Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 2, pp. 385-394, Warsaw 2014

REFLECTION OF PLANE WAVES FROM A FREE SURFACE OF

A GENERALIZED MAGNETO-THERMOELASTIC SOLID HALF-SPACE

WITH DIFFUSION

Baljeet Singh

Post Graduate Government College, Department of Mathematics, Chandigarh, India

e-mail: bsinghgc11@gmail.com

Lakhbir Singh, Sunita Deswal

Guru Jambheshwar University of Science and Technology, Department of Mathematics, Hisar, India

Green-Naghdi’s theory of generalized thermoelasticity is applied to study the reflection of P
and SVwaves from the free surface of a magneto-thermoelastic solid half-space. The boun-
dary conditions are satisfied by appropriate potential functions to obtain a system of four
non-homogeneous equations in reflection coefficients.The reflection coefficients dependupon
the angle of incidence of P and SVwaves,magnetic field, thermal field, diffusion parameters
and other material constants. The numerical values of the modulus of the reflection coeffi-
cients are shown graphically with the angle of incidence of P and SV waves. The effect of
magnetic field is observed significantly on various reflected waves.

Keywords: generalized thermoelasticity, plane waves, reflection, magnetic field, diffusion

1. Introduction

Lord and Shulman (1967) developed a theory of generalized thermoelasticity by including a flux
rate term into the Fourier law of heat conduction, which avoids the unrealistic phenomenon
of the infinite speed of heat propagation in classical model given by Biot (1956). They obta-
ined a hyperbolic heat transport equation which ensures the finite speed of thermal signals.
Green and Lindsay (1972) formulated another theory of generalized thermoelasticity known as
temperature rate dependent thermoelasticity with two relaxation times, which obeys classical
Fourier’s law of heat conduction and also admits a finite speed of heat propagation. Ignaczak
and Ostoja-Starzewski (2009) presented a unified treatment of both Lord-Shulman and Green-
Lindsay theories. Apart from these theories of generalized thermoelasticity, Green and Naghdi
(1991, 1993) formulated a theory of generalized thermoelasticity by including “thermal displa-
cement gradient” among independent constitutive variables, known as the theory of thermoela-
sticity without energy dissipation. Chandrasekharaiah (1986) considered this wave like thermal
phenomenon as ‘second sound’. In a review article, Hetnarski and Ignaczak (1999) presented
these theories of generalized thermoelasticity.
Wave propagation in a generalized thermoelastic media with additional parameters like dif-

fusion, magnetic field, anisotropy, porosity, viscosity, microstructure, temperature and other
parameters provide vital information about the existence of new or modified waves. This in-
formation is useful for experimental seismologists in correcting earthquake estimation. Some
relevant studies on wave propagation in generalized thermoelasticity are studied by various au-
thors. Notable among them are Sinha and Sinha (1974), Sinha and Elsibai (1996), Sinha and
Elsibai (1997), Abd-Alla and Al-dawy (2000), Sharma et al. (2003), Singh (2010), Singh and
Yadav (2012) and Singh (2013).
Thermo-diffusion in an elastic solid is due to the fields of temperature, mass diffusion and

that of strain. There is an exchange of heat and mass in the environment during the process



386 B. Singh et al.

of thermo-diffusion in an elastic solid. The thermo-diffusion phenomenon in solids is used to
describe the process of thermo-mechanical treatment of metals. This phenomenon is of great
concern due to its many geophysical and industrial applications. For example, oil companies
are interested in the process of thermo-diffusion for more efficient extraction of oil from oil
deposits. The thermo-diffusion phenomenon also finds its application in the field associatedwith
the advent of semiconductor devices and the advancement of microelectronics (Oriani, 1969).

Nowacki (1974, 1976) developed the coupled theory of thermoelastic diffusion and studied
some dynamical problems of diffusion in solids. Following Lord and Shulman (1967), Sherief et
al. (2004) developed a theory of generalized thermoelastic diffusion, which allows finite speeds
of propagation of waves. Singh (2005, 2006) studied the wave propagation in a thermoelastic
solid with diffusion in context of Lord-Shulman and Green-Lindsay theories, and showed the
existence of three Coupled Longitudinal waves and a SV wave in a two-dimensional model.
Various other problems in elastic solidswith thermo-diffusionwere studied bymany researchers,
see Abo-Dahab and Singh (2009), Aoudai (2006, 2007, 2008), Choudhary and Deswal (2008),
Deswal and Choudhary (2009), Othman et al. (2009), Singh (2013).

In the present paper, theGreen andNaghdi theory of thermoelasticity without dissipation is
followed to study the reflection from a stress-free surface of a magneto-thermoelastic solid half-
space with diffusion. In Section 2, the basic equations for an isotropic, homogeneous generalized
thermoelasticmediumare formulated in thepresenceofdiffusionandmagneticfield. InSection3,
the basic equations are solved for plane wave solutions in the xz-plane to show the existence of
three P waves and a SVwave. In Section 4, the reflection phenomenon of incident P and SV is
considered.Theappropriate boundaryconditions are satisfiedbyappropriate potential functions
to obtain the reflection coefficients of various reflectedwaves. A particular example of themodel
is chosen in Section 5 to compute the numerical values of the reflection coefficients against the
angle of incidence for different values of themagnetic parameter. The effect of magnetic field on
various reflected waves is depicted graphically.

2. Basic equations

Following Green and Naghdi (1993) and Sherief et al. (2004), the governing equations for an
isotropic, homogenous generalized magneto-thermoelastic solid with diffusion at constant tem-
perature T0 in the absence of body forces are:

(i) The constitutive equations

σij =2µeij +δij(λekk−β1Θ−β2C) ρT0S = ρCEΘ+β1T0ekk+aT0C

P =−β2ekk+ bC −aΘ
(2.1)

(ii) Maxwell’s stress equations

Γij = µe[Hihj +Hjhi−H ·hδij] (2.2)

Assuming that linearized Maxwell’s equations are governing the electromagnetic field and the
medium is a perfect electric conductor in the absence of displacement current, then

curlh= j curlE=−µe
∂h

∂t
divh=0 divE=0 (2.3)

where H is a constant primarymagnetic field acting in the direction y, and

h= curl(u×H0) H=H0+h H0 = [0,H,0] (2.4)



Reflection of plane waves from a free surface... 387

(iii) The equation of motion

σij,j +Γji,j = ρüi (2.5)

(iv) The equation of heat conduction

K∗Θ,ii= ρCEΘ̈+β1T0ëkk+aT0C̈ (2.6)

(v) The equation of mass diffusion

Dβ2ekk,ii+DaΘ,ii+Ċ −DbC,ii=0 (2.7)

where ρ is density of themedium, λ, µ are Lame’s constants, T is absolute temperature, T0 is
temperature of the medium in its natural state, Θ = T −T0 is the change in temperature such
that |Θ/T0| ≪ 1, σij are components of the stress tensor, eij are components of the strain
tensor, ui are components of the displacement vector, S is entropy per unitmass, P is chemical
potential per unit mass, C is mass concentration, CE is specific heat at constant strain, D is
thermo-diffusion constant, which ensures that the equation satisfied by the concentration C
will also predict a finite speed of propagation of matter from one medium to another, a is a
constant tomeasure the thermo-diffusion effects, b is a constant tomeasure the diffusive effects,
δij is the Kronecker delta, β1 = (3λ+2µ)αt, β2 = (3λ+2µ)αc and K

∗ = CE(λ+2µ)/4 are
material constants, αt is a coefficient of linear thermal expansion and αc is a coefficient of linear
diffusion expansion, H0 is the primary constant magnetic field, h is the perturbed magnetic
field over H0.

3. Plane wave solution in the xz-plane

With the help of equations (2.1) to (2.7), the governing field equations for a homogeneous,
isotropic generalized magneto-thermoelastic solid with diffusion in the xz-plane are written as

(λ+2µ)u1,11+(λ+µ)u3,13+µu1,33−β1Θ,1−β2C,1= ρü1

(λ+2µ)u3,33+(λ+µ)u1,13+µu3,11−β1Θ,3−β2C,3= ρü3

K∗∇2Θ = ρCEΘ̈+β1T0(ë11+ ë33)+aT0C̈

Dβ2∇
2e+Da∇2Θ−Db∇2C + Ċ =0

(3.1)

where ∇2 =(∂2/∂x2)+(∂2/∂z2).
Helmholtz’s representations of the displacement components u1 and u3 in terms of scalar

potential functions ϕ and ψ are

u1 =
∂ϕ

∂x
−
∂ψ

∂z
u3 =

∂ϕ

∂z
+
∂ψ

∂x
(3.2)

Using equation (3.2) in equations (3.1), we obtain

µ∇2ψ = ρψ̈ (3.3)

and

(λ+2µ+µeH
2
0)∇

2ϕ−β1Θ−β2C = ρϕ̈

K∗∇2Θ = ρCEΘ̈+β1T0
∂

∂t
∇2ϕ+aT0C̈

Dβ2∇
4ϕ+Da∇2Θ−Db∇2C + Ċ =0

(3.4)



388 B. Singh et al.

Equation (3.3) is uncoupled and equations (3.4) are coupled in potential functions ϕ, Θ and C.
From equations (3.4), it is noticed that thePwave is affected by thermal, diffusion andmagnetic
fields, and the SVwave remains unaffected. The solution to equation (3.3) suggests propagation
of the SVwave with velocity

√

µ/ρ.
The solutions to equations (3.4) are now sought in the form of the harmonic traveling wave

[ϕ,Θ,C] = [ϕ0,Θ0,C0]exp[ik(xsinθ+zcosθ−vt)] (3.5)

where (sinθ,zcosθ) is the projection of the wave normal to the xz-plane, ϕ0, Θ0, C0 are
constants, k is the wave number and v is the phase speed.
By substituting equation (3.5) into equations (3.4), we obtain a homogenous system of equ-

ations in ϕ0, Θ0 and C0 which has a non-trivial solution if ξ satisfies the following cubic
equation

ξ3+Lξ2+Mξ+N =0 (3.6)

Here

ξ = ρv2 L =−[ε+εε2ε
2
1+d1+d2+(λ+2µ+µeH

2
0)]

M =(λ+2µ+µeH
2
0)(d1+d2+εε2ε

2
1)+d1d2+d2ε−2εε1ε2−ε2

N =−d1d2(λ+2µ+µeH
2
0)+ε2d1

where

d1 =
K∗

CE
d2 =

ρDb

τ
ε =

β21T0
ρCE

ε1 =−
a

β1β2
ε2 =

ρDβ22
τ

τ =
i

kv

Cubic equation (3.6) can be solved with the help of Cardano’s method. The three roots of this
equation correspond to three plane longitudinal waves if v2 is real and positive. Following Singh
(2005, 2006), the three real roots v1, v2 and v3, correspond to P1, P2 and P3 waves, where P1
and P2 waves are observed the fastest and slowest, respectively.

4. Reflection

For the incidence of P1 and SVwaves, the boundary conditions at the free surface are satisfied if
the incidentP1 or SVwave gives rise to the reflected SVand three reflected coupled longitudinal
waves (P1, P2 and P3). The complete geometry showing the incident and reflected waves from
the free surface of a magneto-thermoelastic solid half-space with diffusion is shown in Fig. 1.
The appropriate displacement potential functions ϕ and ψ, temperature Θ and concentra-

tion C for the incident and reflected waves are

ϕ = A0exp[ik1(xsinθ0+zcosθ0)− iωt]+
3
∑

i=1

Aiexp[iki(xsinθi−zcosθi)− iωt]

Θ = ς1A0exp[ik1(xsinθ0+zcosθ0)− iωt]+
3
∑

i=1

ςiAiexp[iki(xsinθi−zcosθi)− iωt]

C = η1A0exp[ik1(xsinθ0+zcosθ0)− iωt]+
3
∑

i=1

ηiAiexp[iki(xsinθi−zcosθi)− iωt]

ψ = B0exp[ik4(xsinθ0+zcosθ0)− iωt]+B1exp[ik4(xsinθ4−zcosθ4)− iωt]

(4.1)



Reflection of plane waves from a free surface... 389

Fig. 1. Geometry of the problem showing incident and reflected waves

where the incident P1 or SVwave makes the angle θ0 with the negative direction of the z-axis
and reflected P1, P2, P3, and SV waves makes the angles θ1, θ2, θ3 and θ4, respectively, and
for i =1,2,3

ςi =
k2i
β1
Gi(λ+2µ+µeH

2
0 −ρv

2
i ) ηi =

k2i
β1
Fi(λ+2µ+µeH

2
0 −ρv

2
i ) (4.2)

where

Gi =
ερv2i (ε1ε2−d2+ρv

2
i )

d1ε2+ρv
2
i [ε(d2−ρv

2
i )−ε2−2εε1ε2]

Fi =
ε2[ρv

2
i (εε1+1)−d1]

d1ε2+ρv
2
i [ε(d2−ρv

2
i )−ε2−2εε1ε2]

(4.3)

The required boundary conditions at the free surface z = 0 are the vanishing normal stresses,
tangential stresses, heat flux andmass flux, i.e.

σzz +Γzz =0 σzx+Γzx =0
∂Θ

∂z
=0

∂P

∂z
=0 on z =0

(4.4)

The ratios of amplitudes of the reflected waves to the amplitude of the incident P1wave, name-
ly, A1/A0, A2/A0, A3/A0, B1/A0 give the reflection coefficients for reflected P1, reflected P2,
reflected P3, and reflected SV waves, respectively. Similarly, for the incident SV wave A1/B0,
A2/B0, A3/B0, B1/B0, are the reflection coefficients of P1, P2, P3 and SVwaves, respectively.
The wave numbers k1, k2, k3, k4 are connected by the angles of incidence and reflection as

k1 sinθ1 = k2 sinθ2 = k3 sinθ3 = k4 sinθ4 (4.5)

In order to satisfy the boundary conditions, relation (4.5) is also written as

sinθ1
v1
=
sinθ2
v2
=
sinθ3
v3
=
sinθ4
v4

(4.6)

where v4 =
√

µ/ρ is the velocity of the SVwave and vi (i =1,2,3) are the velocities of P1, P2
and P3 waves.
With the help of equations (2.1), (3.2) and (4.1), boundary condition (4.4) leads to a non-

-homogeneous system of four equations as

4
∑

j=1

aijZj = bi i =1,2,3,4 (4.7)



390 B. Singh et al.

where:
— for the incident P1-wave θ0 = θ1 and j =1,2,3

a1j =−(2µD1j +λ+µeH
2
0 +GjEj +FjEj)

(v1
vj

)2

a14 =2µsinθ0

√

1− sin2θ0
(v4
v1

)2v1
v4

a2j =2sinθ0
√

D1j
v1
vj

a24 =
[

1−2sin2θ0
(v4
v1

)2](v1
v4

)2

a3j =
√

D1j
Gi
β1
Ej
(v1
vj

)3

a34 =0

a4j =
√

D1j
[

β2−a
Gj
β1
Ej + b

Fj
β2
Ej
(v1
vj

)3]

a44 =0

Z1 =
A1
A0

Z2 =
A2
A0

Z3 =
A3
A0

Z4 =
B1
A0

b1 =−a11 b2 = a21 b3 = a31 b4 = a41

where

D1j =1− sin
2θ0
(vj
v1

)2

Ej = λ+2µ+µeH
2
0 −ρv

2
j

— for incident SVwave θ0 = θ4 and j =1,2,3

a1j =−[2µD4j +λ+µeH
2
0 +GjEj +FjEj]

(v4
vj

)2

a14 = µsin2θ0

a2j =2sinθ0
√

D4j
v4
vj

a24 =1−2sin
2θ0

a3j =
√

D4j
Gj
β1
Ej
(v1
vj

)3

a34 =0

a4j =
√

D4j
[

β2−a
Gj
β1
Ej + b

Fj
β2
Ej
(v4
vj

)3]

a44 =0

Z1 =
A1
B0

Z2 =
A2
B0

Z3 =
A3
B0

Z4 =
B1
B0

b1 = a14 b2 =−a24 b3 = a34 b4 =−a44

where

D4j =1− sin
2θ0
(vj
v4

)2

5. Application to a particular model

To study the numerical dependence of the reflection coefficients on various magnetic, thermal,
diffusion, and other material constants, a particular example is chosen with the following phy-
sical constants at T0 = 300K: λ = 5.775 ·10

10N/m2, µ = 2.646 ·1010N/m2, ρ = 2700kg/m3,
CE = 1.415 · 10

4J/(kgK), K = 3.223 · 103W/(mK), τ = 0.04s, αt = 0.137 · 10
2/K,

αc =0.06·10
−3m3/kg,a =0.137·10−3m2/(s2K), b =0.05·10−7m5/(kgs2),D =0.5·103kgs/m3,

ω =20Hz.
Non-homogeneous system (4.7) of four simultaneous equations is solved by a Fortran pro-

gramwith the Gauss elimination method. Here, we concentrate only on observing the effects of
magnetic field on the reflection coefficients, as the diffusion and relaxation effects were already
shown by Singh (2005, 2006) in his papers on L-S and G-L theories.
The reflection coefficients of various reflectedwaves are computed for the range 0 < θ0 < 90

◦

of the angle of incidence of P1 and SV waves when H = 0, 80 and 800A/m2. The variations



Reflection of plane waves from a free surface... 391

of these reflection coefficients against the angle of incidence are shown graphically in Figs. 2
and 3. For the incidence of P1 wave, the variations of reflection coefficients of various reflected
waves against the angle of incidence are shown graphically in Fig. 2. The reflection coefficient
of the SV wave is zero near the normal and grazing incidence. As the angle of incidence varies
from normal to grazing incidence, it first increases to its maximum value and then decreases
sharply as shown by the solid curve. If we compare the solid curve with other dotted curves, it
is observed that the reflection coefficients of the SVwave fall sharply with the increase in value
of the magnetic field at each angle of incidence except for the normal and grazing incidence.
The reflection coefficient of the P1 wave first decreases slowly from its maximum value at the
normal incidence and its starts increasing at angles near to the grazing incidence as shown by
the solid curve. The comparison of the solid curve with other dotted curves shows the effect
of magnetic field on the reflection coefficient of the P1 wave. Similarly, if we see the graphs of
P2 and P3 waves, it is observed that the reflection coefficients of these waves fall sharply with
an increase in the magnetic field at each angle of incidence except for the grazing incidence.
The effect of magnetic field is maximum at the normal incidence, however, it decreases as the
angle of incidence varies from the normal to grazing incidence, and then there is no effect of the
magnetic field at the grazing incidence.

Fig. 2. Variations of reflection coefficients of reflectedwaves against the angle of incidence of the P1wave

For the incidence of the SVwave, the variations of reflection coefficients of various reflected
waves against the angle of incidence are shown graphically in Fig. 3. If we look at the four
plots of reflection coefficients of the reflected SV, P1, P2 and P3 waves in Fig. 3, the effects of
magnetic field are clearly observed at each angle of incidence, except for the normal incidence,
grazing incidence and at the angle 45◦. The angle 45◦ of incidence of the SV wave is observed
as the critical angle. The variations over the angle 45◦ will not appear if we compute the real
parts of the coefficients only.



392 B. Singh et al.

Fig. 3. Variations of reflection coefficients of reflected waves with the angle of incidence of the SVwave

6. Conclusions

The governing equations of a thermoelastic half-space with diffusion and magnetic field are
formulated in context of G-N theory and are solved in a two-dimensional model. Similar to
Singh (2005, 2006), there also exists three coupled longitudinal waves and a shear wave in the
magneto-thermoelastic half-space with diffusion under G-N theory. These waves are considered
for reflection from a thermally insulated half-space to obtain a non-homogeneous system of four
equations in reflection coefficients. Thenumerical example shows that the presence of amagnetic
field significantly changes the reflection coefficients of reflectedwaves for the incidence of bothP
and SVwaves.

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Manuscript received June 2, 2013; accepted for print October 22, 2013