REFLECTION AND TRANSMISSION OF PLANE WAVES AT AN … · REFLECTION AND TRANSMISSION OF PLANE WAVES 379 The origin of the cartesian coordinate system (x1;x2;x3) is taken at any point

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 20, Number 3, Fall 2012

REFLECTION AND TRANSMISSION OF PLANE

WAVES AT AN INTERFACE BETWEEN

ELASTIC AND MICROPOLAR

THERMOELASTIC DIFFUSION MEDIA

RAJNEESH KUMAR AND VIJAY CHAWLA

ABSTRACT. The present investigation is concerned withthe study of reflection and transmission phenomenon due tolongitudinal and transverse waves at a plane interface betweenisotropic elastic solid half-space and micopolar thermoelasticdiffusion half space. It is found that reflections and transmis-sion coefficients are functions of angle of incidence, frequencyof the incident wave and are influenced by the elastic propertiesof the media. Closed form expressions of amplitudes ratio andenergy ratios are derived. Numerical computations have beenperformed for a particular model and the obtained results aredepicted graphically against the angle of incident. It is verifiedthat during transmission there is no dissipation of energy atthe interface. From the present investigation, a special case ofinterest is also deduced to depict the effect of micropolarity.

1 Introduction The classical theory deals with the basic assump-tion that the effect of the microstructural of a material is not essential fordescribing mechanical behavior. In reality, almost all materials possesmicrostructure and in such materials, microstrctural motions (instrin-sic rotations of grain) cannot be ignored at high frequency. Eringen[9] developed the linear theory of micropolar elasticity which takes intoaccount the microstructural motions. The difference between classicaltheory of elasticity and that of micropolar elasticity is the introduction ofan additional kinematic variable corresponding to microrotation. Tomarand Gogna [29] investigated the problem of reflection and refraction oflongitudinal wave at an interface between two micropolar elastic solidsin welded contact. Tomar and Kumar [30] discussed wave propagationat micropolar elastic solid interface. The Generalized theory of ther-

Keywords: Micropolar thermoelastic diffusion, amplitudes, reflection, transmis-sion, energy ratio.

Copyright c©Applied Mathematics Institute, University of Alberta.

375

376 R. KUMAR AND V. CHAWLA

moelasticity is one of the modified version of classical uncoupled andcoupled theory of thermoelasticity and have been developed in order toremove the paradox of physical impossible phenomena of infinite velocityof thermal signals in the classical coupled thermoelasticity.

Lord and Shulman [17] formulated a generalized theory of thermoe-lasticity with one thermal relaxation time, who obtained a wave-typeequation by postulating a new law of heat conduction instead of clas-sical Fouriers law. Green and Lindsay [10] developed a temperaturerate-dependent thermoelasticity that includes two thermal relaxationtimes and does not violate the classical Fouriers law of heat conduction,when the body under consideration has a center of symmetry. One canrefer to Hetnarski and Ignaczak [11] for a review and presentation ofgeneralized theories of thermoelasticity.

The linear theory of micropolar elasticity has been extanded to in-clude thermal effects by Eringen [7, 8] and Nowacki [18, 19, 20]. Boschiand Ieasan [5] extanded a generalized theory of micropolar thermoe-lasticity. When thermal effects are considered, Sinha and Elsibai [27]discussed the reflection and refraction of thermoelastic waves at an in-terface of two semi-infinite media with two relaxation times. Sinha andSinha [28] investigated the problem of reflection of thermoelastic wavesat a solid half space with relaxation times. Kumar and Sarthi [12] dis-cussed the reflection and refraction of thermoelastic plane waves at aninterface of two thermoelastic media without energy dissipation.

Diffusion is defined as the spontaneous movement of the particlesfrom a high concentration region to the low concentration region and itoccurs in response to a concentration gradient expressed as the changein the concentration due to change in position. Thermal diffusion uti-lizes the transfer of heat across a thin liquid or gas to accomplish isotopeseparation. Today, thermal diffusion remains a practical process to sep-arate isotopes of noble gases (e.g., xexon) and other light isotopes (e.g.,carbon) for research purposes.

Nowacki [21–24] developed the theory of thermoelastic diffusion byusing coupled thermoelastic model. Sherief et al. [25] proved the unique-ness and reciprocity theorems for the equations of generalized thermoe-lastic diffusion, in isotropic media. Sherief and Saleh [26] investigatedthe problem of a thermoelastic half-space in the context of the theory ofgeneralized thermoelastic diffusion with one relaxation time. Auodi [2]proved the uniqueness and reciprocity theorems for the generalized prob-lem in anisotropic media, under the restriction that the elastic, thermalconductivity and diffusion tensors are positive definite. Recently, Auodi[13] derived the uniqueness and reciprocity theorems for the general-

REFLECTION AND TRANSMISSION OF PLANE WAVES 377

ized micropolar thermoelastic diffusion problem in anisotropic media.Kumar and Kansal [3] derived the basic equations for generalized ther-moelastic diffusion and discussed the Lamb waves. Kumar and Chawla[14] discussed the plane wave propagation in anisotropic three-phase-lag model. Kumar and Chawla [15] derived the Green’s functions fortwo-dimensional problem in orthotropic thermoelastic diffusion media.However, the important reflection and transmission at an interface be-tween an elastic solid medium and micropolar thermoelastic diffusionsolid medium has not been discussed so far and also verified the law ofconservation of energy at the interface.

Keeping in view of the above applications, we have investigated theproblem of reflection and transmission at a plane interface betweenan elastic solid medium and a micropolar thermoelastic diffusion solidmedium. The graphical representation is given for these energy ratiosfor different direction of propagation. The law of conservation of energyat the interface is verified.

2 Basic equations Following Aouadi [3] and Kumar et al. [16],the basic equations for homogeneous isotropic generalized micropolarthermoelastic diffusion, in the absence of body forces, body couple, heatand mass diffusion sources are

(λ+ µ)∇(∇ · u) + (µ+K)∇2u +K∇× ϕ − β1

(

1 + τ1∂

∂t

)

∇T(1)

− β2

(

1 + τ1 ∂

∂t

)

∇C = ρ∂2u

∂t2,

(α+ β + γ)∇(∇ · ϕ) − γ∇×∇× ϕ +K∇× u− 2Kϕ = ρj∂2

ϕ

∂t2,(2)

ρCE

(

∂

∂t+ τ0

∂2

∂t2

)

T + β1T0

(

∂

∂t+ Ωτ0

∂2

∂t2

)

∇ · u(3)

+ aT0

(

∂

∂t+ ε

∂2

∂t2

)

C = K∗∇2T,

Dβ2∇2u +Da

(

1 + τ1∂

∂t

)

∇2T +

(

∂

∂t+ Ωτ0 ∂

2

∂t2

)

C(4)

−Db

(

1 + τ1 ∂

∂t

)

∇2C = 0,

tij = λur,rδij + µ(ui,j + uj,i) +K(uj,i − εijrφr)(5)


− β1

(

T + τ1∂T

∂t

)

δij − β2

(

C + τ1 ∂C

∂t

)

δij ,

mij = αφr,r + βφi,j + γφj,i,(6)

where β1 = (3λ + 2µ + K)αt, β2 = (3λ + 2µ + K)αc, αt and αc are,respectively, the coefficients of linear thermal expansion and diffusion ex-pansion. λ, µ, α, β, γ and K are material constants, ρ is the density, CEis the specific heat at constant strain, j is the microinertia density, K∗

is the coefficient of thermal conductivity, u = (u1, u2, u3) is the displace-ment vector and ϕ = (φ1, φ2, φ3) is the microrotation vector, T = Θ−T0

is the small temperature increment, Θ is the absolue temperature of themedium, T0 is the reference temperature of the body chosen such that|T/T0| 1, C is the concentration of the diffusive material in the elas-tic body, tij are the components of stress tensor, K∗ is the coefficient ofthermal conductivity, D is the thermoelastic diffusion constant, ∇ and∇2 are, respectively, the gradient and Laplacian operators, δij is Kro-necker delta, τ0 and τ1 are thermal relaxation times with τ1 ≥ τ0 ≥ 0,and τ0 and τ1 are diffusion relaxation times with τ 1 ≥ τ0 ≥ 0. Here,τ1 = τ1 = 0, Ω = 1, ε = τ0 for Lord- Shulman (L-S) model and Ω = 0,ε = τ0 for Green-Lindsay model.

In the above equations, the symbol (“ ′ ”) followed by a suffix denotesdifferentiation with respect to spatial coordinate.

The equations of motion in homogeneous isotropic elastic solid mediumare

(7) (λe + µe)∇(∇ · ue) + µ∇2ue = ρe∂2ue

∂t2,

where λe and µe are Lame’s constants, ue = (ue1, ue2, u

e3) and ρe are,

respectively, the displacement vector and density corresponding to theisotropic elastic solid.

The stress-strain relation in the isotropic elastic medium is given by

(8) σeij = λeeek,kδij + 2µeeeij ,

where σeij and eeij = 12 (uei,j +uej,i) are, respectively, components of stress

and strain tensor, and eek,k is the dilatation.

3 Formulation of the problem We consider an isotropic elasticsolid half-space (medium I) lying over a homogenous isotropic, gener-alized micropolar thermoelastic diffusion solid half-space (medium II).


The origin of the cartesian coordinate system (x1, x2, x3) is taken at anypoint on the plane surface and the x3-axis points vertically downwardsinto the micropolar thermoelastic diffusion solid half-space. The elas-tic solid half-space occupies the region x3 ≤ 0 and the region x3 ≥ 0is occupied by the dissipative micropolar thermoelastic diffusion solidhalf-space as shown in Figure 1.

FIGURE 1: Geometry of the problem.

We consider plane waves in the x1x3-plane with wave front parallelto the x2-axis. For the two-dimensional problem we assume the dis-placement vector ue in medium I and the displacement vector u andmicrorotation vector ϕ in medium II are, respectively, of the form

(9) ue = (ue1, 0, ue3), u = (u1, 0, u3), ϕ = (0, φ2, 0).

We define the following nondimensional quantities

x′i =ω∗xiν1

, u′i =ρω∗ν1β1T0

ui, φ′2 =ρν2

1

β1T0φ2, T ′ =

T

T0,


C ′ =β2C

β1T0, t′ = ω∗t, τ ′1 = ω∗τ1, τ ′0 = ω∗τ0,

τ0′ = ω∗τ0, τ1′ = ω∗τ1, tij =1

β1T0tij , m′

ij =ω∗

ν1β1T0mij ,

P ∗ij′ =

ρν1β2

1T0P ∗ij , P ∗e′ =

ρν1β2

1T0P ∗e, uei

′ =ρω∗ν1β1T0

uei , teij′ =

1

β1T0teij ,

where

(10) ω∗ =ρCEν

21

K∗, ν2

1 =λ+ 2µ+K

ρ.

Making use of equation (10) in equations (1)–(4), with the aid of (9)and after suppressing the primes, we obtain

δ1(u1,11 + u1,33) + δ2(u1,11 + u3,13)(11)

− δ3φ2,3 − τ1t T,1 − τ1

cC,1 = u1,

δ1(u3,11 + u3,33) + δ2(u1,31 + u3,33)(12)

− δ3φ2,1 − τ1t T,3 − τ1

cC,3 = u3,

∇2φ2 + a1(u1,3 − u3,1) − 2a1φ2 = a2φ2,(13)

ζ1τ0tt(u1,1 + u3,3) + τ0

t T + ζ2τ0c C = ∇2T,(14)

q∗1(u1,111 + u3,311 + u1,133 + u3,333) + q∗2τ1t (T,11 + T,33)(15)

− q∗3τ1c (C,11 + C,33) + τ0

ccC = 0,

where

(δ1, δ2, δ3) =1

ρν21

(µ+K,λ+ µ,K),

(a1, a2) =ν21

γ

(

K

ω∗2 , ρ j

)

,

(a3, a4) =1

µ+K(ρν2

1 ,K),

(τ1t , τ

1c , τ

0t , τ

0c , τ

0tt, τ

0cc) =

(

1 + τ1∂

∂t, 1 + τ1 ∂

∂t, 1 + τ0

∂

∂t,

1 + ε∂

∂t, 1 + Ωτ0

∂

∂t, 1 + Ωτ0 ∂

∂t

)

,


(q∗1 , q∗2 , q

∗3) =

Dω∗

ν21

(

β22

ρν21

,β2a

β1, b

)

,

(ζ1, ζ2) =β1T0

K∗ω∗

(

β1

ρ,aν2

1

β2

)

.

We introduce the potential functions φ and ψ through the relations

(16) u1 =∂φ

∂x1−∂ψ

∂x3, u3 =

∂φ

∂x3+∂ψ

∂x1.

Substituting equation (16) in equations (11)–(15), we obtain

(

∇2 −∂2

∂t2

)

φ− τ1t T − τ1

cC = 0,(17)

(

∇2 − a3∂2

∂t2

)

ψ − a4φ2 = 0,(18)

(

∇2 − 2a1 − a2∂2

∂t2

)

φ2 + a1∇2ψ = 0,(19)

(

∇2 − τ0t

∂

∂t

)

T − ζ2τ0c C − ζ1τ

0tt∇

2φ = 0,(20)

q∗1∇4φ+ q∗2τ

1t ∇

2T − q∗3τ1c∇

2C + τ0ccC = 0,(21)

where

∇2 =∂2

∂x21

+∂2

∂x23

.

For harmonic motion, we assume

(22) φ, ψ, φ2, T, C(x1, x3, t) = φ, ψ, φ2, T , Ce−iωt,

where ω is the angular frequency of the vibrations of material particles.Substituting equation (22) in equations (17)–(21), we obtain

(∇2 + ω2)φ− τ11t T − τ11

c C = 0,(23)

(∇2 + a3ω2)ψ − a4φ2 = 0,(24)

(∇2 − 2a1 + a2ω2)φ2 + a1∇

2ψ = 0,(25)

−ζ1τ0tt∇

2φ+ (∇2 − τ10t )T − ζ2τ

10c C = 0,(26)


q∗1∇4φ+ q∗2τ

11t ∇2T − [q∗3τ

11c ∇2 − τ10

cc ]C = 0,(27)

where

(τ11t , τ11

c , τ10t , τ10

c , τ10tt , τ

10cc ) = (1−iωτ1, 1−iωτ

1, 1−iωτ0,−iω(1−iωε),

− iω(1− iωτ0),−iω(1− iωΩτ0)).

Equations (26) and (27) of this system are solved into two relations,given by

[q∗3τ11c ζ1τ

10tt + q∗1τ

10c ζ2∇

4 − ζ1τ10tt τ

10cc∇

2]φ(28)

= [q∗3τ11c ∇4 − q∗3τ

11c τ10

t + τ10cc + q∗2τ

11t τ10

c ζ2∇2 + τ10

t τ10cc ]T ,

[q∗1∇6 + [q∗2τ

11t ζ1τ

10tt − q∗1τ

10t ]∇4φ(29)

= [q∗3τ11c ∇4 − q∗3τ

11c τ10

t + τ10cc + q∗2τ

11t τ10

c ζ2∇2 + τ10

t τ10cc ]C.

Making use of equations (28) and (29) in equation (23), we have

(30) bA1∇6 +A2∇

4 +A3∇2 +A4cφ = 0,

where

A1 = (q∗1 − q∗3)]τ11c ,

A2 = τ10cc + (q∗1 + q∗2)τ10

c τ11t ζ2 + (q∗2 + q∗3)τ10

tt τ11t τ11

c ζ1

A3 = (ω2 − τ10t − ζ1τ

10tt τ

11t )τ10

cc + q∗2ζ2τ11t τ10

c + ω2q∗3τ11c τ10

t ,

A4 = −ω2τ10cc τ

10tt .

Substituting the value of φ2 from equation (25) into equation (24), weobtain

(31) b∇4 +H1∇2 +H2cψ = 0,

where

H1 = a1a4 − 2a1 + (a2 + a3)ω2, H2 = (a2ω

2 − 2a1)ω2.

The general solution of equation (30) can be written as

(32) φ =3

∑

i=1

φi,


where the potentials φi, i = 1, 2, 3 are solutions of wave equations, givenby

(33)

[

∇2 +ω2

V 2i

]

φi = 0, i = 1, 2, 3.

Here, V 2i (i = 1, 2, 3) are the velocities of three longitudinal waves, that

is, longitudinal displacement wave, MD (mass diffusion) and T (thermal)waves and derived from the roots of cubic equation in V 2, given by

(34) A4V6 −A3ω

2V 4 +A2ω4V 2 −A4ω

6 = 0.

The general solution of equation (31) can be written as

(35) ψ =

2∑

i=1

ψi,

where the potentials ψi, i = 4, 5 are solutions of wave equations, givenby

(36)

[

∇2 +ω2

V 2i

]

ψi = 0, i = 4, 5.

Here, V 2i (i = 4, 5) are the velocities of two transverse displacement

waves coupled with transverse microrotation (CD I, CD II) and arederived from the roots of the equation in V 2, given by

(37) H2V4 −H1ω

2V 2 + ω4 = 0.

Making use of equation (32) in the equations (28), (29) with the aidof equations (22) and (33), the general solutions for φ, T and C areobtained as

(38) φ, T, C =

3∑

i=1

1, r1i, r2iφi,

where

r1i =(q∗3τ

11c ζ1τ

10tt + q∗1τ

10c ζ2)ω

4 + ζ1τ10tt τ

10cc ω

2V 2i

p1i,


r2i =−q∗1ω

6 + (q∗2τ11t ζ1τ

10tt − q∗1τ

10t )ω4V 2

i

V 2i pij

,

and

pij = q∗3τ11c ω4+(q∗3τ

11c τ10

t +τ10cc +q∗2τ

11t τ10

c ζ2)ω2V 2i +τ10

t τ10cc V

4i , i = 1, 2, 3.

Substituting equation (35) in equations (25)–(26) with the aid of equa-tions (22) and (36), the general solutions for ψ and φ2 are obtainedas

ψ, φ2 =

5∑

i=4

1, p1iψi,(39)

p1i =a1ω

2

(a2 − 2a1)V 2i − ω2

, i = 4, 5.(40)

Applying the dimensionless quantities defined by (10) in equation (7)and with the aid of equation (9) after suppressing the primes, we obtain

ηe2 − χe2

ν21

[ue1,11 + ue3,13] +χe2

ν21

[ue1,11 + ue1,33] =∂ue1∂t2

,(41)

ηe2 − χe2

ν21

[ue1,13 + ue3,33] +χe2

ν21

[ue3,11 + ue3,33] =∂ue3∂t2

,(42)

where ηe =√

λe+2µe

ρe and χe =√

µe

ρe are, respectively, the velocities of

longitudinal and transverse waves of the medium I.The components ue1 and ue3 are related by the potential functions as

(43) u1 =∂φe

∂x1−∂ψe

∂x3, u3 =

∂φe

∂x3+∂ψe

∂x1,

where φe and ψe are solutions of wave equations

(44) ∇2φe =φ

η′2, ∇2ψe =

ψe

χ′2,

where

η′ =ηe

ν1, χ′ =

χe

ν1.


4 Reflection and transmission We consider a plane wave (P orSV) propagating through the isotropic elastic solid half-space and is in-cident at the interface x3 = 0, as shown in Figure 1. Corresponding tothis incident wave, two homogeneous waves (P and SV) are reflected inisotropic elastic solid half space and five inhomogeneous waves (Longi-tudinal displacement waves, MD, T, CD I and CD II) are transmittedin isotropic micropolar thermoelastic diffusion solid half-space.

The potential functions satisfying equation (44) can be written as

φe = Ae0eiω(x1 sin θ0+x3 cos θ0)/η

′−t(45)

+Ae1eiω(x1 sin θ1+x3 cos θ1)/η

′−t,

ψe = Be0eiω(x1 sin θ0+x3 cos θ0)/χ

′−t(46)

+Be1eiω(x1 sin θ1+x3 cos θ1)/χ

′−t.

The coefficients Ae0, Be0 , A

e1 and Be1 represent the amplitude of the

incident P (or SV), reflected P and reflected SV waves, respectively.Following Borcherdt [4], in an isotropic micropolar thermoelastic dif-

fusion half-space, the potential functions satisfying equations (33) and(36) can be written as

φ, T, C =

3∑

i=1

1, r1i, r2iBie(−→A i,

−→r )ei(−→P i,

−→r −ωt),(47)

ψ, φ2 =

5∑

i=4

1, p1i,Bie(−→A i,

−→r )ei(−→P i,

−→r −ωt).(48)

The coefficients Bi, i = 1, 2, 3, 4, 5, represent the amplitudes of transmit-

ted waves. The propagation vector−→P i, i = 1, 2, 3, 4, 5 and attenuation

factor−→A i, i = 1, 2, 3, 4, 5 are given by

(49)−→P i = ξRx1 + dViRx3, Ai = −ξ1x1 − dViI x3, i = 1, 2, 3, 4, 5,

where

(50) dVi = dViR + idViI = p.v.

√

ω2

V 2i

− ξ2, i = 1, 2, 3, 4, 5.

and ξ = ξR + iξI is the complex wave number. The subscripts R andI denote the real and imaginary parts of the corresponding complex


number and p.v. stands for the principal value of the complex quantityobtained from square root. ξR ≥ 0 ensures propagation in positivex1-direction. The complex wave number ξ in the isotropic micropolarthermoelastic diffusion medium is given by

(51) ξ = |−→P i| sin θ

′i − i|

−→A i| sin(θ′i − γi), i = 1, 2, 3, 4, 5,

where γi, i = 1, 2, 3, 4, 5 is the angle between the propagation and atten-uation vector and, θ′i, i = 1, 2, 3, 4, 5, is the angle of refraction in mediumII.

5 Boundary conditions The boundary conditions to be satisfiedat the interface x3 = 0 as follows:(I) continuity of Stress components

te33 = t33,(52a)

te31 = t31;(52b)

(II) continuity of displacement components

ue3 = u3,(52c)

ue1 = u1;(52d)

(III) vanishing of the tangential couple stress components

m32 = 0;(52e)

(IV) thermally insulated boundary

∂T

∂x3= 0;(52f)

(V) impermeable boundary

∂C

∂x3= 0.(52g)


Making use of equations (45)–(48) in equations (52a)–(52g), we find thatthe boundary conditions are satisfied if and only if

(53) ξR =ω sin θ0V0

=ω sin θ1η′

=ω sin θ2χ′

,

and

(54) ξ1 = 0,

where

(55) V0 =

η′, for incident P -wave

χ′, for incident SV -wave.

It means that waves are attenuating only in x3-direction. From equa-tion (51), it implies that if |Ai| 6= 0, then γi = θ′i, i = 1, 2, 3, 4, 5, thatis, attenuated vectors for the five transmitted waves are directed alongthe x3-axis.

Substituting equations (45)–(48) in equations (52a)–(52g) and withthe aid of the equations (5), (6), (8), (9), (10), (16), (43), (53) and (54),we get a system of seventh nonhomogeneous equations which can bewritten as

(56)

7∑

j=1

fijZj = dj ,

where Zj = |Zj |eiψ∗

j , |Zj | and ψ∗j , j = 1, 2, 3, 4, 5, are respectively,

the ratios of amplitudes and phase shift of reflected P-, reflected SV-transmitted longitudinal displacement wave, MD, T, CDI and CDIIwaves to that of incident wave.

f11 = 2µe(

ξRω

)2

− ρeν21 , f12 = 2µe

ξRω

dVβω,

f16 = (2µ+K)ξRω

dVβω, f17 = (2µ+K)

ξRω

dVβω,

f21 = 2µeξRω

dVη′

ω, f22 = µe

[(

dVχ′

ω

)2

−

(

ξRω

)2]

,

f26 =

[

µ

(

ξRω

)2

− (µ+K)

(

dV4

ω

)2

−Kp14

]

,


f27 =

[

µ

(

ξRω

)2

− (µ+K)

(

dV5

ω

)2

−Kp15

]

,

f31 =ξRω, f32 =

dVχ′

ω, f36 =

dV4

ω, f37 =

dV5

ω,

f41 = −dVη′

ω, f42 =

ξRω, f46 = −

ξRω, f47 = −

ξRω,

f51 = 0, f52 = 0, f56 = 0, f57 = 0, f61 = 0, f62 = 0,

f66 = 0, f67 = 0, f71 = 0, f72 = 0, f76 = 0, f77 = 0,

f2j = 2µξRω

dVjω, f3j = −

ξRω, f4j = −

dVjω,

f5j = r1jdVjω, f6j = r2j

dVjω, f6j = p1j

dVjω,

dVη′

ω=

√

1

η′2−

(

ξRω

)2

=

√

1

η′2−

sin2 θ0V 2

0

,

dVχ′

ω=

√

1

χ′2−

sin2 θ0V 2

0

,

and

dVjω

= p.v.

√

1

V 2j

−sin2 θ0V 2

0

, j = 1, 2, 3, 4, 5.

Here, p.v. is evaluated with restriction dVjI ≥ 0 to satisfy decay condi-tion in the micropolar thermoelastic diffusion medium. The coefficientsdi, i = 1, 2, . . . , 7 on the right side of equation (56) are given by

(I) for incident P-wave

(57) di = (−1)ifi1 for i = 1, 2, 3, 4 and di = 0 for i = 5, 6, 7;

(II) for incident SV-wave

(58) di = (−1)i+1fi2 for i = 1, 2, 3, 4 and di = 0 for i = 5, 6, 7.

We consider a surface element of unit area at the interface betweentwo media. The reason for this consideration is to calculate the partitionof energy of the incident wave among the reflected and refracted waves onthe both sides of surface. Following Achenbach [1], the energy flux across


the surface element, that is, rate at which the energy is communicatedper unit area of the surface is represented as

(59) P ∗ = tqrqruq,

where tqr are the components of stress tensor, qr are the direction cosinesof the unit normal q outward to the surface element and uq are thecomponents of the particle velocity.

The time average of P ∗ over a period, denoted by 〈P ∗〉, represents theaverage energy transmission per unit surface area per unit time. Thus,on the surface with normal along the x3-direction, the average energyintensities of the waves in the elastic solid are given by

(60) 〈P ∗e〉 = Re 〈t〉e31Re (ue1) + Re 〈t〉e33 Re (ue3).

Following Achenbach [1], for any two complex functions f and g, wehave

(61) 〈Re (f) · Re (g)〉 =1

2Re (f · g).

The expressions for the energy ratios Ei (i = 1, 2) for the reflected Pand reflected SV are given by

(62) Ei = −〈P ∗ie〉

〈P ∗0e〉, i = 1, 2,

where

〈P ∗1e〉 =

ω4ρeν21

η′|Z1|

2Re (cos θ1),

〈P ∗2e〉 =

ω4ρeν21

χ′|Z2|

2Re (cos θ2),

and(I) for incident P-wave

(63) 〈P ∗0e〉 = −

ω4ρeν21

η′cos θ0,

(II) for incident SV-wave

(64) 〈P ∗0e〉 = −

ω4ρeν21

χ′cos θ0,


are the average energy intensities of the reflected P-, reflected SV-, inci-dent P- and incident SV-waves, respectively. In equation (62), negativesign is taken because the direction of reflected waves is opposite to thatof incident waves.

For micropolar thermoelastic diffusion solid half-space, the averageenergy intensities of the waves on the surface with normal along x3-direction, are given by

(65) 〈P ∗ij〉 = Re 〈t〉

(i)31 Re (u

(j)1 ) + Re 〈t〉

(i)33 Re (u

(j)3 ) + Re 〈m〉

(i)32 Re (φ

(j)2 ).

The expressions for the energy ratios Eij , i, j = 1, 2, 3, 4, 5 for the trans-mitted waves are given by

(66) Eij =〈P ∗ij〉

〈P ∗0e〉, i, j = 1, 2, . . . , 5,

where

〈P ∗ij〉 = −ω4Re

[

(2µ+K)dViω

ξRω

ξRω

+

λ

(

ξRω

)2

+ ρν21

(

dViω

)2

+ρν2

1 (r1iτ11t + r2iτ

11c )

ω2

dV jω

Zi+2Zj+2

]

,

〈P ∗i4〉 = −ω4Re

[

− (2µ+K)dViω

ξRω

ξRω

+

λ

(

ξRω

)2

+ ρν21

(

dViω

)2

+ρν2

1 (r1iτ11t + r2iτ

11c )

ω2

ξRω

Zi+2Z6

]

,

〈P ∗4j〉 = −ω4Re

[[

µ

(

ξRω

)2

− (µ+K)21

(

dV4

ω

)2

+Kp14

ω2

]

ξRω

+ (2µ+K)ξRω

dV4

ω

dV jω

]

,

〈P ∗5j〉 = −ω4Re

[(

µ

(

ξRω

)2

− (µ+K)

(

dVjω

)2)ξRω

+ (2µ+K)ξRω

dV4

ω

dV jω

Z7Zj+2, i, j = 1, 2, 3,

〈P ∗4j〉 = −ω4Re

[(

(µ+K)

(

dV4

ω

)2

− µ

(

ξRω

)2

−Kp14

ω2

)

dV jω


+ (2µ+K)ξRω

dV4

ω

dξRω

+γω∗2

ρν41ω

2

dV4

ωp14p1j

Z6Zj+2

]

,

〈P ∗5j〉 = −ω4Re

[(

(µ+K)

(

dV5

ω

)2

− µ

(

ξRω

)2

−Kp15

ω2

)

dV jω

+ (2µ+K)ξRω

dV5

ω

dξRω

+γω∗2

ρν41ω

2

dV5

ωp15p1j

Z6Zj+2

]

,

= 4, 5.

The diagonal entries of energy matrix Eij in equation (66) representthe energy ratios of the longitudinal displacement wave, MD, T, CD Iand CD II waves, whereas sum of the nondiagonal entries of Eij givesthe share of interaction energy among all the transmitted waves in themedium and is given by

(67) ERR =

5∑

i=1

( 5∑

j=1

Eij −Eii

)

.

The energy ratios Ei, i = 1, 2, diagonal entries and sum of nondiagonalentries of the energy matrix Eij , that is, E11, E22, E33 E44, E55 andERR yield the conservation of the incident energy across the interface,through the relation

E1 +E2 +E11 +E22 +E33 +E44 +E55 +ERR = 1.

Special case: In the absence of micropolarity effect, that is, if we takeK = 0 in equations (56) and (62), we obtain corresponding expressionsamplitude and energy ratios of reflected P-, reflected SV-, refracted P-,refracted T-, refracted C-, and refracted SV-waves to that of incidentwave. In these expressions, the velocities are derived from equation (34)and is the velocity of the transverse wave and coupling constant givenby equation (38).

6 Numerical results and discussion With the view of illustrat-ing theoretical results obtained in the preceding sections, we now presentsome numerical results; the values of the following relevant parametersare given as

λ = 7.76× 1010 Kgm−1s−2, µ = 3.86× 1010 Kgm−1s−2,


T0 = 0.293× 103 K, CE = 0.3831× 103 JKg−1K−1,

αt = 1.78× 10−5 K−1, αC = 1.98× 10−4 Kg−1m3,

a = 1.2 × 104 m2s−2K−1, b = 9 × 105 m5s−2Kg−1,

D = 0.85× 10−8 m−3sKg, ρ = 8.954× 103 Kgm−3,

K∗ = 0.383× 103 Wm−1K−1, K = 10 dyn/cm2,

j = 0.2 cm2, γ = 7.79 dyn,

and the values of relaxation times are

τ0 = 0.4 s, τ1 = 0.9 s, τ0 = 0.5 s, τ1 = 0.8 s.

Following Bullen [6], the numerical data of granite for elastic medium isgiven by

ρe = 2.65× 103 Kgm−3, ηe = 5.27× 103 ms−1, χe = 3.17× 103 ms−1.

The Matlab software 7.04 has been used to determine the values of en-ergy ratios Ei, i = 1, 2 and energy matrix Eij , i, j = 1, . . . , 5 defined inthe previous section for different values of incident angle (θ0) rangingfrom 0 to 90 for fixed frequency ω = 2 × π × 100 Hz. Correspond-ing to incident P, the variation of energy ratios with respect to angleof incident have been plotted in Figures (2)–(9). Similarly, correspond-ing to SV waves, the variation of energy ratios with respect to angle ofincident have been plotted in Figures (10)–(17). In all figures, micropo-lar thermoelastic diffusion is represented by the word MTED and TDEcorrespond to the thermoelastic diffusion.

Incident P-wave: Figure 2 exhibits the variation of energy ratio E1

with the angle of incidence (θ0). It shows that the values of E1 forboth cases TED and MTED decrease with the increase in θ0 from 0

to 80 and then increase as θ0 increases further. Figure 3 depicts thevariation of energy ratio E2 with θ0 and it indicates that the values ofE2 increase for smaller values of θ0, whereas for higher values of θ0, itincrease for both cases TED and MTED. Figure 4 depicts the variationof energy ratio E11 with θ0 and it shows that the values of E11 for thecase of MTED increase slightly for smaller values of θ0 whereas for thehigher values of θ0 the values of E11 decreases, but for the case TEDthe values of E11 decrease for all values of θ0. Figure 5 exhibits thevariation of energy ratio E22 with θ0. The similar type of behavior and


variation is noticed for E22 as E11 with difference in their magnitudevalues. Figure 6 depicts the variation of energy ratio E33 with θ0 andit indicates the values of E33 for the case of TED show an oscillatingbehavior in the initial stage, but after that it decreases, whereas for thecase of MTED, the values of E33 slightly increase for smaller values ofθ0 although for higher values of θ0 the values of E33 decrease. Figure 7depicts the variation of energy ratioE44 with θ0. It shows that the valuesof E44 for both cases TED and MTED increase with the increase θ0 from0 to 75 and then decrease as θ0 increase further. Figure 8 exhibits thevariation of energy ratio E55 with θ0 and it indicates the values E55 ofslightly increase at the initial stage, but after that it decrease. Figure 9depicts the variation of interaction energy ratio ERR with θ0 and itindicates the values of ERR for the case of MTED decrease, althoughfor the case of TED, the values of ERR decreases for smaller values ofθ0, whereas for higher values of θ0, the values of ERR increase. If wecompare TED and MTED in all figures, we find that the values of E1,E33 and E44 are higher in TED theory in comparison to MTED, but thevalues of E2, E11, E22 and ERR are higher in MTED theory (for highervalues of θ0) in comparison to TED theory.

FIGURE 2: Variation of energy ratio E1 with respect to the angle of

incident P -wave.



incident P -wave.


incident P -wave.



incident P -wave.


incident P -wave.



incident P -wave.


incident P -wave.


FIGURE 9: Variation of energy ratio ERR with respect to the angle of

incident P -wave.

Incident SV-wave: Figures 10–17 depict the variation of energyratios with the angle of incidence (θ0) for SV waves.

Figure 10 represents the variation of energy ratio E1 with θ0 and itindicates that the values of E1 for both cases TED and MTED increasefor smaller values of θ0, whereas for higher values of θ0, the values of E1

decrease. Figure 11 shows the variation of energy ratio E2 with θ0. Thevalues of E2 for both cases TED and MTED decrease with the increaseof θ0 from 0 to 30 and then increase as θ0 increases further. Figure 12shows that the values of E11 for both cases TED and MTED show anoscillatory behavior for initial values of θ0, whereas for higher values ofθ0, the values of E11 decrease. Figure 13 exhibits the variation of energyratio E22 with θ0, and it is noticed that the behavior and variation of E22

is similar to E11 with a difference in their magnitude values. Figure 14exhibits that the values of E33 for the case of MTED oscillate, but forthe case of TED, the values of E33 increase slightly. Figure 15 showsthe variation of energy ratio E44 with θ0, and it found that the behaviorand variation of E44 is similar to E33 with difference in their magnitudevalues. Figure 16 indicates that the values of E55 increase at the initialstage but after that, it decrease. Figure 17 represents the variation ofERR with θ0, and it shows that the values of ERR for both cases TEDand MTED initially decrease, but for higher values of θ0, the values ofERR decrease for the case of MTED, but for the case of TED, the valuesslightly increase. If we compare TED and MTED in all figures, we find


that the values of E1, E2, E33 and E44 are higher in TED theory incomparison to MTED whereas the values of ERR are higher in MTEDtheory (for higher values of θ0) compared to TED theory.


incident SV -wave.


incident SV -wave.



incident SV -wave.


incident SV -wave.



incident SV -wave.


incident SV -wave.



incident SV -wave.

FIGURE 17: Variation of energy ratio ERR with respect to the angle of

incident SV -wave.


7 Conclusion The phenomena of reflection and refraction of ob-liquely incident elastic waves at the interface between an elastic solidhalf space and a micropolar thermoelastic diffusion solid half space hasbeen investigated. The expressions of amplitudes ratio and energy ratiosare presented in compact form. The variations of energy ratios againstthe angle of incident are computed numerically and depicted graphicallyfor a specific model. The law of conservation is verified at the interface.From the present investigation, a special case of interest is also deducedto depict the effect of micropolarity. Appreciable relaxation time andmicropolarity effect are noticed on energy ratios.

From the numerical results, we conclude that the values of E1, E33

and E44 are higher in TED theory in comparison to MTED whereas thevalues of E2, E11, E22 and ERR are higher in MTED theory (for highervalues of θ0) in comparison to TED theory in the case of incident P-wave, although for the case of incident SV-wave we find that the valuesof E1, E2, E33 and E44 are higher in TED theory in comparison toMTED whereas the values ERR are higher in MTED theory (for highervalues of θ0) compared to TED theory. The sum of energy ratios of thereflected waves, transmitted waves and interface between transmittedwaves for both incident P- and SV-waves is verified to be unity, whichshows the law of conservation of energy at the interface.

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Department of Mathematics, Kurukshetra University,

Kurukshetra-136119, Haryana, India

E-mail address: rajneesh [email protected]

E-mail address: vijay1 [email protected]

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