# Reflection of Plane Waves in Generalized Thermoelastic Half Space under the Action of Uniform Magnetic Field

Authors: Narottam Maity; S. P. Barik; P. K. Chaudhuri
Journal: International Journal of Engineering Mathematics
 (2016)

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## Abstract

Reflection of longitudinal displacement waves in a generalized thermoelastic half space under the action of uniform magnetic field has been investigated. The magnetic field is applied in such a direction that the problem can be considered as a two-dimensional one. The discussion is based on the three theories of generalized thermoelasticity: Lord-Shulman (L-S), Green-Lindsay (G-L), and Green-Naghdi (G-N) with energy dissipation. We compute the possible wave velocities for different models. Amplitude ratios have been presented. The effects of magnetic field on various subjects of interest are discussed and shown graphically.


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## Body

## 1. Introduction

It is known that mechanical loadings are not the only cause for deformation in elastic solids; thermal loadings also can play a vital role in producing deformations in structures and machines subject to generation and flow of heat. Thermoelasticity takes care of the deformations and stresses produced due to thermal loadings along with the deformation and stresses produced due to mechanical loadings. Clearly the governing equations of motion in thermoelasticity consist of the coupled equations involving mechanical and thermal stresses and the equation governing heat flow in the solid. The parabolic type of heat conduction equation as was adopted initially in the classical theory encountered a serious drawback in the sense that the speed of the heat propagation in the solid could be infinite, which is absolutely unrealistic. Naturally, scientists endeavoured to develop a new theory compatible with the reality. In 1967 Lord and Shulman [1] employed a modified version of the Fourier law of heat conduction and developed a theory known as generalized theory of thermoelasticity. The new thermoelastic equations yielded a finite velocity of heat propagation by introducing relaxation time into the energy equation, rendering the new thermoelastic field equations fully hyperbolic. In 1972 Green and Lindsay [2] proposed a theory of thermoelasticity with certain special features that contrasts with the previous theory having a thermal relaxation time. In this theory Fourier’s law of heat conduction is unchanged whereas the classical energy equation and stress-strain-temperature relations are modified. The two theories have been developed with independent ideas and one cannot be considered as a particular case of the other. Investigation of various problems characterizing the two theories has been discussed by Chandrasekharaiah [3, 4]. Further modifications in the constitutive equations of thermoelasticity were done by Green and Naghdi [5] to accommodate a wider class of heat flow problems.Another interesting field of recent study is the field of magnetothermoelasticity in which interacting effects of applied magnetic field on elastic and thermal deformations of a solid are studied. Such studies have applications in several areas, particularly in nuclear devices, biomedical engineering, and geomagnetic investigations. Some of the works related to the interaction of the electromagnetic field, the thermal field, and the electric field may be available in literature, namely, Abd-Alla and Al-Dawy [6, 7], Ezzat and Othman [8], Ezzat [9], Wang et al. [10], and Othman [11–13]. A problem of reflection of plane waves in a rotating transversely isotropic magnetothermoelastic half space has been discussed by Singh and Yadav [14]. Roychoudhuri and Banerjee [15] investigated viscoelastic plane waves in a rotating thermoelastic medium following Green-Naghdi model. Thermoelastic plane waves in a rotating isotropic medium have been studied by Ahmad and Khan [16]. The effects of rotation and relaxation times on plane wave propagation in a generalized thermoelastic medium have been investigated by Roychodhuri [17].The present paper aims at the study of reflection of longitudinal wave from the flat boundary of an elastic half space in generalized thermoelasticity. The half space is under the action of a uniform magnetic field and our objective is to study the behavior of reflection of longitudinal plane wave in presence of magnetic field. It is assumed that the direction of the magnetic field is such that the problem can be considered as two-dimensional. It is observed that there are two distinct types of longitudinal plane waves in the medium. The effects of magnetic field on the velocity of propagation of these types of waves have been shown graphically. Magnetic field effects on amplitude ratios of the waves have also been shown.


## 2. Field Equations

In our mathematical expressions we shall use bold letters to represent the vector quantities appearing in our discussion. The field equations governing the displacementu=ui(x,t) and the temperature T(x,t) for a homogeneous isotropic magneto elastic material in generalized magnetothermoelasticity in the absence of body force may be presented in vector form as (1)λ+2μ∇∇·u-μ∇×∇×u-γ1+αδ2kδδt∇T+J×B=ρu¨,(2)δ1k+δ2k+δ3k∂∂tK∇2T+δ3kK∗∇2T=ρCeδ1k+δ2kT˙+δ1k+δ2kα∗+δ3kT¨+γT∗δ2k+δ3k∂∂t+δ1k1+α∗∂∂tΔ˙,where J is the current density, B is the magnetic induction vector, B=μ0H, H is the total magnetic field vector, μ0 is the magnetic permeability of solid, λ,μ are Lame’s elastic constants, α,α∗ are relaxation times, K∗=Ce(λ+2μ)/4 is additional material constant, Ce is the specific heat, T is the temperature, T∗ is the absolute temperature, αt is the coefficient of linear thermal expansion, Δ is the dilatation, γ=(3λ+2μ)αt, K is the coefficient of thermal conductivity, ρ is the mass density, and δij is the Kronecker delta.Equation (2) is a single equation representing the generalized heat conduction equation for three different models depending on the values of suffix k. For k=1, (2) represents the Lord-Shulman (L-S) model, for k=2 it represents the Green-Lindsay (G-L) model, and for k=3 it represents Green-Naghdi (G-N) model. The thermal relaxation time satisfies the inequalities [2]  α≥α∗>0 in the case of G-L theory.The above field equations (1) and (2) are supplemented by the following constitutive equations:(3)τij=λekk+2μeij-γT+δ2kαT˙δij,in which τij are the components of symmetric stress tensor and eij are the components of strain tensor related to displacement components ui, as (4)eij=12ui,j+uj,i,and the indices i,j,k take values 1, 2, and 3.Due to the application of initial magnetic fieldH0, an induced magnetic field h, an induced electric field E, and current density vector J are developed, so the total magnetic field H=H0+h. For a slowly moving homogeneous electrically conducting solid, the linear equations of electrodynamics are(5)∇×h=J+ϵ0E˙,∇×E=-μ0h˙,∇·h=0,E=-μ0u˙×H,where ∇=(∂/∂x,∂/∂y,∂/∂z), ϵ0 is the electrical conductivity, and u˙ is the particle velocity. Here we ignore the small effect of temperature gradient on the current density vector J. The deformation vector is supposed to be small and the displacement vector is measured from the steady state deformed position.Equation (3) when expressed in terms of displacement vector uk becomes (6)τij=λuk,kδij+μui,j+uj,i-γT+δ2kαT˙δij.Equation (1) may be expressed in vector notation in the form(7)c12∇∇·u-c22∇×∇×u-c32∇T+μ0ρJ×H=u¨,where(8)c12=λ+2μρ,c22=μρ,c32=γρ1+αδ2kδδt.It is convenient to write the displacement vector u in terms of a scalar potential q and a vector potential Ψ in the form(9)u=∇q+∇×Ψwith ∇·Ψ=0.


## 3. Formulation of the Problem

We take origin of Cartesian coordinate system on the free surface of the thermoelastic solid occupying a half space.z-axis is drawn along the normal to the free surface pointing into the medium, while x and y axes are taken along free surface of the half space, which is assumed to be thermally insulated (Figure 1). The solid is under the action of a uniform magnetic field H0=(0,H0,0). The applied magnetic field will develop induced magnetic field h=(0,h,0), an induced electric field E=(E1,E2,E3), and current density J=(J1,0,J3).Figure 1
Geometry of the problem.To study the problem of reflection of a longitudinal plane wave at the free surface of the solid under thermal condition and magnetic field effect, we assume that the incident and the reflected waves are all lying on thex-z plane.Thus, for the problem under consideration the total magnetic fieldH=(0,H0+h,0) and the displacement vector u=(u,0,w). Substituting (9) into (7) we obtain the following equations:(10)c12+ca2∇2q-c12χ1+αδ2kδδtT-1+cbq¨=0,(11)c22∇2Ψ=1+cbΨ¨,where ca2=μ0H02/ρ,  cb=ca2/c02,χ=γ/λ+2μ, c0=1/μ0ϵ0. Clearly (10) involves both temperature and magnetic effect but (11) contains magnetic effect only. We also observe that if the medium is such that ϵ0=0 (i.e., no electrical conductivity), then magnetic effect is realisable in (10) only.Substitution of (9) into (2) gives(12)δ1k+δ2kK+δ3kK∗∇2T+δ3kK∇2T˙=ρCeδ1k+δ2kT˙+δ1k+δ2kα∗+δ3kT¨+γT∗δ2k+δ3k∂∂t+δ1k1+α∗∂∂t∇2q˙.Equations (10), (11), and (12) are three equations for the field functions q,Ψ, and T. A look at (11) reveals that the vector potential Ψ is not coupled with q and T, whereas the remaining two equations are coupled in q and T. It is also seen that (12) does not depend on the magnetic field.


## 4. Boundary Conditions

Assuming the surfacez=0 as stress free and considering the motion as two-dimensional in the x-z plane, the boundary conditions are(13)τ33=0=τ13on⁡z=0.Also, as the boundary z=0 of the half space is assumed to be thermally insulated, the boundary condition may be put as(14)∂T∂z=0onz=0.


## 5. Plane Wave Solution

For the propagation of a plane wave in the medium in the direction of the unit vectorν, the field functions q, Ψ, and T may be expressed as(15)q,Ψ,T=q0,Ψ0,T0Ux,t,where(16)Ux,t=Re⁡exp⁡-ζ+iκVν·rexp⁡-iκt,q0 and ϕ0 are constant scalars and Ψ0 is a constant vector, i=-1,ζ (≥0) is the wave attenuation parameter, V is the wave speed, and κ is the frequency.From (9) and (15) it follows that(17)u=νRe⁡q0-ζ+iκVexp⁡-ζ+iκVν·rexp⁡-iκt.This shows that the displacement vector is in the direction of wave propagation. We shall call these waves longitudinal displacement waves.Substituting (15) into (11) yields (18)c22-ζ+iκV2=-κ21+cb.It follows from (18) that if (19)ζ=0thenV2=c221+cb.Equation (19) shows that the speed of the transverse plane wave is affected by the presence of a magnetic field only. In case there is no magnetic field, V=c2=μ/ρ, the classical result.Substituting from (15) into (10) and (12) yields(20)-ζ+iκV2+κ2c62q0-χ∗T0=0,γT∗κ-ζ+iκV2δ1k+δ2ki+δ1kα∗κ+δ3kκq0+-ζ+iκV2δ1kK+δ2kK+δ3kiKκ+K∗+ρCeiκ+α∗κ2δ1k+δ2k+δ3kκ2T0=0.For nonzero q0 and T0 the coefficient determinant of the system of (20) should be zero. This leads to(21)-ζ+iκV4δ1k+δ2k+δ3kiKκ+K∗+κ2c62δ1k+δ2kK+δ3kiKκ+K∗+ρCeδ1k+δ2kiκ+α∗κ2+δ3kκ2+γT∗χ∗κδ1ki+α∗κ+δ2ki+δ3kκ-ζ+iκV2+ρCeκ2c62δ1k+δ2kiκ+α∗κ2+δ3kκ2=0,where c62=c12+ca2/1+cb,χ∗=c12χ/c12+ca2(1-iκαδ2k). From (21) it follows that both the attenuation factor ζ and the wave speed V are functions of the frequency κ, which in turn means that the longitudinal wave suffers attenuation and dispersion due to the thermoelastic character of the medium. The magnetic field also influences dispersion and attenuation as is evident from the terms ca2 and cb present in (21).If we drop the attenuation factorζ and use the angular frequency ω in place of κ, for real velocity, (21) becomes(22)aV4+bV2+c=0,where(23a)a=ρCeδ1k+δ2kα∗K+δ3kK∗,(23b)b=-1+ρCec62δ1k+δ2kα∗K+δ3kK∗+γT∗χ∗c62δ1kK+δ3kK∗,(23c)c=c62.


## 6. Analysis of Dispersion Relation

It is easily observed from (22) that the wave velocity depends upon the frequency and hence longitudinal waves propagated in a magnetothermoelastic material are subject to dispersion of general wave form. Such a dispersion is also in contrast with the classical ideas. In the absence of temperature the roots of (22) become (24)V12=c62=c12+ca21+cb,V22=0.It shows that in absence of temperature in the solid there is only one kind of velocity of propagation, which is influenced by the magnetic field.In case there is no magnetic field,V12=c12=λ+2μ/ρ, the classical result.Now the positive roots of (22) are (25)V1=-b+b2-4ac2a1/2,V2=-b-b2-4ac2a1/2.The discriminant of (22) is(26)b2-4ac=1-pLS2+21+pLSqLS+qLS2forL-Smodel=pGL-12forG-Lmodel=pGN-12+2pGN+1qGN+qGN2forG-Nmodel,where (27)pLS=pGL=ρCeα∗c62K,pGN=ρCec62K∗,qLS=γα∗χ∗T∗c62K,qGN=γα∗χ∗c62K∗.It may be easily checked that the discriminant is greater than or equal to zero.Equation (22) yields a quadratic equation in V2 and hence two distinct types of longitudinal waves are possible in the medium. This information is in contrast with the classical elastic material where there is only one kind of longitudinal wave propagating in the medium for all frequencies with velocity c1=λ+2μ/ρ1/2.The velocity squares which are the roots of (22) for the three different considered models depend on the elastic parameters λ,μ; magnetic parameters ca2,cb, and temperature parameters and are given by (28)V1,22=K2ρCe1+pLS+qLS±1-pLS2+21+pLSqLS+qLS2forL-Smodel=c62,KρCeforG-Lmodel=K∗2ρCe1+pGN+qGN±1-pGN2+21+pGNqGN+qGN2forG-Nmodel.Here we observe that for L-S and G-N models both the velocities are coupled in temperature and magnetic field, whereas for G-L model one velocity purely depends on elastic and magnetic parameters and the other velocity depends only on temperature parameters.


## 7. Reflection of Coupled Longitudinal Displacement Waves

We suppose that a coupled longitudinal displacement wave falls on the surfacez=0 from within the thermoelastic medium. The plane of incidence is assumed to be such that the motion lies in the x-z plane only.We suppose that the incident plane wave is advancing in the direction of the unit vectorν1. After reflection at the stress free surface z=0, we shall have three waves moving in the directions of unit vectors ν2,ν3, and ν4 (Figure 2) such that(a)
one set of longitudinal coupled waves propagates with velocityV1 in the direction of ν3,(b)
another set of longitudinal coupled waves propagates with velocityV2 in the direction of ν2,(c)
a third set of transverse waves propagates with velocityV3 in the direction of ν4.For the incident wave the scalar potential q1 and temperature T1 are expressed in terms of the circular frequency ω1 and wave number k1 in the form (29)q1=a1exp⁡ik1ν1·r-ω1t,T1=b1exp⁡ik1ν1·r-ω1t.Because there will be three types of reflected waves, namely, two distinct longitudinal and one transverse, the corresponding potentials q2,q3, temperature parameters T2,T3, and Ψ are represented as (30)q2=a2exp⁡ik2ν2·r-ω2t,T2=b2exp⁡ik2ν2·r-ω2t,q3=a3exp⁡ik1ν3·r-ω1t,T3=b3exp⁡ik1ν3·r-ω1t,Ψ=ηAexp⁡ik3ν4·r-ω3t,where ω1=k1V1, ω2=k2V2, ω3=k3V3, and η is a unit vector along y-axis.Figure 2
Reflection of longitudinal displacement wave.


## 8. Basic Solutions

We shall use relations (3) and (9) in the boundary condition (13) to get (31)2c22Ψ,xz-q,xx+1+cbq,tt-ca2q,xx+q,zz=0atz=0,2q,xz+Ψ,xx-Ψ,zz=0atz=0.Using (10) we get (32)T=c12+ca2c32q,xx+q,zz-1+cbc32q,tt,where q=q1+q2+q3.The solution of the reflection problem includes the determination of the amplitude and directions of the reflected waves corresponding to an incident wave. Using the potential representation (29) we find that the stress boundary conditions are satisfied at z=0 if(33)ω1=ω2=ω3,(34)k1ν1x=k2ν2x=k1ν3x=k3ν4x,(35)k1ν1y=k2ν2y=k1ν3y=k3ν4y.Since the incident wave is in the x-z plane, ν1y=0 we have from (35)(36)ν2y=ν3y=ν4y=0.Now use of the boundary conditions (13), (14), (31), and (32) yields(37)m1a1+m2a2+m3a3-m4A=0,m1′a1+m2′a2+m3′a3+m4′A=0,m1′′a1+m2′′a2+m3′′a3=0,where(38)m1=2c22ν1x2-1+cbω12k12+ca2,m2=2c22ν1x2-1+cbω22k12+ca2,m3=2c22ν3x2-1+cbω12k12+ca2,m4=2c22ν1x2ν4zν4x,m1′=2ν1xν1z,m2′=2ν1x2ν2zν2x,m3′=2ν3xν3z,m4′=ν1x2-ν1x2ν4z2ν4x2,m1′′=ω12k12ν1z-c62ν1z,m2′′=ω22k12ν1xν2zν2x-c62ν1x2ν2x2ν1xν2zν2x,m3′′=ω12k12ν3z-c62ν3z.Again from (34) we get (39)ν1xV1=ν2xV2=ν3xV1=ν4xV3.Let θi be the angle made by the incident wave (i=1) and three reflected waves (i=2,3,4) with the normal to the half space (Figure 2). Equation (39) implies that(40)cos⁡θ3=cos⁡θ1,cos⁡θ2=V2V1cos⁡θ1,cos⁡θ4=V3V1cos⁡θ1,from which we get θ3=θ1.Solving (37) we obtain the amplitude ratios in the following form:(41)a2a1=2Q12tan⁡θ11-tan2⁡θ4Q22-Q2′2tan2⁡θ11-tan2⁡θ4Q22-Q2′2tan2⁡θ1Q32tan⁡θ2-Q12tan⁡θ1+Q42Q32-Q12tan⁡θ1tan⁡θ2tan⁡θ4-1,a3a1=Q22-Q2′2tan2θ11-tan2⁡θ4-Q12tan⁡θ1-Q32tan⁡θ2+Q52Q72tanθ1tan⁡θ2tan⁡θ41-tan2⁡θ4Q22-Q2′2tan2⁡θ1Q32tan⁡θ2-Q12tan⁡θ1+Q42Q32-Q12tan⁡θ1tan⁡θ2tan⁡θ4-1,Aa1=Q52Q62tan⁡θ1tan⁡θ2Q22-Q2′2tan2⁡θ11-tan2⁡θ4Q22-Q2′2tan2⁡θ1Q32tan⁡θ2-Q12tan⁡θ1+Q42Q32-Q12tan⁡θ1tan⁡θ2tan⁡θ4-1,where(42)Q12=ω2k12-c62,Q22=2c22ω2/k12-Q2′2,Q2′2=1+cbω2k12-ca2ω2/k12,Q32=ω2k12-c62V12V22,Q42=4c22ω2/k12,Q52=1-V12V22,Q62=4c62,Q72=4c22c62ω2/k12,in which ω is the common value of ω1,ω2,ω3.


## 9. Numerical Results and Discussions

The present study investigates the effects of magnetic field on the reflection problem of longitudinal waves in a thermoelastic medium. In our numerical computation we have considered aluminium-epoxy like material to illustrate theoretical results. In the present discussion we have taken the values of elastic and thermal parameters (Sharma et al. [18, 19]), as(43)ρ=2.19gm/cm3,T∗=20°C,λ=7.59×1011dyne/cm2,μ=1.89×1011dyne/cm2,Ce=0.23cal/gm°C,K=0.6×10-2cal/cm°C,α∗=0.5×10-13sec,α=0.75×10-13sec,αt=8.1×10-6K-1,ϵ=0.073.The longitudinal wave velocities V1 and V2 of first and second kind in magnetothermoelastic medium for three different models have been computed numerically based on the above data and have been graphically presented in Figures 3–5. The variations of velocity V1 with applied magnetic field magnitudes in L-S, G-L, and G-N models are shown in Figures 3(a), 4(a), and 5(a) respectively, while the variations of V2 with applied magnetic field magnitudes are presented in Figures 3(b), 4(b), and 5(b). The behaviors of V1 in three different models are different. In L-S model, for a low magnetic field, V1 has some oscillatory character, after which, with the increase of magnetic field, the velocity shows a steady increase. In G-L model, under low magnetic effect, the velocity appears constant, but, with the increase of field magnitude, it increases steadily. The behavior of V1 in G-N model is showing some different character. With the increase of field magnitude from zero value, it shows constant value first, then V1 drops down, and thereafter it increases steadily. The value of V2 remains constant for some low value of the applied field, then it slightly drops down, and thereafter it steadily increases in L-S model. In G-L model V2 remains unaffected by applied magnetic field. In G-L model, the behavior shows that V2 remains unaffected for some low value of applied field but it increases steadily with higher values.Figure 3
(a) Variation of possible wave velocityV1 with H for L-S model. (b) Variation of possible wave velocity V2 with H for L-S model.
(a) (b)Figure 4
(a) Variation of possible wave velocityV1 with H for G-L model. (b) Variation of possible wave velocity V2 with H for G-L model.
(a) (b)Figure 5
(a) Variation of possible wave velocityV1 with H for G-N model. (b) Variation of possible wave velocity V2 with H for G-N model.
(a) (b)Figures6, 7, and 8 show the variation of amplitude ratios a2/a1, a3/a1, and A/a1 with respect to θ1 for different magnetic fields. Figure 6 illustrates the effect of magnetic field on the amplitude ratio a2/a1. It is observed that continuous increase of θ1(0≤θ1<90°) corresponds to a continuous increase of a2/a1, in absence or presence of the magnetic field. The graphs show that the values of amplitude ratio a2/a1 for the presence of magnetic field are always greater than those in absence of the magnetic field. An important observation can be made from Figure 6 that for a particular value of θ1 the effects of magnetic field on the amplitude ratio a2/a1 increase as magnetic field increases.Figure 6
Effect of magnetic field on amplitude ratioa2/a1.Figure 7
Effect of magnetic field on amplitude ratioa3/a1.Figure 8
Effect of magnetic field on the amplitude ratioA/a1.The variations of the amplitude ratioa3/a1 with respect to θ1 are shown in Figure 7. It is clear from the figure that all the values of a3/a1 are negative. The absolute value of a3/a1 at θ1=0 is nonzero. It is observed that the absolute values of amplitude ratio a3/a1 for the presence of magnetic field are always greater than those in absence of magnetic field. It is also noted that for a particular value of θ1 the differences between the absolute values of amplitude ratios a3/a1 increase as magnetic field increases.Figure8 depicts the variation of the amplitude ratio A/a1 with θ1. It is observed that, for all values of magnetic field, A/a1 increases and attains its maximum value in the neighbourhood of θ1=50° approximately. Beyond this value of θ1, A/a1 decreases. The nature A/a1 is almost symmetrical around θ1=50° in 0<θ1<90° for all magnitudes of applied magnetic field. The values of amplitude ratio A/a1 for the presence of magnetic field are always less than those in absence of magnetic field. For a particular value of θ1 the decrement of A/a1 is observed as magnetic field increases.An important observation of our discussed problem due to the presence of temperature is shown in Table1. Here, V0 represents the one type of velocity values in absence of temperature and VLS,VGL,VGN represent the respective velocities for Lord-Shulman, Green-Lindsay, and Green-Naghdi models.Table 1
Values of velocityV1 in absence and presence of temperature for different models.
H V 0 V L S V G L V G N 0.0477 7.2054019 18.069792 7.2054019 2.542775154 0.0573 7.2054024 18.069792 7.2054024 2.542775154 0.1031 7.2054057 18.069794 7.2054057 2.542775154 0.2475 7.2054057 18.069794 7.2054057 2.542775154 0.7425 7.2056208 18.069792 7.2056208 2.542775154 2.6732 7.2082462 18.069794 7.2082462 2.542775154 11.2274 7.2554202 18.069794 7.2554202 2.542775154 53.8916 8.2814789 18.069817 8.2814789 2.542778254


## 10. Conclusion

The study focuses on how a plane longitudinal wave propagating in a generalized thermoelastic medium is affected by an applied magnetic field. The velocity of plane longitudinal waves depends heavily on the thermoelastic character of the medium as well as on the magnetic field. The velocity of the waves also varies in three different generalized thermoelastic models and the nature of the velocities in L-S, G-L, and G-N models is significantly different, particularly at low values of the applied magnetic field. For higher values of applied magnetic field, the nature of the velocities in these models is almost similar.



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