GN-I and GN-III. Here we can interpretate that the wave is nearer to the ideal behaviour because as we have stated that there is no enery loss in GN-II without magnetic field but in the presence of magnetic field, the nature of waves variates slightly in GN-II and shows that the nature of waves variates from ideality and giving a little loss in wave-energy. Moreover there is no the prominent difference in the nature of variation in depth for both qausi-magneto dilational wave and quasi-magneto thermal wave.

Penetration depth of quasi-magneto electrical wave exhibits rapidly decreasing behaviour with respect to frequency and becomes constant with the finite value in both the GN-I and GN-III models of thermoelasticity (see Fig. 4(c)) and there is no significant differences in the results predicted by GN-I and GN-III models.

**7. Summary and observations: **

In the present work, dispersion relation solutions for the plane wave propagating in a magneto-thermoelastic media with finite electrical conductivity have been determined by employing Green and Naghdi theory of thermoelasticity of type-III. We have made a comparative study of GN-I, GN-II and GN-III theory of thermoelasticity in presence of an external magnetic field. From the derived dispersion relation solution, transverse and longitudinal plane waves are investigated. We find that transverse mode elastic wave is uncoupled from the thermal and magnetic field; Further a general dispersion relation associated to the coupled dilatational-thermal and electrical wave is identified and we make attempt to extract three different modes of waves from this coupled dispersion relation. These waves are identified as quasi-magneto dilatational wave, quasi-magneto thermal wave and quasi-magneto electrical wave. The quasi-magneto-electrical wave is found to have similar variation under GN-I, GN-II, GN-III theory. However, significant differences are obtained in other two modes, namely quasi-magneto dilatational and quasi-magneto thermal mode wave predicted by three different models. Hence, we pay attention to these three modes and analyze various wave components like, phase velocity, specific loss and penetration depth. The behaviour of the wave components in limiting cases of frequency values have been investigated with the help of graphical plots. Various features are highlighted. It is believed that this study would be useful due to its various applications in different areas of physics, geophysics etc. The most highlighted features of the present investigation can be summarized as follows:

Significant resemblance and non- resemblance among the results under GN-I, GN-II and GN-III theory of thermoelasticity have been identified. 2. 2.

The phase velocity of thermal mode wave is found to be an increasing function of frequency under GN-I and GN-III models. 3. 3.

Quasi- magneto dilatational and thermal mode waves propagate faster in the theory of type GN- I in comparison to GN-III theory of thermoelasticity. However, phase velocity of quasi magneto-electric wave is unaffected whether we employ GN-I, GN-II or GN-III theory. Quasi-magneto dilatational and thermal wave is found to be nearer to non-dispersive in GN-II model, however due to presence of magnetic field, there is very less specific loss in waves and that is why we are getting a constant limiting value of penetration depth however the constant value is very high but not equal to infinite in the context of GN-II theory of thermoelasiticity.

Penetration depth has a less finite value in the case of GN-I and GN-III theory of thermoelasticity. However in GN-II, we see that penetration depth for both waves namely, quasi-magneto dilatational wave and quasi-magneto thermal wave is very high since waves are propagating with constant speed and there is a very less specific energy loss of waves. This is a very distinct feature of GN-II model.

In view of above points, we can conclude that for coupled magneto-thermoalstic problem, GN-II model exhibits realistic behaviour in comparison to GN-I and GN-III models w.r.t. phase velocity of thermal wave, but when we analyze the behaviour of penetration depth, we find that predictions of GN-I and GN-III theory is more realistic as compared to GN-II model as we obtain in this case a very high nearer to infinite penetration depth which is also physically unrealistic prediction by GN-II model.