Volume 2011, Article ID 763429,47pages doi:10.1155/2011/763429
Research Article
Rayleigh Waves in Generalized
Magneto-Thermo-Viscoelastic Granular Medium under the Influence of Rotation, Gravity Field, and Initial Stress
A. M. Abd-Alla,
1S. M. Abo-Dahab,
1, 2and F. S. Bayones
31Mathematics Department, Faculty of Science, Taif University, Taif 21974, Saudi Arabia
2Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
3Mathematics Department, Faculty of Science, Umm Al-Qura University, P.O. Box 10109, Makkah 13401, Saudi Arabia
Correspondence should be addressed to S. M. Abo-Dahab,sdahb@yahoo.com Received 4 December 2010; Revised 14 January 2011; Accepted 25 February 2011 Academic Editor: Ezzat G. Bakhoum
Copyrightq2011 A. M. Abd-Alla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 Oshima1. The dynamical problem of a generalized thermoelastic granular infinite cylinder under initial stress has been illustrated by El-Naggar2. 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.4and El-Naggar et al.5.
Rayleigh waves in a thermoelastic granular medium under initial stress on the prop- agation of waves in granular medium are discussed by Ahmed6. 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 Mahmoud8. 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.13investigated 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, thez-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 stressPalong thex-direction and the primary magnetic field−−→
H0acting ony-axis, as well as the gravity field and incremental thermal stresses, as shown inFigure 1.
The state of deformation in the granular medium is described by the displacement vector
−
→Uu, o, wof the center of gravity of a grain and the rotation vector→−
ξξ, η, ζof 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
z=K O
z Ω Granular
layer
y P P
−−→ x H0
z=0
Granular half space
g
Figure 1: Depiction of the problem.
is the centripetal acceleration due to time varying motion only, and 2→− Ω×→−•
u is the Coriolis acceleration, andΩ 0,Ω,0.
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 currentSI see the work of Mukhopadhyay14:
−
→J curl→− h,
−μe∂→− h
∂t curl→− E, div→−
h0, div→−
E0,
−
→E−μe
∂→−u
∂t ×−→
H
,
2.1
where
−
→hcurl→−u×−−→
H0
, −→
H−−→
H0→− h, −−→
H0 0, H0,0, 2.2
where→−
h is the perturbed magnetic field over the primary magnetic field vector,→− E is the electric intensity,→−
J is the electric current density,μeis the magnetic permeability,−−→
H0 is the constant primary magnetic field vector, and→−uis the displacement vector.
The stress and stress couple may be taken to be nonsymmetric, that is, τij/τji, Mij/Mji. The stress tensorτijcan be expressed as the sum of symmetric and antisymmetric tensors
τijσijσij, 2.3
where
σij 1 2
τijτji
, σij 1 2
τij−τji
. 2.4
The symmetric tensorσijσjiis related to the symmetric strain tensor
eijeji 1 2
∂ui
∂xj ∂uj
∂xi
. 2.5
The antisymmetric stressσij are given by
σ23 −F∂ξ
∂t, σ31 −F∂η
∂t, σ12 −F∂ζ
∂t, σ11 σ22σ33 0, 2.6 whereF is the coefficient of friction between the individual grains. The stress coupleMij is given by
MijMνij, 2.7
where,Mis the third elastic constant,M11, M13, M33, and so forth, are the components of the resultant acting on a surface.
The non-symmetric strain tensorνijis defined as
ν11 ∂ξ
∂x, ν31 ∂ξ
∂z, ν33 ∂ζ
∂z, ν21 ν22ν230, ν12 ∂
∂x
ω2η
, ν32 ∂
∂z
ω2η
, ν13 ∂ζ
∂x,
2.8
whereω2 1/2∂u/∂z−∂w/∂x.
The dynamic equation of motion, if the magnetic field and rotation are added, can be written as15
τji,jFiρ ••
ui →− Ω×→−
Ω× −→u
i
2→− Ω×→−u•
i
, i, j1,2,3. 2.9
The heat conduction equation is given by16
K∇2T ρs∂
∂t
1τ2∂
∂t
Tγ T0∂
∂t
1τ2δ∂
∂t
∇ · −→u, 2.10
whereρis density of the material,Kis 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 andμare Lame’s elastic constants,θis the absolute temperature,γαt3λ2μ,
T0 is reference temperature solid, T is temperature differenceθ−T0,τ0 is the mechanical relaxation time due to the viscosity, andτm 1τ0∂/∂t.
The components of stress in generalized thermoelastic medium are given by
σ11 τm
λ2μ p∂u
∂x τmλP∂w
∂z −γ
1τ1
∂
∂t
T, σ33τmλ∂u
∂xτm
λ2μ∂w
∂z −γ
1τ1∂
∂t
T, σ13τmμ
∂u
∂z ∂w
∂x
.
2.11
If we neglect the thermal relaxation time, then2.11tends to Nowacki17and Biot18.
The Maxwell’s electro-magnetic stress tensorτijis given by τijμe
HihjHjhi−Hk·hkδij
, i, j1,2,3, 2.12
which takes the form
τ11−μeH02∇2φ, τ13τ230, τ33 μeH02∇2φ, ∇2φ ∂u
∂x∂w
∂z. 2.13 The dynamic equations of motion are
∂τ11
∂x ∂τ31
∂z P 2
∂ω2
∂z −ρg∂w
∂x Fxρ ∂2u
∂t2 2Ω∂w
∂t −Ω2u
,
∂τ12
∂x ∂τ32
∂z Fy 0,
∂τ13
∂x ∂τ33
∂z P 2
∂ω2
∂x ρg∂w
∂x Fzρ ∂2w
∂t2 −2Ω∂u
∂t −Ω2w
,
2.14
wheregis the Earth’s gravity and F
−μeH02∇2φ,0, μeH02∇2φ
, 2.15
τ23−τ32∂M11
∂x ∂M31
∂z 0, τ31−τ13∂M12
∂x ∂M32
∂z 0, τ12−τ21∂M13
∂x ∂M33
∂z 0.
2.16
From2.3–2.8and2.11, we have
τ11 τm
λ2μ p∂u
∂x τmλP∂w
∂z −γ
1τ1
∂
∂t
T,
τ33 τmλ∂u
∂xτm
λ2μ∂w
∂z −γ
1τ1 ∂
∂t
T,
τ13 τmμ ∂u
∂z ∂w
∂x
F∂η
∂t,
τ12 −F∂ζ
∂t, τ23 −F∂ξ
∂t, M11 M∂ξ
∂x, M31M∂ξ
∂z, M33 M∂ζ
∂z, M21M22 M23 0, M12 M ∂
∂x
ω2η
, M32 M ∂
∂z
ω2η
, M13M∂ζ
∂x.
2.17
Substituting2.17into2.14and2.16tends to
τm λ2μ
P∂2u
∂x2 τmλP ∂2w
∂x ∂z−γ
1τ1 ∂
∂t ∂T
∂xτmμ ∂2u
∂z2 ∂2w
∂x ∂z
P 2
∂2u
∂z2 − ∂2w
∂x ∂z
−ρg∂w
∂x F ∂2η
∂z ∂tμeH02 ∂2u
∂x2 ∂2w
∂x ∂z
ρ ∂2u
∂t22Ω∂w
∂t −Ω2u
, 2.18
then
τm
λ2μ
PμeH02∂2u
∂x2
τm λμ
P
2 μeH02 ∂2w
∂x ∂z
τmμP 2
∂2u
∂z2
−γ
1τ1∂
∂t ∂T
∂x −ρg∂w
∂x F ∂2η
∂z ∂t ρ ∂2u
∂t2 2Ω∂w
∂t −Ω2u
.
2.19
Also,
∂
∂t ∂ζ
∂x− ∂ξ
∂z
0, 2.20
τmμ ∂2u
∂x ∂z∂2w
∂x2
−F ∂2η
∂x ∂tτmλ ∂2u
∂x ∂zτm
λ2μ∂2w
∂z2 −γ
1τ1 ∂
∂t ∂T
∂z P
2 ∂2u
∂x ∂z− ∂2w
∂x2
ρg∂u
∂xμeH02 ∂2u
∂x ∂z∂2w
∂z2
ρ ∂2w
∂t2 −2Ω∂u
∂t −Ω2w
, 2.21
then
τm λμ
P
2 μeH02 ∂2u
∂x ∂z
τmμ−P 2
∂2w
∂x2 τm
λ2μ
μeH02∂2w
∂z2
−γ
1τ1∂
∂t ∂T
∂z ρg∂u
∂x −F ∂2η
∂x ∂tρ ∂2w
∂t2 −2Ω∂u
∂t −Ω2w
,
2.22
and, from2.16, we have
∇2ξ−s2∂ξ
∂t 0, 2.23
∇2 ω2η
−s2
∂η
∂t 0, 2.24
∇2ζ−s2∂ζ
∂t 0, 2.25
where
s2 2F
M. 2.26
3. Solution of the Problem
By Helmholtz’s theorem19, the displacement vector→−u can be written in the displacement potentialsφandψform, as
−
→u gradφcurl→−ψ , →−ψ 0, ψ,0
, 3.1
which reduces to
u ∂φ
∂x−∂ψ
∂z, w ∂φ
∂z ∂ψ
∂x. 3.2
Substituting3.2into2.19,2.22, and2.24, the wave equations tend to
α2∇2φ− γ ρ
1τ1∂
∂t
T−g∂ψ
∂x ∂2φ
∂t2 2Ω∂ψ
∂t −Ω2φ, 3.3 β2∇2ψ−s1∂η
∂t g∂φ
∂x ∂2ψ
∂t2 −2Ω∂φ
∂t −Ω2ψ, 3.4
∇2η−s2∂η
∂t − ∇4ψ0, 3.5
where
s1 F
ρ, α2 τm
λ2μ
PμeH02
ρ , β2 2τmμ−P
2ρ . 3.6
Substituting3.2into2.10, we obtain
K∇2T ρs∂
∂t
1τ2∂
∂t
Tγ T0 ∂
∂t
1τ2δ∂
∂t
∇2φ. 3.7
From3.3and3.7, by eliminatingT, we obtain
∇2− 1 χ
∂
∂t
1τ2 ∂
∂t
α2∇2φ−g∂ψ
∂x− ∂2φ
∂t2 −2Ω∂ψ
∂t Ω2φ
−ε∂
∂t
1τ1 ∂
∂t
1τ2δ∂
∂t
∇2φ0,
3.8
where
χ K
ρs, ε γ2T0
ρK. 3.9
From3.4and3.5by eliminatingη, we obtain
∇2−s2∂
∂t
β2∇2ψ−∂2ψ
∂t2 g∂φ
∂x2Ω∂φ
∂t Ω2ψ
−s1∇4∂ψ
∂t 0. 3.10
For a plane harmonic wave propagation in thex-direction, we assume
φφ1eikx−ct, ψψ1eikx−ct, 3.11
ξ, η, ζ
ξ1, η1, ζ1
eikx−ct. 3.12
From3.12into2.20,2.23, and2.25, we get
Dξ1−ikζ10, 3.13
D2ξ1q2ξ10, 3.14
D2ζ1q2ζ10, 3.15
where
q2ikcs2−k2, D≡ d
dz. 3.16
The solution of3.14and3.15takes the form
ξ1 A1eiqzA2e−iqz, ζ1B1eiqzB2e−iqz, 3.17
whereA1, A2, B1, andB2are arbitrary constants.
From3.13and3.17, we obtain
q
A1eiqz−A2e−iqz
−k
B1eiqzB2e−iqz
0, 3.18
then
qA1−kB10, qA2−kB20
⇒Aj −1j−1k
q Bj, j 1,2. 3.19
Substituting3.11into3.8and3.10, we obtain
α2∗D4G1D2G2
φ1−
G3D2G4
ψ10,
R1D4R2D2R3
ψ1
R4D2R5
φ10,
3.20
where
Γ01−ikcτ0, Γ11−ikcτ1, Γ21−ikcτ2, Γ31−ikcτ2δ,
α2∗ Γ0
λ2μ
PμeH02
ρ , β2∗ 2Γ0μ−P 2ρ . G1k2
c2−2α2∗ ikc
χ
α2∗Γ2χεΓ1Γ3
Ω2,
G2k4
α2∗−c2
ikcΓ2
χ
k2 1−α2∗
Ω2
−k2
Ω2ikεcΓ1Γ3
,
G3 ik
g−2Ωc
, G4−
g−2Ωc
ik3k2cΓ2
χ
,
R1β2∗ikcs1, R2k2
c2−2β2∗ ikc
s2β2∗−2k2s1
Ω2,
R3k2
k2−ikcs2
β2∗−c2 ikc
s2Ω2k4s1
, R4ik
g−2Ωc
, R5
2Ωc−g
ik3−k2cs2
.
3.21 The solution of3.20takes the form
φ14
j1
CjeikNjzDje−ikNjz ,
ψ14
j1
EjeikNjzFje−ikNjz ,
3.22
where the constantsEjandFjare related to the constantsCjandDjin the form Ej mjCj, FjmjDj, j1,2,3,4,
mj 1
g−2Ωc
ikNj2−ik−
εΓ2/χ
×
α2∗k2Nj4−
k2
c2−2α2∗ ikc
χ
α2∗Γ2χεΓ1Γ3
Ω2
Nj2
ikcΓ2
χ
1−α2∗Ω2 k2
−
Ω2ikεcΓ1Γ3
.
3.23
Substituting3.22into3.11, we obtain
φ4
j1
CjeikNjzDje−ikNjz
eikx−ct,
ψ 4
j1
EjeikNjzFje−ikNjz
eikx−ct,
3.24
and values of displacement componentsuandware
uik 4 j1
1−Njmj
CjeikNjz
1Njmj
Dje−ikNjz
eikx−ct,
wik 4 j1
Njmj
CjeikNjz
mj−Nj
Dje−ikNjz
eikx−ct,
3.25
whereN1, N2, N3, andN4are taken to be the complex roots of the following equation
N8t1N6t2N4t3N2t40, 3.26
where
t1 k2 α2∗
c2−2α2∗ ikc
α2∗χ
α2∗Γ2χεΓ1Γ3
Ω2 1 β∗2ikcs1
× k2
c2−2β2∗ ikc
s2β2∗−2k2s1
Ω2 ,
3.27
t2 1 α2∗
k4
α2∗−c2
ikcΓ2
χ
k2 1−α2∗
Ω2
−k2
Ω2ikεcΓ1Γ3
1 α2∗
β2∗ikcs1
k2
c2−2β2∗ ikc
s2β2∗−2k2s1
Ω2
×
k2
c2−2α2∗ ikc
χ
α2∗Γ2χεΓ1Γ3
Ω2
1 β2∗ikcs1
k2
k2−ikcs2
β∗2−c2
ikc
s2Ω2k4s1
− 1
α2∗
β2∗ikcs1
k2
g−2Ωc2 ,
3.28
t3 1 α2∗
β2∗ikcs1
× k2
c2−2β2∗ ikc
s2β2∗−2k2s1
Ω2
×
k4
α2∗−c2
ikcΓ2
χ
k2 1−α2∗
Ω2
−k2
Ω2ikεcΓ1Γ3
k2
k2−ikcs2
β2∗−c2 ikc
s2Ω2k4s1
×
k2
c2−2α2∗ ikc
χ
α2∗Γ2χεΓ1Γ3
Ω2
−
ik
g−2Ωc2
ik3k2cΓ2
χ
−ik3
g−2Ωc2
ik−cs2 ,
3.29 t4 1
α2∗
β2∗ikcs
× k2
k2−ikcs2
β2∗−c2 ikc
s2Ω2k4s1
× ik
g−2Ωc
2Ωc−g2
ik3−k2cs2
ik3 k2cΓ2
χ
.
3.30
From3.4,3.11,3.12,3.22, and3.23, we obtain
η14
j1
1
ikcs1 k2β2∗mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
×
CjeikNjzDje−ikNjz . 3.31
Using3.22and3.11into3.3, we obtain
T ρ γΓ1
4 j1
−α2∗k2
1Nj2
k2c2−ikgmj
CjeikNjzDje−ikNjz
eikx−ct. 3.32
With the lower medium, we use the symbols with primes, forξ1, ζ1, η1, T, φ, ψ, andq, forz > K,
ξ1 −k
qB2e−iqz, ζ1 B2e−iqz, η1 4
j1
1
ikcs1 k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
Dje−ikNjz,
T ρ γΓ1
4 j1
−α∗2k2
1Nj2
k2c2−ikgmj
Dje−ikNjzeikx−ct,
φ4
j1
Dje−ikNjzeikx−ct,
ψ4
j1
Fje−ikNjzeikx−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 interfacezK, that is,
iuu, iiww, iiiξξ, ivηη,
vζζ, viM33M33, viiM31M31, viiiM32M32,
ixτ33τ33 τ33 τ33 xτ31τ31 τ31 τ31, xiτ32τ32 τ32 τ32, xiiT T,
xiii ∂T/∂z θT ∂T/∂z θT.
The boundary conditions on the free surfacez0 are xivM330,
xvM310, xviM320, xviiτ33τ33 0, xviiiτ31τ31 0, xixτ32τ32 0,
xx ∂T/∂z θT 0.
From conditionsiii,v,vi, andvii, we obtain
B1eiqK−B2e−iqK−B2e−iqK, B1eiqKB2e−iqKB2e−iqK, M
B1eiqK−B2e−iqK
−MB2e−iqK,
M
B1eiqKB2e−iqK
−MB2e−iqK.
4.1
Hence,
B1B2B20, ξζξζ0. 4.2
The other significant boundary conditions are responsible for the following relations:
i
4 j1
1−Njmj
CjeikNjK
1Njmj
Dje−ikNjK−
1Njmj
Dje−ikNjK0, 4.3
ii
4 j1
Njmj
CjeikNjK
mj−Nj
Dje−ikNjK−
mj−Nj
Dje−ikNjK0, 4.4
iv
4 j1
1 cs1
k2β2∗mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
×
CjeikNjKDje−ikNjK
−4
j1
1
cs1 k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
Dje−ikNjK0,
4.5
viii MNj
4 j1
k2mj
Nj21 1
ikcs1
k2β2∗mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
×
CjeikNjK−Dje−ikNjK MNj
4 j1
k2mj
Nj21 1
ikcs1
×
k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
Dje−ikNjK0,
4.6
ix
4 j1
Γ0λμeH02
1−Njmj
Γ0
λ2μ
μeH02
Nj2mjNj
×CjeikNjK
Γ0λμeH02
1Njmj
Γ0
λ2μ
μeH02
Nj2−mjNj
×Dje−ikNjKρ
−α2∗
1Nj2
c2−ig kmj
CjeikNjKDje−ikNjK
−
Γ0λμeH02
1Njmj
Γ0
λ2μ μeH02
Nj2−mjNj
Dje−ikNjK
−ρ
−α∗2
1Nj2
c2−ig kmj
Dje−ikNjK
0,
4.7
x
4 j1
−2k2Γ0μNj
CjeikNjK−Dje−ikNjK
−k2Γ0μmj
1−Nj2 F
s1 k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
×
CjeikNjKDje−ikNjK
−2k2Γ0μNjDje−ikNjK
−
−k2Γ0μmj
1−Nj2 F
s1 k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
×Dje−ikNjK 0,
4.8
xii
4 j1
ρ γ
−α2∗k2
Nj21
k2c2−igkmj
CjeikNjKDje−ikNjK
−ρ γ
−α∗2k2
Nj21
k2c2−igkmj
Dje−ikNjK0,
4.9
xiii
4 j1
ρ γ
−α2∗k2
Nj21
k2c2−igkmj
θikNj
CjeikNjK
θ−ikNj
Dje−ikNjK
− ρ γ
−α∗2k2
Nj21
k2c2−igkmj
×
θ−ikNj
Dje−ikNjK0,
4.10
xvi MNj
4 j1
k2mj
Nj21 1
ikcs1
k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
× Cj−Dj
0,
4.11
xvii
4 j1
Γ0λμeH02
1−Njmj
Γ0
λ2μ
μeH02
Nj2mjNj Cj
Γ0λμeH02
1Njmj
Γ0
λ2μ
μeH02
Nj2−mjNj
Dj
ρ
−α2∗ 1Nj2
c2−ig kmj
CjDj
0,
4.12
xviii
4 j1
−2k2Γ0μNj
Cj−Dj
−k2Γ0μmj
1−Nj2 F
s1 k2β∗2mj
1Nj2
−mj
k2c2 Ω2 ik
2Ωc−g
× CjDj
0,
4.13
