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

Authors: A. M. Abd-Alla; S. M. Abo-Dahab; F. S. Bayones
Journal: Mathematical Problems in Engineering
 (2011)

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## 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.


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## Body

## 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 planesz=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 H0⃗ 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 ξ⃗(ξ,η,ζ) of the grain about its center of gravity. The elastic medium is rotating uniformly with an angular velocity  Ω̲=Ωn̲, where n̲ 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, Ω̲×(Ω̲×u̲) is the centripetal acceleration due to time varying motion only, and 2Ω⃗×u•⃗ is the Coriolis acceleration, and Ω̲=(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]):
(2.1)J⃗=curl⁡h⃗,-μe∂h⃗∂t=curl⁡E⃗,div⁡h⃗=0,div⁡E⃗=0,E⃗=-μe(∂u⃗∂t×H⃗),
where(2.2)h⃗=curl⁡(u⃗×H0⃗),H⃗=H0⃗+h⃗,H0⃗=(0,H0,0),
where  h⃗ is the perturbed magnetic field over the primary magnetic field vector, E⃗ is the electric intensity, J⃗ is the electric current density, μe is the magnetic permeability, H0⃗ 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,τij≠τji, Mij≠Mji. The stress tensor τij can be expressed as the sum of symmetric and antisymmetric tensors (2.3)τij=σij+σij′,
where
(2.4)σij=12(τij+τji),σ′ij=12(τij-τji).The symmetric tensorσij=σji is related to the symmetric strain tensor (2.5)eij=eji=12(∂ui∂xj+∂uj∂xi).
The antisymmetric stress σij′ are given by(2.6)σ23′=-F∂ξ∂t,σ31′=-F∂η∂t,σ12′=-F∂ζ∂t,σ11′=σ22′=σ33′=0,
where F is the coefficient of friction between the individual grains. The stress couple Mij is given by(2.7)Mij=Mνij,
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 tensorνij is defined as(2.8)ν11=∂ξ∂x,ν31=∂ξ∂z,ν33=∂ζ∂z,ν21=ν22=ν23=0,ν12=∂∂x(ω2+η),ν32=∂∂z(ω2+η),ν13=∂ζ∂x,
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](2.9)τji,j+Fi=ρ[u••i+{Ω⃗×(Ω⃗×u⃗)}i+(2Ω⃗×u⃗•)i],i,j=1,2,3.
The heat conduction equation is given by [16]   (2.10)K∇2T=ρs∂∂t(1+τ2∂∂t)T+γT0∂∂t(1+τ2δ∂∂t)∇̲⋅u⃗,
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 (θ-T0), 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 by(2.11)σ11=[τm(λ+2μ)+p]∂u∂x+(τmλ+P)∂w∂z-γ(1+τ1∂∂t)T,σ33=τmλ∂u∂x+τm(λ+2μ)∂w∂z-γ(1+τ1∂∂t)T,σ13=τmμ(∂u∂z+∂w∂x).
If we neglect the thermal relaxation time, then (2.11) tends to Nowacki [17] and Biot [18].The Maxwell's electro-magnetic stress tensorτ¯ij is given by (2.12)τ¯ij=μe[Hihj+Hjhi-(Hk⋅hk)δij],i,j=1,2,3,
which takes the form(2.13)τ¯11=-μeH02∇2ϕ,τ¯13=τ¯23=0,τ¯33=μeH02∇2ϕ,∇2ϕ=∂u∂x+∂w∂z.The dynamic equations of motion are(2.14)∂τ11∂x+∂τ31∂z+P2∂ω2∂z-ρg∂w∂x+Fx=ρ[∂2u∂t2+2Ω∂w∂t-Ω2u],∂τ12∂x+∂τ32∂z+Fy=0,∂τ13∂x+∂τ33∂z+P2∂ω2∂x+ρg∂w∂x+Fz=ρ[∂2w∂t2-2Ω∂u∂t-Ω2w],
where g is the Earth's gravity and(2.15)F→=(-μeH02∇2ϕ,0,μeH02∇2ϕ),(2.16)τ23-τ32+∂M11∂x+∂M31∂z=0,τ31-τ13+∂M12∂x+∂M32∂z=0,τ12-τ21+∂M13∂x+∂M33∂z=0.From (2.3)–(2.8) and (2.11), we have(2.17)τ11=[τm(λ+2μ)+p]∂u∂x+(τmλ+P)∂w∂z-γ(1+τ1∂∂t)T,τ33=τmλ∂u∂x+τm(λ+2μ)∂w∂z-γ(1+τ1∂∂t)T,τ13=τmμ(∂u∂z+∂w∂x)+F∂η∂t,τ12=-F∂ζ∂t,τ23=-F∂ξ∂t,M11=M∂ξ∂x,M31=M∂ξ∂z,M33=M∂ζ∂z,M21=M22=M23=0,M12=M∂∂x(ω2+η),M32=M∂∂z(ω2+η),M13=M∂ζ∂x.
Substituting (2.17) into (2.14) and (2.16) tends to(2.18)[τm(λ+2μ)+P]∂2u∂x2+(τmλ+P)∂2w∂x∂z-γ(1+τ1∂∂t)∂T∂x+τmμ(∂2u∂z2+∂2w∂x∂z)+P2(∂2u∂z2-∂2w∂x∂z)-ρg∂w∂x+F∂2η∂z∂t+μeH02(∂2u∂x2+∂2w∂x∂z)=ρ[∂2u∂t2+2Ω∂w∂t-Ω2u],
then(2.19)[τm(λ+2μ)+P+μeH02]∂2u∂x2+[τm(λ+μ)+P2+μeH02]∂2w∂x∂z+(τmμ+P2)∂2u∂z2-γ(1+τ1∂∂t)∂T∂x-ρg∂w∂x+F∂2η∂z∂t=ρ[∂2u∂t2+2Ω∂w∂t-Ω2u].Also,(2.20)∂∂t(∂ζ∂x-∂ξ∂z)=0,(2.21)τmμ(∂2u∂x∂z+∂2w∂x2)-F∂2η∂x∂t+τmλ∂2u∂x∂z+τm(λ+2μ)∂2w∂z2-γ(1+τ1∂∂t)∂T∂z+P2(∂2u∂x∂z-∂2w∂x2)+ρg∂u∂x+μeH02(∂2u∂x∂z+∂2w∂z2)=ρ[∂2w∂t2-2Ω∂u∂t-Ω2w],
then(2.22)[τm(λ+μ)+P2+μeH02]∂2u∂x∂z+(τmμ-P2)∂2w∂x2+[τm(λ+2μ)+μeH02]∂2w∂z2-γ(1+τ1∂∂t)∂T∂z+ρg∂u∂x-F∂2η∂x∂t=ρ[∂2w∂t2-2Ω∂u∂t-Ω2w],
and, from (2.16), we have(2.23)∇2ξ-s2∂ξ∂t=0,(2.24)∇2(ω2+η)-s2∂η∂t=0,(2.25)∇2ζ-s2∂ζ∂t=0,
where
(2.26)s2=2FM.


## 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, as(3.1)u⃗=gradϕ+curl⁡ψ⃗,ψ⃗=(0,ψ,0),
which reduces to(3.2)u=∂ϕ∂x-∂ψ∂z,w=∂ϕ∂z+∂ψ∂x.Substituting (3.2) into (2.19), (2.22), and (2.24), the wave equations tend to(3.3)α2∇2ϕ-γρ(1+τ1∂∂t)T-g∂ψ∂x=∂2ϕ∂t2+2Ω∂ψ∂t-Ω2ϕ,(3.4)β2∇2ψ-s1∂η∂t+g∂ϕ∂x=∂2ψ∂t2-2Ω∂ϕ∂t-Ω2ψ,(3.5)∇2η-s2∂η∂t-∇4ψ=0,
where(3.6)s1=Fρ,α2=τm(λ+2μ)+P+μeH02ρ,β2=2τmμ-P2ρ.Substituting (3.2) into (2.10), we obtain(3.7)K∇2T=ρs∂∂t(1+τ2∂∂t)T+γT0∂∂t(1+τ2δ∂∂t)∇2ϕ.From (3.3) and  (3.7), by eliminating T, we obtain (3.8)[∇2-1χ∂∂t(1+τ2∂∂t)][α2∇2ϕ-g∂ψ∂x-∂2ϕ∂t2-2Ω∂ψ∂t+Ω2ϕ]-ε∂∂t(1+τ1∂∂t)(1+τ2δ∂∂t)∇2ϕ=0,
where(3.9)χ=Kρs,ε=γ2T0ρK.From (3.4) and (3.5) by eliminating η, we obtain (3.10)(∇2-s2∂∂t)(β2∇2ψ-∂2ψ∂t2+g∂ϕ∂x+2Ω∂ϕ∂t+Ω2ψ)-s1∇4∂ψ∂t=0.For a plane harmonic wave propagation in thex-direction, we assume(3.11)ϕ=ϕ1eik(x-ct),ψ=ψ1eik(x-ct),(3.12)(ξ,η,ζ)=(ξ1,η1,ζ1)eik(x-ct).From (3.12) into (2.20), (2.23), and (2.25), we get(3.13)Dξ1-ikζ1=0,(3.14)D2ξ1+q2ξ1=0,(3.15)D2ζ1+q2ζ1=0,
where(3.16)q2=ikcs2-k2,D≡ddz.The solution of (3.14) and (3.15) takes the form(3.17)ξ1=A1eiqz+A2e-iqz,ζ1=B1eiqz+B2e-iqz,
where A1,A2,B1, and B2 are arbitrary constants.From (3.13) and (3.17), we obtain (3.18)q(A1eiqz-A2e-iqz)-k(B1eiqz+B2e-iqz)=0,
then(3.19)qA1-kB1=0,qA2-kB2=0}⟹Aj=(-1)j-1kqBj,j=1,2.Substituting (3.11) into (3.8) and (3.10), we obtain(3.20)[α*2D4+G1D2+G2]ϕ1-[G3D2+G4]ψ1=0,[R1D4+R2D2+R3]ψ1+[R4D2+R5]ϕ1=0,
where(3.21)Γ0=1-ikcτ0,Γ1=1-ikcτ1,Γ2=1-ikcτ2,Γ3=1-ikcτ2δ,α*2=Γ0(λ+2μ)+P+μeH02ρ,β*2=2Γ0μ-P2ρ.G1=k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2,G2=k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3),G3=ik(g-2Ωc),G4=-(g-2Ωc)(ik3+k2cΓ2χ),R1=β*2+ikcs1,R2=k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2,R3=k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1),R4=ik(g-2Ωc),R5=(2Ωc-g)(ik3-k2cs2).The solution of (3.20) takes the form(3.22)ϕ1=∑j=14[CjeikNjz+Dje-ikNjz],ψ1=∑j=14[EjeikNjz+Fje-ikNjz],
where the constants Ej and Fj are related to the constants Cj and Dj in the form(3.23)Ej=mjCj,Fj=mjDj,j=1,2,3,4,mj=1(g-2Ωc)(ikNj2-ik-(εΓ2/χ))×{ikcΓ2χ(1-α*2+Ω2k2)α*2k2Nj4-[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]Nj2+ikcΓ2χ(1-α*2+Ω2k2)-(Ω2+ikεcΓ1Γ3)}.
Substituting (3.22) into (3.11), we obtain (3.24)ϕ=∑j=14[CjeikNjz+Dje-ikNjz]eik(x-ct),ψ=∑j=14[EjeikNjz+Fje-ikNjz]eik(x-ct),
and values of displacement components u and w are(3.25)u=ik∑j=14[(1-Njmj)CjeikNjz+(1+Njmj)Dje-ikNjz]eik(x-ct),w=ik∑j=14[(Nj+mj)CjeikNjz+(mj-Nj)Dje-ikNjz]eik(x-ct),
where N1,N2,N3, and N4  are taken to be the complex roots of the following equation(3.26)N8+t1N6+t2N4+t3N2+t4=0,
where
(3.27)t1=k2α*2(c2-2α*2)+ikcα*2χ(α*2Γ2+χεΓ1Γ3)+Ω2+1β*2+ikcs1×[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2],(3.28)t2=1α*2[k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3)]+1α*2(β*2+ikcs1)[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]+1(β*2+ikcs1)[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]-1α*2(β*2+ikcs1)[k2(g-2Ωc)2],(3.29)t3=1α*2(β*2+ikcs1)×{[ik(g-2Ωc)2(ik3+k2cΓ2χ)][k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3)]+[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]-[ik(g-2Ωc)2(ik3+k2cΓ2χ)]-ik3[(g-2Ωc)2(ik-cs2)]},(3.30)t4=1α*2(β*2+ikcs)×{[(2Ωc-g)2(ik3-k2cs2)(ik3+k2cΓ2χ)][k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[ik(g-2Ωc)]+[(2Ωc-g)2(ik3-k2cs2)(ik3+k2cΓ2χ)]}.From (3.4), (3.11), (3.12), (3.22), and (3.23), we obtain
(3.31)η1=∑j=141ikcs1{k2β*2mj(1+Nj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}×[CjeikNjz+Dje-ikNjz].Using (3.22) and (3.11) into (3.3), we obtain(3.32)T=ργΓ1∑j=14[-α*2k2(1+Nj2)+k2c2-ikgmj](CjeikNjz+Dje-ikNjz)eik(x-ct).With the lower medium, we use the symbols with primes, forξ1,ζ1,η1,T,ϕ,ψ, and q, for z>K,(3.33)ξ1′=-kq′B2′e-iq′z,ζ1′=B2′e-iq′z,η1′=∑j=141ikcs1′{k2β*′2mj′(1+Nj′2)-mj′(k2c2+Ω′2)+ik(2Ω′c-g)}Dj′e-ikNj′z,T′=ρ′γ′Γ1∑j=14[-α*′2k2(1+Nj′2)+k2c2-ikgmj′]Dj′e-ikNj′zeik(x-ct),ϕ′=∑j=14Dj′e-ikNj′zeik(x-ct),ψ′=∑j=14Fj′e-ikNj′zeik(x-ct).


## 4. Boundary Conditions and Frequency Equation

In this section, we obtain the frequency equation for the boundary conditions which are specific to the interfacez=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+τ¯33=τ33′+τ¯33′(x)
τ31+τ¯31=τ31′+τ¯31′,(xi)
τ32+τ¯32=τ32′+τ¯32′,(xii)
T=T′,(xiii)
(∂T/∂z)+θT=(∂T′/∂z)+θT′.The boundary conditions on the free surfacez=0 are(xiv)
M33=0,(xv)
M31=0,(xvi)
M32=0,(xvii)
τ33+τ¯33=0,(xviii)
τ31+τ¯31=0,(xix)
τ32+τ¯32=0,(xx)
(∂T/∂z)+θT=0.From conditions (iii), (v), (vi), and (vii), we obtain(4.1)B1eiqK-B2e-iqK=-B2′e-iq′K,B1eiqK+B2e-iqK=B2′e-iq′K,M(B1eiqK-B2e-iqK)=-M′B2′e-iq′K,M(B1eiqK+B2e-iqK)=-M′B2′e-iq′K.
Hence,
(4.2)B1=B2=B2′=0,ξ=ζ=ξ′=ζ′=0.The other significant boundary conditions are responsible for the following relations:(i)
(4.3)∑j=14(1-Njmj)CjeikNjK+(1+Njmj)Dje-ikNjK-(1+Nj′mj′)Dj′e-ikNj′K=0,(ii)
(4.4)∑j=14(Nj+mj)CjeikNjK+(mj-Nj)Dje-ikNjK-(mj′-Nj′)Dj′e-ikNj′K=0,(iv)
(4.5)∑j=141cs1{k2β*2mj(1+Nj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}×[CjeikNjK+Dje-ikNjK]-∑j=141cs1′{k2β*′2mj′(1+Nj′2)-mj′(k2c2+Ω′2)+ik(2Ω′c-g)}Dj′e-ikNj′K=0,(viii)
(4.6)MNj∑j=14{k2mj(Nj2+1)+1ikcs1(k2β*2mj(1+Nj2)-mj(k2c2+Ω2)+ik(2Ωc-g))}×[CjeikNjK-Dje-ikNjK]+M′Nj′∑j=14{k2mj′(Nj′2+1)+1ikcs1′×(k2β*′2mj′(1+Nj′2)-mj′(k2c2+Ω′2)+ik(2Ω′c-g))+1ikcs1′}Dj′e-ikNj′K=0,(ix)
(4.7)∑j=14{[-α*′2(1+Nj′2)+c2-igkmj′][(Γ0λ+μeH02)(1-Njmj)+(Γ0(λ+2μ)+μeH02)(Nj2+mjNj)]×CjeikNjK+[(Γ0λ+μeH02)(1+Njmj)+(Γ0(λ+2μ)+μeH02)(Nj2-mjNj)]×Dje-ikNjK+ρ[-α*2(1+Nj2)+c2-igkmj][CjeikNjK+Dje-ikNjK]-[(Γ0′λ′+μe′H02)(1+Nj′mj′)+(Γ0′(λ′+2μ′)+μe′H02)(Nj′2-mj′Nj′)]Dj′e-ikNj′K-ρ′[-α*′2(1+Nj′2)+c2-igkmj′]Dj′e-ikNj′K}=0,(x)
(4.8)∑j=14{-2k2Γ0μNj[CjeikNjK-Dje-ikNjK]+[-k2Γ0μmj(1-Nj2)+Fs1{k2β*2mj(1+Nj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}]×[CjeikNjK+Dje-ikNjK]-2k2Γ0′μ′Nj′Dj′e-ikNj′K-[-k2Γ0′μ′mj′(1-Nj′2)+Fs1′{k2β*′2mj′(1+Nj′2)-mj′(k2c2+Ω′2)+ik(2Ω′c-g)}]×Dj′e-ikNj′K}=0,(xii)
(4.9)∑j=14ργ[-α*2k2(Nj2+1)+k2c2-igkmj][CjeikNjK+Dje-ikNjK]-ρ′γ′[-α*′2k2(Nj′2+1)+k2c2-igkmj′]Dj′e-ikNj′K=0,(xiii)
(4.10)∑j=14ργ[-α*2k2(Nj2+1)+k2c2-igkmj][(θ+ikNj)CjeikNjK+(θ-ikNj)Dje-ikNjK]-ρ′γ′[-α*′2k2(Nj′2+1)+k2c2-igkmj′]×(θ-ikNj′)Dj′e-ikNj′K=0,(xvi)
(4.11)MNj∑j=14{k2mj(Nj2+1)+1ikcs1(k2β*2mj(1+Nj2)-mj(k2c2+Ω2)+ik(2Ωc-g))}×[Cj-Dj]=0,(xvii)
(4.12)∑j=14{[-α*2(1+Nj2)+c2-igkmj][(Γ0λ+μeH02)(1-Njmj)+(Γ0(λ+2μ)+μeH02)(Nj2+mjNj)]Cj+[(Γ0λ+μeH02)(1+Njmj)+(Γ0(λ+2μ)+μeH02)(Nj2-mjNj)]Dj+ρ[-α*2(1+Nj2)+c2-igkmj][Cj+Dj]}=0,(xviii)
(4.13)∑j=14{-2k2Γ0μNj[Cj-Dj]+[-k2Γ0μmj(1-Nj2)+Fs1{k2β*2mj(1+Nj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}]×[Cj+Dj]}=0,(xx)
(4.14)∑j=14[-α*2k2(Nj2+1)+k2c2-igkmj][(θ+ikNj)Cj+(θ-ikNj)Dj]=0.


## 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 (5.1)V8+h1V6+h2V4+h3V2+h4=0,
where(5.2)α*2=Γ0(λ+2μ)ρ,β*2=Γ0μρ,mj=1(g-2Ωc)(ikVj2-ik-(ε/χ))×{α*2k2Vj4-[k2(c2-2α*2)+ikcχ(α*2+χε)+Ω2]Vj2+ikcχ(1-α*2+Ω2k2)-(Ω2+ikεc)},h1=k2α*2(c2-2α*2)+ikcα*2χ(α*2+χε)+Ω2+1β*2+ikcs1×[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2],h2=1α*2[k4(α*2-c2)+ikcχ(k2(1-α*2)+Ω2)-k2(Ω2+ikεc)]+1α*2(β*2+ikcs1)[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k2(c2-2α*2)+ikcχ(α*2+χε)+Ω2]+1(β*2+ikcs1)[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]-1α*2(β*2+ikcs1)[k2(g-2Ωc)2],h3=1α*2(β*2+ikcs1){[ik(g-2Ωc)2(ik3+k2cχ)][k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k4(α*2-c2)+ikcχ(k2(1-α*2)+Ω2)-k2(Ω2+ikεc)]+[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[k2(c2-2α*2)+ikcχ(α*2+χε)+Ω2]-[ik(g-2Ωc)2(ik3+k2cχ)]-ik3[(g-2Ωc)2(ik-cs2)]},h4=1α*2(β*2+ikcs){[(2Ωc-g)2(ik3-k2cs2)(ik3+k2cχ)][k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[ik(g-2Ωc)]+[(2Ωc-g)2(ik3-k2cs2)(ik3+k2cχ)]}.Also,(5.3)η1=∑j=141ikcs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}×[CjeikVjz+Dje-ikVjz].T=ργ∑j=14[-α*2k2(1+Vj2)+k2c2-ikgmj](CjeikVjz+Dje-ikVjz)eik(x-ct),ξ1′=-kq′B2′e-iq′z,ζ1′=B2′e-iq′z,η1′=∑j=141ikcs1′{k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω′2)+ik(2Ω′c-g)}Dj′e-ikVj′z,T′=ρ′γ′∑j=14[-α*′2k2(1+Vj′2)+k2c2-ikgmj′]Dj′e-ikVj′zeik(x-ct),ϕ1′=∑j=14Dj′e-ikVj′z,ψ1′=∑j=14Fj′e-ikVj′z,
Using the boundary conditions, we obtain(5.4)[d11d12⋯d18d15′d16′⋯d18′d21d22⋯d28d25′d26′⋯d28′⋮⋮⋮⋮⋮⋮⋮⋮d121d122⋯d128d125′d126′⋯d128′][C1C2C3C4D1D2D3D4D1′D2′D3′D4′]=[0],
where(5.5)d1j=(1-Vjmj)CjeikVjΚ+(1+Vjmj)Dje-ikVjΚ,d1j′=(1+Vj′mj′)Dj′e-ikVj′Κ,d2j=(Vj+mj)CjeikVjΚ+(mj-Vj)Dje-ikVjΚ,d2j′=(mj′-Vj′)Dj′e-ikVj′Κ,d3j=1cs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}[CjeikVjz+Dje-ikVjz],d3j′=1cs1′{k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω2)+ik(2Ωc-g)}Dj′e-ikVj′z,d4j=MVj{k2mj(Vj2+1)+1ikcs1(k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g))×[CjeikVjK-Dje-ikVjK],d4j′=M′Vj′{k2mj′(Vj′2+1)+1ikcs1′(k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω2)+ik(2Ωc-g))}×Dj′e-ikVj′K,d5j=[Γ0λ(1-Vjmj)+Γ0(λ+2μ)(Vj2+mjVj)]CjeikVjK+[Γ0λ(1+Vjmj)+Γ0(λ+2μ)(Vj2-mjVj)]Dje-ikVjK+ρ[-α*2(1+Vj2)+c2-igkmj][CjeikVjK+Dje-ikVjK],d5j′=[Γ0′λ′(1+Vj′mj′)+Γ0′(λ′+2μ′)(Vj′2-mj′Vj′)]Dj′e-ikVj′K+ρ′[-α*′2(1+Vj′2)+c2-igkmj′]×Dj′e-ikVj′K},d6j=-2k2Γ0μVj[CjeikVjK-Dje-ikVjK]+[-k2Γ0μmj(1-Vj2)+Fs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}]×[CjeikVjK+Dje-ikVjK],d6j′=2k2Γ0′μ′Vj′Dj′e-ikVj′K+[-k2Γ′0μ′mj′(1-Vj′2)+Fs1′{k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω2)+ik(2Ωc-g)}]×Dj′e-ikVj′K,d7j=ργ[-α*2k2(Vj2+1)+k2c2-igkmj][CjeikVjK+Dje-ikVjK],d7j′=ρ′γ′[-α*′2k2(Vj′2+1)+k2c2-igkmj′]Dj′e-ikVj′K,d8j=ργ[-α*2k2(Vj2+1)+k2c2-igkmj][(θ+ikVj)CjeikVjK+(θ-ikVj)Dje-ikVjK],d8j′=ρ′γ′[-α*′2k2(Vj′2+1)+k2c2-igkmj′](θ-ikVj′)Dj′e-ikVj′K,d9j=MVj{k2mj(Vj2+1)+1ikcs1(k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g))}×[Cj-Dj],d10j=[Γ0λ(1-Vjmj)+Γ0(λ+2μ)(Vj2+mjVj)]Cj+[Γ0λ(1+Vjmj)+Γ0(λ+2μ)(Vj2-mjVj)]Dj+ρ[-α*2(1+Vj2)+c2-igkmj][Cj+Dj],d11j=-2k2Γ0μVj[Cj-Dj]+[-k2Γ0μmj(1-Vj2)+Fs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}]×[Cj+Dj],d12j=[-α*2k2(Vj2+1)+k2c2-igkmj][(θ+ikVj)Cj+(θ-ikVj)Dj],d9j′=d10j′=d11j′=d12j′=0,j=1,2,3,4.


### 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 putg=0, then (3.20) becomes(5.6)[α*2D4+G1D2+G2]ϕ1-[G3*D2+G4*]ψ1=0,[R1D4+R2D2+R3]ψ1+[R4*D2+R5*]ϕ1=0,
where(5.7)G3*=-2ikΩc,G4*=2Ωc(ik3+k2cΓ2χ),R4*=-2ikΩc,R5*=2Ωc(ik3-k2cs2),
and  G1,G2,R1,R2, and R3 are as in (3.21).The solution of (5.6) take the form(5.8)ϕ=∑j=14[Cj*eikXjz+Dj*e-ikXjz]eik(x-ct),ψ=∑j=14[Ej*eikXjz+Fj*e-ikXjz]eik(x-ct),
where(5.9)Ej*=mj*Cj*,Fj*=mj*Dj*,j=1,2,3,4,(5.10)mj*=1-2Ωc(ikXj2-ik-((εΓ2)/χ))×{(1-α*2+Ω2k2)α*2k2Xj4-[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]Xj2+ikcΓ2χ(1-α*2+Ω2k2)-(Ω2+ikεcΓ1Γ3)},
and X1,X2,X3,and X4 are taken to be the complex roots of equation(5.11)X8+t1*X6+t2*X4+t3*X2+t4*=0,
where(5.12)t1*=k2α*2(c2-2α*2)+ikcα*2χ(α*2Γ2+χεΓ1Γ3)+Ω2+1β*2+ikcs1×[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2],t2*=1α*2[k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3)]+1α*2(β*2+ikcs1)[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]+1(β*2+ikcs1)[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]-1α*2(β*2+ikcs1)[4k2Ω2c2],t3*=1α*2(β*2+ikcs1)×{[ik(ik3+k2cΓ2χ)+ik3(ik-cs2)][k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3)]+[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]-4Ω2c2[ik(ik3+k2cΓ2χ)+ik3(ik-cs2)]},t4*=1α*2(β*2+ikcs){[4Ω2c2(ik3-k2cs2)(ik3+k2cΓ2χ)][k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)][-2ikΩc]+[4Ω2c2(ik3-k2cs2)(ik3+k2cΓ2χ)]},u=ik∑j=14[(1-Xjmj*)Cj*eikXjz+(1+Xjmj*)Dj*e-ikXjz]eik(x-ct),w=ik∑j=14[(Xj+mj*)Cj*eikXjz+(mj*-Xj)Dj*e-ikXjz]eik(x-ct),η1=∑j=141ikcs1{k2β*2mj*(1+Xj2)-mj*(k2c2+Ω2)+2ikΩc}×[Cj*eikXjz+Dj*e-ikXjz],T=ργΓ1∑j=14[-α*2k2(1+Xj2)+k2c2](Cj*eikXjz+Dj*e-ikXjz)eik(x-ct).With the lower medium, we use the symbols with primes, forξ1,ζ1,η1,T,ϕ,ψ, and q, for z>Κ,(5.13)ξ1′=-kq′B2′e-iq′z,ζ1′=B2′e-iq′z,η1′=∑j=141ikcs1′{k2β*′2mj′*(1+Xj′2)-mj′*(k2c2+Ω′2)+2ikΩ′c}Dj′*e-ikXj′z,T′=ρ′γ′Γ1∑j=14[-α*′2k2(1+Xj′2)+k2c2]Dj′*e-ikXj′zeik(x-ct),ϕ′=∑j=14Dj′e-ikNj′zeik(x-ct),ψ′=∑j=14Fj′*e-ikXj′zeik(x-ct).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)
(5.14)q1C1*eikX1K+q2C2*eikX2K+q3C3*eikX3K+q4C4*eikX4K+q5D1*e-ikX1K+q6D2*e-ikX2K+q7D3*e-ikX3K+q8D4*e-ikX4K=q9D1′*e-ikX1′K+q10D2′*e-ikX2′K+q11D3′*e-ikX3′K+q12D4′*e-ikX4′K,(ii)
(5.15)q13C1*eikX1Κ+q14C2*eikX2Κ+q15C3*eikX3Κ+q16C4*eikX4Κ+q17D1*e-ikX1Κ+q18D2*e-ikX2Κ+q19D3*e-ikX3Κ+q20D4*e-ikX4Κ=q21D1′*e-ikX1′Κ+q22D2′*e-ikX2′Κ+q23D3′*e-ikX3′Κ+q24D4′*e-ikX4′Κ,(iv)
(5.16)q25C1*eikX1Κ+q26C2*eikX2Κ+q27C3*eikX3Κ+q28C4*eikX4Κ+q25D1*e-ikX1Κ+q26D2*e-ikX2Κ+q27D3*e-ikX3Κ+q28D4*e-ikX4Κ=q29D1′*e-ikX1′Κ+q30D2′*e-ikX2′Κ+q31D3′*e-ikX3′Κ+q32D4′*e-ikX4′Κ,(viii)
(5.17)q33C1*eikX1Κ+q34C2*eikX2Κ+q35C3*eikX3Κ+q36C4*eikX4Κ-q33D1*e-ikX1Κ-q34D2*e-ikX2Κ-q35D3*e-ikX3Κ-q36D4*e-ikX4Κ=-q37D1′*e-ikX1′Κ-q38D2′*e-ikX2′Κ-q39D3′*e-ikX3′Κ-q40D4′*e-ikX4′Κ,(ix)
(5.18)q41C1*eikX1Κ+q42C2*eikX2Κ+q43C3*eikX3Κ+q44C4*eikX4Κ+q45D1*e-ikX1Κ+q46D2*e-ikX2Κ+q47D3*e-ikX3Κ+q48D4*e-ikX4Κ=q49D1′*e-ikX1′Κ+q50D2′*e-ikX2′Κ+q51D3′*e-ikX3′Κ+q52D4′*e-ikX4′Κ,(x)
(5.19)q53C1*eikX1Κ+q54C2*eikX2Κ+q55C3*eikX3Κ+q56C4*eikX4Κ+q57D1*e-ikX1Κ+q58D2*e-ikX2Κ+q59D3*e-ikX3Κ+q60D4*e-ikX4Κ=q61D1′*e-ikX1′Κ+q62D2′*e-ikX2′Κ+q63D3′*e-ikX3′Κ+q64D4′*e-ikX4′Κ,(xii)
(5.20)q65C1*eikX1Κ+q66C2*eikX2Κ+q67C3*eikX3Κ+q68C4*eikX4Κ+q65D1*e-ikX1Κ+q66D2*e-ikX2Κ+q67D3*e-ikX3Κ+q68D4*e-ikX4Κ=q69D1′*e-ikX1′Κ+q70D2′*e-ikX2′Κ+q71D3′*e-ikX3′Κ+q72D4′*e-ikX4′Κ,(xiii)
(5.21)q73C1*eikX1Κ+q74C2*eikX2Κ+q75C3*eikX3Κ+q76C4*eikX4Κ+q77D1*e-ikX1Κ+q78D2*e-ikX2Κ+q79D3*e-ikX3Κ+q80D4*e-ikX4Κ=q81D1′*e-ikX1′Κ+q82D2′*e-ikX2′Κ+q83D3′*e-ikX3′Κ+q84D4′*e-ikX4′Κ,(xvi)
(5.22)q85C1*eikX1Κ+q86C2*eikX2Κ+q87C3*eikX3Κ+q88C4*eikX4Κ-[q85D1*e-ikX1Κ+q86D2*e-ikX2Κ+q87D3*e-ikX3Κ+q88D4*e-ikX4Κ]=0,(xvii)
(5.23)q89C1*eikX1Κ+q90C2*eikX2Κ+q91C3*eikX3Κ+q92C4*eikX4Κ+q93D1*e-ikX1Κ+q94D2*e-ikX2Κ+q95D3*e-ikX3Κ+q96D4*e-ikX4Κ=0,(xviii)
(5.24)q97C1*eikX1Κ+q98C2*eikX2Κ+q99C3*eikX3Κ+q100C4*eikX4Κ+q101D1*e-ikX1Κ+q102D2*e-ikX2Κ+q103D3*e-ikX3Κ+q104D4*e-ikX4Κ=0,(xx)
(5.25)q105C1*eikX1Κ+q106C2*eikX2Κ+q107C3*eikX3Κ+q108C4*eikX4Κ+q109D1*e-ikX1Κ+q110D2*e-ikX2Κ+q111D3*e-ikX3Κ+q112D4*e-ikX4Κ=0,where(5.26)q1=(1-X1m1*),q2=(1-X2m2*),q3=(1-X3m3*),q4=(1-X4m4*),q5=(1+X1m1*),q6=(1+X2m2*),q7=(1+X3m3*),q8=(1+X4m4*),q9=(1+X1′m1′*),q10=(1+X2′m2′*),q11=(1+X3′m3′*),q12=(1+X4′m4′*),q13=(X1+m1*),q14=(X2+m2*),q15=(X3+m3*),q16=(X4+m4*),q17=(m1*-X1),q18=(m2*-X2),q19=(m3*-X3),q20=(m4*-X4),q21=(m1′*-X1′),q22=(m′2*-X′2),q23=(m′3*-X′3),q24=(m′4*-X′4),q25=1cs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q26=1cs1{k2β*2m4*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q27=1cs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q28=1cs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q29=1cs1′{k2β′*2m′1*(1+X′12)-m′1*(k2c2+Ω2)+2ikΩc},q30=1cs1′{k2β′*2m′2*(1+X′22)-m′2*(k2c2+Ω2)+2ikΩc},q31=1cs1′{k2β′*2m′3*(1+X′32)-m′3*(k2c2+Ω2)+2ikΩc},q32=1cs1′{k2β′*2m′4*(1+X′42)-m′4*(k2c2+Ω2)+2ikΩc},q33=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q34=MX2{k2m2*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q35=MX3{k2m3*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q36=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q37=M′X1′{k2m′1(X′12+1)+1ikcs1′(k2β′*2m′1*(1+X′12)-m′1*(k2c2+Ω2)+2ikΩc)},q38=M′X2′{k2m′2(X′22+1)+1ikcs1′(k2β′*2m′2*(1+X′22)-m′2*(k2c2+Ω2)+2ikΩc)},q39=M′X3′{k2m′3(X′32+1)+1ikcs′1(k2β′*2m′3*(1+X′32)-m′3*(k2c2+Ω2)+2ikΩc)},q40=M′X4′{k2m′4(X′42+1)+1ikcs′1(k2β′*2m′4*(1+X′12)-m′4*(k2c2+Ω2)+2ikΩc)},q41=(Γ0λ+μeH02)(1-X1m1*)+(Γ0(λ+2μ)+μeH02)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q42=(Γ0λ+μeH02)(1-X2m2*)+(Γ0(λ+2μ)+μeH02)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q43=(Γ0λ+μeH02)(1-X3m3*)+(Γ0(λ+2μ)+μeH02)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q64=2k2Γ0′μ′X4′-k2Γ0′μ′m′4*(1-X′42)+Fs1′{k2β′*2m′4*(1+X′42)-m′4*(k2c2+Ω2)+2ikΩc},q44=(Γ0λ+μeH02)(1-X4m4*)+(Γ0(λ+2μ)+μeH02)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q45=(Γ0λ+μeH02)(1+X1m1*)+(Γ0(λ+2μ)+μeH02)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q46=(Γ0λ+μeH02)(1+X2m2*)+(Γ0(λ+2μ)+μeH02)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q47=(Γ0λ+μeH02)(1+X3m3*)+(Γ0(λ+2μ)+μeH02)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q48=(Γ0λ+μeH02)(1+X4m4*)+(Γ0(λ+2μ)+μeH02)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q49=(Γ′0λ′+μ′eH02)(1+X′1m′1*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′12-m′1*X′1)+ρ′[-α′*2(1+X′12)+c2],q50=(Γ′0λ′+μ′eH02)(1+X′2m′2*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′22-m′2*X′2)+ρ′[-α′*2(1+X′22)+c2],q51=(Γ′0λ′+μ′eH02)(1+X′3m′3*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′32-m′3*X′3)+ρ′[-α′*2(1+X′32)+c2],q52=(Γ′0λ′+μ′eH02)(1+X′4m′4*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′42-m′4*X′4)+ρ′[-α′*2(1+X′42)+c2],q53=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q54=-2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q55=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q56=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q57=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q58=2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q59=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q60=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q61=2k2Γ0′μ′X1′-k2Γ0′μ′m′1*(1-X′12)+Fs1′{k2β′*2m′1*(1+X′12)-m′1*(k2c2+Ω2)+2ikΩc},q62=2k2Γ0′μ′X2′-k2Γ0′μ′m′2*(1-X′22)+Fs1′{k2β′*2m′2*(1+X′22)-m′2*(k2c2+Ω2)+2ikΩc},q63=2k2Γ0′μ′X3′-k2Γ0′μ′m′3*(1-X′32)+Fs1′{k2β′*2m′3*(1+X′32)-m′3*(k2c2+Ω2)+2ikΩc},q65=ργ[-α*2k2(X12+1)+k2c2],q66=ργ[-α*2k2(X22+1)+k2c2],q67=ργ[-α*2k2(X32+1)+k2c2],q68=ργ[-α*2k2(X42+1)+k2c2],q69=ρ′γ′[-α′*2k2(X′12+1)+k2c2],q70=ρ′γ′[-α′*2k2(X′22+1)+k2c2],q71=ρ′γ′[-α′*2k2(X′32+1)+k2c2],q72=ρ′γ′[-α′*2k2(X′42+1)+k2c2],q73=ργ[-α*2k2(X12+1)+k2c2](θ+ikX1),q74=ργ[-α*2k2(X22+1)+k2c2](θ+ikX2),q75=ργ[-α*2k2(X32+1)+k2c2](θ+ikX3),q76=ργ[-α*2k2(X42+1)+k2c2](θ+ikX4),q77=ργ[-α*2k2(X12+1)+k2c2](θ-ikX1),q78=ργ[-α*2k2(X22+1)+k2c2](θ-ikX2),q79=ργ[-α*2k2(X32+1)+k2c2](θ-ikX3),q80=ργ[-α*2k2(X42+1)+k2c2](θ-ikX4),q81=ρ′γ′[-α′*2k2(X′12+1)+k2c2](θ-ikX′1),q82=ρ′γ′[-α′*2k2(X′22+1)+k2c2](θ-ikX′2),q83=ρ′γ′[-α′*2k2(X′32+1)+k2c2](θ-ikX′3),q84=ρ′γ′[-α′*2k2(X′32+1)+k2c2](θ-ikX′3),q85=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q86=MX2{k2mj*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q87=MX3{k2mj*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q88=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q89=(Γ0λ+μeH02)(1-X1m1*)+(Γ0(λ+2μ)+μeH02)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q90=(Γ0λ+μeH02)(1-X2m2*)+(Γ0(λ+2μ)+μeH02)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q91=(Γ0λ+μeH02)(1-X3m3*)+(Γ0(λ+2μ)+μeH02)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q92=(Γ0λ+μeH02)(1-X4m4*)+(Γ0(λ+2μ)+μeH02)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q93=(Γ0λ+μeH02)(1+X1m1*)+(Γ0(λ+2μ)+μeH02)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q94=(Γ0λ+μeH02)(1+X2m2*)+(Γ0(λ+2μ)+μeH02)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q95=(Γ0λ+μeH02)(1+X3m3*)+(Γ0(λ+2μ)+μeH02)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q96=(Γ0λ+μeH02)(1+X4m4*)+(Γ0(λ+2μ)+μeH02)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q97=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q98=-2k2Γ0μX2-k2Γ0μm2*(1-Xj2)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q99=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q100=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q101=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q102=2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q103=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q104=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q105=[-α*2k2(X12+1)+k2c2](θ+ikX1),q106=[-α*2k2(X22+1)+k2c2](θ+ikX2),q107=[-α*2k2(X32+1)+k2c2](θ+ikX3),q108=[-α*2k2(X42+1)+k2c2](θ+ikX4),q109=[-α*2k2(X12+1)+k2c2](θ-ikX1),q110=[-α*2k2(X22+1)+k2c2](θ-ikX2),q111=[-α*2k2(X32+1)+k2c2](θ-ikX3),q112=[-α*2k2(X42+1)+k2c2](θ-ikX4).Elimination ofCj*,Dj*, and D′j* gives the wave velocity equation in the determinant form (5.27)det⁡⁡dij=0.
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  dij is given by(5.28)|q85e-ikX1Kq86e-ikX2Kq87e-ikX3Kq88e-ikX4Kq85eikX1Kq86eikX2Kq87eikX3Kq88eikX4K0000q89e-ikX1Kq90e-ikX2Kq91e-ikX3Kq92e-ikX4Kq93eikX1Kq94eikX2Kq95eikX3Kq96eikX4K0000q97e-ikX1Kq98e-ikX2Kq99e-ikX3Kq100e-ikX4Kq101eikX1Kq102eikX2Kq103eikX3Kq104eikX4K0000q105e-ikX1Kq106e-ikX2Kq107e-ikX3Kq108e-ikX4Kq109eikX1Kq110eikX2Kq111eikX3Kq112eikX4K0000q13q14q15q16q17q18q19q20q21q22q23q24q25q26q27q28q25q26q27q28q29q30q31q32q33q34q35q36-q33-q34-q35-q36-q37-q38-q39-q40q41q42q43q44q45q46q47q48q49q50q51q52q53q54q55q56q57q58q59q60q61q62q63q64q65q66q67q68q65q66q67q68q69q70q71q72q73q74q75q76q77q78q79q80q81q82q83q84q1q2q3q4q5q6q7q8q9q10q11q12|=0.


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

In this caseg=0,P=0,H0=0, and θ=0, we obtain(5.29)α*2=Γ0(λ+2μ)ρ,β*2=Γ0μρ,lim⁡ε→0mj*=1-2Ωc(ikXj2-ik)×{α*2k2Xj4-[k2(c2-2α*2)+ikcχα*2Γ2+Ω2]Xj2+ikcΓ2χ(1-α*2+Ω2k2)-Ω2},lim⁡γ→0|q22q23q24q30q31q32-q38-q39-q40q50q51q52q62q63q64|=0.
Multiplying the rows 10, 11, and 12 of the determinant |dij| by γ and then taking limγ→0, (5.28) reduces, after some computation, to the following ninth-order determinant  equation:(5.30)|q85e-ikX1Kq86e-ikX2Kq87e-ikX3Kq88e-ikX4Kq85eikX1Kq86eikX2Kq87eikX3Kq88eikX4K0q89e-ikX1Kq90e-ikX2Kq91e-ikX3Kq92e-ikX4Kq93eikX1Kq94eikX2Kq95eikX3Kq96eikX4K0q97e-ikX1Kq98e-ikX2Kq99e-ikX3Kq100e-ikX4Kq101eikX1Kq102eikX2Kq103eikX3Kq104eikX4K0q105e-ikX1Kq106e-ikX2Kq107e-ikX3Kq108e-ikX4Kq109eikX1Kq110eikX2Kq111eikX3Kq112eikX4K0q13q14q15q16q17q18q19q20q21q25q26q27q28q25q26q27q28q29q33q34q35q36-q33-q34-q35-q36-q37q41q42q43q44q45q46q47q48q49q53q54q55q56q57q58q59q60q61|=0,where (5.31)q1=(1-X1m1*),q2=(1-X2m2*),q3=(1-X3m3*),q4=(1-X4m4*),q5=(1+X1m1*),q6=(1+X2m2*),q7=(1+X3m3*),q8=(1+X4m4*),q9=(1+X1′m1′*),q10=(1+X2′m2′*),q11=(1+X3′m3′*),q12=(1+X4′m4′*),q13=(X1+m1*),q14=(X2+m2*),q15=(X3+m3*),q16=(X4+m4*),q17=(m1*-X1),q18=(m2*-X2),q19=(m3*-X3),q20=(m4*-X4),q21=(m1′*-X1′),q22=(m2′*-X2′),q23=(m3′*-X3′),q24=(m4′*-X4′),q25=1cs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q26=1cs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q27=1cs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q28=1cs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q29=1cs1′{k2β*′2m1′*(1+X12′)-m1′*(k2c2+Ω2)+2ikΩc},q30=1cs1′{k2β*′2m2′*(1+X22′)-m2′*(k2c2+Ω2)+2ikΩc},q31=1cs1′{k2β*′2m3′*(1+X32′)-m3′*(k2c2+Ω2)+2ikΩc},q32=1cs1′{k2β*′2m4′*(1+X42′)-m4′*(k2c2+Ω2)+2ikΩc},q33=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q34=MX2{k2m2*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q35=MX3{k2m3*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q36=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q37=M′X1′{k2m1′(X1′2+1)+1ikcs1′(k2β*′2m1′*(1+X1′2)-m1′*(k2c2+Ω2)+2ikΩc)},q38=M′X2′{k2m2′(X2′2+1)+1ikcs1′(k2β*′2m2′*(1+X2′2)-m2′*(k2c2+Ω2)+2ikΩc)},q39=M′X3′{k2m3′(X3′2+1)+1ikcs1′(k2β*′2m3′*(1+X3′2)-m3′*(k2c2+Ω2)+2ikΩc)},q40=M′X4′{k2m4′(X4′2+1)+1ikcs1′(k2β*′2m4′*(1+X4′2)-m4′*(k2c2+Ω2)+2ikΩc)},q41=Γ0λ(1-X1m1*)+Γ0(λ+2μ)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q42=Γ0λ(1-X2m2*)+Γ0(λ+2μ)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q43=Γ0λ(1-X3m3*)+Γ0(λ+2μ)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q44=Γ0λ(1-X4m4*)+Γ0(λ+2μ)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q45=Γ0λ(1+X1m1*)+Γ0(λ+2μ)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q46=Γ0λ(1+X2m2*)+Γ0(λ+2μ)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q47=Γ0λ(1+X3m3*)+Γ0(λ+2μ)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q48=Γ0λ(1+X4m4*)+Γ0(λ+2μ)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q49=Γ0′λ′(1+X1′m1′*)+Γ0′(λ′+2μ′)(X1′2-m1′*X1′)+ρ′[-α*′2(1+X1′2)+c2′],q50=Γ0′λ′(1+X2′m2′*)+Γ0′(λ′+2μ′)(X2′2-m2′*X2′)+ρ′[-α*′2(1+X2′2)+c2′],q51=Γ0′λ′(1+X3′m3′*)+Γ0′(λ′+2μ′)(X3′2-m3′*X3′)+ρ′[-α*′2(1+X3′2)+c2′],q52=Γ0′λ′(1+X4′m4′*)+Γ0′(λ′+2μ′)(X4′2-m4′*X4′)+ρ′[-α*′2(1+X4′2)+c2′],q53=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q54=-2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q55=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q56=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q57=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q58=2k2Γ0X2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q59=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q60=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q61=2k2Γ0′μ′X1′-k2Γ0′μ′m1′*(1-X1′2)+Fs1′{k2β*′2m1′*(1+X1′2)-m1′*(k2c2+Ω2)+2ikΩc},q62=2k2Γ0′μ′X2′-k2Γ0′μ′m2′*(1-X2′2)+Fs1′{k2β*′2m2′*(1+X2′2)-m2′*(k2c2+Ω2)+2ikΩc},q63=2k2Γ0′μ′X3′-k2Γ0′μ′m3′*(1-X3′2)+Fs1′{k2β*′2m3′*(1+X3′2)-m3′*(k2c2+Ω2)+2ikΩc},q64=2k2Γ0′μ′X4′-k2Γ0′μ′m4′*(1-X4′2)+Fs1′{k2β*′2m4′*(1+X4′2)-m4′*(k2c2+Ω2)+2ikΩc},q65=ργ[-α*2k2(X12+1)+k2c2],q66=ργ[-α*2k2(X22+1)+k2c2],q67=ργ[-α*2k2(X32+1)+k2c2],q68=ργ[-α*2k2(X42+1)+k2c2],q69=ρ′γ′[-α*′2k2(X1′2+1)+k2c2],q70=ρ′γ′[-α*′2k2(X2′2+1)+k2c2],q71=ρ′γ′[-α*′2k2(X3′2+1)+k2c2],q72=ρ′γ′[-α*′2k2(X4′2+1)+k2c2],q73=ργ[-α*2k2(X12+1)+k2c2]ikX1,q74=ργ[-α*2k2(X22+1)+k2c2]ikX2,q75=ργ[-α*2k2(X32+1)+k2c2]ikX3,q76=ργ[-α*2k2(X42+1)+k2c2]ikX4,q77=-ργ[-α*2k2(X12+1)+k2c2]ikX1,q78=-ργ[-α*2k2(X22+1)+k2c2]ikX2,q79=-ργ[-α*2k2(X32+1)+k2c2]ikX3,q80=-ργ[-α*2k2(X42+1)+k2c2]ikX4,q81=-ρ′γ′[-α*′2k2(X1′2+1)+k2c2]ikX1′,q82=-ρ′γ′[-α*′2k2(X2′2+1)+k2c2]ikX2′,q83=-ρ′γ′[-α*′2k2(X3′2+1)+k2c2]ikX3′,q84=-ρ′γ′[-α*′2k2(X3′2+1)+k2c2]ikX3′,q85=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q86=MX2{k2mj*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q87=MX3{k2mj*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q88=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q89=Γ0λ(1-X1m1*)+Γ0(λ+2μ)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q90=Γ0λ(1-X2m2*)+Γ0(λ+2μ)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q91=Γ0λ(1-X3m3*)+Γ0(λ+2μ)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q92=Γ0λ(1-X4m4*)+Γ0(λ+2μ)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q93=Γ0λ(1+X1m1*)+Γ0(λ+2μ)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q94=Γ0λ(1+X2m2*)+Γ0(λ+2μ)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q95=Γ0λ(1+X3m3*)+Γ0(λ+2μ)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q96=Γ0λ(1+X4m4*)+Γ0(λ+2μ)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q97=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q98=-2k2Γ0μX2-k2Γ0μm2*(1-Xj2)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q99=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q100=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q101=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q102=2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q103=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q104=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q105=[-α*2k2(X12+1)+k2c2]ikX1,q106=[-α*2k2(X22+1)+k2c2]ikX2,q107=[-α*2k2(X32+1)+k2c2]ikX3,q108=[-α*2k2(X42+1)+k2c2]ikX4,q109=[α*2k2(X12+1)-k2c2]ikX1,q110=[α*2k2(X22+1)-k2c2]ikX2,q111=[α*2k2(X32+1)-k2c2]ikX3,q112=[α*2k2(X42+1)-k2c2]ikX4.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 Figures2 and 3.Effects ofH0,Ω,g,τ1, and p on Rayleigh wave velocity and attenuation coefficient with respect to the initial stress.Effects ofH0,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.Effects ofH0,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.Effects ofΩ,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 ofΩ,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 ofΩ,τ1, and K on Rayleigh wave velocity and attenuation coefficient with respect to the wave number.


## 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 (5.1)V8+h1V6+h2V4+h3V2+h4=0,
where(5.2)α*2=Γ0(λ+2μ)ρ,β*2=Γ0μρ,mj=1(g-2Ωc)(ikVj2-ik-(ε/χ))×{α*2k2Vj4-[k2(c2-2α*2)+ikcχ(α*2+χε)+Ω2]Vj2+ikcχ(1-α*2+Ω2k2)-(Ω2+ikεc)},h1=k2α*2(c2-2α*2)+ikcα*2χ(α*2+χε)+Ω2+1β*2+ikcs1×[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2],h2=1α*2[k4(α*2-c2)+ikcχ(k2(1-α*2)+Ω2)-k2(Ω2+ikεc)]+1α*2(β*2+ikcs1)[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k2(c2-2α*2)+ikcχ(α*2+χε)+Ω2]+1(β*2+ikcs1)[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]-1α*2(β*2+ikcs1)[k2(g-2Ωc)2],h3=1α*2(β*2+ikcs1){[ik(g-2Ωc)2(ik3+k2cχ)][k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k4(α*2-c2)+ikcχ(k2(1-α*2)+Ω2)-k2(Ω2+ikεc)]+[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[k2(c2-2α*2)+ikcχ(α*2+χε)+Ω2]-[ik(g-2Ωc)2(ik3+k2cχ)]-ik3[(g-2Ωc)2(ik-cs2)]},h4=1α*2(β*2+ikcs){[(2Ωc-g)2(ik3-k2cs2)(ik3+k2cχ)][k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[ik(g-2Ωc)]+[(2Ωc-g)2(ik3-k2cs2)(ik3+k2cχ)]}.Also,(5.3)η1=∑j=141ikcs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}×[CjeikVjz+Dje-ikVjz].T=ργ∑j=14[-α*2k2(1+Vj2)+k2c2-ikgmj](CjeikVjz+Dje-ikVjz)eik(x-ct),ξ1′=-kq′B2′e-iq′z,ζ1′=B2′e-iq′z,η1′=∑j=141ikcs1′{k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω′2)+ik(2Ω′c-g)}Dj′e-ikVj′z,T′=ρ′γ′∑j=14[-α*′2k2(1+Vj′2)+k2c2-ikgmj′]Dj′e-ikVj′zeik(x-ct),ϕ1′=∑j=14Dj′e-ikVj′z,ψ1′=∑j=14Fj′e-ikVj′z,
Using the boundary conditions, we obtain(5.4)[d11d12⋯d18d15′d16′⋯d18′d21d22⋯d28d25′d26′⋯d28′⋮⋮⋮⋮⋮⋮⋮⋮d121d122⋯d128d125′d126′⋯d128′][C1C2C3C4D1D2D3D4D1′D2′D3′D4′]=[0],
where(5.5)d1j=(1-Vjmj)CjeikVjΚ+(1+Vjmj)Dje-ikVjΚ,d1j′=(1+Vj′mj′)Dj′e-ikVj′Κ,d2j=(Vj+mj)CjeikVjΚ+(mj-Vj)Dje-ikVjΚ,d2j′=(mj′-Vj′)Dj′e-ikVj′Κ,d3j=1cs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}[CjeikVjz+Dje-ikVjz],d3j′=1cs1′{k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω2)+ik(2Ωc-g)}Dj′e-ikVj′z,d4j=MVj{k2mj(Vj2+1)+1ikcs1(k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g))×[CjeikVjK-Dje-ikVjK],d4j′=M′Vj′{k2mj′(Vj′2+1)+1ikcs1′(k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω2)+ik(2Ωc-g))}×Dj′e-ikVj′K,d5j=[Γ0λ(1-Vjmj)+Γ0(λ+2μ)(Vj2+mjVj)]CjeikVjK+[Γ0λ(1+Vjmj)+Γ0(λ+2μ)(Vj2-mjVj)]Dje-ikVjK+ρ[-α*2(1+Vj2)+c2-igkmj][CjeikVjK+Dje-ikVjK],d5j′=[Γ0′λ′(1+Vj′mj′)+Γ0′(λ′+2μ′)(Vj′2-mj′Vj′)]Dj′e-ikVj′K+ρ′[-α*′2(1+Vj′2)+c2-igkmj′]×Dj′e-ikVj′K},d6j=-2k2Γ0μVj[CjeikVjK-Dje-ikVjK]+[-k2Γ0μmj(1-Vj2)+Fs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}]×[CjeikVjK+Dje-ikVjK],d6j′=2k2Γ0′μ′Vj′Dj′e-ikVj′K+[-k2Γ′0μ′mj′(1-Vj′2)+Fs1′{k2β*′2mj′(1+Vj′2)-mj′(k2c2+Ω2)+ik(2Ωc-g)}]×Dj′e-ikVj′K,d7j=ργ[-α*2k2(Vj2+1)+k2c2-igkmj][CjeikVjK+Dje-ikVjK],d7j′=ρ′γ′[-α*′2k2(Vj′2+1)+k2c2-igkmj′]Dj′e-ikVj′K,d8j=ργ[-α*2k2(Vj2+1)+k2c2-igkmj][(θ+ikVj)CjeikVjK+(θ-ikVj)Dje-ikVjK],d8j′=ρ′γ′[-α*′2k2(Vj′2+1)+k2c2-igkmj′](θ-ikVj′)Dj′e-ikVj′K,d9j=MVj{k2mj(Vj2+1)+1ikcs1(k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g))}×[Cj-Dj],d10j=[Γ0λ(1-Vjmj)+Γ0(λ+2μ)(Vj2+mjVj)]Cj+[Γ0λ(1+Vjmj)+Γ0(λ+2μ)(Vj2-mjVj)]Dj+ρ[-α*2(1+Vj2)+c2-igkmj][Cj+Dj],d11j=-2k2Γ0μVj[Cj-Dj]+[-k2Γ0μmj(1-Vj2)+Fs1{k2β*2mj(1+Vj2)-mj(k2c2+Ω2)+ik(2Ωc-g)}]×[Cj+Dj],d12j=[-α*2k2(Vj2+1)+k2c2-igkmj][(θ+ikVj)Cj+(θ-ikVj)Dj],d9j′=d10j′=d11j′=d12j′=0,j=1,2,3,4.


## 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 putg=0, then (3.20) becomes(5.6)[α*2D4+G1D2+G2]ϕ1-[G3*D2+G4*]ψ1=0,[R1D4+R2D2+R3]ψ1+[R4*D2+R5*]ϕ1=0,
where(5.7)G3*=-2ikΩc,G4*=2Ωc(ik3+k2cΓ2χ),R4*=-2ikΩc,R5*=2Ωc(ik3-k2cs2),
and  G1,G2,R1,R2, and R3 are as in (3.21).The solution of (5.6) take the form(5.8)ϕ=∑j=14[Cj*eikXjz+Dj*e-ikXjz]eik(x-ct),ψ=∑j=14[Ej*eikXjz+Fj*e-ikXjz]eik(x-ct),
where(5.9)Ej*=mj*Cj*,Fj*=mj*Dj*,j=1,2,3,4,(5.10)mj*=1-2Ωc(ikXj2-ik-((εΓ2)/χ))×{(1-α*2+Ω2k2)α*2k2Xj4-[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]Xj2+ikcΓ2χ(1-α*2+Ω2k2)-(Ω2+ikεcΓ1Γ3)},
and X1,X2,X3,and X4 are taken to be the complex roots of equation(5.11)X8+t1*X6+t2*X4+t3*X2+t4*=0,
where(5.12)t1*=k2α*2(c2-2α*2)+ikcα*2χ(α*2Γ2+χεΓ1Γ3)+Ω2+1β*2+ikcs1×[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2],t2*=1α*2[k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3)]+1α*2(β*2+ikcs1)[k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]+1(β*2+ikcs1)[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]-1α*2(β*2+ikcs1)[4k2Ω2c2],t3*=1α*2(β*2+ikcs1)×{[ik(ik3+k2cΓ2χ)+ik3(ik-cs2)][k2(c2-2β*2)+ikc(s2β*2-2k2s1)+Ω2]×[k4(α*2-c2)+ikcΓ2χ(k2(1-α*2)+Ω2)-k2(Ω2+ikεcΓ1Γ3)]+[k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)]×[k2(c2-2α*2)+ikcχ(α*2Γ2+χεΓ1Γ3)+Ω2]-4Ω2c2[ik(ik3+k2cΓ2χ)+ik3(ik-cs2)]},t4*=1α*2(β*2+ikcs){[4Ω2c2(ik3-k2cs2)(ik3+k2cΓ2χ)][k2(k2-ikcs2)(β*2-c2)+ikc(s2Ω2+k4s1)][-2ikΩc]+[4Ω2c2(ik3-k2cs2)(ik3+k2cΓ2χ)]},u=ik∑j=14[(1-Xjmj*)Cj*eikXjz+(1+Xjmj*)Dj*e-ikXjz]eik(x-ct),w=ik∑j=14[(Xj+mj*)Cj*eikXjz+(mj*-Xj)Dj*e-ikXjz]eik(x-ct),η1=∑j=141ikcs1{k2β*2mj*(1+Xj2)-mj*(k2c2+Ω2)+2ikΩc}×[Cj*eikXjz+Dj*e-ikXjz],T=ργΓ1∑j=14[-α*2k2(1+Xj2)+k2c2](Cj*eikXjz+Dj*e-ikXjz)eik(x-ct).With the lower medium, we use the symbols with primes, forξ1,ζ1,η1,T,ϕ,ψ, and q, for z>Κ,(5.13)ξ1′=-kq′B2′e-iq′z,ζ1′=B2′e-iq′z,η1′=∑j=141ikcs1′{k2β*′2mj′*(1+Xj′2)-mj′*(k2c2+Ω′2)+2ikΩ′c}Dj′*e-ikXj′z,T′=ρ′γ′Γ1∑j=14[-α*′2k2(1+Xj′2)+k2c2]Dj′*e-ikXj′zeik(x-ct),ϕ′=∑j=14Dj′e-ikNj′zeik(x-ct),ψ′=∑j=14Fj′*e-ikXj′zeik(x-ct).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)
(5.14)q1C1*eikX1K+q2C2*eikX2K+q3C3*eikX3K+q4C4*eikX4K+q5D1*e-ikX1K+q6D2*e-ikX2K+q7D3*e-ikX3K+q8D4*e-ikX4K=q9D1′*e-ikX1′K+q10D2′*e-ikX2′K+q11D3′*e-ikX3′K+q12D4′*e-ikX4′K,(ii)
(5.15)q13C1*eikX1Κ+q14C2*eikX2Κ+q15C3*eikX3Κ+q16C4*eikX4Κ+q17D1*e-ikX1Κ+q18D2*e-ikX2Κ+q19D3*e-ikX3Κ+q20D4*e-ikX4Κ=q21D1′*e-ikX1′Κ+q22D2′*e-ikX2′Κ+q23D3′*e-ikX3′Κ+q24D4′*e-ikX4′Κ,(iv)
(5.16)q25C1*eikX1Κ+q26C2*eikX2Κ+q27C3*eikX3Κ+q28C4*eikX4Κ+q25D1*e-ikX1Κ+q26D2*e-ikX2Κ+q27D3*e-ikX3Κ+q28D4*e-ikX4Κ=q29D1′*e-ikX1′Κ+q30D2′*e-ikX2′Κ+q31D3′*e-ikX3′Κ+q32D4′*e-ikX4′Κ,(viii)
(5.17)q33C1*eikX1Κ+q34C2*eikX2Κ+q35C3*eikX3Κ+q36C4*eikX4Κ-q33D1*e-ikX1Κ-q34D2*e-ikX2Κ-q35D3*e-ikX3Κ-q36D4*e-ikX4Κ=-q37D1′*e-ikX1′Κ-q38D2′*e-ikX2′Κ-q39D3′*e-ikX3′Κ-q40D4′*e-ikX4′Κ,(ix)
(5.18)q41C1*eikX1Κ+q42C2*eikX2Κ+q43C3*eikX3Κ+q44C4*eikX4Κ+q45D1*e-ikX1Κ+q46D2*e-ikX2Κ+q47D3*e-ikX3Κ+q48D4*e-ikX4Κ=q49D1′*e-ikX1′Κ+q50D2′*e-ikX2′Κ+q51D3′*e-ikX3′Κ+q52D4′*e-ikX4′Κ,(x)
(5.19)q53C1*eikX1Κ+q54C2*eikX2Κ+q55C3*eikX3Κ+q56C4*eikX4Κ+q57D1*e-ikX1Κ+q58D2*e-ikX2Κ+q59D3*e-ikX3Κ+q60D4*e-ikX4Κ=q61D1′*e-ikX1′Κ+q62D2′*e-ikX2′Κ+q63D3′*e-ikX3′Κ+q64D4′*e-ikX4′Κ,(xii)
(5.20)q65C1*eikX1Κ+q66C2*eikX2Κ+q67C3*eikX3Κ+q68C4*eikX4Κ+q65D1*e-ikX1Κ+q66D2*e-ikX2Κ+q67D3*e-ikX3Κ+q68D4*e-ikX4Κ=q69D1′*e-ikX1′Κ+q70D2′*e-ikX2′Κ+q71D3′*e-ikX3′Κ+q72D4′*e-ikX4′Κ,(xiii)
(5.21)q73C1*eikX1Κ+q74C2*eikX2Κ+q75C3*eikX3Κ+q76C4*eikX4Κ+q77D1*e-ikX1Κ+q78D2*e-ikX2Κ+q79D3*e-ikX3Κ+q80D4*e-ikX4Κ=q81D1′*e-ikX1′Κ+q82D2′*e-ikX2′Κ+q83D3′*e-ikX3′Κ+q84D4′*e-ikX4′Κ,(xvi)
(5.22)q85C1*eikX1Κ+q86C2*eikX2Κ+q87C3*eikX3Κ+q88C4*eikX4Κ-[q85D1*e-ikX1Κ+q86D2*e-ikX2Κ+q87D3*e-ikX3Κ+q88D4*e-ikX4Κ]=0,(xvii)
(5.23)q89C1*eikX1Κ+q90C2*eikX2Κ+q91C3*eikX3Κ+q92C4*eikX4Κ+q93D1*e-ikX1Κ+q94D2*e-ikX2Κ+q95D3*e-ikX3Κ+q96D4*e-ikX4Κ=0,(xviii)
(5.24)q97C1*eikX1Κ+q98C2*eikX2Κ+q99C3*eikX3Κ+q100C4*eikX4Κ+q101D1*e-ikX1Κ+q102D2*e-ikX2Κ+q103D3*e-ikX3Κ+q104D4*e-ikX4Κ=0,(xx)
(5.25)q105C1*eikX1Κ+q106C2*eikX2Κ+q107C3*eikX3Κ+q108C4*eikX4Κ+q109D1*e-ikX1Κ+q110D2*e-ikX2Κ+q111D3*e-ikX3Κ+q112D4*e-ikX4Κ=0,where(5.26)q1=(1-X1m1*),q2=(1-X2m2*),q3=(1-X3m3*),q4=(1-X4m4*),q5=(1+X1m1*),q6=(1+X2m2*),q7=(1+X3m3*),q8=(1+X4m4*),q9=(1+X1′m1′*),q10=(1+X2′m2′*),q11=(1+X3′m3′*),q12=(1+X4′m4′*),q13=(X1+m1*),q14=(X2+m2*),q15=(X3+m3*),q16=(X4+m4*),q17=(m1*-X1),q18=(m2*-X2),q19=(m3*-X3),q20=(m4*-X4),q21=(m1′*-X1′),q22=(m′2*-X′2),q23=(m′3*-X′3),q24=(m′4*-X′4),q25=1cs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q26=1cs1{k2β*2m4*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q27=1cs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q28=1cs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q29=1cs1′{k2β′*2m′1*(1+X′12)-m′1*(k2c2+Ω2)+2ikΩc},q30=1cs1′{k2β′*2m′2*(1+X′22)-m′2*(k2c2+Ω2)+2ikΩc},q31=1cs1′{k2β′*2m′3*(1+X′32)-m′3*(k2c2+Ω2)+2ikΩc},q32=1cs1′{k2β′*2m′4*(1+X′42)-m′4*(k2c2+Ω2)+2ikΩc},q33=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q34=MX2{k2m2*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q35=MX3{k2m3*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q36=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q37=M′X1′{k2m′1(X′12+1)+1ikcs1′(k2β′*2m′1*(1+X′12)-m′1*(k2c2+Ω2)+2ikΩc)},q38=M′X2′{k2m′2(X′22+1)+1ikcs1′(k2β′*2m′2*(1+X′22)-m′2*(k2c2+Ω2)+2ikΩc)},q39=M′X3′{k2m′3(X′32+1)+1ikcs′1(k2β′*2m′3*(1+X′32)-m′3*(k2c2+Ω2)+2ikΩc)},q40=M′X4′{k2m′4(X′42+1)+1ikcs′1(k2β′*2m′4*(1+X′12)-m′4*(k2c2+Ω2)+2ikΩc)},q41=(Γ0λ+μeH02)(1-X1m1*)+(Γ0(λ+2μ)+μeH02)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q42=(Γ0λ+μeH02)(1-X2m2*)+(Γ0(λ+2μ)+μeH02)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q43=(Γ0λ+μeH02)(1-X3m3*)+(Γ0(λ+2μ)+μeH02)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q64=2k2Γ0′μ′X4′-k2Γ0′μ′m′4*(1-X′42)+Fs1′{k2β′*2m′4*(1+X′42)-m′4*(k2c2+Ω2)+2ikΩc},q44=(Γ0λ+μeH02)(1-X4m4*)+(Γ0(λ+2μ)+μeH02)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q45=(Γ0λ+μeH02)(1+X1m1*)+(Γ0(λ+2μ)+μeH02)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q46=(Γ0λ+μeH02)(1+X2m2*)+(Γ0(λ+2μ)+μeH02)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q47=(Γ0λ+μeH02)(1+X3m3*)+(Γ0(λ+2μ)+μeH02)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q48=(Γ0λ+μeH02)(1+X4m4*)+(Γ0(λ+2μ)+μeH02)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q49=(Γ′0λ′+μ′eH02)(1+X′1m′1*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′12-m′1*X′1)+ρ′[-α′*2(1+X′12)+c2],q50=(Γ′0λ′+μ′eH02)(1+X′2m′2*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′22-m′2*X′2)+ρ′[-α′*2(1+X′22)+c2],q51=(Γ′0λ′+μ′eH02)(1+X′3m′3*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′32-m′3*X′3)+ρ′[-α′*2(1+X′32)+c2],q52=(Γ′0λ′+μ′eH02)(1+X′4m′4*)+(Γ′0(λ′+2μ′)+μ′eH02)(X′42-m′4*X′4)+ρ′[-α′*2(1+X′42)+c2],q53=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q54=-2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q55=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q56=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q57=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q58=2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q59=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q60=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q61=2k2Γ0′μ′X1′-k2Γ0′μ′m′1*(1-X′12)+Fs1′{k2β′*2m′1*(1+X′12)-m′1*(k2c2+Ω2)+2ikΩc},q62=2k2Γ0′μ′X2′-k2Γ0′μ′m′2*(1-X′22)+Fs1′{k2β′*2m′2*(1+X′22)-m′2*(k2c2+Ω2)+2ikΩc},q63=2k2Γ0′μ′X3′-k2Γ0′μ′m′3*(1-X′32)+Fs1′{k2β′*2m′3*(1+X′32)-m′3*(k2c2+Ω2)+2ikΩc},q65=ργ[-α*2k2(X12+1)+k2c2],q66=ργ[-α*2k2(X22+1)+k2c2],q67=ργ[-α*2k2(X32+1)+k2c2],q68=ργ[-α*2k2(X42+1)+k2c2],q69=ρ′γ′[-α′*2k2(X′12+1)+k2c2],q70=ρ′γ′[-α′*2k2(X′22+1)+k2c2],q71=ρ′γ′[-α′*2k2(X′32+1)+k2c2],q72=ρ′γ′[-α′*2k2(X′42+1)+k2c2],q73=ργ[-α*2k2(X12+1)+k2c2](θ+ikX1),q74=ργ[-α*2k2(X22+1)+k2c2](θ+ikX2),q75=ργ[-α*2k2(X32+1)+k2c2](θ+ikX3),q76=ργ[-α*2k2(X42+1)+k2c2](θ+ikX4),q77=ργ[-α*2k2(X12+1)+k2c2](θ-ikX1),q78=ργ[-α*2k2(X22+1)+k2c2](θ-ikX2),q79=ργ[-α*2k2(X32+1)+k2c2](θ-ikX3),q80=ργ[-α*2k2(X42+1)+k2c2](θ-ikX4),q81=ρ′γ′[-α′*2k2(X′12+1)+k2c2](θ-ikX′1),q82=ρ′γ′[-α′*2k2(X′22+1)+k2c2](θ-ikX′2),q83=ρ′γ′[-α′*2k2(X′32+1)+k2c2](θ-ikX′3),q84=ρ′γ′[-α′*2k2(X′32+1)+k2c2](θ-ikX′3),q85=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q86=MX2{k2mj*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q87=MX3{k2mj*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q88=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q89=(Γ0λ+μeH02)(1-X1m1*)+(Γ0(λ+2μ)+μeH02)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q90=(Γ0λ+μeH02)(1-X2m2*)+(Γ0(λ+2μ)+μeH02)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q91=(Γ0λ+μeH02)(1-X3m3*)+(Γ0(λ+2μ)+μeH02)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q92=(Γ0λ+μeH02)(1-X4m4*)+(Γ0(λ+2μ)+μeH02)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q93=(Γ0λ+μeH02)(1+X1m1*)+(Γ0(λ+2μ)+μeH02)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q94=(Γ0λ+μeH02)(1+X2m2*)+(Γ0(λ+2μ)+μeH02)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q95=(Γ0λ+μeH02)(1+X3m3*)+(Γ0(λ+2μ)+μeH02)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q96=(Γ0λ+μeH02)(1+X4m4*)+(Γ0(λ+2μ)+μeH02)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q97=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q98=-2k2Γ0μX2-k2Γ0μm2*(1-Xj2)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q99=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q100=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q101=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q102=2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q103=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q104=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q105=[-α*2k2(X12+1)+k2c2](θ+ikX1),q106=[-α*2k2(X22+1)+k2c2](θ+ikX2),q107=[-α*2k2(X32+1)+k2c2](θ+ikX3),q108=[-α*2k2(X42+1)+k2c2](θ+ikX4),q109=[-α*2k2(X12+1)+k2c2](θ-ikX1),q110=[-α*2k2(X22+1)+k2c2](θ-ikX2),q111=[-α*2k2(X32+1)+k2c2](θ-ikX3),q112=[-α*2k2(X42+1)+k2c2](θ-ikX4).Elimination ofCj*,Dj*, and D′j* gives the wave velocity equation in the determinant form (5.27)det⁡⁡dij=0.
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  dij is given by(5.28)|q85e-ikX1Kq86e-ikX2Kq87e-ikX3Kq88e-ikX4Kq85eikX1Kq86eikX2Kq87eikX3Kq88eikX4K0000q89e-ikX1Kq90e-ikX2Kq91e-ikX3Kq92e-ikX4Kq93eikX1Kq94eikX2Kq95eikX3Kq96eikX4K0000q97e-ikX1Kq98e-ikX2Kq99e-ikX3Kq100e-ikX4Kq101eikX1Kq102eikX2Kq103eikX3Kq104eikX4K0000q105e-ikX1Kq106e-ikX2Kq107e-ikX3Kq108e-ikX4Kq109eikX1Kq110eikX2Kq111eikX3Kq112eikX4K0000q13q14q15q16q17q18q19q20q21q22q23q24q25q26q27q28q25q26q27q28q29q30q31q32q33q34q35q36-q33-q34-q35-q36-q37-q38-q39-q40q41q42q43q44q45q46q47q48q49q50q51q52q53q54q55q56q57q58q59q60q61q62q63q64q65q66q67q68q65q66q67q68q69q70q71q72q73q74q75q76q77q78q79q80q81q82q83q84q1q2q3q4q5q6q7q8q9q10q11q12|=0.


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

In this caseg=0,P=0,H0=0, and θ=0, we obtain(5.29)α*2=Γ0(λ+2μ)ρ,β*2=Γ0μρ,lim⁡ε→0mj*=1-2Ωc(ikXj2-ik)×{α*2k2Xj4-[k2(c2-2α*2)+ikcχα*2Γ2+Ω2]Xj2+ikcΓ2χ(1-α*2+Ω2k2)-Ω2},lim⁡γ→0|q22q23q24q30q31q32-q38-q39-q40q50q51q52q62q63q64|=0.
Multiplying the rows 10, 11, and 12 of the determinant |dij| by γ and then taking limγ→0, (5.28) reduces, after some computation, to the following ninth-order determinant  equation:(5.30)|q85e-ikX1Kq86e-ikX2Kq87e-ikX3Kq88e-ikX4Kq85eikX1Kq86eikX2Kq87eikX3Kq88eikX4K0q89e-ikX1Kq90e-ikX2Kq91e-ikX3Kq92e-ikX4Kq93eikX1Kq94eikX2Kq95eikX3Kq96eikX4K0q97e-ikX1Kq98e-ikX2Kq99e-ikX3Kq100e-ikX4Kq101eikX1Kq102eikX2Kq103eikX3Kq104eikX4K0q105e-ikX1Kq106e-ikX2Kq107e-ikX3Kq108e-ikX4Kq109eikX1Kq110eikX2Kq111eikX3Kq112eikX4K0q13q14q15q16q17q18q19q20q21q25q26q27q28q25q26q27q28q29q33q34q35q36-q33-q34-q35-q36-q37q41q42q43q44q45q46q47q48q49q53q54q55q56q57q58q59q60q61|=0,where (5.31)q1=(1-X1m1*),q2=(1-X2m2*),q3=(1-X3m3*),q4=(1-X4m4*),q5=(1+X1m1*),q6=(1+X2m2*),q7=(1+X3m3*),q8=(1+X4m4*),q9=(1+X1′m1′*),q10=(1+X2′m2′*),q11=(1+X3′m3′*),q12=(1+X4′m4′*),q13=(X1+m1*),q14=(X2+m2*),q15=(X3+m3*),q16=(X4+m4*),q17=(m1*-X1),q18=(m2*-X2),q19=(m3*-X3),q20=(m4*-X4),q21=(m1′*-X1′),q22=(m2′*-X2′),q23=(m3′*-X3′),q24=(m4′*-X4′),q25=1cs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q26=1cs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q27=1cs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q28=1cs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q29=1cs1′{k2β*′2m1′*(1+X12′)-m1′*(k2c2+Ω2)+2ikΩc},q30=1cs1′{k2β*′2m2′*(1+X22′)-m2′*(k2c2+Ω2)+2ikΩc},q31=1cs1′{k2β*′2m3′*(1+X32′)-m3′*(k2c2+Ω2)+2ikΩc},q32=1cs1′{k2β*′2m4′*(1+X42′)-m4′*(k2c2+Ω2)+2ikΩc},q33=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q34=MX2{k2m2*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q35=MX3{k2m3*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q36=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q37=M′X1′{k2m1′(X1′2+1)+1ikcs1′(k2β*′2m1′*(1+X1′2)-m1′*(k2c2+Ω2)+2ikΩc)},q38=M′X2′{k2m2′(X2′2+1)+1ikcs1′(k2β*′2m2′*(1+X2′2)-m2′*(k2c2+Ω2)+2ikΩc)},q39=M′X3′{k2m3′(X3′2+1)+1ikcs1′(k2β*′2m3′*(1+X3′2)-m3′*(k2c2+Ω2)+2ikΩc)},q40=M′X4′{k2m4′(X4′2+1)+1ikcs1′(k2β*′2m4′*(1+X4′2)-m4′*(k2c2+Ω2)+2ikΩc)},q41=Γ0λ(1-X1m1*)+Γ0(λ+2μ)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q42=Γ0λ(1-X2m2*)+Γ0(λ+2μ)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q43=Γ0λ(1-X3m3*)+Γ0(λ+2μ)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q44=Γ0λ(1-X4m4*)+Γ0(λ+2μ)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q45=Γ0λ(1+X1m1*)+Γ0(λ+2μ)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q46=Γ0λ(1+X2m2*)+Γ0(λ+2μ)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q47=Γ0λ(1+X3m3*)+Γ0(λ+2μ)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q48=Γ0λ(1+X4m4*)+Γ0(λ+2μ)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q49=Γ0′λ′(1+X1′m1′*)+Γ0′(λ′+2μ′)(X1′2-m1′*X1′)+ρ′[-α*′2(1+X1′2)+c2′],q50=Γ0′λ′(1+X2′m2′*)+Γ0′(λ′+2μ′)(X2′2-m2′*X2′)+ρ′[-α*′2(1+X2′2)+c2′],q51=Γ0′λ′(1+X3′m3′*)+Γ0′(λ′+2μ′)(X3′2-m3′*X3′)+ρ′[-α*′2(1+X3′2)+c2′],q52=Γ0′λ′(1+X4′m4′*)+Γ0′(λ′+2μ′)(X4′2-m4′*X4′)+ρ′[-α*′2(1+X4′2)+c2′],q53=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q54=-2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q55=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q56=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q57=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q58=2k2Γ0X2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q59=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q60=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q61=2k2Γ0′μ′X1′-k2Γ0′μ′m1′*(1-X1′2)+Fs1′{k2β*′2m1′*(1+X1′2)-m1′*(k2c2+Ω2)+2ikΩc},q62=2k2Γ0′μ′X2′-k2Γ0′μ′m2′*(1-X2′2)+Fs1′{k2β*′2m2′*(1+X2′2)-m2′*(k2c2+Ω2)+2ikΩc},q63=2k2Γ0′μ′X3′-k2Γ0′μ′m3′*(1-X3′2)+Fs1′{k2β*′2m3′*(1+X3′2)-m3′*(k2c2+Ω2)+2ikΩc},q64=2k2Γ0′μ′X4′-k2Γ0′μ′m4′*(1-X4′2)+Fs1′{k2β*′2m4′*(1+X4′2)-m4′*(k2c2+Ω2)+2ikΩc},q65=ργ[-α*2k2(X12+1)+k2c2],q66=ργ[-α*2k2(X22+1)+k2c2],q67=ργ[-α*2k2(X32+1)+k2c2],q68=ργ[-α*2k2(X42+1)+k2c2],q69=ρ′γ′[-α*′2k2(X1′2+1)+k2c2],q70=ρ′γ′[-α*′2k2(X2′2+1)+k2c2],q71=ρ′γ′[-α*′2k2(X3′2+1)+k2c2],q72=ρ′γ′[-α*′2k2(X4′2+1)+k2c2],q73=ργ[-α*2k2(X12+1)+k2c2]ikX1,q74=ργ[-α*2k2(X22+1)+k2c2]ikX2,q75=ργ[-α*2k2(X32+1)+k2c2]ikX3,q76=ργ[-α*2k2(X42+1)+k2c2]ikX4,q77=-ργ[-α*2k2(X12+1)+k2c2]ikX1,q78=-ργ[-α*2k2(X22+1)+k2c2]ikX2,q79=-ργ[-α*2k2(X32+1)+k2c2]ikX3,q80=-ργ[-α*2k2(X42+1)+k2c2]ikX4,q81=-ρ′γ′[-α*′2k2(X1′2+1)+k2c2]ikX1′,q82=-ρ′γ′[-α*′2k2(X2′2+1)+k2c2]ikX2′,q83=-ρ′γ′[-α*′2k2(X3′2+1)+k2c2]ikX3′,q84=-ρ′γ′[-α*′2k2(X3′2+1)+k2c2]ikX3′,q85=MX1{k2m1*(X12+1)+1ikcs1(k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc)},q86=MX2{k2mj*(X22+1)+1ikcs1(k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc)},q87=MX3{k2mj*(X32+1)+1ikcs1(k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc)},q88=MX4{k2m4*(X42+1)+1ikcs1(k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc)},q89=Γ0λ(1-X1m1*)+Γ0(λ+2μ)(X12+m1*X1)+ρ[-α*2(1+X12)+c2],q90=Γ0λ(1-X2m2*)+Γ0(λ+2μ)(X22+m2*X2)+ρ[-α*2(1+X22)+c2],q91=Γ0λ(1-X3m3*)+Γ0(λ+2μ)(X32+m3*X3)+ρ[-α*2(1+X32)+c2],q92=Γ0λ(1-X4m4*)+Γ0(λ+2μ)(X42+m4*X4)+ρ[-α*2(1+X42)+c2],q93=Γ0λ(1+X1m1*)+Γ0(λ+2μ)(X12-m1*X1)+ρ[-α*2(1+X12)+c2],q94=Γ0λ(1+X2m2*)+Γ0(λ+2μ)(X22-m2*X2)+ρ[-α*2(1+X22)+c2],q95=Γ0λ(1+X3m3*)+Γ0(λ+2μ)(X32-m3*X3)+ρ[-α*2(1+X32)+c2],q96=Γ0λ(1+X4m4*)+Γ0(λ+2μ)(X42-m4*X4)+ρ[-α*2(1+X42)+c2],q97=-2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q98=-2k2Γ0μX2-k2Γ0μm2*(1-Xj2)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q99=-2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q100=-2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q101=2k2Γ0μX1-k2Γ0μm1*(1-X12)+Fs1{k2β*2m1*(1+X12)-m1*(k2c2+Ω2)+2ikΩc},q102=2k2Γ0μX2-k2Γ0μm2*(1-X22)+Fs1{k2β*2m2*(1+X22)-m2*(k2c2+Ω2)+2ikΩc},q103=2k2Γ0μX3-k2Γ0μm3*(1-X32)+Fs1{k2β*2m3*(1+X32)-m3*(k2c2+Ω2)+2ikΩc},q104=2k2Γ0μX4-k2Γ0μm4*(1-X42)+Fs1{k2β*2m4*(1+X42)-m4*(k2c2+Ω2)+2ikΩc},q105=[-α*2k2(X12+1)+k2c2]ikX1,q106=[-α*2k2(X22+1)+k2c2]ikX2,q107=[-α*2k2(X32+1)+k2c2]ikX3,q108=[-α*2k2(X42+1)+k2c2]ikX4,q109=[α*2k2(X12+1)-k2c2]ikX1,q110=[α*2k2(X22+1)-k2c2]ikX2,q111=[α*2k2(X32+1)-k2c2]ikX3,q112=[α*2k2(X42+1)-k2c2]ikX4.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 Figures2 and 3.Effects ofH0,Ω,g,τ1, and p on Rayleigh wave velocity and attenuation coefficient with respect to the initial stress.Effects ofH0,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.Effects ofH0,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.Effects ofΩ,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 ofΩ,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 ofΩ,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:(6.1)ρ=2kgm-3,λ=9.3×1010Nm-1,μ=8.4×1010Nm-1,T0=293.1k,K=50Wm-1k-1,s=6.4×102JKg-1,αt=13.2×10-6deg⁡-1.


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

Figure2 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.Figure3 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

Figure4  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

Figure5 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

Figure6 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

Figure7 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.


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

Figure2 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.Figure3 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

Figure4  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

Figure5 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

Figure6 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

Figure7 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 ofΩ, 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 ofg, 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 ofK, 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.



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*Source: 763429-2011-06-02.xml*