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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 763429, 47 pages
http://dx.doi.org/10.1155/2011/763429
Research Article

Rayleigh Waves in Generalized Magneto-Thermo-Viscoelastic Granular Medium under the Influence of Rotation, Gravity Field, and Initial Stress

1Mathematics Department, Faculty of Science, Taif University, Taif 21974, Saudi Arabia
2Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
3Mathematics Department, Faculty of Science, Umm Al-Qura University, P.O. Box 10109, Makkah 13401, Saudi Arabia

Received 4 December 2010; Revised 14 January 2011; Accepted 25 February 2011

Academic Editor: Ezzat G. Bakhoum

Copyright © 2011 A. M. Abd-Alla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The surface waves propagation in generalized magneto-thermo-viscoelastic granular medium subjected to continuous boundary conditions has been investigated. In addition, it is also subjected to thermal boundary conditions. The solution of the more general equations are obtained for thermoelastic coupling. The frequency equation of Rayleigh waves is obtained in the form of a determinant containing a term involving the coefficient of friction of a granular media which determines Rayleigh waves velocity as a real part and the attenuation coefficient as an imaginary part, and the effects of rotation, magnetic field, initial stress, viscosity, and gravity field on Rayleigh waves velocity and attenuation coefficient of surface waves have been studied in detail. Dispersion curves are computed numerically for a specific model and presented graphically. Some special cases have also been deduced. The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field is very pronounced.

1. Introduction

The dynamical problem in granular media of generalized magneto-thermoelastic waves has been studied in recent times, necessitated by its possible applications in soil mechanics, earthquake science, geophysics, mining engineering, and plasma physics, and so forth. The granular medium under consideration is a discontinuous one and is composed of numerous large or small grains. Unlike a continuous body each element or grain cannot only translate but also rotate about its center of gravity. This motion is the characteristic of the medium and has an important effect upon the equations of motion to produce internal friction. It was assumed that the medium contains so many grains that they will never be separated from each other during the deformation and that each grain has perfect thermoelasticity. The effect of the granular media on dynamics was pointed out by Oshima [1]. The dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress has been illustrated by El-Naggar [2]. Rayleigh wave propagation of thermoelasticity or generalized thermoelasticity was pointed out by Dawan and Chakraporty [3]. Rayleigh waves in a magnetoelastic material under the influence of initial stress and a gravity field were discussed by Abd-Alla et al. [4] and El-Naggar et al. [5].

Rayleigh waves in a thermoelastic granular medium under initial stress on the propagation of waves in granular medium are discussed by Ahmed [6]. Abd-Alla and Ahmed [7] discussed the problem of Rayleigh wave propagation in an orthotropic medium under gravity and initial stress. Magneto-thermoelastic problem in rotating nonhomogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model is discussed by Abd-Alla and Mahmoud [8]. Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross-section is discussed by Venkatesan and Ponnusamy [9]. Some problems discussed the effect of rotation of different materials. Thermoelastic wave propagation in a rotating elastic medium without energy dissipation was studied by Roychoudhuri and Bandyopadhyay [10]. Sharma and Grover [11] studied the body wave propagation in rotating thermoelastic media. Thermal stresses in a rotating nonhomogeneous orthotropic hollow cylinder were discussed by El-Naggar et al. [12]. Abd-El-Salam et al. [13] investigated the numerical solution of magneto-thermoelastic problem nonhomogeneous isotropic material.

In this paper, the effect of magnetic field, rotation, thermal relaxation time, gravity field, viscosity, and initial stress on propagation of Rayleigh waves in a thermoelastic granular medium is discussed. General solution is obtained by using Lame’s potential. The frequency equation of Rayleigh waves is obtained in the form of a determinant. Some special cases have also been deduced. Dispersion curves are computed numerically for a specific model and presented graphically. The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field are very pronounced.

2. Formulation of the Problem

Let us consider a system of orthogonal Cartesian axes, Oxyz, with the interface and the free surface of the granular layer resting on the granular half space of different materials being the planes 𝑧=𝐾 and 𝑧=0, respectively. The origin 𝑂 is any point on the free surface, the 𝑧-axis is positive along the direction towards the exterior of the half space, and the 𝑥-axis is positive along the direction of Rayleigh waves propagation. Both media are under initial compression stress 𝑃 along the 𝑥-direction and the primary magnetic field 𝐻0 acting on 𝑦-axis, as well as the gravity field and incremental thermal stresses, as shown in Figure 1. The state of deformation in the granular medium is described by the displacement vector 𝑈(𝑢,𝑜,𝑤) of the center of gravity of a grain and the rotation vector 𝜉(𝜉,𝜂,𝜁) of the grain about its center of gravity. The elastic medium is rotating uniformly with an angular velocity Ω=Ω𝑛, where 𝑛 is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame has two additional terms, Ω×(Ω×𝑢) is the centripetal acceleration due to time varying motion only, and 2Ω×𝑢 is the Coriolis acceleration, and Ω=(0,Ω,0).

763429.fig.001
Figure 1: Depiction of the problem.

The electromagnetic field is governed by Maxwell equations, under the consideration that the medium is a perfect electric conductor taking into account the absence of the displacement current (SI) (see the work of Mukhopadhyay [14]): 𝐽=curl,𝜇𝑒𝜕𝜕𝑡=curl𝐸,div=0,div𝐸=0,𝐸=𝜇𝑒𝜕𝑢×𝐻,𝜕𝑡(2.1) where𝐻=curl𝑢×0,𝐻𝐻=0+𝐻,0=0,𝐻0,0,(2.2) where is the perturbed magnetic field over the primary magnetic field vector, 𝐸 is the electric intensity, 𝐽 is the electric current density, 𝜇𝑒 is the magnetic permeability, 𝐻0 is the constant primary magnetic field vector, and 𝑢 is the displacement vector.

The stress and stress couple may be taken to be nonsymmetric, that is, 𝜏𝑖𝑗𝜏𝑗𝑖, 𝑀𝑖𝑗𝑀𝑗𝑖. The stress tensor 𝜏𝑖𝑗 can be expressed as the sum of symmetric and antisymmetric tensors 𝜏𝑖𝑗=𝜎𝑖𝑗+𝜎𝑖𝑗,(2.3) where 𝜎𝑖𝑗=12𝜏𝑖𝑗+𝜏𝑗𝑖,𝜎𝑖𝑗=12𝜏𝑖𝑗𝜏𝑗𝑖.(2.4)

The symmetric tensor 𝜎𝑖𝑗=𝜎𝑗𝑖 is related to the symmetric strain tensor 𝑒𝑖𝑗=𝑒𝑗𝑖=12𝜕𝑢𝑖𝜕𝑥𝑗+𝜕𝑢𝑗𝜕𝑥𝑖.(2.5) The antisymmetric stress 𝜎𝑖𝑗 are given by𝜎23=𝐹𝜕𝜉𝜕𝑡,𝜎31=𝐹𝜕𝜂𝜕𝑡,𝜎12=𝐹𝜕𝜁𝜕𝑡,𝜎11=𝜎22=𝜎33=0,(2.6) where 𝐹 is the coefficient of friction between the individual grains. The stress couple 𝑀𝑖𝑗 is given by𝑀𝑖𝑗=𝑀𝜈𝑖𝑗,(2.7) where 𝑀 is the third elastic constant, 𝑀11,𝑀13,𝑀33, and so forth, are the components of the resultant acting on a surface.

The non-symmetric strain tensor 𝜈𝑖𝑗 is defined as𝜈11=𝜕𝜉𝜕𝑥,𝜈31=𝜕𝜉𝜕𝑧,𝜈33=𝜕𝜁𝜕𝑧,𝜈21=𝜈22=𝜈23𝜈=0,12=𝜕𝜔𝜕𝑥2+𝜂,𝜈32=𝜕𝜔𝜕𝑧2+𝜂,𝜈13=𝜕𝜁,𝜕𝑥(2.8) where 𝜔2=(1/2)((𝜕𝑢/𝜕𝑧)(𝜕𝑤/𝜕𝑥)).

The dynamic equation of motion, if the magnetic field and rotation are added, can be written as [15]𝜏𝑗𝑖,𝑗+𝐹𝑖=𝜌𝑢𝑖+Ω×Ω×𝑢𝑖+2Ω×𝑢𝑖,𝑖,𝑗=1,2,3.(2.9) The heat conduction equation is given by [16] 𝐾2𝜕𝑇=𝜌𝑠𝜕𝑡1+𝜏2𝜕𝜕𝑡𝑇+𝛾𝑇0𝜕𝜕𝑡1+𝜏2𝛿𝜕𝜕𝑡𝑢,(2.10) where 𝜌 is density of the material, 𝐾 is thermal conductivity, s is specific heat of the material per unit mass, 𝜏1,𝜏2 are thermal relaxation parameter, 𝛼𝑡 is coefficient of linear thermal expansion, 𝜆 and 𝜇 are Lame’s elastic constants, 𝜃 is the absolute temperature, 𝛾=𝛼𝑡(3𝜆+2𝜇), 𝑇0 is reference temperature solid, 𝑇 is temperature difference (𝜃𝑇0), 𝜏0 is the mechanical relaxation time due to the viscosity, and 𝜏𝑚=(1+𝜏0(𝜕/𝜕𝑡)).

The components of stress in generalized thermoelastic medium are given by𝜎11=𝜏𝑚(𝜆+2𝜇)+𝑝𝜕𝑢+𝜏𝜕𝑥𝑚𝜆+𝑃𝜕𝑤𝜕𝑧𝛾1+𝜏1𝜕𝜎𝜕𝑡𝑇,33=𝜏𝑚𝜆𝜕𝑢𝜕𝑥+𝜏𝑚(𝜆+2𝜇)𝜕𝑤𝜕𝑧𝛾1+𝜏1𝜕𝜎𝜕𝑡𝑇,13=𝜏𝑚𝜇𝜕𝑢+𝜕𝑧𝜕𝑤.𝜕𝑥(2.11) If we neglect the thermal relaxation time, then (2.11) tends to Nowacki [17] and Biot [18].

The Maxwell's electro-magnetic stress tensor 𝜏𝑖𝑗 is given by 𝜏𝑖𝑗=𝜇𝑒𝐻𝑖𝑗+𝐻𝑗𝑖𝐻𝑘𝑘𝛿𝑖𝑗,𝑖,𝑗=1,2,3,(2.12) which takes the form𝜏11=𝜇𝑒𝐻202𝜙,𝜏13=𝜏23=0,𝜏33=𝜇𝑒𝐻202𝜙,2𝜙=𝜕𝑢+𝜕𝑥𝜕𝑤𝜕𝑧.(2.13)

The dynamic equations of motion are𝜕𝜏11+𝜕𝑥𝜕𝜏31+𝑃𝜕𝑧2𝜕𝜔2𝜕𝑧𝜌𝑔𝜕𝑤𝜕𝑥+𝐹𝑥𝜕=𝜌2𝑢𝜕𝑡2+2Ω𝜕𝑤𝜕𝑡Ω2𝑢,𝜕𝜏12+𝜕𝑥𝜕𝜏32𝜕𝑧+𝐹𝑦=0,𝜕𝜏13+𝜕𝑥𝜕𝜏33+𝑃𝜕𝑧2𝜕𝜔2𝜕𝑥+𝜌𝑔𝜕𝑤𝜕𝑥+𝐹𝑧𝜕=𝜌2𝑤𝜕𝑡22Ω𝜕𝑢𝜕𝑡Ω2𝑤,(2.14) where 𝑔 is the Earth's gravity and𝐹=𝜇𝑒𝐻202𝜙,0,𝜇𝑒𝐻202𝜙,𝜏(2.15)23𝜏32+𝜕𝑀11+𝜕𝑥𝜕𝑀31𝜏𝜕𝑧=0,31𝜏13+𝜕𝑀12+𝜕𝑥𝜕𝑀32𝜏𝜕𝑧=0,12𝜏21+𝜕𝑀13+𝜕𝑥𝜕𝑀33𝜕𝑧=0.(2.16)

From (2.3)–(2.8) and (2.11), we have𝜏11=𝜏𝑚(𝜆+2𝜇)+𝑝𝜕𝑢+𝜏𝜕𝑥𝑚𝜆+𝑃𝜕𝑤𝜕𝑧𝛾1+𝜏1𝜕𝜏𝜕𝑡𝑇,33=𝜏𝑚𝜆𝜕𝑢𝜕𝑥+𝜏𝑚(𝜆+2𝜇)𝜕𝑤𝜕𝑧𝛾1+𝜏1𝜕𝜏𝜕𝑡𝑇,13=𝜏𝑚𝜇𝜕𝑢+𝜕𝑧𝜕𝑤𝜕𝑥+𝐹𝜕𝜂,𝜏𝜕𝑡12=𝐹𝜕𝜁,𝜏𝜕𝑡23=𝐹𝜕𝜉,𝑀𝜕𝑡11=𝑀𝜕𝜉𝜕𝑥,𝑀31=𝑀𝜕𝜉𝜕𝑧,𝑀33=𝑀𝜕𝜁𝜕𝑧,𝑀21=𝑀22=𝑀23𝑀=0,12𝜕=𝑀𝜔𝜕𝑥2+𝜂,𝑀32𝜕=𝑀𝜔𝜕𝑧2+𝜂,𝑀13=𝑀𝜕𝜁.𝜕𝑥(2.17) Substituting (2.17) into (2.14) and (2.16) tends to𝜏𝑚𝜕(𝜆+2𝜇)+𝑃2𝑢𝜕𝑥2+𝜏𝑚𝜕𝜆+𝑃2𝑤𝜕𝑥𝜕𝑧𝛾1+𝜏1𝜕𝜕𝑡𝜕𝑇𝜕𝑥+𝜏𝑚𝜇𝜕2𝑢𝜕𝑧2+𝜕2𝑤+𝑃𝜕𝑥𝜕𝑧2𝜕2𝑢𝜕𝑧2𝜕2𝑤𝜕𝑥𝜕𝑧𝜌𝑔𝜕𝑤𝜕𝜕𝑥+𝐹2𝜂𝜕𝑧𝜕𝑡+𝜇𝑒𝐻20𝜕2𝑢𝜕𝑥2+𝜕2𝑤𝜕𝜕𝑥𝜕𝑧=𝜌2𝑢𝜕𝑡2+2Ω𝜕𝑤𝜕𝑡Ω2𝑢,(2.18) then𝜏𝑚(𝜆+2𝜇)+𝑃+𝜇𝑒𝐻20𝜕2𝑢𝜕𝑥2+𝜏𝑚𝑃(𝜆+𝜇)+2+𝜇𝑒𝐻20𝜕2𝑤+𝜏𝜕𝑥𝜕𝑧𝑚𝑃𝜇+2𝜕2𝑢𝜕𝑧2𝛾1+𝜏1𝜕𝜕𝑡𝜕𝑇𝜕𝑥𝜌𝑔𝜕𝑤𝜕𝜕𝑥+𝐹2𝜂𝜕𝜕𝑧𝜕𝑡=𝜌2𝑢𝜕𝑡2+2Ω𝜕𝑤𝜕𝑡Ω2𝑢.(2.19)

Also,𝜕𝜕𝑡𝜕𝜁𝜕𝑥𝜕𝜉𝜏𝜕𝑧=0,(2.20)𝑚𝜇𝜕2𝑢+𝜕𝜕𝑥𝜕𝑧2𝑤𝜕𝑥2𝜕𝐹2𝜂𝜕𝑥𝜕𝑡+𝜏𝑚𝜆𝜕2𝑢𝜕𝑥𝜕𝑧+𝜏𝑚𝜕(𝜆+2𝜇)2𝑤𝜕𝑧2𝛾1+𝜏1𝜕𝜕𝑡𝜕𝑇+𝑃𝜕𝑧2𝜕2𝑢𝜕𝜕𝑥𝜕𝑧2𝑤𝜕𝑥2+𝜌𝑔𝜕𝑢𝜕𝑥+𝜇𝑒𝐻20𝜕2𝑢+𝜕𝜕𝑥𝜕𝑧2𝑤𝜕𝑧2𝜕=𝜌2𝑤𝜕𝑡22Ω𝜕𝑢𝜕𝑡Ω2𝑤,(2.21) then𝜏𝑚𝑃(𝜆+𝜇)+2+𝜇𝑒𝐻20𝜕2𝑢+𝜏𝜕𝑥𝜕𝑧𝑚𝑃𝜇2𝜕2𝑤𝜕𝑥2+𝜏𝑚(𝜆+2𝜇)+𝜇𝑒𝐻20𝜕2𝑤𝜕𝑧2𝛾1+𝜏1𝜕𝜕𝑡𝜕𝑇𝜕𝑧+𝜌𝑔𝜕𝑢𝜕𝜕𝑥𝐹2𝜂𝜕𝜕𝑥𝜕𝑡=𝜌2𝑤𝜕𝑡22Ω𝜕𝑢𝜕𝑡Ω2𝑤,(2.22) and, from (2.16), we have2𝜉𝑠2𝜕𝜉𝜕𝑡=0,(2.23)2𝜔2+𝜂𝑠2𝜕𝜂𝜕𝑡=0,(2.24)2𝜁𝑠2𝜕𝜁𝜕𝑡=0,(2.25) where 𝑠2=2𝐹𝑀.(2.26)

3. Solution of the Problem

By Helmholtz's theorem [19], the displacement vector 𝑢 can be written in the displacement potentials 𝜙 and 𝜓 form, as𝑢=grad𝜙+curl𝜓,𝜓=(0,𝜓,0),(3.1) which reduces to𝑢=𝜕𝜙𝜕𝑥𝜕𝜓𝜕𝑧,𝑤=𝜕𝜙+𝜕𝑧𝜕𝜓𝜕𝑥.(3.2)

Substituting (3.2) into (2.19), (2.22), and (2.24), the wave equations tend to𝛼22𝛾𝜙𝜌1+𝜏1𝜕𝜕𝑡𝑇𝑔𝜕𝜓=𝜕𝜕𝑥2𝜙𝜕𝑡2+2Ω𝜕𝜓𝜕𝑡Ω2𝛽𝜙,(3.3)22𝜓𝑠1𝜕𝜂𝜕𝑡+𝑔𝜕𝜙=𝜕𝜕𝑥2𝜓𝜕𝑡22Ω𝜕𝜙𝜕𝑡Ω2𝜓,(3.4)2𝜂𝑠2𝜕𝜂𝜕𝑡4𝜓=0,(3.5) where𝑠1=𝐹𝜌,𝛼2=𝜏𝑚(𝜆+2𝜇)+𝑃+𝜇𝑒𝐻20𝜌,𝛽2=2𝜏𝑚𝜇𝑃2𝜌.(3.6)

Substituting (3.2) into (2.10), we obtain𝐾2𝜕𝑇=𝜌𝑠𝜕𝑡1+𝜏2𝜕𝜕𝑡𝑇+𝛾𝑇0𝜕𝜕𝑡1+𝜏2𝛿𝜕𝜕𝑡2𝜙.(3.7)

From (3.3) and (3.7), by eliminating 𝑇, we obtain 21𝜒𝜕𝜕𝑡1+𝜏2𝜕𝛼𝜕𝑡22𝜙𝑔𝜕𝜓𝜕𝜕𝑥2𝜙𝜕𝑡22Ω𝜕𝜓𝜕𝑡+Ω2𝜙𝜕𝜀𝜕𝑡1+𝜏1𝜕𝜕𝑡1+𝜏2𝛿𝜕𝜕𝑡2𝜙=0,(3.8) where𝐾𝜒=𝛾𝜌𝑠,𝜀=2𝑇0𝜌𝐾.(3.9)

From (3.4) and (3.5) by eliminating 𝜂, we obtain 2𝑠2𝜕𝛽𝜕𝑡22𝜕𝜓2𝜓𝜕𝑡2+𝑔𝜕𝜙𝜕𝑥+2Ω𝜕𝜙𝜕𝑡+Ω2𝜓𝑠14𝜕𝜓𝜕𝑡=0.(3.10)

For a plane harmonic wave propagation in the 𝑥-direction, we assume𝜙=𝜙1𝑒𝑖𝑘(𝑥𝑐𝑡),𝜓=𝜓1𝑒𝑖𝑘(𝑥𝑐𝑡),𝜉(3.11)(𝜉,𝜂,𝜁)=1,𝜂1,𝜁1𝑒𝑖𝑘(𝑥𝑐𝑡).(3.12)

From (3.12) into (2.20), (2.23), and (2.25), we get𝐷𝜉1𝑖𝑘𝜁1𝐷=0,(3.13)2𝜉1+𝑞2𝜉1𝐷=0,(3.14)2𝜁1+𝑞2𝜁1=0,(3.15) where𝑞2=𝑖𝑘𝑐𝑠2𝑘2𝑑,𝐷.𝑑𝑧(3.16)

The solution of (3.14) and (3.15) takes the form𝜉1=𝐴1𝑒𝑖𝑞𝑧+𝐴2𝑒𝑖𝑞𝑧,𝜁1=𝐵1𝑒𝑖𝑞𝑧+𝐵2𝑒𝑖𝑞𝑧,(3.17) where 𝐴1,𝐴2,𝐵1, and 𝐵2 are arbitrary constants.

From (3.13) and (3.17), we obtain 𝑞𝐴1𝑒𝑖𝑞𝑧𝐴2𝑒𝑖𝑞𝑧𝐵𝑘1𝑒𝑖𝑞𝑧+𝐵2𝑒𝑖𝑞𝑧=0,(3.18) then𝑞𝐴1𝑘𝐵1=0,𝑞𝐴2𝑘𝐵2=0𝐴𝑗=(1)𝑗1𝑘𝑞𝐵𝑗,𝑗=1,2.(3.19)

Substituting (3.11) into (3.8) and (3.10), we obtain𝛼2𝐷4+𝐺1𝐷2+𝐺2𝜙1𝐺3𝐷2+𝐺4𝜓1𝑅=0,1𝐷4+𝑅2𝐷2+𝑅3𝜓1+𝑅4𝐷2+𝑅5𝜙1=0,(3.20) whereΓ0=1𝑖𝑘𝑐𝜏0,Γ1=1𝑖𝑘𝑐𝜏1,Γ2=1𝑖𝑘𝑐𝜏2,Γ3=1𝑖𝑘𝑐𝜏2𝛼𝛿,2=Γ0(𝜆+2𝜇)+𝑃+𝜇𝑒𝐻20𝜌,𝛽2=2Γ0𝜇𝑃.𝐺2𝜌1=𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2,𝐺2=𝑘4𝛼2𝑐2+𝑖𝑘𝑐Γ2𝜒𝑘21𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3,𝐺3=𝑖𝑘(𝑔2Ω𝑐),𝐺4=(𝑔2Ω𝑐)𝑖𝑘3+𝑘2𝑐Γ2𝜒,𝑅1=𝛽2+𝑖𝑘𝑐𝑠1,𝑅2=𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2,𝑅3=𝑘2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠1,𝑅4=𝑖𝑘(𝑔2Ω𝑐),𝑅5=(2Ω𝑐𝑔)𝑖𝑘3𝑘2𝑐𝑠2.(3.21)

The solution of (3.20) takes the form𝜙1=4𝑗=1𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧,𝜓1=4𝑗=1𝐸𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝐹𝑗𝑒𝑖𝑘𝑁𝑗𝑧,(3.22) where the constants 𝐸𝑗 and 𝐹𝑗 are related to the constants 𝐶𝑗 and 𝐷𝑗 in the form𝐸𝑗=𝑚𝑗𝐶𝑗,𝐹𝑗=𝑚𝑗𝐷𝑗𝑚,𝑗=1,2,3,4,𝑗=1(𝑔2Ω𝑐)𝑖𝑘𝑁2𝑗𝑖𝑘𝜀Γ2×𝛼/𝜒2𝑘2𝑁4𝑗𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2𝑁2𝑗+𝑖𝑘𝑐Γ2𝜒1𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3.(3.23) Substituting (3.22) into (3.11), we obtain 𝜙=4𝑗=1𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝜓=4𝑗=1𝐸𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝐹𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),(3.24) and values of displacement components 𝑢 and 𝑤 are𝑢=𝑖𝑘4𝑗=11𝑁𝑗𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝑧+1+𝑁𝑗𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝑤=𝑖𝑘4𝑗=1𝑁𝑗+𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝑚𝑗𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),(3.25) where 𝑁1,𝑁2,𝑁3, and 𝑁4 are taken to be the complex roots of the following equation𝑁8+𝑡1𝑁6+𝑡2𝑁4+𝑡3𝑁2+𝑡4=0,(3.26) where 𝑡1=𝑘2𝛼2𝑐22𝛼2+𝑖𝑘𝑐𝛼2𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2+1𝛽2+𝑖𝑘𝑐𝑠1×𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2,𝑡(3.27)2=1𝛼2𝑘4𝛼2𝑐2+𝑖𝑘𝑐Γ2𝜒𝑘21𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3+1𝛼2𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2×𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2+1𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠11𝛼2𝛽2+𝑖𝑘𝑐𝑠1𝑘2(𝑔2Ω𝑐)2,𝑡(3.28)3=1𝛼2𝛽2+𝑖𝑘𝑐𝑠1×𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2×𝑘4𝛼2𝑐2+𝑖𝑘𝑐Γ2𝜒𝑘21𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3+𝑘2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠1×𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2𝑖𝑘(𝑔2Ω𝑐)2𝑖𝑘3+𝑘2𝑐Γ2𝜒𝑖𝑘3(𝑔2Ω𝑐)2𝑖𝑘𝑐𝑠2,𝑡(3.29)4=1𝛼2𝛽2×𝑘+𝑖𝑘𝑐𝑠2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠1×[]+𝑖𝑘(𝑔2Ω𝑐)(2Ω𝑐𝑔)2𝑖𝑘3𝑘2𝑐𝑠2𝑖𝑘3+𝑘2𝑐Γ2𝜒.(3.30)

From (3.4), (3.11), (3.12), (3.22), and (3.23), we obtain 𝜂1=4𝑗=11𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑁2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧.(3.31)

Using (3.22) and (3.11) into (3.3), we obtain𝜌𝑇=𝛾Γ14𝑗=1𝛼2𝑘21+𝑁2𝑗+𝑘2𝑐2𝑖𝑘𝑔𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡).(3.32)

With the lower medium, we use the symbols with primes, for 𝜉1,𝜁1,𝜂1,𝑇,𝜙,𝜓, and 𝑞, for 𝑧>𝐾,𝜉1𝑘=𝑞𝐵2𝑒𝑖𝑞𝑧,𝜁1=𝐵2𝑒𝑖𝑞𝑧,𝜂1=4𝑗=11𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑁𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘2Ω𝐷𝑐𝑔𝑗𝑒𝑖𝑘𝑁𝑗𝑧,𝑇=𝜌𝛾Γ14𝑗=1𝛼2𝑘21+𝑁𝑗2+𝑘2𝑐2𝑖𝑘𝑔𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝜙=4𝑗=1𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝜓=4𝑗=1𝐹𝑗𝑒𝑖𝑘𝑁𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡).(3.33)

4. Boundary Conditions and Frequency Equation

In this section, we obtain the frequency equation for the boundary conditions which are specific to the interface 𝑧=𝐾, that is,(i)𝑢=𝑢, (ii)𝑤=𝑤, (iii)𝜉=𝜉, (iv)𝜂=𝜂, (v)𝜁=𝜁, (vi)𝑀33=𝑀33, (vii)𝑀31=𝑀31, (viii)𝑀32=𝑀32, (ix)𝜏33+𝜏33=𝜏33+𝜏33(x)𝜏31+𝜏31=𝜏31+𝜏31, (xi)𝜏32+𝜏32=𝜏32+𝜏32, (xii)𝑇=𝑇, (xiii)(𝜕𝑇/𝜕𝑧)+𝜃𝑇=(𝜕𝑇/𝜕𝑧)+𝜃𝑇.

The boundary conditions on the free surface 𝑧=0 are(xiv)𝑀33=0, (xv)𝑀31=0, (xvi)𝑀32=0, (xvii)𝜏33+𝜏33=0, (xviii)𝜏31+𝜏31=0, (xix)𝜏32+𝜏32=0, (xx)(𝜕𝑇/𝜕𝑧)+𝜃𝑇=0.

From conditions (iii), (v), (vi), and (vii), we obtain𝐵1𝑒𝑖𝑞K𝐵2𝑒𝑖𝑞K=𝐵2𝑒𝑖𝑞K,𝐵1𝑒𝑖𝑞K+𝐵2𝑒𝑖𝑞K=𝐵2𝑒𝑖𝑞K,𝑀𝐵1𝑒𝑖𝑞K𝐵2𝑒𝑖𝑞K=𝑀𝐵2𝑒𝑖𝑞K,𝑀𝐵1𝑒𝑖𝑞K+𝐵2𝑒𝑖𝑞K=𝑀𝐵2𝑒𝑖𝑞K.(4.1) Hence, 𝐵1=𝐵2=𝐵2=0,𝜉=𝜁=𝜉=𝜁=0.(4.2)

The other significant boundary conditions are responsible for the following relations:(i)4𝑗=11𝑁𝑗𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾+1+𝑁𝑗𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾1+𝑁𝑗𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.3)(ii)4𝑗=1𝑁𝑗+𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝑚𝑗𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝑚𝑗𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.4)(iv)4𝑗=11𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑁2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾4𝑗=11𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑁𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘2Ω𝐷𝑐𝑔𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.5)(viii)𝑀𝑁𝑗4𝑗=1𝑘2𝑚𝑗𝑁2𝑗+1+1𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑁2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝑀𝑁𝑗4𝑗=1𝑘2𝑚𝑗𝑁𝑗2+1+1𝑖𝑘𝑐𝑠1×𝑘2𝛽2𝑚𝑗1+𝑁𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘2Ω𝐷𝑐𝑔𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.6)(ix)4𝑗=1Γ0𝜆+𝜇𝑒𝐻201𝑁𝑗𝑚𝑗+Γ0(𝜆+2𝜇)+𝜇𝑒𝐻20𝑁2𝑗+𝑚𝑗𝑁𝑗×𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾+Γ0𝜆+𝜇𝑒𝐻201+𝑁𝑗𝑚𝑗+Γ0(𝜆+2𝜇)+𝜇𝑒𝐻20𝑁2𝑗𝑚𝑗𝑁𝑗×𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝜌𝛼21+𝑁2𝑗+𝑐2𝑖𝑔𝑘𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾Γ0𝜆+𝜇𝑒𝐻201+𝑁𝑗𝑚𝑗+Γ0𝜆+2𝜇+𝜇𝑒𝐻20𝑁𝑗2𝑚𝑗𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝜌𝛼21+𝑁𝑗2+𝑐2𝑖𝑔𝑘𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.7)(x)4𝑗=12𝑘2Γ0𝜇𝑁𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝑘2Γ0𝜇𝑚𝑗1𝑁2𝑗+𝐹𝑠1𝑘2𝛽2𝑚𝑗1+𝑁2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾2𝑘2Γ0𝜇𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝑘2Γ0𝜇𝑚𝑗1𝑁𝑗2+𝐹𝑠1𝑘2𝛽2𝑚𝑗1+𝑁𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘2Ω𝑐𝑔×𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.8)(xii)4𝑗=1𝜌𝛾𝛼2𝑘2𝑁2𝑗+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝜌𝛾𝛼2𝑘2𝑁𝑗2+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.9)(xiii)4𝑗=1𝜌𝛾𝛼2𝑘2𝑁2𝑗+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝜃+𝑖𝑘𝑁𝑗𝐶𝑗𝑒𝑖𝑘𝑁𝑗𝐾+𝜃𝑖𝑘𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾𝜌𝛾𝛼2𝑘2𝑁𝑗2+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗×𝜃𝑖𝑘𝑁𝑗𝐷𝑗𝑒𝑖𝑘𝑁𝑗𝐾=0,(4.10)(xvi)𝑀𝑁𝑗4𝑗=1𝑘2𝑚𝑗𝑁2𝑗+1+1𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑁2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝐷𝑗=0,(4.11)(xvii)4𝑗=1Γ0𝜆+𝜇𝑒𝐻201𝑁𝑗𝑚𝑗+Γ0(𝜆+2𝜇)+𝜇𝑒𝐻20𝑁2𝑗+𝑚𝑗𝑁𝑗𝐶𝑗+Γ0𝜆+𝜇𝑒𝐻201+𝑁𝑗𝑚𝑗+Γ0(𝜆+2𝜇)+𝜇𝑒𝐻20𝑁2𝑗𝑚𝑗𝑁𝑗𝐷𝑗+𝜌𝛼21+𝑁2𝑗+𝑐2𝑖𝑔𝑘𝑚𝑗𝐶𝑗+𝐷𝑗=0,(4.12)(xviii)4𝑗=12𝑘2Γ0𝜇𝑁𝑗𝐶𝑗𝐷𝑗+𝑘2Γ0𝜇𝑚𝑗1𝑁2𝑗+𝐹𝑠1𝑘2𝛽2𝑚𝑗1+𝑁2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗+𝐷𝑗=0,(4.13)(xx)4𝑗=1𝛼2𝑘2𝑁2𝑗+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝜃+𝑖𝑘𝑁𝑗𝐶𝑗+𝜃𝑖𝑘𝑁𝑗𝐷𝑗=0.(4.14)

5. Special Cases and Discussion

5.1. The Magnetic Field, Initial Stress, and Thermal Relaxation Time Are Neglected

In this case (i.e., 𝐻0=0,𝑝=0, and 𝜏1=𝜏2=0), (3.26) tends to 𝑉8+1𝑉6+2𝑉4+3𝑉2+4=0,(5.1) where𝛼2=Γ0(𝜆+2𝜇)𝜌,𝛽2=Γ0𝜇𝜌,𝑚𝑗=1(𝑔2Ω𝑐)𝑖𝑘𝑉2𝑗×𝛼𝑖𝑘(𝜀/𝜒)2𝑘2𝑉4𝑗𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2+𝜒𝜀+Ω2𝑉2𝑗+𝑖𝑘𝑐𝜒1𝛼2+Ω2𝑘2Ω2,+𝑖𝑘𝜀𝑐1=𝑘2𝛼2𝑐22𝛼2+𝑖𝑘𝑐𝛼2𝜒𝛼2+𝜒𝜀+Ω2+1𝛽2+𝑖𝑘𝑐𝑠1×𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2,2=1𝛼2𝑘4𝛼2𝑐2+𝑖𝑘𝑐𝜒𝑘21𝛼2+Ω2𝑘2Ω2+1+𝑖𝑘𝜀𝑐𝛼2𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2×𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2+𝜒𝜀+Ω2+1𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠11𝛼2𝛽2+𝑖𝑘𝑐𝑠1𝑘2(𝑔2Ω𝑐)2,3=1𝛼2𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2×𝑘4𝛼2𝑐2+𝑖𝑘𝑐𝜒𝑘21𝛼2+Ω2𝑘2Ω2+𝑘+𝑖𝑘𝜀𝑐2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠1×𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2+𝜒𝜀+Ω2𝑖𝑘(𝑔2Ω𝑐)2𝑖𝑘3+𝑘2𝑐𝜒𝑖𝑘3(𝑔2Ω𝑐)2𝑖𝑘𝑐𝑠2,4=1𝛼2𝛽2𝑘+𝑖𝑘𝑐𝑠2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠1×[]+(𝑖𝑘(𝑔2Ω𝑐)2Ω𝑐𝑔)2𝑖𝑘3𝑘2𝑐𝑠2𝑖𝑘3+𝑘2𝑐𝜒.(5.2)

Also, 𝜂1=4𝑗=11𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑉𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝑧.𝜌𝑇=𝛾4𝑗=1𝛼2𝑘21+𝑉2𝑗+𝑘2𝑐2𝑖𝑘𝑔𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝜉1𝑘=𝑞𝐵2𝑒𝑖𝑞𝑧,𝜁1=𝐵2𝑒𝑖𝑞𝑧,𝜂1=4𝑗=11𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘2Ω𝐷𝑐𝑔𝑗𝑒𝑖𝑘𝑉𝑗𝑧,𝑇=𝜌𝛾4𝑗=1𝛼2𝑘21+𝑉𝑗2+𝑘2𝑐2𝑖𝑘𝑔𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝜙1=4𝑗=1𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝑧,𝜓1=4𝑗=1𝐹𝑗𝑒𝑖𝑘𝑉𝑗𝑧,(5.3) Using the boundary conditions, we obtain𝑑11𝑑12𝑑18𝑑15𝑑16𝑑18𝑑21𝑑22𝑑28𝑑25𝑑26𝑑28𝑑121𝑑122𝑑128𝑑125𝑑126𝑑128𝐶1𝐶2𝐶3𝐶4𝐷1𝐷2𝐷3𝐷4𝐷1𝐷2𝐷3𝐷4=[0],(5.4) where𝑑1𝑗=1𝑉𝑗𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗Κ+1+𝑉𝑗𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗Κ,𝑑1𝑗=1+𝑉𝑗𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗Κ,𝑑2𝑗=𝑉𝑗+𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗Κ+𝑚𝑗𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗Κ,𝑑2𝑗=𝑚𝑗𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗Κ,𝑑3𝑗=1𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉2𝑗𝑚𝑗𝑘2𝑐2+Ω2𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑉𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝑧,𝑑3𝑗=1𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉𝑗2𝑚𝑗𝑘2𝑐2+Ω2𝐷+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑉𝑗𝑧,𝑑4𝑗=𝑀𝑉𝑗𝑘2𝑚𝑗𝑉2𝑗+1+1𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑉𝑗𝐾𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑4𝑗=𝑀𝑉𝑗𝑘2𝑚𝑗𝑉𝑗2+1+1𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘(2Ω𝑐𝑔)×𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑5𝑗=Γ0𝜆1𝑉𝑗𝑚𝑗+Γ0𝑉(𝜆+2𝜇)2𝑗+𝑚𝑗𝑉𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗𝐾+Γ0𝜆1+𝑉𝑗𝑚𝑗+Γ0(𝑉𝜆+2𝜇)2𝑗𝑚𝑗𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝜌𝛼21+𝑉2𝑗+𝑐2𝑖𝑔𝑘𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑5𝑗=Γ0𝜆1+𝑉𝑗𝑚𝑗+Γ0𝜆+2𝜇𝑉𝑗2𝑚𝑗𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝜌𝛼21+𝑉𝑗2+𝑐2𝑖𝑔𝑘𝑚𝑗×𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑6𝑗=2𝑘2Γ0𝜇𝑉𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗𝐾𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝑘2Γ0𝜇𝑚𝑗1𝑉2𝑗+𝐹𝑠1𝑘2𝛽2𝑚𝑗1+𝑉2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑6𝑗=2𝑘2Γ0𝜇𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝑘2Γ0𝜇𝑚𝑗1𝑉𝑗2+𝐹𝑠1𝑘2𝛽2𝑚𝑗1+𝑉𝑗2𝑚𝑗𝑘2𝑐2+Ω2+𝑖𝑘(2Ω𝑐𝑔)×𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑7𝑗=𝜌𝛾𝛼2𝑘2𝑉2𝑗+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑7𝑗=𝜌𝛾𝛼2𝑘2𝑉𝑗2+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑8𝑗=𝜌𝛾𝛼2𝑘2𝑉2𝑗+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝜃+𝑖𝑘𝑉𝑗𝐶𝑗𝑒𝑖𝑘𝑉𝑗𝐾+𝜃𝑖𝑘𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑8𝑗=𝜌𝛾𝛼2𝑘2𝑉𝑗2+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝜃𝑖𝑘𝑉𝑗𝐷𝑗𝑒𝑖𝑘𝑉𝑗𝐾,𝑑9𝑗=𝑀𝑉𝑗𝑘2𝑚𝑗𝑉2𝑗+1+1𝑖𝑘𝑐𝑠1𝑘2𝛽2𝑚𝑗1+𝑉2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗𝐷𝑗,𝑑10𝑗=Γ0𝜆1𝑉𝑗𝑚𝑗+Γ0𝑉(𝜆+2𝜇)2𝑗+𝑚𝑗𝑉𝑗𝐶𝑗+Γ0𝜆1+𝑉𝑗𝑚𝑗+Γ0𝑉(𝜆+2𝜇)2𝑗𝑚𝑗𝑉𝑗𝐷𝑗+𝜌𝛼21+𝑉2𝑗+𝑐2𝑖𝑔𝑘𝑚𝑗𝐶𝑗+𝐷𝑗,𝑑11𝑗=2𝑘2Γ0𝜇𝑉𝑗𝐶𝑗𝐷𝑗+𝑘2Γ0𝜇𝑚𝑗1𝑉2𝑗+𝐹𝑠1𝑘2𝛽2𝑚𝑗1+𝑉2𝑗𝑚𝑗𝑘2𝑐2+Ω2×𝐶+𝑖𝑘(2Ω𝑐𝑔)𝑗+𝐷𝑗,𝑑12𝑗=𝛼2𝑘2𝑉2𝑗+1+𝑘2𝑐2𝑖𝑔𝑘𝑚𝑗𝜃+𝑖𝑘𝑉𝑗𝐶𝑗+𝜃𝑖𝑘𝑉𝑗𝐷𝑗,𝑑9𝑗=𝑑10𝑗=𝑑11𝑗=𝑑12𝑗=0,𝑗=1,2,3,4.(5.5)

5.2. The Magnetic Field, Initial Stress, Rotation, and Thermal Relaxation Time Are Neglected and in Viscoelastic Medium

In this case (i.e., 𝐻0=0, 𝑃=0, Ω=0,and𝜏0=𝜏1=𝜏2=0), the previous results obtained as in Abd-Alla et al. [20].

5.3. Absence of the Gravity Field

In this case, we put 𝑔=0, then (3.20) becomes𝛼2𝐷4+𝐺1𝐷2+𝐺2𝜙1𝐺3𝐷2+𝐺4𝜓1𝑅=0,1𝐷4+𝑅2𝐷2+𝑅3𝜓1+𝑅4𝐷2+𝑅5𝜙1=0,(5.6) where𝐺3=2𝑖𝑘Ω𝑐,𝐺4=2Ω𝑐𝑖𝑘3+𝑘2𝑐Γ2𝜒,𝑅4=2𝑖𝑘Ω𝑐,𝑅5=2Ω𝑐𝑖𝑘3𝑘2𝑐𝑠2,(5.7) and 𝐺1,𝐺2,𝑅1,𝑅2, and 𝑅3 are as in (3.21).

The solution of (5.6) take the form𝜙=4𝑗=1𝐶𝑗𝑒𝑖𝑘𝑋𝑗𝑧+𝐷𝑗𝑒𝑖𝑘𝑋𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),𝜓=4𝑗=1𝐸𝑗𝑒𝑖𝑘𝑋𝑗𝑧+𝐹𝑗𝑒𝑖𝑘𝑋𝑗𝑧𝑒𝑖𝑘(𝑥𝑐𝑡),(5.8) where𝐸𝑗=𝑚𝑗𝐶𝑗,𝐹𝑗=𝑚𝑗𝐷𝑗𝑚,𝑗=1,2,3,4,(5.9)𝑗=12Ω𝑐𝑖𝑘𝑋2𝑗𝑖𝑘𝜀Γ2×𝛼/𝜒2𝑘2𝑋4𝑗𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2𝑋2𝑗+𝑖𝑘𝑐Γ2𝜒1𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3,(5.10) and 𝑋1,𝑋2,𝑋3,and 𝑋4 are taken to be the complex roots of equation𝑋8+𝑡1𝑋6+𝑡2𝑋4+𝑡3𝑋2+𝑡4=0,(5.11) where𝑡1=𝑘2𝛼2𝑐22𝛼2+𝑖𝑘𝑐𝛼2𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2+1𝛽2+𝑖𝑘𝑐𝑠1×𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2,𝑡2=1𝛼2𝑘4𝛼2𝑐2+𝑖𝑘𝑐Γ2𝜒𝑘21𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3+1𝛼2𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2×𝑘2𝑐22𝛼2+𝑖𝑘𝑐𝜒𝛼2Γ2+𝜒𝜀Γ1Γ3+Ω2+1𝛽2+𝑖𝑘𝑐𝑠1𝑘2𝑘2𝑖𝑘𝑐𝑠2𝛽2𝑐2𝑠+𝑖𝑘𝑐2Ω2+𝑘4𝑠11𝛼2𝛽2+𝑖𝑘𝑐𝑠14𝑘2Ω2𝑐2,𝑡3=1𝛼2𝛽2+𝑖𝑘𝑐𝑠1×𝑘2𝑐22𝛽2𝑠+𝑖𝑘𝑐2𝛽22𝑘2𝑠1+Ω2×𝑘4𝛼2𝑐2+𝑖𝑘𝑐Γ2𝜒𝑘21𝛼2+Ω2𝑘2Ω2+𝑖𝑘𝜀𝑐Γ1Γ3+𝑘2𝑘