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

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 𝐻₀ 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 𝑈⃗(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 2Ω⃗×𝑢^•⃗ is the Coriolis acceleration, and Ω=(0,Ω,0).

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]): −𝜇𝐽_𝑒⃗=curlℎ⃗,𝜕ℎ⃗𝐸⃗=−dd𝜇𝜕iivv_𝑒𝑡𝐸ℎ=⃗⃗==cu00r,,l𝐸⃗,𝜕𝜕𝑢𝑡⃗×𝐻,(2.1) whereℎ⃗=curl𝑢⃗×𝐻₀,𝐻=𝐻⁰+ℎ⃗,𝐻₀=0,𝐻₀,0,(2.2) where ℎ⃗ is the perturbed magnetic field over the primary magnetic field vector, 𝐸⃗ is the electric intensity, 𝐽⃗ is the electric current density, 𝜇_𝑒 is the magnetic permeability, 𝐻₀ is the constant primary magnetic field vector, and 𝑢⃗ is the displacement vector.

The stress and stress couple may be taken to be nonsymmetric, that is, 𝜏_𝑖𝑗≠𝜏_𝑗𝑖, 𝑀_𝑖𝑗≠𝑀_𝑗𝑖. The stress tensor 𝜏_𝑖𝑗 can be expressed as the sum of symmetric and antisymmetric tensors 𝜏_𝑖𝑗=𝜎_𝑖𝑗+𝜎^𝑖𝑗,(2.3) where 𝜎_𝑖𝑗=12𝜏^𝑖𝑗+𝜏_𝑗𝑖,𝜎′_𝑖𝑗=12𝜏^𝑖𝑗−𝜏_𝑗𝑖.(2.4)

The symmetric tensor 𝜎_𝑖𝑗=𝜎_𝑗𝑖 is related to the symmetric strain tensor 𝑒_𝑖𝑗=𝑒_𝑗𝑖=12𝜕𝜕𝑥𝑢^𝑖_𝑗+𝜕𝑢^𝑗𝜕𝑥_𝑖.(2.5) The antisymmetric stress 𝜎^𝑖𝑗 are given by𝜎^23=−𝐹𝜕𝜉𝜕𝑡,𝜎^31=−𝐹𝜕𝜂𝜕𝑡,𝜎^12=−𝐹𝜕𝜁𝜕𝑡,𝜎^11=𝜎^22=𝜎^33=0,(2.6) where 𝐹 is the coefficient of friction between the individual grains. The stress couple 𝑀_𝑖𝑗 is given by𝑀_𝑖𝑗=𝑀𝜈_𝑖𝑗,(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 𝜈_𝑖𝑗 is defined as𝜈₁₁=𝜕𝜉𝜕𝑥,𝜈³¹=𝜕𝜉𝜕𝑧,𝜈³³=𝜕𝜁𝜕𝑧,𝜈²¹𝜈=𝜈₂₂=𝜈₂₃=0,₁₂=𝜕𝜕𝑥𝜔²+𝜂,𝜈₃₂=𝜕𝜕𝑧𝜔²+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]𝜏_{𝑗𝑖,𝑗}+𝐹_𝑖=𝜌𝑢^••_𝑖+Ω⃗×Ω⃗×𝑢⃗_𝑖+2Ω⃗×𝑢^•⃗_𝑖,𝑖,𝑗=1,2,3.(2.9) The heat conduction equation is given by [16] 𝐾∇²𝑇=𝜌𝑠𝜕𝜕𝑡1+𝜏²𝜕𝜕𝑡𝑇+𝛾𝑇⁰𝜕𝜕𝑡1+𝜏²𝛿𝜕𝜕𝑡∇⋅𝑢,⃗(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, 𝛼_𝑡 is coefficient of linear thermal expansion, 𝜆 and 𝜇 are Lame’s elastic constants, 𝜃 is the absolute temperature, 𝛾=𝛼_𝑡(3𝜆+2𝜇), 𝑇₀ is reference temperature solid, 𝑇 is temperature difference (𝜃−𝑇₀), 𝜏₀ is the mechanical relaxation time due to the viscosity, and 𝜏_𝑚=(1+𝜏₀(𝜕/𝜕𝑡)).

The components of stress in generalized thermoelastic medium are given by𝜎₁₁=𝜏_𝑚(𝜆+2𝜇)+𝑝𝜕𝑢𝜕𝑥+𝜏^𝑚𝜆+𝑃𝜕𝑤𝜕𝑧−𝛾1+𝜏¹𝜕𝜎𝜕𝑡𝑇,₃₃=𝜏_𝑚𝜆𝜕𝑢𝜕𝑥+𝜏^𝑚(𝜆+2𝜇)𝜕𝑤𝜕𝑧−𝛾1+𝜏¹𝜕𝜎𝜕𝑡𝑇,₁₃=𝜏_𝑚𝜇𝜕𝑢𝜕𝑧+𝜕𝜕𝑤𝑥.(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 𝜏_𝑖𝑗 is given by 𝜏_𝑖𝑗=𝜇_𝑒𝐻_𝑖ℎ_𝑗+𝐻_𝑗ℎ_𝑖−𝐻_𝑘⋅ℎ_𝑘𝛿_𝑖𝑗,𝑖,𝑗=1,2,3,(2.12) which takes the form𝜏₁₁=−𝜇_𝑒𝐻²⁰∇²𝜙,𝜏₁₃=𝜏₂₃=0,𝜏₃₃=𝜇_𝑒𝐻²⁰∇²𝜙,∇²𝜙=𝜕𝑢𝜕𝑥+𝜕𝜕𝑤𝑧.(2.13)

The dynamic equations of motion are𝜕𝜏₁₁𝜕𝑥+𝜕𝜕𝜏𝑧³¹+𝑃2𝜕𝜕𝜔𝑧²−𝜌𝑔𝜕𝜕𝑤𝑥+𝐹^𝑥=𝜌𝜕²𝑢𝜕𝑡²+2Ω𝜕𝑤𝜕𝑡−Ω²𝜕𝜏₁₂𝑢,𝜕𝑥+𝜕𝜕𝜏𝑧³²+𝐹^𝑦𝜕𝜏₁₃=0,𝜕𝑥+𝜕𝜕𝜏𝑧³³+𝑃2𝜕𝜕𝜔𝑥²+𝜌𝑔𝜕𝜕𝑤𝑥+𝐹^𝑧=𝜌𝜕²𝑤𝜕𝑡²−2Ω𝜕𝑢𝜕𝑡−Ω²𝑤,(2.14) where 𝑔 is the Earth's gravity and𝜏𝐹^→=−𝜇_𝑒𝐻²⁰∇²𝜙,0,𝜇_𝑒𝐻²⁰∇²𝜙,(2.15)₂₃−𝜏₃₂+𝜕𝑀¹¹𝜏𝜕𝑥+𝜕𝑀𝜕𝑧³¹=0,₃₁−𝜏₁₃+𝜕𝑀¹²𝜏𝜕𝑥+𝜕𝑀𝜕𝑧³²=0,₁₂−𝜏₂₁+𝜕𝑀¹³𝜕𝑥+𝜕𝑀𝜕𝑧³³=0.(2.16)

From (2.3)–(2.8) and (2.11), we have𝜏₁₁=𝜏_𝑚(𝜆+2𝜇)+𝑝𝜕𝑢𝜕𝑥+𝜏^𝑚𝜆+𝑃𝜕𝑤𝜕𝑧−𝛾1+𝜏¹𝜕𝜏𝜕𝑡𝑇,₃₃=𝜏_𝑚𝜆𝜕𝑢𝜕𝑥+𝜏^𝑚(𝜆+2𝜇)𝜕𝑤𝜕𝑧−𝛾1+𝜏¹𝜕𝜏𝜕𝑡𝑇,₁₃=𝜏_𝑚𝜇𝜕𝑢𝜏𝜕𝑧+𝜕𝜕𝑤𝑥+𝐹𝜕𝜕𝜂𝑡₁₂,=−𝐹𝜕𝜁𝜏𝜕𝑡₂₃,=−𝐹𝜕𝜉𝑀𝜕𝑡₁₁,=𝑀𝜕𝜉𝜕𝑥,𝑀³¹=𝑀𝜕𝜉𝜕𝑧,𝑀³³=𝑀𝜕𝜁𝜕𝑧,𝑀²¹𝑀=𝑀₂₂=𝑀₂₃=0,₁₂=𝑀𝜕𝜕𝑥𝜔²+𝜂,𝑀₃₂=𝑀𝜕𝜕𝑧𝜔²+𝜂,𝑀₁₃=𝑀𝜕𝜁𝜕𝑥.(2.17) Substituting (2.17) into (2.14) and (2.16) tends to𝜏_𝑚(𝜆+2𝜇)+𝑃𝜕²𝑢𝜕𝑥²+𝜏_𝑚𝜆+𝑃𝜕²𝑤𝜕𝑥𝜕𝑧−𝛾1+𝜏¹𝜕𝜕𝑡𝜕𝜕𝑇𝑥+𝜏^𝑚𝜇𝜕²𝑢𝜕𝑧²+𝜕²𝑤+𝑃𝜕𝑥𝜕𝑧2𝜕²𝑢𝜕𝑧²−𝜕²𝑤𝜕𝑥𝜕𝑧−𝜌𝑔𝜕𝜕𝑤𝑥+𝐹𝜕²𝜂𝜕𝑧𝜕𝑡+𝜇^𝑒𝐻²⁰𝜕²𝑢𝜕𝑥²+𝜕²𝑤𝜕𝑥𝜕𝑧=𝜌𝜕²𝑢𝜕𝑡²+2Ω𝜕𝑤𝜕𝑡−Ω²𝑢,(2.18) then𝜏_𝑚(𝜆+2𝜇)+𝑃+𝜇_𝑒𝐻²⁰𝜕²𝑢𝜕𝑥²+𝜏_𝑚(𝜆+𝜇)+𝑃2+𝜇^𝑒𝐻²⁰𝜕²𝑤𝜕𝑥𝜕𝑧+𝜏^𝑚𝜇+𝑃2𝜕²𝑢𝜕𝑧²−𝛾1+𝜏₁𝜕𝜕𝑡𝜕𝜕𝑇𝑥−𝜌𝑔𝜕𝜕𝑤𝑥+𝐹𝜕²𝜂𝜕𝑧𝜕𝑡=𝜌𝜕²𝑢𝜕𝑡²+2Ω𝜕𝑤𝜕𝑡−Ω²𝑢.(2.19)

Also,𝜕𝜏𝜕𝑡𝜕𝜕𝑥𝜁−𝜕𝜕𝑧𝜉=0,(2.20)_𝑚𝜇𝜕²𝑢𝜕𝑥𝜕𝑧+𝜕²𝑤𝜕𝑥²−𝐹𝜕²𝜂𝜕𝑥𝜕𝑡+𝜏^𝑚𝜆𝜕²𝑢𝜕𝑥𝜕𝑧+𝜏^𝑚(𝜆+2𝜇)𝜕²𝑤𝜕𝑧²−𝛾1+𝜏₁𝜕+𝑃𝜕𝑡𝜕𝜕𝑇𝑧2𝜕²𝑢𝜕𝑥𝜕𝑧−𝜕²𝑤𝜕𝑥²+𝜌𝑔𝜕𝜕𝑥𝑢+𝜇^𝑒𝐻²⁰𝜕²𝑢𝜕𝑥𝜕𝑧+𝜕²𝑤𝜕𝑧²=𝜌𝜕²𝑤𝜕𝑡²−2Ω𝜕𝑢𝜕𝑡−Ω²𝑤,(2.21) then𝜏^𝑚(𝜆+𝜇)+𝑃2+𝜇^𝑒𝐻²⁰𝜕²𝑢𝜕𝑥𝜕𝑧+𝜏^𝑚𝜇−𝑃2𝜕²𝑤𝜕𝑥²+𝜏_𝑚(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝜕²𝑤𝜕𝑧²−𝛾1+𝜏₁𝜕𝜕𝑡𝜕𝜕𝑇𝑧+𝜌𝑔𝜕𝜕𝑥𝑢−𝐹𝜕²𝜂𝜕𝑥𝜕𝑡=𝜌𝜕²𝑤𝜕𝑡²−2Ω𝜕𝑢𝜕𝑡−Ω²𝑤,(2.22) and, from (2.16), we have∇²𝜉−𝑠₂𝜕𝜉∇𝜕𝑡=0,(2.23)₂𝜔₂+𝜂−𝑠₂𝜕𝜂∇𝜕𝑡=0,(2.24)₂𝜁−𝑠₂𝜕𝜁𝜕𝑡=0,(2.25) where 𝑠₂=2𝐹𝑀.(2.26)

3. Solution of the Problem

By Helmholtz's theorem [19], the displacement vector 𝑢⃗ can be written in the displacement potentials 𝜙 and 𝜓 form, as𝑢⃗=grad𝜙+curl𝜓⃗,𝜓⃗=(0,𝜓,0),(3.1) which reduces to𝑢=𝜕𝜙𝜕𝑥−𝜕𝜕𝜓𝑧,𝑤=𝜕𝜕𝜙𝑧+𝜕𝜕𝜓𝑥.(3.2)

Substituting (3.2) into (2.19), (2.22), and (2.24), the wave equations tend to𝛼²∇²𝜙−𝛾𝜌1+𝜏¹𝜕𝜕𝑡𝑇−𝑔𝜕𝜕𝜓𝑥=𝜕²𝜙𝜕𝑡²+2Ω𝜕𝜓𝜕𝑡−Ω²𝛽𝜙,(3.3)₂∇²𝜓−𝑠₁𝜕𝜂𝜕𝑡+𝑔𝜕𝜕𝜙𝑥=𝜕²𝜓𝜕𝑡²−2Ω𝜕𝜙𝜕𝑡−Ω²∇𝜓,(3.4)₂𝜂−𝑠₂𝜕𝜂𝜕𝑡−∇⁴𝜓=0,(3.5) where𝑠₁=𝐹𝜌,𝛼²=𝜏^𝑚(𝜆+2𝜇)+𝑃+𝜇_𝑒𝐻²⁰𝜌,𝛽²=2𝜏^𝑚𝜇−𝑃2𝜌.(3.6)

Substituting (3.2) into (2.10), we obtain𝐾∇²𝑇=𝜌𝑠𝜕𝜕𝑡1+𝜏²𝜕𝜕𝑡𝑇+𝛾𝑇⁰𝜕𝜕𝑡1+𝜏²𝛿𝜕𝜕𝑡∇²𝜙.(3.7)

From (3.3) and (3.7), by eliminating 𝑇, we obtain ∇²−1𝜒𝜕𝜕𝑡1+𝜏²𝜕𝜕𝑡𝛼²∇²𝜙−𝑔𝜕𝜓𝜕𝑥−𝜕²𝜙𝜕𝑡²−2Ω𝜕𝜓𝜕𝑡+Ω²−𝜀𝜕𝜙𝜕𝑡1+𝜏¹𝜕𝜕𝑡1+𝜏²𝛿𝜕𝜕𝑡∇²𝜙=0,(3.8) where𝜒=𝐾𝜌𝑠,𝜀=𝛾²𝑇₀𝜌𝐾.(3.9)

From (3.4) and (3.5) by eliminating 𝜂, we obtain ∇²−𝑠₂𝜕𝜕𝑡𝛽²∇²𝜓−𝜕²𝜓𝜕𝑡²+𝑔𝜕𝜙𝜕𝑥+2Ω𝜕𝜕𝜙𝑡+Ω²𝜓−𝑠₁∇⁴𝜕𝜓𝜕𝑡=0.(3.10)

For a plane harmonic wave propagation in the 𝑥-direction, we assume𝜙=(𝜉,𝜙𝜂₁,𝑒𝜁)=^{𝑖𝑘(𝑥−𝑐𝑡)}𝜉,₁𝜓=𝜓₁𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},(3.11),𝜂₁,𝜁₁𝑒^{𝑖𝑘(𝑥−𝑐𝑡)}.(3.12)

From (3.12) into (2.20), (2.23), and (2.25), we get𝐷𝐷𝜉₁−𝑖𝑘𝜁₁=0,(3.13)₂𝐷𝜉₁+𝑞²𝜉₁=0,(3.14)₂𝜁₁+𝑞²𝜁₁=0,(3.15) where𝑞²=𝑖𝑘𝑐𝑠₂−𝑘²,𝐷≡𝑑𝑑𝑧.(3.16)

The solution of (3.14) and (3.15) takes the form𝜉₁=𝐴₁𝑒^𝑖𝑞𝑧+𝐴₂𝑒^{−𝑖𝑞𝑧},𝜁₁=𝐵₁𝑒^𝑖𝑞𝑧+𝐵₂𝑒^{−𝑖𝑞𝑧},(3.17) where 𝐴₁,𝐴₂,𝐵₁, and 𝐵₂ are arbitrary constants.

From (3.13) and (3.17), we obtain 𝑞𝐴₁𝑒^𝑖𝑞𝑧−𝐴₂𝑒^{−𝑖𝑞𝑧}−𝑘𝐵₁𝑒^𝑖𝑞𝑧+𝐵₂𝑒^{−𝑖𝑞𝑧}=0,(3.18) then𝑞𝑞𝐴𝐴₁₂−𝑘𝐵₁=0,−𝑘𝐵₂=0⟹𝐴^𝑗=(−1)^𝑗−1𝑘𝑞𝐵^𝑗,𝑗=1,2.(3.19)

Substituting (3.11) into (3.8) and (3.10), we obtain𝑅𝛼^2∗𝐷⁴+𝐺₁𝐷²+𝐺₂𝜙₁−𝐺₃𝐷²+𝐺₄𝜓₁=0,₁𝐷⁴+𝑅₂𝐷²+𝑅₃𝜓₁+𝑅₄𝐷²+𝑅₅𝜙₁=0,(3.20) whereΓ₀=1−𝑖𝑘𝑐𝛼𝜏₀,Γ₁=1−𝑖𝑘𝑐𝜏₁,Γ₂=1−𝑖𝑘𝑐𝜏₂,Γ₃=1−𝑖𝑘𝑐𝜏₂𝛿,_2∗=Γ⁰(𝜆+2𝜇)+𝑃+𝜇_𝑒𝐻²⁰𝜌,𝛽^2∗=2Γ⁰𝜇−𝑃𝐺2𝜌₁.=𝑘²𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝜒𝛼^2∗𝐺₂Γ₂+𝜒𝜀Γ₁Γ₃+Ω²,=𝑘⁴𝛼^2∗−𝑐²+𝑖𝑘𝑐Γ²𝜒𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²𝐺₃+𝑖𝑘𝜀𝑐Γ₁Γ₃,=𝑖𝑘(𝑔−2Ω𝑐),𝐺₄=−(𝑔−2Ω𝑐)𝑖𝑘³+𝑘²𝑐Γ₂𝑅₁𝜒,=𝛽^2∗𝑅+₃𝑖𝑘𝑐𝑠₁,𝑅₂=𝑘²𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²,𝑅₄=𝑘²𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁,=𝑖𝑘(𝑔−2Ω𝑐),𝑅₅=(2Ω𝑐−𝑔)𝑖𝑘³−𝑘²𝑐𝑠₂.(3.21)

The solution of (3.20) takes the form𝜙₁=⁴_𝑗=1𝐶_𝑗𝜓₁𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧},=⁴_𝑗=1𝐸_𝑗𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝐹_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧},(3.22) where the constants 𝐸_𝑗 and 𝐹_𝑗 are related to the constants 𝐶_𝑗 and 𝐷_𝑗 in the form𝑚𝐸_𝑗=𝑚_𝑗𝐶_𝑗,𝐹_𝑗=𝑚_𝑗𝐷_𝑗,𝑗=1,2,3,4,_𝑗=1×(𝑔−𝛼2Ω𝑐)𝑖𝑘𝑁^2𝑗−𝑖𝑘−𝜀Γ₂/𝜒_2∗𝑘²𝑁^4𝑗−𝑘²𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝜒𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²𝑁^2𝑗+𝑖𝑘𝑐Γ²𝜒1−𝛼^2∗+Ω²𝑘²−Ω²+𝑖𝑘𝜀𝑐Γ₁Γ₃.(3.23) Substituting (3.22) into (3.11), we obtain 𝜙=⁴_𝑗=1𝐶_𝑗𝜓=₄𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},_𝑗=1𝐸_𝑗𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝐹_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},(3.24) and values of displacement components 𝑢 and 𝑤 are𝑢=𝑖𝑘⁴_𝑗=11−𝑁_𝑗𝑤=𝑖𝑘₄𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝑧}+1+𝑁_𝑗𝑚_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},_𝑗=1𝑁_𝑗+𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝑚_𝑗−𝑁_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},(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=𝑘²𝛼^2∗𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝛼^2∗𝜒𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²+1𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑡×𝑘²𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²,(3.27)₂=1𝛼^2∗𝑘⁴+𝛼1^2∗−𝑐²+𝑖𝑘𝜒𝑐Γ²𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²+𝑖𝑘𝜀𝑐Γ₁Γ₃𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²×𝑘₂𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²+𝑐1²−2𝛼^2∗+𝑖𝑘𝜒𝑐𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²−1𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²𝑡(𝑔−2Ω𝑐)²,(3.28)₃=1×𝛼^2∗𝛽^2∗𝑘₂+𝑖𝑘𝑐𝑠₁×𝑘₄𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²+𝑘₂𝛼^2∗−𝑐²+𝑖𝑘𝜒𝑐Γ²𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²+𝑖𝑘𝜀𝑐Γ₁Γ₃×𝑘𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁₂−𝑖𝑘(𝑔𝑐−²−2Ω2𝛼𝑐)^2∗₂+𝑖𝑘𝜒𝑐𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²𝑖𝑘³+𝑘²𝑐Γ₂𝑡𝜒−𝑖𝑘³(𝑔−2Ω𝑐)²𝑖𝑘−𝑐𝑠₂,(3.29)₄=1×𝛼^2∗𝛽^2∗𝑘+𝑖𝑘𝑐𝑠₂+(2Ω𝑘²𝑐−−𝑖𝑔𝑘)𝑐₂𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁×[𝑖𝑘(𝑔−2Ω𝑐)]𝑖𝑘³−𝑘²𝑐𝑠₂𝑖𝑘³+𝑘²𝑐Γ₂𝜒.(3.30)

From (3.4), (3.11), (3.12), (3.22), and (3.23), we obtain 𝜂₁=⁴_𝑗=11𝑖𝑘𝑐𝑠₁𝑘²𝛽^2∗𝑚_𝑗1+𝑁^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)×𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧}.(3.31)

Using (3.22) and (3.11) into (3.3), we obtain𝑇=𝜌𝛾Γ_14𝑗=1−𝛼^2∗𝑘²1+𝑁^2𝑗+𝑘²𝑐²−𝑖𝑘𝑔𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)}.(3.32)

With the lower medium, we use the symbols with primes, for 𝜉₁,𝜁₁,𝜂₁,𝑇,𝜙,𝜓, and 𝑞, for 𝑧>𝐾,𝜉^1=−𝑘𝑞^𝜂_1𝐵^2𝑒^{−𝑖𝑞′𝑧},𝜁^1=𝐵^2𝑒^{−𝑖𝑞′𝑧},=⁴_𝑗=11𝑖𝑘𝑐𝑠^1𝑇_𝑘²𝛽^∗2𝑚^𝑗1+𝑁^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω^2+𝑖𝑘2Ω^𝑐−𝑔𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝑧},=𝜌^𝛾^Γ_14𝑗=1−𝛼^∗2𝑘𝜙²_1+𝑁^𝑗2+𝑘²𝑐²−𝑖𝑘𝑔𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=⁴_𝑗=1𝜓_𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=⁴_𝑗=1𝐹^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)}.(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=𝑀^33, (vii)𝑀₃₁=𝑀^31, (viii)𝑀₃₂=𝑀^32, (ix)𝜏₃₃+𝜏₃₃=𝜏^33+𝜏^33(x)𝜏₃₁+𝜏₃₁=𝜏^31+𝜏^31, (xi)𝜏₃₂+𝜏₃₂=𝜏^32+𝜏^32, (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 obtain𝐵𝐵₁₁𝑒^𝑖𝑞K−𝐵₂𝑒^−𝑖𝑞K=−𝐵^2𝑒^{−𝑖𝑞′K},𝑀𝐵₁𝑒^𝑖𝑞K+𝐵₂𝑒^−𝑖𝑞K=𝐵^2𝑒^{−𝑖𝑞′K},𝑀𝐵₁𝑒^𝑖𝑞K−𝐵₂𝑒^−𝑖𝑞K=−𝑀^𝐵^2𝑒^{−𝑖𝑞′K},𝑒^𝑖𝑞K+𝐵₂𝑒^−𝑖𝑞K=−𝑀^𝐵^2𝑒^{−𝑖𝑞′K}.(4.1) Hence, 𝐵₁=𝐵₂=𝐵^2=0,𝜉=𝜁=𝜉^=𝜁^=0.(4.2)

The other significant boundary conditions are responsible for the following relations:(i)⁴_𝑗=11−𝑁_𝑗𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+1+𝑁_𝑗𝑚_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}−1+𝑁^𝑗𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.3)(ii)⁴_𝑗=1𝑁_𝑗+𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+𝑚_𝑗−𝑁_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}−𝑚^𝑗−𝑁^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.4)(iv)⁴_𝑗=11𝑐𝑠₁𝑘²−₄𝛽^2∗𝑚_𝑗1+𝑁^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)×𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}_𝑗=11𝑐𝑠^1𝑘²𝛽^∗2𝑚^𝑗1+𝑁^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω^2+𝑖𝑘2Ω^𝑐−𝑔𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.5)(viii)𝑀𝑁_𝑗4_𝑗=1𝑘²𝑚_𝑗𝑁^2𝑗+1+1𝑖𝑘𝑐𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑁^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗+𝑀_𝑒^{𝑖𝑘𝑁𝑗𝐾}−𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}𝑁^𝑗4_𝑗=1𝑘²𝑚^𝑗𝑁^𝑗2+1+1𝑖𝑘𝑐𝑠^1×𝑘²𝛽^∗2𝑚^𝑗1+𝑁^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω^2+𝑖𝑘2Ω^𝑐−𝑔𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.6)(ix)⁴_𝑗=1Γ⁰×𝐶_𝑗𝜆+𝜇_𝑒𝐻²⁰1−𝑁_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑁^2𝑗+𝑚_𝑗𝑁_𝑗×𝐷_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑁_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑁^2𝑗−𝑚_𝑗𝑁_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}+𝜌−𝛼^2∗1+𝑁^2𝑗+𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐶^𝑗−Γ_0𝑒^{𝑖𝑘𝑁𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}𝜆^+𝜇^𝑒𝐻²⁰1+𝑁^𝑗𝑚^𝑗+Γ^0𝜆^+2𝜇^+𝜇^𝑒𝐻²⁰𝑁^𝑗2−𝑚^𝑗𝑁^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}−𝜌^−𝛼^∗21+𝑁^𝑗2+𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.7)(x)⁴_𝑗=1−2𝑘²+−Γ𝑘₀₂𝜇𝑁_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}−𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}Γ₀𝜇𝑚_𝑗1−𝑁^2𝑗+𝐹𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑁^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}−2𝑘²Γ^0𝜇^𝑁^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}−−𝑘²Γ^0𝜇^𝑚^𝑗1−𝑁^𝑗2+𝐹𝑠^1×𝐷_𝑗𝑘²𝛽^∗2𝑚^𝑗1+𝑁^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω^2+𝑖𝑘2Ω^𝑐−𝑔𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.8)(xii)⁴_𝑗=1𝜌𝛾−𝛼^2∗−𝜌_𝑘²𝑁^2𝑗+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}𝛾^−𝛼^∗2𝑘²𝑁^𝑗2+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝐾}=0,(4.9)(xiii)⁴_𝑗=1𝜌𝛾−𝛼^2∗−𝜌_𝑘²𝑁^2𝑗+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚_𝑗𝜃+𝑖𝑘𝑁_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑁𝑗𝐾}+𝜃−𝑖𝑘𝑁_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}𝛾^−𝛼^∗2𝑘²𝑁^𝑗2+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚^𝑗×𝜃−𝑖𝑘𝑁^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑁𝑗𝐾}=0,(4.10)(xvi)𝑀𝑁_𝑗4_𝑗=1𝑘²𝑚_𝑗𝑁^2𝑗+1+1𝑖𝑘𝑐𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑁^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗−𝐷_𝑗=0,(4.11)(xvii)⁴_𝑗=1Γ⁰𝜆+𝜇_𝑒𝐻²⁰1−𝑁_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑁^2𝑗+𝑚_𝑗𝑁_𝑗𝐶_𝑗+Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑁_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑁^2𝑗−𝑚_𝑗𝑁_𝑗𝐷_𝑗+𝜌−𝛼^2∗1+𝑁^2𝑗+𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐶^𝑗+𝐷_𝑗=0,(4.12)(xviii)⁴_𝑗=1−2𝑘²+−𝑘Γ₀₂𝜇𝑁_𝑗𝐶_𝑗−𝐷_𝑗Γ₀𝜇𝑚_𝑗1−𝑁^2𝑗+𝐹𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑁^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗+𝐷_𝑗=0,(4.13)(xx)⁴_𝑗=1−𝛼^2∗𝑘²𝑁^2𝑗+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚_𝑗𝜃+𝑖𝑘𝑁_𝑗𝐶_𝑗+𝜃−𝑖𝑘𝑁_𝑗𝐷_𝑗=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+ℎ₁𝑉⁶+ℎ₂𝑉⁴+ℎ₃𝑉²+ℎ₄=0,(5.1) where𝛼^2∗=Γ⁰(𝜆+2𝜇)𝜌,𝛽^2∗=Γ⁰𝜇𝑚_𝑗𝜌,=1×(𝑔−𝛼2Ω𝑐)𝑖𝑘𝑉^2𝑗−𝑖𝑘−(𝜀/𝜒)_2∗𝑘²𝑉^4𝑗−𝑘²𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝜒𝛼^2∗+𝜒𝜀+Ω²𝑉^2𝑗+𝑖𝑘𝑐𝜒1−𝛼^2∗+Ω²𝑘²−Ω²ℎ+𝑖𝑘𝜀𝑐₁,=𝑘²𝛼^2∗𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝛼^2∗𝜒𝛼^2∗+𝜒𝜀+Ω²+1𝛽^2∗+𝑖𝑘𝑐𝑠₁ℎ₂×𝑘²𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²,=1𝛼^2∗𝑘⁴+𝛼1^2∗−𝑐²+𝑖𝑘𝜒𝑐𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²+𝑖𝑘𝜀𝑐𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²×𝑘₂𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²𝑐²−2𝛼^2∗+𝑖𝑘𝜒𝑐𝛼^2∗+𝜒𝜀+Ω²+1𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²−1𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²ℎ₃(𝑔−2Ω𝑐)²,=1𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²×𝑘₄𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²+𝑘𝛼^2∗−𝑐²+𝑖𝑘𝜒𝑐𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²+𝑖𝑘𝜀𝑐₂×𝑘𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁₂−𝑖𝑘(𝑔𝑐−²−2Ω2𝛼𝑐)^2∗₂+𝑖𝑘𝜒𝑐𝛼^2∗+𝜒𝜀+Ω²𝑖𝑘³+𝑘²𝑐ℎ𝜒−𝑖𝑘³(𝑔−2Ω𝑐)²𝑖𝑘−𝑐𝑠₂₄,=1𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠𝑘²+(2Ω𝑐−𝑘²𝑔)−₂𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁×[𝑖𝑘(𝑔−2Ω𝑐)]𝑖𝑘³−𝑘²𝑐𝑠₂𝑖𝑘³+𝑘²𝑐𝜒.(5.2)

Also, 𝜂₁=⁴_𝑗=11𝑖𝑘𝑐𝑠₁𝑘²𝑇=𝜌𝛽^2∗𝑚_𝑗1+𝑉^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)×𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝑧}.𝛾⁴_𝑗=1−𝛼^2∗𝑘²1𝜉_1+𝑉^2𝑗+𝑘²𝑐²−𝑖𝑘𝑔𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=−𝑘𝑞^𝜂_1𝐵^2𝑒^{−𝑖𝑞′𝑧},𝜁^1=𝐵^2𝑒^{−𝑖𝑞′𝑧},=⁴_𝑗=11𝑖𝑘𝑐𝑠^1𝑇_𝑘²𝛽^∗2𝑚^𝑗1+𝑉^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω^2+𝑖𝑘2Ω^𝑐−𝑔𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝑧},=𝜌^𝛾^4_𝑗=1−𝛼^∗2𝑘²1𝜙+_1𝑉^𝑗2+𝑘²𝑐²−𝑖𝑘𝑔𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=⁴_𝑗=1𝜓_1𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝑧},=⁴_𝑗=1𝐹^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝑧},(5.3) Using the boundary conditions, we obtain⎡⎢⎢⎢⎢⎣𝑑¹¹𝑑₁₂⋯𝑑₁₈𝑑^15𝑑^16⋯𝑑^18𝑑₂₁𝑑₂₂⋯𝑑₂₈𝑑^25𝑑^26⋯𝑑^28𝑑⋮⋮⋮⋮⋮⋮⋮⋮₁₂₁𝑑₁₂₂⋯𝑑₁₂₈𝑑^125𝑑^126⋯𝑑^128⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐶¹𝐶₂𝐶₃𝐶₄𝐷₁𝐷₂𝐷₃𝐷₄𝐷^1𝐷^2𝐷^3𝐷^4⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=[0],(5.4) where𝑑𝑑_1𝑗2𝑗=1−𝑉_𝑗𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗Κ}+1+𝑉_𝑗𝑚_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗Κ},𝑑^1𝑗=1+𝑉^𝑗𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗Κ},𝑑_3𝑗=𝑉_𝑗+𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗Κ}+𝑚_𝑗−𝑉_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗Κ},𝑑^2𝑗=𝑚^𝑗−𝑉^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗Κ},=1𝑐𝑠₁𝑘²𝑑_3𝑗𝛽^2∗𝑚_𝑗1+𝑉^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝑧}+𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝑧},=1𝑐𝑠^1𝑑_4𝑗𝑘²𝛽^∗2𝑚^𝑗1+𝑉^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝑧},=𝑀𝑉_𝑗𝑘²𝑚_𝑗𝑉^2𝑗+1+1𝑖𝑘𝑐𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑉^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗𝑑_4𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}−𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾},=𝑀^𝑉^𝑗𝑘²𝑚^𝑗𝑉^𝑗2+1+1𝑖𝑘𝑐𝑠^1×𝐷_𝑗𝑘²𝛽^∗2𝑚^𝑗1+𝑉^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)𝑑_5𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾},=Γ₀𝜆1−𝑉_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)𝑉^2𝑗+𝑚_𝑗𝑉_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}+Γ₀𝜆1+𝑉_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)𝑉^2𝑗−𝑚_𝑗𝑉_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾}+𝜌−𝛼^2∗1+𝑉^2𝑗+𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐶^𝑗𝑑_5𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾},=Γ^0𝜆^1+𝑉^𝑗𝑚^𝑗+Γ^0𝜆^+2𝜇^𝑉^𝑗2−𝑚^𝑗𝑉^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾}+𝜌^−𝛼^∗21+𝑉^𝑗2+𝑐²−𝑖𝑔×𝐷_𝑗𝑘𝑚^𝑗𝑑_6𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾},=+−2−𝑘𝑘²₂Γ₀𝜇𝑉_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}−𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾}Γ₀𝜇𝑚_𝑗1−𝑉^2𝑗+𝐹𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑉^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗𝑑_6𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾},=2𝑘²Γ^0𝜇^𝑉^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾}+−𝑘²Γ^0𝜇^𝑚^𝑗1−𝑉^𝑗2+𝐹𝑠^1×𝐷_𝑗𝑘²𝛽^∗2𝑚^𝑗1+𝑉^𝑗2−𝑚^𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)𝑑_7𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾},=𝜌𝛾−𝛼^2∗𝑑_7𝑗𝑘²𝑉^2𝑗+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}+𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾},=𝜌^𝛾^−𝛼^∗2𝑑_8𝑗𝑘²𝑉^𝑗2+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾},=𝜌𝛾−𝛼^2∗𝑑_8𝑗𝑘²𝑉^2𝑗+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚_𝑗𝜃+𝑖𝑘𝑉_𝑗𝐶_𝑗𝑒^{𝑖𝑘𝑉𝑗𝐾}+𝜃−𝑖𝑘𝑉_𝑗𝐷_𝑗𝑒^{−𝑖𝑘𝑉𝑗𝐾},=𝜌^𝛾^−𝛼^∗2𝑑_9𝑗𝑘²𝑉^𝑗2+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚^𝑗𝜃−𝑖𝑘𝑉^𝑗𝐷^𝑗𝑒^{−𝑖𝑘𝑉′𝑗𝐾},=𝑀𝑉_𝑗𝑘²𝑚_𝑗𝑉^2𝑗+1+1𝑖𝑘𝑐𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑉^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗𝑑_10𝑗−𝐷_𝑗,=Γ₀𝜆1−𝑉_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)𝑉^2𝑗+𝑚_𝑗𝑉_𝑗𝐶_𝑗+Γ₀𝜆1+𝑉_𝑗𝑚_𝑗+Γ₀(𝜆+2𝜇)𝑉^2𝑗−𝑚_𝑗𝑉_𝑗𝐷_𝑗+𝜌−𝛼^2∗1+𝑉^2𝑗+𝑐²−𝑖𝑔𝑘𝑚^𝑗𝐶^𝑗𝑑_11𝑗+𝐷_𝑗,+=−−2𝑘𝑘²₂Γ₀𝜇𝑉_𝑗𝐶_𝑗−𝐷_𝑗Γ₀𝜇𝑚_𝑗1−𝑉^2𝑗+𝐹𝑠₁𝑘²×𝐶𝛽^2∗𝑚_𝑗1+𝑉^2𝑗−𝑚_𝑗𝑘²𝑐²+Ω²+𝑖𝑘(2Ω𝑐−𝑔)_𝑗𝑑_12𝑗+𝐷_𝑗,𝑑_9𝑗=−𝛼^2∗𝑘²𝑉^2𝑗+1+𝑘²𝑐²−𝑖𝑔𝑘𝑚_𝑗𝜃+𝑖𝑘𝑉_𝑗𝐶_𝑗+𝜃−𝑖𝑘𝑉_𝑗𝐷_𝑗,=𝑑^10𝑗=𝑑^11𝑗=𝑑^12𝑗=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) becomes𝑅𝛼^2∗𝐷⁴+𝐺₁𝐷²+𝐺₂𝜙₁−𝐺^∗3𝐷²+𝐺^∗4𝜓₁=0,₁𝐷⁴+𝑅₂𝐷²+𝑅₃𝜓₁+𝑅^∗4𝐷²+𝑅^∗5𝜙₁=0,(5.6) where𝐺^∗3=−2𝑖𝑘Ω𝑐,𝐺^∗4=2Ω𝑐𝑖𝑘³+𝑘²𝑐Γ₂𝑅_∗4𝜒,=−2𝑖𝑘Ω𝑐,𝑅^∗5=2Ω𝑐𝑖𝑘³−𝑘²𝑐𝑠₂,(5.7) and 𝐺₁,𝐺₂,𝑅₁,𝑅₂, and 𝑅₃ are as in (3.21).

The solution of (5.6) take the form𝜙=⁴_𝑗=1𝐶^∗𝑗𝜓=₄𝑒^{𝑖𝑘𝑋𝑗𝑧}+𝐷^∗𝑗𝑒^{−𝑖𝑘𝑋𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},_𝑗=1𝐸^∗𝑗𝑒^{𝑖𝑘𝑋𝑗𝑧}+𝐹^∗𝑗𝑒^{−𝑖𝑘𝑋𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},(5.8) where𝑚𝐸^∗𝑗=𝑚^∗𝑗𝐶^∗𝑗,𝐹^∗𝑗=𝑚^∗𝑗𝐷^∗𝑗,𝑗=1,2,3,4,(5.9)_∗𝑗=1×−2Ω𝛼𝑐𝑖𝑘𝑋^2𝑗−𝑖𝑘−𝜀Γ₂/𝜒_2∗𝑘²𝑋^4𝑗−𝑘²𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝜒𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²𝑋^2𝑗+𝑖𝑘𝑐Γ²𝜒1−𝛼^2∗+Ω²𝑘²−Ω²+𝑖𝑘𝜀𝑐Γ₁Γ₃,(5.10) and 𝑋₁,𝑋₂,𝑋₃,and 𝑋₄ are taken to be the complex roots of equation𝑋⁸+𝑡^∗1𝑋⁶+𝑡^∗2𝑋⁴+𝑡^∗3𝑋²+𝑡^∗4=0,(5.11) where𝑡^∗1=𝑘²𝛼^2∗𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝛼^2∗𝜒𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²+1𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑡_∗2×𝑘²𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²,=1𝛼^2∗𝑘⁴+𝛼1^2∗−𝑐²+𝑖𝑘𝜒𝑐Γ²𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²+𝑖𝑘𝜀𝑐Γ₁Γ₃𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²×𝑘₂𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²+𝑐1²−2𝛼^2∗+𝑖𝑘𝜒𝑐𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²𝛽^2∗+𝑖𝑘𝑐𝑠₁𝑘²−1𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠₁4𝑘²𝑡_∗3Ω²𝑐²,=1×𝛼^2∗𝛽^2∗𝑘₂+𝑖𝑘𝑐𝑠₁×𝑘₄𝑐²−2𝛽^2∗+𝑖𝑘𝑐𝑠₂𝛽^2∗−2𝑘²𝑠₁+Ω²+𝑘₂𝛼^2∗−𝑐²+𝑖𝑘𝜒𝑐Γ²𝑘²1−𝛼^2∗+Ω²−𝑘²Ω²+𝑖𝑘𝜀𝑐Γ₁Γ₃×𝑘𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁₂−4Ω₂𝑐²−2𝛼^2∗+𝑖𝑘𝜒𝑐𝛼^2∗Γ₂+𝜒𝜀Γ₁Γ₃+Ω²𝑐²𝑖𝑘𝑖𝑘³+𝑘²𝑐Γ₂𝑡𝜒+𝑖𝑘³𝑖𝑘−𝑐𝑠₂_∗4,=1𝛼^2∗𝛽^2∗+𝑖𝑘𝑐𝑠𝑘²+4Ω₂𝑘²−𝑖𝑘𝑐𝑠₂𝛽^2∗−𝑐²+𝑖𝑘𝑐𝑠₂Ω²+𝑘⁴𝑠₁[−2𝑖𝑘Ω𝑐]𝑐²𝑖𝑘³−𝑘²𝑐𝑠₂𝑖𝑘³+𝑘²𝑐Γ₂𝑢=𝑖𝑘₄𝜒,_𝑗=11−𝑋_𝑗𝑤=𝑖𝑘₄𝑚^∗𝑗𝐶^∗𝑗𝑒^{𝑖𝑘𝑋𝑗𝑧}+1+𝑋_𝑗𝑚^∗𝑗𝐷^∗𝑗𝑒^{−𝑖𝑘𝑋𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},_𝑗=1𝑋_𝑗𝜂₁+𝑚^∗𝑗𝐶^∗𝑗𝑒^{𝑖𝑘𝑋𝑗𝑧}+𝑚^∗𝑗−𝑋_𝑗𝐷^∗𝑗𝑒^{−𝑖𝑘𝑋𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=⁴_𝑗=11𝑖𝑘𝑐𝑠₁𝑘²𝑇=𝜌𝛽^2∗𝑚^∗𝑗1+𝑋^2𝑗−𝑚^∗𝑗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐×𝐶^∗𝑗𝑒^{𝑖𝑘𝑋𝑗𝑧}+𝐷^∗𝑗𝑒^{−𝑖𝑘𝑋𝑗𝑧},𝛾Γ_14𝑗=1−𝛼^2∗𝑘²1+𝑋^2𝑗+𝑘²𝑐²𝐶^∗𝑗𝑒^{𝑖𝑘𝑋𝑗𝑧}+𝐷^∗𝑗𝑒^{−𝑖𝑘𝑋𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)}.(5.12)

With the lower medium, we use the symbols with primes, for 𝜉₁,𝜁₁,𝜂₁,𝑇,𝜙,𝜓, and 𝑞, for 𝑧>Κ,𝜉^1=−𝑘𝑞^𝜂_1𝐵^2𝑒^{−𝑖𝑞′𝑧},𝜁^1=𝐵^2𝑒^{−𝑖𝑞′𝑧},=⁴_𝑗=11𝑖𝑘𝑐𝑠^1𝑘²𝑇_𝛽^∗2𝑚^𝑗∗1+𝑋^𝑗2−𝑚^𝑗∗𝑘²𝑐²+Ω^2+2𝑖𝑘Ω^𝑐𝐷^𝑗∗𝑒^{−𝑖𝑘𝑋′𝑗𝑧},=𝜌^𝛾^Γ_14𝑗=1−𝛼^∗2𝜙_𝑘²1+𝑋^𝑗2+𝑘²𝑐²𝐷^𝑗∗𝑒^{−𝑖𝑘𝑋′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=⁴_𝑗=1𝜓_𝐷^𝑗𝑒^{−𝑖𝑘𝑁′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)},=⁴_𝑗=1𝐹^𝑗∗𝑒^{−𝑖𝑘𝑋′𝑗𝑧}𝑒^{𝑖𝑘(𝑥−𝑐𝑡)}.(5.13)

From conditions (iii), (v), (vi), (vii), we get the same equations (4.1) and (4.2): the other significant boundary conditions are responsible for the following relations: (i)𝑞₁𝐶^∗1𝑒^{𝑖𝑘𝑋1𝐾}+𝑞₂𝐶^∗2𝑒^{𝑖𝑘𝑋2𝐾}+𝑞₃𝐶^∗3𝑒^{𝑖𝑘𝑋3𝐾}+𝑞₄𝐶^∗4𝑒^{𝑖𝑘𝑋4𝐾}+𝑞₅𝐷^∗1𝑒^{−𝑖𝑘𝑋1𝐾}+𝑞₆𝐷^∗2𝑒^{−𝑖𝑘𝑋2𝐾}+𝑞₇𝐷^∗3𝑒^{−𝑖𝑘𝑋3𝐾}+𝑞₈𝐷^∗4𝑒^{−𝑖𝑘𝑋4𝐾}=𝑞₉𝐷^1∗𝑒^{−𝑖𝑘𝑋′1𝐾}+𝑞₁₀𝐷^2∗𝑒^{−𝑖𝑘𝑋′2𝐾}+𝑞₁₁𝐷^3∗𝑒^{−𝑖𝑘𝑋′3𝐾}+𝑞₁₂𝐷^4∗𝑒^{−𝑖𝑘𝑋′4𝐾},(5.14)(ii)𝑞₁₃𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₁₄𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₁₅𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₁₆𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₁₇𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₁₈𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₁₉𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₂₀𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=𝑞₂₁𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}+𝑞₂₂𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}+𝑞₂₃𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}+𝑞₂₄𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.15)(iv)𝑞₂₅𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₂₆𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₂₇𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₂₈𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₂₅𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₂₆𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₂₇𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₂₈𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=𝑞₂₉𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}+𝑞₃₀𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}+𝑞₃₁𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}+𝑞₃₂𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.16)(viii)𝑞₃₃𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₃₄𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₃₅𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₃₆𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}−𝑞₃₃𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}−𝑞₃₄𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}−𝑞₃₅𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}−𝑞₃₆𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=−𝑞₃₇𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}−𝑞₃₈𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}−𝑞₃₉𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}−𝑞₄₀𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.17)(ix)𝑞₄₁𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₄₂𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₄₃𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₄₄𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₄₅𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₄₆𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₄₇𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₄₈𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=𝑞₄₉𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}+𝑞₅₀𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}+𝑞₅₁𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}+𝑞₅₂𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.18)(x)𝑞₅₃𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₅₄𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₅₅𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₅₆𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₅₇𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₅₈𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₅₉𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₆₀𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=𝑞₆₁𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}+𝑞₆₂𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}+𝑞₆₃𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}+𝑞₆₄𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.19)(xii)𝑞₆₅𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₆₆𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₆₇𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₆₈𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₆₅𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₆₆𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₆₇𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₆₈𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=𝑞₆₉𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}+𝑞₇₀𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}+𝑞₇₁𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}+𝑞₇₂𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.20)(xiii)𝑞₇₃𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₇₄𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₇₅𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₇₆𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₇₇𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₇₈𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₇₉𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₈₀𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=𝑞₈₁𝐷^1∗𝑒^{−𝑖𝑘𝑋′1Κ}+𝑞₈₂𝐷^2∗𝑒^{−𝑖𝑘𝑋′2Κ}+𝑞₈₃𝐷^3∗𝑒^{−𝑖𝑘𝑋′3Κ}+𝑞₈₄𝐷^4∗𝑒^{−𝑖𝑘𝑋′4Κ},(5.21)(xvi)𝑞₈₅𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₈₆𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₈₇𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₈₈𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}−𝑞₈₅𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₈₆𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₈₇𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₈₈𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=0,(5.22)(xvii)𝑞₈₉𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₉₀𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₉₁𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₉₂𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₉₃𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₉₄𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₉₅𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₉₆𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=0,(5.23)(xviii)𝑞₉₇𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₉₈𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₉₉𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₁₀₀𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₁₀₁𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₁₀₂𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₁₀₃𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₁₀₄𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=0,(5.24)(xx)𝑞₁₀₅𝐶^∗1𝑒^{𝑖𝑘𝑋1Κ}+𝑞₁₀₆𝐶^∗2𝑒^{𝑖𝑘𝑋2Κ}+𝑞₁₀₇𝐶^∗3𝑒^{𝑖𝑘𝑋3Κ}+𝑞₁₀₈𝐶^∗4𝑒^{𝑖𝑘𝑋4Κ}+𝑞₁₀₉𝐷^∗1𝑒^{−𝑖𝑘𝑋1Κ}+𝑞₁₁₀𝐷^∗2𝑒^{−𝑖𝑘𝑋2Κ}+𝑞₁₁₁𝐷^∗3𝑒^{−𝑖𝑘𝑋3Κ}+𝑞₁₁₂𝐷^∗4𝑒^{−𝑖𝑘𝑋4Κ}=0,(5.25)

where 𝑞𝑞₁₅=1−𝑋₁𝑚^∗1,𝑞₂=1−𝑋₂𝑚^∗2,𝑞₃=1−𝑋₃𝑚^∗3,𝑞₄=1−𝑋₄𝑚^∗4,𝑞₉=1+𝑋₁𝑚^∗1,𝑞₆=1+𝑋₂𝑚^∗2,𝑞₇=1+𝑋₃𝑚^∗3,𝑞₈=1+𝑋₄𝑚^∗4,𝑞₁₃=1+𝑋^1𝑚^1∗,𝑞₁₀=1+𝑋^2𝑚^2∗,𝑞₁₁=1+𝑋^3𝑚^3∗,𝑞₁₂=1+𝑋^4𝑚^4∗,𝑞₁₇=𝑋₁+𝑚^∗1,𝑞₁₄=𝑋₂+𝑚^∗2,𝑞₁₅=𝑋₃+𝑚^∗3,𝑞₁₆=𝑋₄+𝑚^∗4,𝑞₂₁=𝑚^∗1−𝑋₁,𝑞₁₈=𝑚^∗2−𝑋₂,𝑞₁₉=𝑚^∗3−𝑋₃,𝑞₂₀=𝑚^∗4−𝑋₄,𝑞₂₅=𝑚^1∗−𝑋^1,𝑞₂₂=𝑚^∗2−𝑋^2,𝑞₂₃=𝑚^∗3−𝑋^3,𝑞₂₄=𝑚^∗4−𝑋^4,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₆,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₇,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₈,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₉,=1𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋^21−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₀,=1𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋^22−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₁,=1𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋^23−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₂,=1𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋^24−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₃,=𝑀𝑋₁𝑘²𝑚^∗1𝑋²¹+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₄,=𝑀𝑋₂𝑘²𝑚^∗2𝑋²²+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₅,=𝑀𝑋₃𝑘²𝑚^∗3𝑋²³+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₆,=𝑀𝑋₄𝑘²𝑚^∗4𝑋²⁴+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₇,=𝑀^𝑋^1𝑘²𝑚^1𝑋^21+1+1𝑖𝑘𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋^21−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₈,=𝑀^𝑋^2𝑘²𝑚^2𝑋^22+1+1𝑖𝑘𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋^22−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₉,=𝑀^𝑋^3𝑘²𝑚^3𝑋^23+1+1𝑖𝑘𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋^23−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₄₀,=𝑀^𝑋^4𝑘²𝑚^4𝑋^24+1+1𝑖𝑘𝑐𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋^21−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₄₁,𝑞₄₂=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²¹+𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₄₃=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²²+𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₆₄=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²³+𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,=2𝑘²Γ^0𝜇^𝑋^4−𝑘²Γ^0𝜇^𝑚^∗41−𝑋^24+𝐹𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋^24−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₄₄,𝑞₄₅=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²⁴+𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,𝑞₄₆=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²¹−𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₄₇=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²²−𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₄₈=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²³−𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₄₉=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²⁴−𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,=+Γ𝜌^0_𝜆^+𝜇^𝑒𝐻²⁰1+𝑋^1𝑚^∗1+Γ^0𝜆^+2𝜇^+𝜇^𝑒𝐻²⁰𝑋^21−𝑚^∗1𝑋^1𝑞₅₀−𝛼^2∗1+𝑋^21+𝑐²,=+Γ𝜌^0_𝜆^+𝜇^𝑒𝐻²⁰1+𝑋^2𝑚^∗2+Γ^0𝜆^+2𝜇^+𝜇^𝑒𝐻²⁰𝑋^22−𝑚^∗2𝑋^2𝑞₅₁−𝛼′^2∗1+𝑋′²²+𝑐²,=+Γ𝜌^0_𝜆^+𝜇^𝑒𝐻²⁰1+𝑋^3𝑚^∗3+Γ^0𝜆^+2𝜇^+𝜇^𝑒𝐻²⁰𝑋^23−𝑚^∗3𝑋^3𝑞₅₂−𝛼′^2∗1+𝑋′²³+𝑐²,=+Γ𝜌^0_𝜆^+𝜇^𝑒𝐻²⁰1+𝑋^4𝑚^∗4+Γ^0𝜆^+2𝜇^+𝜇^𝑒𝐻²⁰𝑋^24−𝑚^∗4𝑋^4𝑞₅₃−𝛼′^2∗1+𝑋′²⁴+𝑐²,=−2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₄,=−2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋²²+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₅,=−2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₆,=−2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₇,=2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₈,=2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋²²+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₉,=2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₀,=2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₁,=2𝑘²Γ^0𝜇^𝑋^1−𝑘²Γ^0𝜇^𝑚^∗11−𝑋^21+𝐹𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋^21−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₂,=2𝑘²Γ^0𝜇^𝑋^2−𝑘²Γ^0𝜇^𝑚^∗21−𝑋^22+𝐹𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋^22−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₃,=2𝑘²Γ^0𝜇^𝑋^3−𝑘²Γ^0𝜇^𝑚^∗31−𝑋^23+𝐹𝑠^1𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋^23−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₅,=𝜌𝛾−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²,𝑞₆₆=𝜌𝛾−𝛼^2∗𝑞₆₇𝑘²𝑋²²+1+𝑘²𝑐²,=𝜌𝛾−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²,𝑞₆₈=𝜌𝛾−𝛼^2∗𝑞₆₉𝑘²𝑋²⁴+1+𝑘²𝑐²,=𝜌′𝛾′−𝛼^2∗𝑘²𝑋^21+1+𝑘²𝑐²,𝑞⁷⁰=𝜌′𝛾′−𝛼′^2∗𝑞₇₁𝑘²𝑋′²²+1+𝑘²𝑐²,=𝜌′𝛾′−𝛼^2∗𝑘²𝑋^23+1+𝑘²𝑐²,𝑞⁷²=𝜌′𝛾′−𝛼′^2∗𝑞₇₃𝑘²𝑋′²⁴+1+𝑘²𝑐²,=𝜌𝛾−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₁,𝑞₇₄=𝜌𝛾−𝛼^2∗𝑞₇₅𝑘²𝑋²²+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₂,=𝜌𝛾−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₃,𝑞₇₆=𝜌𝛾−𝛼^2∗𝑞₇₇𝑘²𝑋²⁴+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₄,=𝜌𝛾−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₁,𝑞₇₈=𝜌𝛾−𝛼^2∗𝑞₇₉𝑘²𝑋²²+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₂,=𝜌𝛾−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₃,𝑞₈₀=𝜌𝛾−𝛼^2∗𝑞₈₁𝑘²𝑋²⁴+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₄,=𝜌^𝛾^−𝛼^2∗𝑘²𝑋^21+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋^1,𝑞₈₂=𝜌^𝛾^−𝛼^2∗𝑞₈₃𝑘²𝑋^22+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋^2,=𝜌^𝛾^−𝛼^2∗𝑘²𝑋^23+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋^3,𝑞₈₄=𝜌^𝛾^−𝛼^2∗𝑞₈₅𝑘²𝑋^23+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋^3,=𝑀𝑋₁𝑘²𝑚^∗1𝑋²¹+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₆,=𝑀𝑋₂𝑘²𝑚^∗𝑗𝑋²²+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₇,=𝑀𝑋₃𝑘²𝑚^∗𝑗𝑋²³+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₈,=𝑀𝑋₄𝑘²𝑚^∗4𝑋²⁴+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₉,𝑞₉₀=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²¹+𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₉₁=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²²+𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₉₂=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²³+𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₉₃=Γ₀𝜆+𝜇_𝑒𝐻²⁰1−𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²⁴+𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,𝑞₉₄=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²¹−𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₉₅=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²²−𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₉₆=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²³−𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₉₇=Γ₀𝜆+𝜇_𝑒𝐻²⁰1+𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)+𝜇_𝑒𝐻²⁰𝑋²⁴−𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,=−2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₉₈,=−2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋^2𝑗+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₉₉,=−2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₀,=−2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₁,=2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₂,=2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋²²+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₃,=2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₄,=2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₅,𝑞₁₀₇=−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₁,𝑞₁₀₆=−𝛼^2∗𝑘²𝑋²²+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₂,𝑞₁₀₉=−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₃,𝑞₁₀₈=−𝛼^2∗𝑘²𝑋²⁴+1+𝑘²𝑐²𝜃+𝑖𝑘𝑋₄,𝑞₁₁₁=−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₁,𝑞₁₁₀=−𝛼^2∗𝑘²𝑋²²+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₂,=−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₃,𝑞₁₁₂=−𝛼^2∗𝑘²𝑋²⁴+1+𝑘²𝑐²𝜃−𝑖𝑘𝑋₄.(5.26)

Elimination of 𝐶^∗𝑗,𝐷^∗𝑗, and 𝐷′^∗𝑗 gives the wave velocity equation in the determinant form det𝑑_𝑖𝑗=0.(5.27) This equation has complex roots: the real part (Re) gives the Rayleigh wave velocity, and the imaginary part (Im) gives the attenuation coefficient due to the friction of the granular nature of the medium, where the nonvanishing of the twelfth-order determinant of 𝑑_𝑖𝑗 is given by|||||||||||||||||_{𝑞85𝑒1𝐾−𝑖𝑘𝑋𝑞86𝑒2𝐾−𝑖𝑘𝑋𝑞87𝑒3𝐾−𝑖𝑘𝑋𝑞88𝑒4𝐾−𝑖𝑘𝑋𝑞85𝑒1𝐾𝑖𝑘𝑋𝑞86𝑒2𝐾𝑖𝑘𝑋𝑞87𝑒3𝐾𝑖𝑘𝑋𝑞88𝑒4𝐾𝑖𝑘𝑋𝑞000089𝑒1𝐾−𝑖𝑘𝑋𝑞90𝑒2𝐾−𝑖𝑘𝑋𝑞91𝑒3𝐾−𝑖𝑘𝑋𝑞92𝑒4𝐾−𝑖𝑘𝑋𝑞93𝑒1𝐾𝑖𝑘𝑋𝑞94𝑒2𝐾𝑖𝑘𝑋𝑞95𝑒3𝐾𝑖𝑘𝑋𝑞96𝑒4𝐾𝑖𝑘𝑋𝑞000097𝑒1𝐾−𝑖𝑘𝑋𝑞98𝑒2𝐾−𝑖𝑘𝑋𝑞99𝑒3𝐾−𝑖𝑘𝑋𝑞100𝑒4𝐾−𝑖𝑘𝑋𝑞101𝑒1𝐾𝑖𝑘𝑋𝑞102𝑒2𝐾𝑖𝑘𝑋𝑞103𝑒3𝐾𝑖𝑘𝑋𝑞104𝑒4𝐾𝑖𝑘𝑋𝑞0000105𝑒1𝐾−𝑖𝑘𝑋𝑞106𝑒2𝐾−𝑖𝑘𝑋𝑞107𝑒3𝐾−𝑖𝑘𝑋𝑞108𝑒4𝐾−𝑖𝑘𝑋𝑞109𝑒1𝐾𝑖𝑘𝑋𝑞110𝑒2𝐾𝑖𝑘𝑋𝑞111𝑒3𝐾𝑖𝑘𝑋𝑞112𝑒4𝐾𝑖𝑘𝑋𝑞000013𝑞14𝑞15𝑞16𝑞17𝑞18𝑞19𝑞20𝑞21𝑞22𝑞23𝑞24𝑞25𝑞26𝑞27𝑞28𝑞25𝑞26𝑞27𝑞28𝑞29𝑞30𝑞31𝑞32𝑞33𝑞34𝑞35𝑞36−𝑞33−𝑞34−𝑞35−𝑞36−𝑞37−𝑞38−𝑞39−𝑞40𝑞41𝑞42𝑞43𝑞44𝑞45𝑞46𝑞47𝑞48𝑞49𝑞50𝑞51𝑞52𝑞53𝑞54𝑞55𝑞56𝑞57𝑞58𝑞59𝑞60𝑞61𝑞62𝑞63𝑞64𝑞65𝑞66𝑞67𝑞68𝑞65𝑞66𝑞67𝑞68𝑞69𝑞70𝑞71𝑞72𝑞73𝑞74𝑞75𝑞76𝑞77𝑞78𝑞79𝑞80𝑞81𝑞82𝑞83𝑞84𝑞1𝑞2𝑞3𝑞4𝑞5𝑞6𝑞7𝑞8𝑞9𝑞10𝑞11𝑞12}|||||||||||||||||=0.(5.28)

5.4. The Gravity Field, Initial Stress, and Magnetic Field Are Neglected and There Is Uncoupling between the Temperature and Strain Field

In this case 𝑔=0,𝑃=0,𝐻₀=0, and 𝜃=0, we obtain𝛼^2∗=Γ⁰(𝜆+2𝜇)𝜌,𝛽^2∗=Γ⁰𝜇lim_𝜀→0𝜌,𝑚^∗𝑗=1×−2Ω𝛼𝑐𝑖𝑘𝑋^2𝑗−𝑖𝑘_2∗𝑘²𝑋^4𝑗−𝑘²𝑐²−2𝛼^2∗+𝑖𝑘𝑐𝜒𝛼^2∗Γ₂+Ω²𝑋^2𝑗+𝑖𝑘𝑐Γ²𝜒1−𝛼^2∗+Ω²lim_𝛾→0𝑘²−Ω²,||||||||||𝑞₂₂𝑞₂₃𝑞₂₄𝑞₃₀𝑞₃₁𝑞₃₂−𝑞₃₈−𝑞₃₉−𝑞₄₀𝑞₅₀𝑞₅₁𝑞₅₂𝑞₆₂𝑞₆₃𝑞₆₄||||||||||=0.(5.29) Multiplying the rows 10, 11, and 12 of the determinant |𝑑_𝑖𝑗| by 𝛾 and then taking lim_𝛾→0, (5.28) reduces, after some computation, to the following ninth-order determinant equation:|||||||||||||||||||𝑞₈₅𝑞₈₉𝑒^{−𝑖𝑘𝑋1𝐾}𝑞₈₆𝑒^{−𝑖𝑘𝑋2𝐾}𝑞₈₇𝑒^{−𝑖𝑘𝑋3𝐾}𝑞₈₈𝑒^{−𝑖𝑘𝑋4𝐾}𝑞₈₅𝑒^{𝑖𝑘𝑋1𝐾}𝑞₈₆𝑒^{𝑖𝑘𝑋2𝐾}𝑞₈₇𝑒^{𝑖𝑘𝑋3𝐾}𝑞₈₈𝑒^{𝑖𝑘𝑋4𝐾}0𝑞₉₇𝑒^{−𝑖𝑘𝑋1𝐾}𝑞₉₀𝑒^{−𝑖𝑘𝑋2𝐾}𝑞₉₁𝑒^{−𝑖𝑘𝑋3𝐾}𝑞₉₂𝑒^{−𝑖𝑘𝑋4𝐾}𝑞₉₃𝑒^{𝑖𝑘𝑋1𝐾}𝑞₉₄𝑒^{𝑖𝑘𝑋2𝐾}𝑞₉₅𝑒^{𝑖𝑘𝑋3𝐾}𝑞₉₆𝑒^{𝑖𝑘𝑋4𝐾}0𝑞₁₀₅𝑒^{−𝑖𝑘𝑋1𝐾}𝑞₉₈𝑒^{−𝑖𝑘𝑋2𝐾}𝑞₉₉𝑒^{−𝑖𝑘𝑋3𝐾}𝑞₁₀₀𝑒^{−𝑖𝑘𝑋4𝐾}𝑞₁₀₁𝑒^{𝑖𝑘𝑋1𝐾}𝑞₁₀₂𝑒^{𝑖𝑘𝑋2𝐾}𝑞₁₀₃𝑒^{𝑖𝑘𝑋3𝐾}𝑞₁₀₄𝑒^{𝑖𝑘𝑋4𝐾}0𝑒𝑞^{−𝑖𝑘𝑋1𝐾}₁₃𝑞₁₀₆𝑒^{−𝑖𝑘𝑋2𝐾}𝑞₁₀₇𝑒^{−𝑖𝑘𝑋3𝐾}𝑞₁₀₈𝑒^{−𝑖𝑘𝑋4𝐾}𝑞₁₀₉𝑒^{𝑖𝑘𝑋1𝐾}𝑞₁₁₀𝑒^{𝑖𝑘𝑋2𝐾}𝑞₁₁₁𝑒^{𝑖𝑘𝑋3𝐾}𝑞₁₁₂𝑒^{𝑖𝑘𝑋4𝐾}0𝑞₁₄𝑞₁₅𝑞₁₆𝑞₁₇𝑞₁₈𝑞₁₉𝑞₂₀𝑞₂₁𝑞₂₅𝑞₂₆𝑞₂₇𝑞₂₈𝑞₂₅𝑞₂₆𝑞₂₇𝑞₂₈𝑞₂₉𝑞₃₃𝑞₃₄𝑞₃₅𝑞₃₆−𝑞₃₃−𝑞₃₄−𝑞₃₅−𝑞₃₆−𝑞₃₇𝑞₄₁𝑞₄₂𝑞₄₃𝑞₄₄𝑞₄₅𝑞₄₆𝑞₄₇𝑞₄₈𝑞₄₉𝑞₅₃𝑞₅₄𝑞₅₅𝑞₅₆𝑞₅₇𝑞₅₈𝑞₅₉𝑞₆₀𝑞₆₁|||||||||||||||||||=(50.,30)where 𝑞𝑞₁₅=1−𝑋₁𝑚^∗1,𝑞₂=1−𝑋₂𝑚^∗2,𝑞₃=1−𝑋₃𝑚^∗3,𝑞₄=1−𝑋₄𝑚^∗4,𝑞₉=1+𝑋₁𝑚^∗1,𝑞₆=1+𝑋₂𝑚^∗2,𝑞₇=1+𝑋₃𝑚^∗3,𝑞₈=1+𝑋₄𝑚^∗4,=𝑞₁₃1+𝑋^1𝑚^1∗,𝑞₁₀=1+𝑋^2𝑚^2∗,𝑞₁₁=1+𝑋^3𝑚^3∗,𝑞₁₂=1+𝑋^4𝑚^4∗,𝑞₁₇=𝑋₁+𝑚^∗1,𝑞₁₄=𝑋₂+𝑚^∗2,𝑞₁₅=𝑋₃+𝑚^∗3,𝑞₁₆=𝑋₄+𝑚^∗4,𝑞₂₁=𝑚^∗1−𝑋₁,𝑞₁₈=𝑚^∗2−𝑋₂,𝑞₁₉=𝑚^∗3−𝑋₃,𝑞₂₀=𝑚^∗4−𝑋₄,=𝑚^1∗−𝑋𝑞^1₂₅,𝑞₂₂=𝑚^2∗−𝑋^2,𝑞₂₃=𝑚^3∗−𝑋^3,𝑞₂₄=𝑚^4∗−𝑋^4,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₆,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₇,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₈,=1𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₂₉,=1𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^1∗1+𝑋^12−𝑚^1∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₀,=1𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^2∗1+𝑋^22−𝑚^2∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₁,=1𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^3∗1+𝑋^32−𝑚^3∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₂,=1𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^4∗1+𝑋^42−𝑚^4∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₃,=𝑀𝑋₁𝑘²𝑚^∗1𝑋²¹+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₄,=𝑀𝑋₂𝑘²𝑚^∗2𝑋²²+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₅,=𝑀𝑋₃𝑘²𝑚^∗3𝑋²³+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₆,=𝑀𝑋₄𝑘²𝑚^∗4𝑋²⁴+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₇,=𝑀^𝑋^1𝑘²𝑚^1𝑋^12+1+1𝑖𝑘𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^1∗1+𝑋^12−𝑚^1∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₈,=𝑀^𝑋^2𝑘²𝑚^2𝑋^22+1+1𝑖𝑘𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^2∗1+𝑋^22−𝑚^2∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₃₉,=𝑀^𝑋^3𝑘²𝑚^3𝑋^32+1+1𝑖𝑘𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^3∗1+𝑋^32−𝑚^3∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₄₀,=𝑀^𝑋^4𝑘²𝑚^4𝑋^42+1+1𝑖𝑘𝑐𝑠^1𝑞𝑘²𝛽^∗2𝑚^4∗1+𝑋^42−𝑚^4∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₄₁,𝑞₄₂=Γ₀𝜆1−𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)𝑋²¹+𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₄₃=Γ₀𝜆1−𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)𝑋²²+𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₄₄=Γ₀𝜆1−𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)𝑋²³+𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₄₅=Γ₀𝜆1−𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)𝑋²⁴+𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,𝑞₄₆=Γ₀𝜆1+𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)𝑋²¹−𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₄₇=Γ₀𝜆1+𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)𝑋²²−𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₄₈=Γ₀𝜆1+𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)𝑋²³−𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₄₉=Γ₀𝜆1+𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)𝑋²⁴−𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,𝑞₅₀=Γ^0𝜆^1+𝑋^1𝑚^1∗+Γ^0𝜆^+2𝜇^𝑋^12−𝑚^1∗𝑋^1+𝜌^−𝛼^∗21+𝑋^12+𝑐^2,𝑞₅₁=Γ^0𝜆^1+𝑋^2𝑚^2∗+Γ^0𝜆^+2𝜇^𝑋^22−𝑚^2∗𝑋^2+𝜌^−𝛼^∗21+𝑋^22+𝑐^2,𝑞₅₂=Γ^0𝜆^1+𝑋^3𝑚^3∗+Γ^0𝜆^+2𝜇^𝑋^32−𝑚^3∗𝑋^3+𝜌^−𝛼^∗21+𝑋^32+𝑐^2,𝑞₅₃=Γ^0𝜆^1+𝑋^4𝑚^4∗+Γ^0𝜆^+2𝜇^𝑋^42−𝑚^4∗𝑋^4+𝜌^−𝛼^∗21+𝑋^42+𝑐^2,=−2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₄,=−2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋²²+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₅,=−2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₆,=−2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₇,=2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₈,=2𝑘²Γ₀𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋²²+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₅₉,=2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₀,=2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₁,=2𝑘²Γ^0𝜇^𝑋^1−𝑘²Γ^0𝜇^𝑚^1∗1−𝑋^12+𝐹𝑠^1𝑞𝑘²𝛽^∗2𝑚^1∗1+𝑋^12−𝑚^1∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₂,=2𝑘²Γ^0𝜇^𝑋^2−𝑘²Γ^0𝜇^𝑚^2∗1−𝑋^22+𝐹𝑠^1𝑞𝑘²𝛽^∗2𝑚^2∗1+𝑋^22−𝑚^2∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₃,=2𝑘²Γ^0𝜇^𝑋^3−𝑘²Γ^0𝜇^𝑚^3∗1−𝑋^32+𝐹𝑠^1𝑞𝑘²𝛽^∗2𝑚^3∗1+𝑋^32−𝑚^3∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₄,=2𝑘²Γ^0𝜇^𝑋^4−𝑘²Γ^0𝜇^𝑚^4∗1−𝑋^42+𝐹𝑠^1𝑞𝑘²𝛽^∗2𝑚^4∗1+𝑋^42−𝑚^4∗𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₆₅,=𝜌𝛾−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²,𝑞₆₆=𝜌𝛾−𝛼^2∗𝑞₆₇𝑘²𝑋²²+1+𝑘²𝑐²,=𝜌𝛾−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²,𝑞₆₈=𝜌𝛾−𝛼^2∗𝑞₆₉𝑘²𝑋²⁴+1+𝑘²𝑐²,=𝜌^𝛾^−𝛼^∗2𝑘²𝑋^12+1+𝑘²𝑐²,𝑞₇₀=𝜌^𝛾^−𝛼^∗2𝑞₇₁𝑘²𝑋^22+1+𝑘²𝑐²,=𝜌^𝛾^−𝛼^∗2𝑘²𝑋^32+1+𝑘²𝑐²,𝑞₇₂=𝜌^𝛾^−𝛼^∗2𝑞₇₃𝑘²𝑋^42+1+𝑘²𝑐²,=𝜌𝛾−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝑖𝑘𝑋₁,𝑞₇₄=𝜌𝛾−𝛼^2∗𝑞₇₅𝑘²𝑋²²+1+𝑘²𝑐²𝑖𝑘𝑋₂,=𝜌𝛾−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝑖𝑘𝑋₃,𝑞₇₆=𝜌𝛾−𝛼^2∗𝑞₇₇𝑘²𝑋²⁴+1+𝑘²𝑐²𝑖𝑘𝑋₄,=−𝜌𝛾−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝑖𝑘𝑋₁,𝑞₇₈=−𝜌𝛾−𝛼^2∗𝑞₇₉𝑘²𝑋²²+1+𝑘²𝑐²𝑖𝑘𝑋₂,=−𝜌𝛾−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝑖𝑘𝑋₃,𝑞₈₀=−𝜌𝛾−𝛼^2∗𝑞₈₁𝑘²𝑋²⁴+1+𝑘²𝑐²𝑖𝑘𝑋₄,=−𝜌^𝛾^−𝛼^∗2𝑘²𝑋^12+1+𝑘²𝑐²𝑖𝑘𝑋^1,𝑞₈₂=−𝜌^𝛾^−𝛼^∗2𝑞₈₃𝑘²𝑋^22+1+𝑘²𝑐²𝑖𝑘𝑋^2,=−𝜌^𝛾^−𝛼^∗2𝑘²𝑋^32+1+𝑘²𝑐²𝑖𝑘𝑋^3,𝑞₈₄=−𝜌^𝛾^−𝛼^∗2𝑞₈₅𝑘²𝑋^32+1+𝑘²𝑐²𝑖𝑘𝑋^3,=𝑀𝑋₁𝑘²𝑚^∗1𝑋²¹+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₆,=𝑀𝑋₂𝑘²𝑚^∗𝑗𝑋²²+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₇,=𝑀𝑋₃𝑘²𝑚^∗𝑗𝑋²³+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₈,=𝑀𝑋₄𝑘²𝑚^∗4𝑋²⁴+1+1𝑖𝑘𝑐𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₈₉,𝑞₉₀=Γ₀𝜆1−𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)𝑋²¹+𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₉₁=Γ₀𝜆1−𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)𝑋²²+𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₉₂=Γ₀𝜆1−𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)𝑋²³+𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₉₃=Γ₀𝜆1−𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)𝑋²⁴+𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,𝑞₉₄=Γ₀𝜆1+𝑋₁𝑚^∗1+Γ₀(𝜆+2𝜇)𝑋²¹−𝑚^∗1𝑋₁+𝜌−𝛼^2∗1+𝑋²¹+𝑐²,𝑞₉₅=Γ₀𝜆1+𝑋₂𝑚^∗2+Γ₀(𝜆+2𝜇)𝑋²²−𝑚^∗2𝑋₂+𝜌−𝛼^2∗1+𝑋²²+𝑐²,𝑞₉₆=Γ₀𝜆1+𝑋₃𝑚^∗3+Γ₀(𝜆+2𝜇)𝑋²³−𝑚^∗3𝑋₃+𝜌−𝛼^2∗1+𝑋²³+𝑐²,𝑞₉₇=Γ₀𝜆1+𝑋₄𝑚^∗4+Γ₀(𝜆+2𝜇)𝑋²⁴−𝑚^∗4𝑋₄+𝜌−𝛼^2∗1+𝑋²⁴+𝑐²,=−2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₉₈,=−2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋^2𝑗+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₉₉,=−2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₀,=−2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₁,=2𝑘²Γ₀𝜇𝑋₁−𝑘²Γ₀𝜇𝑚^∗11−𝑋²¹+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗11+𝑋²¹−𝑚^∗1𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₂,=2𝑘²Γ₀𝜇𝑋₂−𝑘²Γ₀𝜇𝑚^∗21−𝑋²²+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗21+𝑋²²−𝑚^∗2𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₃,=2𝑘²Γ₀𝜇𝑋₃−𝑘²Γ₀𝜇𝑚^∗31−𝑋²³+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗31+𝑋²³−𝑚^∗3𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₄,=2𝑘²Γ₀𝜇𝑋₄−𝑘²Γ₀𝜇𝑚^∗41−𝑋²⁴+𝐹𝑠₁𝑘²𝑞𝛽^2∗𝑚^∗41+𝑋²⁴−𝑚^∗4𝑘²𝑐²+Ω²+2𝑖𝑘Ω𝑐₁₀₅,𝑞₁₀₇=−𝛼^2∗𝑘²𝑋²¹+1+𝑘²𝑐²𝑖𝑘𝑋₁,𝑞₁₀₆=−𝛼^2∗𝑘²𝑋²²+1+𝑘²𝑐²𝑖𝑘𝑋₂,𝑞=₁₀₉−𝛼^2∗𝑘²𝑋²³+1+𝑘²𝑐²𝑖𝑘𝑋₃,𝑞₁₀₈=−𝛼^2∗𝑘²𝑋²⁴+1+𝑘²𝑐²𝑖𝑘𝑋₄,𝑞₁₁₁=𝛼^2∗𝑘²𝑋²¹+1−𝑘²𝑐²𝑖𝑘𝑋₁,𝑞₁₁₀=𝛼^2∗𝑘²𝑋²²+1−𝑘²𝑐²𝑖𝑘𝑋₂,=𝛼^2∗𝑘²𝑋²³+1−𝑘²𝑐²𝑖𝑘𝑋₃,𝑞₁₁₂=𝛼^2∗𝑘²𝑋²⁴+1−𝑘²𝑐²𝑖𝑘𝑋₄.(5.31)

From (5.30), we can determine by numerical effects the initial stress, gravity field, friction coefficient, magnetic field, and rotation, for a computation using the maple program; we use sandstone as a granular medium and nephiline as a granular layer taking into consideration that the relaxation times 𝜏₀=0.1,𝜏₁=0.4, and 𝜏₂=0.5, the friction coefficient 𝐹=0.4, and the third elastic constant 𝑀=0.2.

(i) Effects of the initial stress, gravity field, friction coefficient, magnetic field, relaxation time, and rotation are discussed in Figures 2 and 3.

Figure 2 

Effects of 𝐻0,Ω,𝑔,𝜏1, and 𝑝 on Rayleigh wave velocity and attenuation coefficient with respect to the initial stress.

Figure 3 

Effects of 𝐻0,Ω,𝑔,𝜏1, and 𝐾,𝑃 on Rayleigh wave velocity and Attenuation coefficient with respect to the wave number.

(ii) From (5.30), if the initial stress are neglected, we can discuss the effects of the gravity field, friction coefficient, magnetic field, relaxation time, and rotation, and the discussion is clear up from Figure 4.

Figure 4 

Effects of 𝐻0,Ω,𝑔,𝜏1, and 𝐾 on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.

(iii) From (5.30), if the initial stress and magnetic field are neglected, we can discuss the effects of the gravity field, friction coefficient, relaxation time and rotation, and the discussion is clear up from Figure 5.

Figure 5 

Effects of Ω,𝑔,𝜏1, and 𝐾 on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.

(iv) From (5.30), if the initial stress, magnetic field, and gravity field are neglected, we can discuss the effects of the friction coefficient, relaxation time, and rotation, and the discussion is clear up from Figure 6.

Figure 6 

Effects of Ω,𝜏1, and 𝐾 on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.

(v) From (5.30), if the initial stress, magnetic field, and gravity field are neglected and there is uncoupling between the temperature and strain field, we can discuss the effects the friction coefficient, relaxation time, rotation, and the discussion is clear up from Figure 7.

Figure 7 

Effects of Ω,𝜏1, and 𝐾 on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.

6. Numerical Results and Discussion

In order to illustrate theoretical results obtained in the proceeding section, we now present some numerical results. The material chosen for this purpose of Carbon steel, the physical data is given [21] as follows:𝜌=2k𝐾gm=5⁻³,0Wm𝜆=₋₁9.3×10¹⁰Nm⁻¹,𝜇=8.4×10¹⁰Nm⁻¹,𝑇₀=293.1k,k⁻¹,𝑠=6.4×10²JKg⁻¹,𝛼_𝑡=13.2×10⁻⁶deg⁻¹.(6.1)

6.1. Effects of the Initial Stress, Gravity Field, Friction Coefficient, Magnetic Field, Relaxation Time, and Rotation

Figure 2 shows the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under the effect of gravity field, friction coefficient, magnetic field, relaxation time, and rotation with respect to the initial stress; we found that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) increased with increasing values of 𝑝 and 𝐻₀, and the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) decreased and increased with increasing values of 𝑔 and 𝐾, respectively; while the values of (Re) and (Im) take one curve at another value of the relaxation time 𝜏₁ increased with increasing values of initial stresse 𝑃.

Figure 3 shows the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under effect of initial stress, gravity field, friction coefficient, magnetic field, relaxation time and rotation with respect to the wave number, we find that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) decreased and increased with increasing values of 𝐻₀, respectively, and the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) increased and decreased with increasing values of Ω and 𝑔, respectively; also, the values of (Re) and (Im) increased with increasing values of 𝐾, while the values of (Re) and (Im) take one curve at another value of the relaxation time 𝜏₁, decreased with increasing values of wave number 𝑘.

6.2. If the Initial Stresses Are Neglected

Figure 4 shows the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under effect of gravity field, friction coefficient, magnetic field, relaxation time, and rotation with respect to the wave number, we find that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) decreased with increasing values of 𝐻₀ and Ω, while that contrary with increasing values of 𝑔; also, the values of (Re) and (Im) increased and decreased with increasing values of 𝐾, respectively, while the values of (Re) and (Im) take one curve at another value of the relaxation time 𝜏₁ decreased, then increased with increasing values of wave number 𝑘.

6.3. If the Initial Stresses and Magnetic Field Are Neglected

Figure 5 shows that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under effect of gravity field, friction coefficient, relaxation time, and rotation with respect to the wave number; we find that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) decreased and increased with increasing values of Ω, and the values of (Re) and (Im) increased with increasing values of g, while that contrary with increasing values of 𝐾; also, the values of (Re) and (Im) take one curve at another value of the relaxation time 𝜏₁, increased, then decreased with increasing values of wave number 𝑘.

6.4. If the Initial Stresses, Magnetic Field, and Gravity Field Are Neglected

Figure 6 shows the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under the effect of friction coefficient, relaxation time, and rotation with respect to the wave number; we find that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) decreased and increased with increasing Ω, and the values of (Re) and (Im) decreased with increasing values of 𝐾, while the values of (Re) and (Im) take one curve at another value of the relaxation time 𝜏₁, decreased and increased with increasing values of the wave number 𝑘, respectively.

6.5. If the Initial Stresses, Magnetic Field, and Gravity Field Are Neglected and There Is Uncoupling between the Temperature and Strain Field

Figure 7 shows the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) under the effect of friction coefficient, relaxation time, and rotation with respect to the wave number; we find that the velocity of Rayleigh waves (Re) and attenuation coefficient (Im) increased and decreased with increasing of Ω, respectively, while that contrary with increasing values of 𝐾; finally, the values of (Re) and (Im) take one curve at another value of the relaxation time 𝜏₁, decreased with increasing, the values of the wave number 𝑘.

7. Conclusions

The problem of the Rayleigh waves in generalized magneto-thermo-viscoelastic granular medium under the influence of rotation, gravity field, and initial stress is considered, and the frequency equation of the wave motion in the explicit form is derived, by considering various special cases. The numerical results are obtained for carbon-steel material, although the effect of the rotation, magnetic field, relaxation times, initial stress, gravity field, and friction coefficient is observed to be quite large on wave propagation of Rayleigh wave velocity (Re) and attenuation coefficient (Im).

The problem of the Rayleigh waves in generalized magneto-thermo-viscoelastic granular medium under the influence of rotation, gravity field, and initial stress is considered, and the frequency equation of the wave motion in the explicit form is derived, by considering various special cases. The numerical results are obtained for carbon-steel material, although the effect of the rotation, magnetic field, relaxation times, initial stress, gravity field, and friction coefficient is observed to be quite large on wave propagation of Rayleigh wave velocity (Re) and attenuation coefficient (Im).

It is easy to see that the values of (Re) and (Im) with respect to the initial stress are increased with increasing values of Ω, while that contrary if the initial stress are neglected and with respect to the wave number; also, if the initial stress are constant with respect to the wave number the values of (Re) and (Im) increased and decreased with increasing values of Ω, respectively, and if the initial stress and the magnetic field are neglected, the values of (Re) and (Im) decreased and increased with increasing values of Ω, respectively, while that contrary if 𝑃,𝐻₀,𝑔,𝜃, and 𝜀 are neglected and with respect to the wave number; finally, if 𝑃,𝐻₀, and 𝑔 are neglected and with respect to the wave number, the values of (Re) and (Im) increased with increasing values of Ω.

It is easy to see that the values of (Re) and (Im) with respect to the initial stress are decreased and increased with increasing values of 𝑔, respectively, while that contrary if the initial stress are constant and with respect to the wave number; also, if the initial stress are neglected and if the initial stress and the magnetic field are neglected with respect to the wave number, the values of (Re) and (Im) increased with increasing values of 𝑔.

It is easy to see that the values of (Re) and (Im) with respect to the initial stress are decreased and increased with increasing values of 𝐾, respectively, while that contrary if the initial stress are neglected and with respect to the wave number; also, if the initial stress and the magnetic field are neglected and if 𝑃,𝐻₀, and 𝑔 are neglected with respect the wave number, the values of (Re) and (Im) decreased with increasing values of 𝐾; finally, if 𝑃,𝐻₀,𝑔,𝜃, and 𝜀 are neglected and with respect to the wave number, the values of (Re) and (Im) decreased and increased with increasing values of 𝐾.

Finally, the frequency equation has been discussed under effect of rotation, gravity field, and initial stress and in case of various classical and nonclassical theories of thermoelasticity. The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field is very pronounced. The frequency equations derived in this paper may be useful in practical applications. It is concluded from the above analyses and results that the present solution is accurate and reliable and the method is simple and effective. So it may be as a reference to solve other problems of Rayleigh waves in generalized magneto-thermoelastic granular medium.