A Comparative Study of magneto-thermo-elastic wave propagation in a finitely conducting medium under thermoelasticity of type I, II, III
aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b A Comparative Study of magneto-thermo-elastic wavepropagation in a finitely conducting medium underthermoelasticity of type I, II, III *Dr.Rakhi Tiwari, Prof. J. C. Mishra *Department of Mathematics, Babasaheb Bhimrao Ambedakar Bihar UniversityDepartment of Mathematics, Indian Institute of Engineering, Science and Technology*E-mail: [email protected]
Abstract:
The present work is concerned with the propagation of electro-magneto-thermoelastic planewaves of assigned frequency in a homogeneous isotropic and finitely conducting elastic medium permeatedby a primary uniform external magnetic field. We formulate our problem under the theory of Greenand Naghdi of type-III (GN-III) to account for the interactions between the elastic, thermal as wellas magnetic fields. A general dispersion relation for coupled waves is deduced to ascertain the natureof waves propagating through the medium. Perturbation technique has been employed to obtain thesolution of dispersion relation for small thermo-elastic coupling parameter and identify three differenttypes 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 wavecomponents numerically to observe their variations with frequency. The effect of presence of magneticfield is analyzed. Comparative results under theories of type GN-I, II and III have been presentednumerically 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-Iand type-III. Furthermore, the thermal mode wave is observed to propagate with finite phase velocityin case of GN-II model, whereas the phase velocity of thermal mode wave is found to be an increasingfunction of frequency in other two cases. We achieve significant variations among the results predictedby all three theories.
Keywords:
Generalized electro-magneto-thermoelasticity; Green and Naghdi theory; Thermoelasticityof 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 inter-actions between magnetic, thermal and mechanical fields in a thermo-elastic solid in presence of magneticfield. The magnetic field originating inside nuclear reactors and the extremely high temperature affecttheir 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, plasmaphysics, damping of acoustic wave in magnetic fields and other related topics involving sensing and actua-tion. 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. Theconcept of elasto-magnetic coupling effects in materials have applications in health monitoring of civilstructures. Basically, the coupled theory of electro-magneto-thermoelasticity is the combination of two1ifferent disciplines namely theory of electro-magnetism and theory of thermoelasticity. A systematicpresentation 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 fieldhas 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 tocharacterize harmonically time dependent plane waves of assigned frequency in a homogeneous, isotropicand unbounded solid. But unfortunately, the classical theory of thermoelasticity by Biot [6] is unrealisticfrom 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 thistheory, some theories have been developed as extended theory of classical thermoelasticity. The theoriesgiven by Lord and Shulman [7] and Green and Lindsay [8] are two theories that have been extensivelyemployed 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 GreenNaghdi -I, II and III (GN-I, GN-II and GN-III) theory of thermoelasticty. The linearized version of GN-Itheory is identical to the classical theory of thermoelasticity. GN-II theory accounts for finite speed ofthermal wave, but with no energy dissipation. This implies that undamped waves of second sound effectis identified. However GN-III theory does not follow GN-I and GN-II but shows GN-I & GN-II theoriesas 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 effectsof thermal relaxation parameters. Roychoudhuri [14,15] and Roychoudhuri and Debnath [16,17] stud-ied propagation of magneto-thermoelastic plane waves in rotating thermoelastic media permeated by aprimary 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 byemploying GN-II theory of thermoelasticity. Puri and Jordan [19] and later on, Kothari and Mukhopad-hyay [20] used GN-III theory to explain the propagation of thermoelastic plane waves. Roychoudhuriand Banerjee (Chattopadhyay) [21] discussed magnetoelastic plane waves in rotating media under ther-moelasticity of type-II model. Das and Kanoria [22] also worked on magneto-thermo-elastic waves in amedium with infinite conductivity in the context of GN-III theory. Abbas and Alzahrani [23] studiedGreen-Naghdi Model in a 2D problem of a mode I crack in an Isotropic Thermoelastic Plate. MoreoverEzzat, El-Karamany and El-Bary [24] presented Two-temperature theory in Green–Naghdi thermoelas-ticity with fractional phase-lag heat transfer. Tiwari and Mukhopadhyay [25] studied electromagneticwave propagation in GN-II model of thermoelasticity.At present, our motive of this work is to investigate the propagation of magneto-thermo-elastic planewaves of assigned frequency in an infinite, homogeneous, isotropic, thermally and electrically conductingsolid in the context of GN-I, GN-II and GN-III theory of thermoelasticity. We have taken the media havingfinite conductivity permeated by a primary uniform external magnetic field. The basic governing equationsare 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 ofplane waves are identified by solving the general dispersion relation analytically and various physicalcharacterizations of plane waves are deduced. The problem is illustrated with numerical results of variouswave characterizations and the nature of waves of different modes are critically analyzed. Since GN-I andGN-II an be obtained as a special cases of GN-III model of thermoelasticity. Therefore, the nature ofelectro-magnetic harmonic plane waves have been characterized under the theories of Green and Naghdiof types I, II and III (GN-I, GN-II and GN-III) using numerical results and conclusions are explained tohighlight 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 solid2ermeated by a primary magnetic field ~B = ( B , B , B ) is considered. The media is characterized bythe density ρ and Lame’s elastic constants λ and µ .Using a fixed rectangular cartesian coordinate system (x,y,z ) and employing the thermoelasticity theoryof Green and Naghdi [[9]-[11]], the equations of motion and the equation of heat conduction in the presenceof magnetic field in the absence of external body force (mechanical) and heat sources can be representedin the following manner:Equation of motion: µ ∇ ~u + ( λ + µ ) ~ ∇ ( ~ ∇ .~u ) + ~J × ~B − ν ~ ∇ θ = ρ ¨ ~u (1)Equation of heat conduction [9]: K ∗ ∇ θ + K ∇ ˙ θ = ρC v ¨ θ + νT ¨ u i,i (2)where ~J is the current density vector and ~J × ~B is the electromagnetic body force (Lorentz force).Here, ~u is the displacement field, θ is the temperature above reference temperature T . Total magneticfield ~B = ~B + ~b is assumed to be small so that the products with ~b and ˙ ~u and their derivatives can beneglected for the linearization of the field equations. ~b = ( b x , b y , b z ) is the perturbed magnetic field. λ, µ are Lame’s constants and ν = (3 λ + 2 µ ) α t , where α 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 constantvolume, 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 ther-moelasticity and if we substitute K = 0 in equation (2) i.e. the thermal conductivity rate isabsent, 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 supplementedby 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 ~ ∇ × ~H = ~J (3) ~ ∇ × ~E = − ∂ −→ B∂t (4)where ~B = µ e ~H and µ e is the magnetic permeability. ~ ∇ . ~B = 0 (5)Generalized Ohm’s law is given by ~J = σ [ ~E + ∂u∂t × ~B ] (6)where σ is the electrical conductivity, ∂u∂t is the particle velocity of the medium. Here the small effect oftemperature gradient on ~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 fieldquantities are proportional to e i ( kx − ωt ) , where k is the wave number and ω is the angular frequency ofplane waves. Here, we have assumed that ω is real and k may be complex quantity such that Im ( k ) ≤ ~u = ( u, v, w ) = ( u , v , w ) e i ( kx − ωt ) , θ = θ e i ( kx − ωt ) ; ~E = ( E x , E y , E z ) , ~J = ( j , j , j ) e i ( kx − ωt ) ; b = ( b x , b y , b z ) = ( b , b , b ) e i ( kx − ωt ) where u , v , w ; j , j , j , b , b , b , θ are constants.With the help of Max well’s relations, we achieve the following relation: div −→ b = 0 (7)Above relation implies that b x = 0 . Using equation (3) we obtain µ e −→ J = ~ ∇ × −→ b . (8)which is in agreement with the following value of ~J~J = [0 , − ikb z µ e , ikb y µ e ] (9)and ~J × ~B = [ − ik ( b z B + b y B ) µ e , ikb y B µ e , ikb z B µ e ] (10)Thus the term −→ J × −→ B can be replaced by the term −→ J × −→ B .Substituting values of the quantities ~u and θ in the equation of heat conduction , we achieve the followingrelation: θ = αu (11)where α = iνθ kω K ∗ k − iKk ω − ρC v ω (12)We obtain ~E = ( E x , E y , E z ) = ( E x , ωb z k , − ωb y k )Now, making use of the field quantities ~u , ~J and ~E and comparing both sides of the modified Ohm’s law(in which ~B is replaced by ~B ), we achieve the following three relations: σ [ E x − iω ( qB − rB )] = 0 (13) σ [ ωb z k − iω ( rB − pB )] = − ikb z µ e (14) σ [ − ωb y k − iω ( pB − qB )] = ikb y µ e (15)Further, substituting the values of field quantities −→ u , −→ J , ~B , ~θ in the equation of motion, we achieve thefollowing relations: u ( − ρω + ( λ + 2 µ ) k + iναk ) + ikµ e ( b B + b B ) = 0 (16) v ( − ρω + µk ) − ikµ e ( b B ) = 0 (17) w ( − ρω + µk ) − ikµ e ( b B ) = 0 (18)Equations (14) and (15) can be rewritten as: u σ ( iωB ) + w ( − σiωB ) + b [ ikµ e − σωk ] = 0 (19)4 σ ( − iωB ) + v ( σiωB ) − b [ ikµ e + σωk ] = 0 (20)Equations [(16)-(20)] constitute a system of five equations with five unknowns namely u , v , w , b , b .We can make further assumptions: we have taken that ~b is directed along y-axis and we consider w = 0provided that ( µk − ρω = 0) so that b = 0 (equation (18)). Hence, applied and perturbed magnetic fieldare taken as ( −→ B , −→ B ,
0) and (0 , −→ b , u , v and b as u [ − ρω + ( λ + 2 µ ) k + iναk ] + ikB b µ e = 0 (21) v ( − ρω + µk ) − ikB b µ e = 0 (22) u ( σiωB ) − v σ ( iωB ) + b ( ikµ e + σωk ) = 0 (23)In order to find the solution for the system of equations (21), (22) and (23) for u ,v and b equations,we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ρω + ( λ + 2 µ ) k + iναk ikB µ e µk − ρω − ikB µ e iσωB ikµ e + σωk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0Further, we assume that the initial magnetic field is directed towards y axis i.e. ~B = (0 , B , B = 0 in the determinent. For simplifying the system for achieving the solution, we introducethe following non- dimensional quantities in above: χ = ωω ∗ , ξ = kc ω ∗ , ǫ H = ω ∗ ν H c ǫ θ = T ν ρ C v c , ν H = µ e σ ,k = K ∗ ρC v c , k . Kω ∗ ρC v c c = q ( λ +2 µ ) ρ , c = q µρ where c is the longitudinal elastic wave velocity and c is the transverse elastic wave velocity. We furtherassume that s = c c , R H = B ρc µ e .where R H is the magnetic pressure number. Substituting the value of α from eqn. (12) and employingall dimensionless quantities in the expansion of the above determinant, we find the following dispersionrelation: ( s ξ − χ )[( ξ − χ )( ξ k − χ ) − ik ξ χ ( ξ − χ ) − ǫ θ ξ χ ]( χ + iξ ǫ H )+ R H χξ [( ξ k − χ ) − ik ξ χ ] = 0 (24)The first part ( s ξ − χ ) = 0 in (24) corresponds to a transverse elastic wave which is clearly found tobe uncoupled by thermal and magnetic field. Hence, we take the second part of (24) as[( ξ − χ )( ξ k − χ ) − ik ξ χ ( ξ − χ ) − ǫ θ ξ χ ]( χ + iξ ǫ H ) + R H χξ [( ξ k − χ ) − ik ξ χ ] = 0 (25)Equation (25) is clearly identified as the dispersion relation for coupled thermal-dilatational-electricalwaves propagating in the medium in the present context. With the help of this relation, we will char-acterize the behaviour of plane waves propagating in the present context. We will specially concentrate5n the analysis of the important wave characterizations like, phase velocity, specific loss and penetrationdepth. For solving equation (25), we attempt to find the perturbation solution of the dispersion equationfor small values of thermoelastic coupling constant ǫ θ . Therefore, substituting ǫ θ = 0 in the dispersionrelation, we achieve the following solutions: ξ = Aχ (26)where A = k + ik χk + k χ = a + ib [( ξ − χ )( χ + iξ ǫ H ) + R H χξ ] = 0 (27)Equation (27) being an equation of degree 4, we consider the roots of above equation (27) as ± α and ± α .where α , = iχ ǫ H − χ (1 + R H ) ± (cid:8) χ (1 + R H ) − χ ǫ H − χ (1 + R H ) ǫ H i + 4 iǫ H χ (cid:9) iǫ H (28)Which implies: α , = iχ ǫ H − χ (1 + R H ) ± ( a + ib )2 iǫ H (29)Therefore α = iχ ǫ H − χ (1 + R H ) + ( a + ib )2 iǫ H (30) α = iχ ǫ H − χ (1 + R H ) − ( a + ib )2 iǫ H (31)where a = r χ (1+ R H ) − χ ǫ H + χ [ { (1+ R H ) + χ ǫ H − χǫ H √ R H }{ (1+ R H ) + χ ǫ H +4 χǫ H √ R H } ] b = r − χ (1+ R H ) + χ ǫ H + χ [ { (1+ R H ) + χ ǫ H − χǫ H √ R H }{ (1+ R H ) + χ ǫ H +4 χǫ H √ R H } ] As we are attempting to find the perturbation solution for small values of thermoelastic couplingconstant ǫ θ . Therefore ξ can be written in the following form- ξ = α + n ǫ θ + O ( ǫ θ ) (32) ξ = α + n ǫ θ + O ( ǫ θ ) (33) ξ = Aχ + n ǫ θ + O ( ǫ θ ) (34)Substituting equations (32), (33) and (34) in equation (25) and comparing the lowest power of ǫ θ onboth sides of equation and neglecting the terms of O ( ǫ θ ), we obtain the following solution: ξ = α [1 + Aǫ θ χ ( χ + iα ǫ H ) G ] (35) ξ = α [1 + Aǫ θ χ ( χ + iα ǫ H ) G ] (36)6 = Aχ [1 + Aǫ θ χ (1 + iAχǫ H )( A − χ + iAǫ H χ ) + R H Aχ ] (37)where G , = (( χ + iα , ǫ H )( α , − χ ) + χR H α , ) + ( α , − χ )((1 + R H ) χ + 2 iα , ǫ H − iǫ H χ ) (38)Clearly, the above expressions for ξ i , i = 1 , , Im ( ξ i ≤
0) correspond to three modes of plane wavespropagating 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 depthwe use the following formulae:
Phase velocity: V , , = χRe [ ξ , , ] (39) Specific loss: S , , = 4 π | Im [ ξ , , ] Re [ ξ , , ] | (40) Penetration depth: D , , = 1 | Im [ ξ , , ] | (41)From the solutions obtained as above, we can clearly observe that there are three modes of waves which aredissimilar 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) electricalwave. It is observed that both the elastic and thermal mode wave are influenced by the thermoelasticcoupling constants ǫ θ , magneto-elastic coupling constant ǫ H , as well as magnetic pressure number R H . Wewill specially concentrate on these two modes. Since it is very difficult to characterize the nature of wavesfrom analytical solutions obtained above. Therefore, the various components of magneto-thermoelasticplane wave like, phase velocity, specific loss and penetration depth for all three kinds of waves regardingGN-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 havebeen 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 orderto illustrate the analytical solution and to have a critical analysis of the nature of waves with the variationof frequency, we will make an attempt to show the variations of various wave components of the identifiedwaves in three different models of thermolelasticity with the help of numerical results. We will also assessthe limiting behaviour of wave components.We make an attempt to represent the plane wave characterizations numerically with the help ofcomputational work by using programming on Mathematica. Copper material has been chosen for thepurpose of numerical evaluations. The physical data for our problem are taken as ǫ T = 0 . λ = 7 . × N m − µ = 3 . × N m − µ e = 4 ∗ − N A − σ = 5 . ∗ Sm − ω ∗ = 1 . ∗ sec − = 8954 Kgm − We assume: k = 1and k = 1.We will specially analyze the physical parameters of waves like, phase velocity, specific loss andpenetration depth. Using the formulae given by equations (39)-(41), we compute these components ofwaves 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 acomparison between GN-I, GN-II & GN-III models, we carry out computational work for these two specialcases too (GN-I and GN-II). Due to this reason, we have considered the variations in the propagationof waves for GN-I and GN-III model of thermoelasticity in each figure. Since we are obtaining a verydifferent nature of waves in GN-II theory of thermoelasticity as compare to GN-I and GN-II theories ofthermoelasticity; therefore, nature of waves in GN-II model of thermoelasticity has been represented inseperate plots. We have taken the values of the non dimensional thermal conductivity rate i.e. k = 1 ,k = 1 in GN-III model. k = 0 represents the case of GN-I model. k = 0 represents the case of GN-IImodel. In each Figure, the thin solid line is used for k = 0(GN-I), thick dashed line is for k = 1and k = 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, aprominent difference in the results under GN-I model, GN-II model and GN-III model is indicated for thequasi-magneto elastic wave and quasi-magneto thermal mode wave. We highlight several specific featuresarisen out of the numerical results in the following sections: :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 phasevelocity of modified quasi-magneto thermal wave, respectively in GN-I and GN-III model of thermoe-lasticity. Fig. 1(c) represents the variation of phase velocities of quasi-magneto dilatational wave andquasi-magneto thermal wavein GN-II model of thermoelasticity. We observe from Fig.1( a ) that the start-ing from a intial value, phase velocity of quasi-magneto dilatational wave propagates in increasing modeand after giving a extreme value it becomes constant very soon. In GN-I model, after achieving a localmaximum value, wave goes down and becomes constant. But in GN-III model, wave is achieving the localmaximum value and at local maximum value it becomes constant. We obtain a slight difference betweenthe extreme values as well as constant values in phase velocity in both models. Surprisingly in all caseseach 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-IIImodels of thermoelasticity. In the present context, waves are propagating from a constant limiting valuegreater than 0 when frequency tends to zero value and then starts increasing with respect to frequencyand goes towards infinity. Here, we are obtaining the similar nature of wave under GN-I and GN-IIImodels 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-magnetodilatational wave is similar in nature ion GN-I and GN-III models of thermoelasticity i.e. quasi-magnetothermal mode wave exhibits a remarkable difference between two models (GN-I and GN-III) as compareto quasi-magneto dilatational wave.As we have already stated above that when k = 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 weobtain that phase velocity of quasi-magneto dilatational wave as well as of quasi-magneto thermal modewave show almost similar trend of variation and have constant limiting value. Phase-velocity of thermalwave is greater that the phase velocity of elastic mode wave and both of them reaches to constant value aswe increase the frequency. In this case, we achieve completely different nature of quasi-magneto dilationalwave and quasi-magneto thermal wave in comparison to the results predicted by two models GN-I and8N-III. We have already identified that the speed of modified (quasi magneto) thermal wave increaseswith the increases of frequency in cases of GN-I and GN-III models and goes towards infinity, but it isconstant in GN-II model and this is the special feature of GN-II model of magneto-thermoelasticity thatthe speed of magneto-thermal wave is always finite in this case.In the present case, wave is unaffected with the values of k and k and the predictions of GN-I, GN-IIand GN-III model is almost similar in nature. Therefore, Fig. 4( a ) represents the behaviour of phasevelocity of quasi-magneto electrical wave in all models . Here, phase velocity of waves increases rapidlywith respect to frequency. :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-magnetothermal wave, respectively. In quasi-magneto dilatational wave, initially specific loss is 0 but it startsincreasing with respect to frequency and giving a maximum value of specific loss it starts decreasing andgoes towards 0 value. This maximum value is different for GN-I and GN-III however the difference is notprominent.Nature of quasi-magneto thermal wave is completely different from quasi-magneto dilatational wave. Itstarts 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 modelexhibits 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 ascompare to GN-I and GN-III model of thermoelasticity. Here, the value of loss is very less nearer to 0. Aswe have studied that there is no specific loss in GN-II model of thermoelasticity without magnetis fieldBut here we see that magnetic field is effective which means that in presence of magnetic field, the idealnature of GN-II variates and we are finding wave-energy-loss. Nature of loss for both type of waves arealmost similar.Fig.4(b) expresses the behaviour of specific loss for quasi-magneto-electrical wave. Here we are achiev-ing a constant value of loss and the value is appromately 1. Furthermore, we obtain the result that thereis no variation in the behaviour of quasi-magneto electrical wave for GN-I, GN-II and GN-III model ofthermoelasticity i.e. we obtain the same result in all three models. :The behaviour of penetration depth of quasi-magneto dilatational wave and quasi-magneto thermalwave can be seen from Figs. 3( a,b ), respectively which show that the penetration depth of quasi-magnetodilatational wave decreases from infinite value as the frequency increases and reaches to constant limitingvalue nearer to119. But we observe significant differences among the plots of this field for differentGN-I and GN-III. We see that in GN-III model, penetration depth of quasi-magneto dilatational wavedecreases from infinite value and after giving a local minimum it starts increases and become constant atthe limiting value nearer to 119 However in GN-I model, penetration depth decreases from infinite andbecomes constant that in GN-III, is more prominent for lower frequency values, although in all cases, itfinally reaches constant value nearer to 119. Here, there is no significicant difference between the constantlimiting values under two models (GN-I and GN-III).For quasi-magneto thermal wave, penetration depth decreases ferom infinite and goes down to a verylow 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-thermalwave in GN-II model. Here we are getting completely different results as compare to GN-I and GN-III. Inthe present context, penetration depth decreases from infinite and becoms constant at a very high value9earer to in power of 10 which is very large as compare to the constant limiting value in case of GN-Iand GN-III. Here we can interpretate that the wave is nearer to the ideal behaviour because as we havestated 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 idealityand giving a little loss in wave-energy. Moreover there is no the prominent difference in the nature ofvariation 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 respectto frequency and becomes constant with the finite value in both the GN-I and GN-III models of thermoe-lasticity (see Fig. 4(c)) and there is no significant differences in the results predicted by GN-I and GN-IIImodels.
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 andNaghdi theory of thermoelasticity of type-III. We have made a comparative study of GN-I, GN-II andGN-III theory of thermoelasticity in presence of an external magnetic field. From the derived disper-sion relation solution, transverse and longitudinal plane waves are investigated. We find that transversemode elastic wave is uncoupled from the thermal and magnetic field; Further a general dispersion relationassociated to the coupled dilatational-thermal and electrical wave is identified and we make attempt toextract three different modes of waves from this coupled dispersion relation. These waves are identi-fied 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 andquasi-magneto thermal mode wave predicted by three different models. Hence, we pay attention to thesethree 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 withthe help of graphical plots. Various features are highlighted. It is believed that this study would be usefuldue to its various applications in different areas of physics, geophysics etc. The most highlighted featuresof the present investigation can be summarized as follows:1. Significant resemblance and non- resemblance among the results under GN-I, GN-II and GN-IIItheory of thermoelasticity have been identified.2. The phase velocity of thermal mode wave is found to be an increasing function of frequency underGN-I and GN-III models.3. Quasi- magneto dilatational and thermal mode waves propagate faster in the theory of type GN- I incomparison to GN-III theory of thermoelasticity. However, phase velocity of quasi magneto-electricwave is unaffected whether we employ GN-I, GN-II or GN-III theory. Quasi-magneto dilatationaland thermal wave is found to be nearer to non-dispersive in GN-II model, however due to presenceof magnetic field, there is very less specific loss in waves and that is why we are getting a constantlimiting value of penetration depth however the constant value is very high but not equal to infinitein 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 waveand quasi-magneto thermal wave is very high since waves are propagating with constant speed and thereis a very less specific energy loss of waves. This is a very distinct feature of GN-II model.1. In view of above points, we can conclude that for coupled magneto-thermoalstic problem, GN-IImodel exhibits realistic behaviour in comparison to GN-I and GN-III models w.r.t. phase velocityof thermal wave, but when we analyze the behaviour of penetration depth, we find that predictions10f GN-I and GN-III theory is more realistic as compared to GN-II model as we obtain in this casea very high nearer to infinite penetration depth which is also physically unrealistic prediction byGN-II model.