Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space

$Page 1: Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space$
Arch Appl Mech (2013) 83:1109–1128DOI 10.1007/s00419-013-0737-6

ORIGINAL

Rajneesh Kumar · Vandana Gupta

Reflection and transmission of plane waves at the interfaceof an elastic half-space and a fractional order thermoelastichalf-space

Received: 24 January 2012 / Accepted: 1 February 2013 / Published online: 27 February 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract The problem of reflection and transmission due to longitudinal and transverse waves incidentobliquely at a plane interface between uniform elastic solid half-space and fractional order thermoelasticsolid half-space has been studied. It is found that the amplitude ratios of various reflected and refracted wavesare functions of angle of incidence and frequency of incident wave and are influenced by the fractional orderthermoelastic properties of media. The expressions of amplitude ratios and energy ratios have been computednumerically for a particular model. The variation of amplitude and energy ratios with angle of incidence isshown graphically. The conservation of energy at the interface is verified.

Keywords Fractional order thermoelastic solid · Elastic waves · Reflection · Transmission ·Amplitude and energy ratios

1 Introduction

During recent years, several interesting models have been developed by using fractional calculus to study thephysical processes particularly in the area of heat conduction, diffusion, viscoelasticity, mechanics of solids,control theory, electricity, etc. It has been realized that the use of fractional order derivatives and integrals leadsto the formulation of certain physical problems which is more economical and useful than the classical approach.There exist many material and physical situations like amorphous media, colloids, glassy and porous materials,man-made and biological materials/polymers and transient loading, where the classical thermoelasticity basedon Fourier type heat conduction breaks down. In such cases, one needs to use a generalized thermoelasticitytheory based on an anomalous heat conduction model involving time-fractional (noninteger order) derivatives.

Povstenko [13] proposed a quasi-static uncoupled theory of thermoelasticity based on the heat conductionequation with a time-fractional derivative of order α. Because the heat conduction equation in case 1 ≤α ≤ 2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), this theory interpolatesa classical thermoelasticity and a thermoelasticity without energy dissipation. He also obtained the stressescorresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equationfor one-dimensional and two-dimensional cases.

Povstenko [14] investigated the nonlocal generalizations of the Fourier law and heat conduction by usingtime and space fractional derivatives. Youssef [22] proposed a new model of thermoelasticity theory in thecontext of a new consideration of heat conduction with fractional order and proved the uniqueness theorem.Jiang and Xu [10] obtained a fractional heat conduction equation with a time-fractional derivative in the general

R. Kumar (B) · V. GuptaDepartment of Mathematics, Kurukshetra University, Kurukshetra 136119, IndiaE-mail: [email protected]

V. GuptaE-mail: [email protected]

1110 R. Kumar, V. Gupta

orthogonal curvilinear coordinate and also in other orthogonal coordinate system. Povstenko [15] investigatedthe fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermalstresses.

Ezzat [8] constructed a new model of the magneto-thermoelasticity theory in the context of a new consid-eration of heat conduction with fractional derivative. Ezzat [9] studied the problem of state-space approachto thermoelectric fluid with fractional order heat transfer. The Laplace transform and state-space techniqueswere used to solve a one-dimensional application for a conducting half-space of thermoelectric elastic mater-ial. Povstenko [16] investigated the generalized Cattaneo-type equations with time-fractional derivatives andformulated the theory of thermal stresses. Biswas and Sen [2] proposed a scheme for optimal control and apseudo-state-space representation for a particular type of fractional dynamical equation.

Borejko [4] discussed the reflection and transmission coefficients for three-dimensional plane waves inelastic media. Wu and Lundberg [21] investigated the problem of reflection and transmission of the energy ofharmonic elastic waves in a bent bar. Sinha and Elsibai [19] discussed the reflection and refraction of ther-moelastic waves at an interface of two semi-infinite media with two relaxation times. Sharma and Gogna [17]discussed the problem of reflection and transmission of plane harmonic waves at an interface between elasticsolid and porous solid saturated by viscous liquid. Tomar and Arora [20]studied reflection and transmissionof elastic waves at an elastic/porous solid saturated by immiscible fluids. Kumar and Sarthi [11] discussed thereflection and transmission of thermoelastic plane waves at an interface of thermoelastic media without energydissipation.

In the present paper, the reflection and transmission phenomenon at a plane interface between an elasticsolid medium and a fractional thermoelastic solid medium has been analyzed. In fractional thermoelastic solidmedium, potential functions are introduced to represent two longitudinal waves and one transverse wave. Theamplitude ratios of various reflected and refracted waves to that of incident wave are derived. The amplituderatios are further used to find the expressions of energy ratios of various reflected and refracted waves tothat of incident wave. The graphical representation is given for these energy ratios for different direction ofpropagation and different fractional orders. Also, the variation of amplitude ratios with frequency is showngraphically. The law of conservation of energy at the interface is verified.

2 Governing equations

Following Youssef [22], the basic equations of fractional order theory of thermoelasticity for an isotropic andhomogeneous elastic medium in the absence of body forces and heat sources are considered as follows.

The constitutive equation (stress–strain and temperature relations) is given by

(λ+ μ)∇∇ · u + μ∇2u − γ∇T = ρ∂2u∂t2 , (1)

The heat conduction equations

K∇2T = ∂

∂t

(1 + (τ0)

α

α!∂α

∂tα

)(ρCE T + γ T0∇ · u) , (2)

Following Caputo [6], the fractional derivative of order α ∈ (0, 1] of the absolutely continuous function f(t) is

dα

dtαf (t) = I 1−α f ′(t),

and the fractional integral

I α f (t) =t∫

0

(t − τ)α−1

� (α)f (τ ) dτ, α > 0

where I α is the fractional integral of the function f (t) of order α defined by Miller and Ross [12].The strain displacement relations

ei j = 1

2

(ui, j + u j,i

), (3)

Reflection and transmission of plane waves 1111

where γ = (3λ+ 2μ)αt ; λ,μ are the Lame’s constants, αt is the coefficient of thermal linear expansion, uiare the components of displacement vector u, T = θ − T0 is small temperature increment, T0 is the reference

temperature of the body chosen such that∣∣∣ T

T0

∣∣∣ � 1, θ is the absolute temperature of the medium, ρ is the

density assumed to be independent of time, σi j , ei j are the components of the stress and strain, respectively,ekk is the dilatation, CE is the specific heat at constant strain, K is the coefficient of thermal conductivity, α isthe fractional order parameter, and τ0 is the relaxation time.

3 Formulation of the problem

We consider an isotropic elastic solid half-space (medium I) lying over a homogeneous isotropic, frac-tional order generalized thermoelastic half-space (medium II). The origin of the Cartesian coordinate system(x1, x2, x3) is taken at any point on the plane surface (interface), and x3-axis points vertically downward intothe fractional order thermoelastic half-space which is thus represented by x3 ≥ 0. We consider plane waves inthe x1 − x3 plane with wavefront parallel to the x2-axis. For two-dimensional problem, we take

u = (u1, 0, u3) . (4)

We define the following dimensionless quantities:

x ′i = C0ηxi , u′

i = C0ηui , i = 1, 2, 3

t ′ = C20ηt, T ′ = T

T0, σ ′

i j = σi j

ρC20

, σ ′ei j = σ e

i j

ρC20

(5)

P∗e′ = λ+ 2μ

C0P∗e, P∗′

i j = λ+ 2μ

C0P∗

i j , u′ei = C0ηue

i

where

C20 = λ+ 2μ

ρ, η = ρCE

K(6)

Equations (1) and (2) with the aid of (4), (5) and (6), after suppressing the primes, take the form:

(β2 − 1

) ∂

∂x1

(∂u1

∂x1+ ∂u3

∂x3

)+ ∇2u1 − b

∂

∂x1T = β2 ∂

2u1

∂t2 , (7)

(β2 − 1

) ∂

∂x3

(∂u1

∂x1+ ∂u3

∂x3

)+ ∇2u3 − b

∂

∂x3T = β2 ∂

2u3

∂t2 , (8)

∇2T =(∂

∂t+ (τ0)

α

α!∂α+1

∂tα+1

)(T + ∈

(∂u1

∂x1+ ∂u3

∂x3

)). (9)

where

∈= γ

ρCE, β2 = λ+ 2μ

μ, b = γ T0

μ. (10)

We introduce the potential functions φ and ψ through the relations

u1 = ∂φ

∂x1− ∂ψ

∂x3, u3 = ∂φ

∂x3+ ∂ψ

∂x1, (11)

where φ and ψ are the displacement potentials of longitudinal and transverse waves, respectively.Substituting Eq. (11) in the Eqs. (8) and (9), we obtain

β2(

∇2 − ∂2

∂t2

)φ − bT = 0, (12)

∇2ψ − β2 ∂2ψ

∂t2 = 0, (13)

∇2T =(∂

∂t+ (τ0)

α

α!∂α+1

∂tα+1

) (T + ∈ ∇2φ

), (14)


For the propagation of harmonic waves in x1 − x3 plane, we assume

{φ,ψ, T } (x1, x3, t) = {φ, ψ, T

}e−iωt , (15)

where ω is the angular frequency of vibrations of material particles.Substituting the value of φ,ψ, T from Eq. (15) into the Eqs. (12)–(14) after simplification, we obtain

[A∇4 + B∇2 + C

]φ = 0, (16)

where

A = β2,

B = β2(ω2 + iω − (τ0)

α

α! (−iω)α+1)

− b ∈(

−iω + (τ0)α

α! (−iω)α+1),

C = ω2(

iω − (τ0)α

α! (−iω)α+1).

The general solution of Eq. (16) can be written as

φ = φ1 + φ2 (17)

where the potentials φi , i = 1, 2 are solutions of wave equations, given by[∇2 + ω2

V 2i

]φi = 0, i = 1, 2 (18)

Here V1, V2 are the velocities of the longitudinal waves, that is, P-wave and T-wave, and derived from the rootsof quadratic equations in V 2, given by

CV 4 − Bω2V 2 + Aω4 = 0. (19)

From Eq. (13) with the aid of (15), we obtain[∇2 + ω2

V 23

]ψ = 0, (20)

where V3 = 1β

is the velocity of transverse wave (SV-wave).Using Eqs. (14), (18), (15) and (17), we obtain

{φ, T } =2∑

i=1

{1, ni }φi , (21)

where

ni =(

iω − (τ0)α

α! (−iω)α+1)

∈ ω2

(∇2 + iω + (τ0)

α

α! (−iω)α+1)

V 2i

, i = 1, 2 (22)

The basic equations of homogeneous isotropic elastic solid are written as

(λe + μe)∇ (∇ · ue) + μ∇2ue = ρe ∂

2ue

∂t2 , (23)

where λe, μe are the Lame’s constants, uei are the components of displacement vector, ue, ρe is the density

corresponding to medium I, and ee1 = ∂ue

1∂x1

+ ∂ue3

∂x3


contact surface x3= 0

Fractional order thermoelastic half-space (medium II) x

3> 0

Elastic half-space (medium I) x

3< 0 θ1

θ2

θ0

Incident (P or SV) SV

P

A3

A2

A1

P3

P2

P1

x3-axis

x1-axis

Fig. 1 Geometry of the problem

For two-dimensional problem, the components of ue = (ue

1, 0, ue3

)can be written as

ue1 = ∂φe

∂x1− ∂ψe

∂x3, ue

3 = ∂φe

∂x3+ ∂ψe

∂x1, (24)

where φe and ψe are the potential functions satisfying the wave equations

∇2φe = 1

α2 φe, (25)

∇2ψe = 1

β2 ψe, (26)

where

α = αe

C0, β = βe

C0, αe =

√λe + 2μe

ρe, βe = μe

ρe. (27)

The stress tensor σ ei j and strain tensor ee

i j in the isotropic elastic medium are given by

σ ei j = 2μeee

i j + λeeekkδi j ,

eei j = 1

2

(ue

i, j + uej,i

). (28)

where eeii is the dilatation.

4 Reflection and transmission

We consider a harmonic wave (P or SV) propagating through the isotropic elastic solid half-space, which isincident at the interface x3 = 0. Corresponding to this incident wave, two homogeneous waves (P and SV)are reflected in isotropic elastic solid and three inhomogeneous waves (P, T and SV) are refracted in isotropicfractional order thermoelastic solid (Fig. 1).


In elastic solid half-space, the potential functions satisfying Eqs. (25) and (26) can be written as

φe = Ae0e

[iω

{(x1sinθ0+x3cosθ0

)/α−t

}]+ Ae

1e[

iω{(

x1sinθ1+x3cosθ1

)/α−t

}], (29)

ψe = Be0e

[iω

{(x1sinθ0+x3cosθ0

)/β−t

}]+ Be

1e[

iω{(

x1sinθ2+x3cosθ2

)/β−t

}]. (30)

The coefficients Ae0(B

e0), Ae

1 and Be1 are amplitudes of the incident P (or SV), reflected P and reflected

SV-waves, respectively.Following Borcherdt [3], in isotropic fractional order thermoelastic half-space, the potential functions

satisfying Eqs. (18) and (20) can be written as

{φ, T

} =2∑

i=1

{1, ni

}Bi e

( Ai ·r)e{

i( Pi ·r−ωt

)}, (31)

ψ = B3e( A3·r

)e{

i( P3·r−ωt

)}. (32)

The coefficients Bi , i = 1, 2, 3 are the amplitudes of refracted P, T and SV-waves, respectively. The propaga-tion vector Pi , i = 1, 2, 3 and attenuation factor Ai , i = 1, 2, 3 are given by

Pi = ξR x1 + dViR x3,Ai = −ξI x1 − dViI x3, i = 1, 2, 3. (33)

where

dVi = dViR + idViI = p.v.

(ω2

V 2i

− ξ2

), i = 1, 2, 3, (34)

x1 and x3 denote the unit vectors propagating in the x1 and x3 directions, respectively, and ξ = ξR + iξI isthe complex wave number. The subscripts R and I denote the real and imaginary parts of the correspondingcomplex number, and p.v. stands for the principal value of the complex quantity derived from square root. Thecomplex wave number ξ in the isotropic fractional order thermoelastic medium is given by

ξ =∣∣∣Pi

∣∣∣ sin θ ′i − ι

∣∣∣ Ai

∣∣∣ sin(θ ′

i − γi), i = 1, 2, 3 (35)

where γi , i = 1, 2, 3 is the angle between the propagation and attenuation vector and θ ′i , i = 1, 2, 3 is the

angle of transmission in medium II.

5 Boundary conditions

The boundary conditions to be satisfied at the interface x3 = 0 are as follows:

(a) Mechanical conditions

σ e33 = σ33, (36)

σ e31 = σ31, (37)

(b) Displacement conditions

ue1 = u1, (38)

ue3 = u3, (39)

(c) Thermal condition

∂T

∂x3+ hT = 0. (40)

where h is the heat transfer coefficient.h → 0 corresponds to insulated boundary.h → ∞ corresponds to isothermal boundary.


Making use of potentials given by Eqs. (29)–(32), we find that the boundary conditions are satisfied if andonly if

ξR = ω sin θ0

V0= ω sin θ1

α= ω sin θ2

β, (41)

where

V0 ={α, for incident P-waveβ, for incident SV-wave (42)

and

ξI = 0

It means that waves are attenuating only in x3-direction. From Eq. (35), it implies that if |Ai | �= 0, thenγi = θ ′

i , i = 1, 2, 3, that is, attenuated vectors for the four refracted waves are directed along the x3-axis.Using Eqs. (29)–(32) in the boundary conditions (36)–(40) with the aid of Eqs. (11), (24), (41) and (42),

we get a system of five nonhomogeneous equations which can be written as

5∑j=1

di j Z j = gi , (43)

where Z j , j = 1, 2, 3, 4, 5 are the ratios of amplitudes of reflected P, reflected SV, refracted P, refractedT and refracted SV-waves to that of amplitude of incident wave.

d11 = 2μe(ξR

ω

)2

− ρec20, d12 = 2μe ξR

ω

dVβω, d15 = 2μ

ξR

ω

dV3

ω, d21 = 2μe ξR

ω

dVαω,

d22 = μe

[(dVβω

)2

−(ξR

ω

)2], d25 = μ

[(ξR

ω

)2

−(

dV3

ω

)2], d31 = ξR

ω, d32 = dVβ

ω

d35 = dV3

ω, d41 = −dVα

ω, d42 = ξR

ω, d45 = −ξR

ω, d51 = d52 = d55 = 0,

d1j = λ

(ξR

ω

)2

+ ρc20

(dVj

ω

)2

+ γ n jT0

ω2 , d2 j = 2μξR

ω

dVj

ω, d3 j = −ξR

ω, d4 j = −dVj

ω

d5j = in jdVj

ω+ n j

h

ω, j = 3, 4.

dVαω

=(

1

α2 −(ξR

ω

)2) 1

2

=(

1

α2 − sin2θ0

V 20

) 12

,dVβω

=(

1

β2 − sin2θ0

V 20

) 12

.

and

dVj

ω= p.v.

(1

V 2j

− sin2θ0

V 20

) 12

, j = 1, 2, 3.

Here p.v. is calculated with restriction dVj I ≥ 0 to satisfy decay condition in fractional order thermoelasticmedium. The coefficients gi , i = 1, 2, 3 on the right side of the Eq. (43) are given by

(a) For incident P-wave

g1 = −d11, g2 = d21, g3 = −d31, g4 = d41, g5 = 0. (44)

(b) For incident SV-wave

g1 = d12, g2 = −d22, g3 = d32, g4 = −d42, g5 = 0. (45)


0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

α=0.5α=1.0

E1

θ0

Fig. 2 Variation of energy ratio E1 w.r.t. angle of incidence θ0

0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

α=0.5

α=1.0

E2

θ0


Now we consider a surface element of unit area at the interface between two media. The reason for thisconsideration is to calculate the partition of energy of the incident wave among the reflected and refractedwaves on both sides of the surface. Following Achenbach [1], the energy flux across the surface element, thatis, rate at which the energy is communicated per unit area of the surface, is represented as

P∗ = σlmlmul , (46)

where σlm is the stress tensor, lm are the direction cosines of the unit normal l outward to the surface element,and ul are the components of the particle velocity.


0 20 40 60 80 100

0

0.01

0.02

0.03

0.04

α=0.5α=1.0

E11

θ0


0 20 40 60 80 100

0

0.2

0.4

0.6

α=0.5α=1.0

E22

θ0


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

⟨P∗e⟩ = Re 〈σ 〉e

13 · Re⟨ue

1

⟩ + Re 〈σ 〉e33 · Re

⟨ue

3

⟩. (47)

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

〈Re ( f ) · Re (g)〉 = 1

2Re ( f · g) . (48)


0 20 40 60 80 100

0

0.2

0.4

0.6

α=0.5α=1.0

E33

θ0


0 20 40 60 80 100

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0α=0.5α=1.0

ERR

θ0

Fig. 7 Variation of energy ratio ERR w.r.t. angle of incidence θ0

The expressions for energy ratios Ei , i = 1, 2, 3 for the reflected P and reflected SV-waves are given by

Ei = −⟨P∗e

i

⟩⟨P∗e

0

⟩ , i = 1, 2. (49)

where

⟨P∗e

1

⟩ = 1

2

ω4ρec20

α

∣∣Ae1

∣∣2 cosθ1,⟨P∗e

2

⟩ = 1

2

ω4ρec20

β

∣∣Be1

∣∣2 cosθ2, (50)

and

(a) For incident P-wave

⟨P∗e

0

⟩ = −1

2

ω4ρec20

α

∣∣Ae0

∣∣ cosθ0, (51)


0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

E1

θ0

α=0.5α=1.0


0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

α=0.5

α=1.0

E2

θ0


(b) For incident SV-wave

⟨P∗e

0

⟩ = −1

2

ω4ρec20

β

∣∣Be0

∣∣ cosθ0, (52)

are the average energy intensities of the reflected P, reflected SV, incident P, and incident SV-waves, respectively.In Eq. (49), negative sign is taken because the direction of reflected waves is opposite to that of incident wave.


0 20 40 60 80 100

0

0.0001

0.0002

0.0003

0.0004

α=0.5α=1.0

E11

θ0


0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

α=0.5α=1.0

E22

θ0


For fractional order thermoelastic solid, the average intensities of the waves on the surface with normalalong x3-direction are given by

⟨P∗

ij

⟩= Re 〈σ 〉(i)13 · Re

⟨u( j)

1

⟩+ Re 〈σ 〉(i)33 · Re

⟨u( j)

3

⟩. (53)

The expressions for energy ratios for the reflected P, reflected T and reflected SV-waves are given by

Ei j =⟨P∗

ij

⟩⟨P∗e

0

⟩ , i, j = 1, 2, 3 (54)


0 20 40 60 80

0

0.2

0.4

0.6

0.8

1 α=0.5α=1.0

100

E33

θ0


0 20 40 60 80 100

-0.6

-0.4

-0.2

0

0.2 α=0.5α=1.0

ERR

θ0

Fig. 13 Variation of energy ratio ERR w.r.t. angle of incidence θ0

where

⟨P∗

ij

⟩= −ω

4

2Re

[{2μ

dVi

ω

ξR

ω

ξR

ω+ λ

(ξR

ω

)2(

dV j

ω

)+ ρc2

0

(dVi

ω

)2(

dV j

ω

)+ γ ni T0

ω2

(dV j

ω

)}Bi B j

],

⟨P∗

i3

⟩ = −ω4

2Re

[{−2μ

dVi

ω

dV3

ω

ξR

ω+ λ

(ξR

ω

)2 (ξR

ω

)+ ρc2

0

(dVi

ω

)2 (ξR

ω

)+ γ ni T0

ω2

(ξR

ω

)}Bi B3

],


Fig. 14 Variation of amplitude ratio |Z1| w.r.t. Frequency(ω)

⟨P∗

3 j

⟩= −ω

4

2Re

[{μ

((ξR

ω

)2ξR

ω− ξR

ω

(dV3

ω

)2)

− λξR

ω

dV3

ω

dV j

ω+ ρc2

0ξR

ω

dV3

ω

dV j

ω

}B j B3

],

⟨P∗

33

⟩ = −ω4

2Re

[{μ

((dV3

ω

)2

−(ξR

ω

)2)

dV3

ω− 2μ

ξR

ω

ξR

ω

dV3

ω

}B3 B3

], i, j = 1, 2.

The diagonal entries of energy matrix Ei j in Eq. (54) represent the energy ratios of P, T and SV-waves,respectively, whereas sum of the nondiagonal entries of Ei j gives the share of interaction energy among allrefracted waves in the medium and is given by

ERR =3∑

i=1

⎛⎝ 3∑

j=1

Ei j − Eii

⎞⎠. (55)

The energy ratios Ei , i = 1, 2, diagonal entries and nondiagonal entries of energy matrix Ei j , that is,E11, E22, E33 and ERR, yield the conservation of incident energy across the interface, through the relation

E1 + E2 + E11 + E22 + E33 + ERR = 1. (56)

6 Numerical results and discussion

We now represent some numerical results for copper material [18], for which the physical data are given below:

λ = 7.76 × 1010 Kg m−1 s−2, µ = 3.86 × 1010 Kg m−1 s−2, T0 = 0.293 × 103 K,

CE = .3831 × 103 J Kg−1 K−1, αt = 1.78 × 10−5 K−1

h = 0, ρ = 8.954 × 103 Kgm−3, K = 0.383 × 103 Wm−1 K−1.

Following Bullen [5], the numerical data of granite in elastic medium are given by

ρe = 2.65 × 103Kgm−3, αe = 5.27 × 103ms−1, βe = 3.17 × 103ms−1

The software Matlab 7.0.4 has been used to determine the values of energy ratios Ei , i = 1, 2 and an energymatrix Ei j , i, j = 1, 2, 3 defined in the previous section for different values of incident angle (θ0) ranging


Fig. 15 Variation of amplitude ratio |Z2| w.r.t. frequency (ω)|

1 2 3 4 5

Frequency

0.032

0.036

0.04

0.044

0.048

0.052

0.056

|Z3|

α=0.5α=1.0

Fig. 16 Variation of amplitude ratio |Z3| w.r.t. Frequency

from 0 to 90◦ for fixed frequencyω = 2×π×100 Hz corresponding to incident P and SV-waves; the variationsof these energy ratios with respect to angle of incidence have been plotted in Figs. 2, 3, 4, 5, 6, and 7 andFigs. 8, 9, 10, 11, 12 and 13. The variation of magnitude values of amplitude ratios is drawn w.r.t. frequencyin Figs. 14, 15, 16, 17 and 18 and Figs. 19, 20, 21, 22 and 23 for incident P and SV-waves, respectively, for afixed angle of incidence θ0 = 60◦. In all the figures, the dark and dotted lines correspond to fractional ordersα = 0.5 and α = 1.0, respectively.


1 2 3 4 5

Frequency

0.0012

0.0016

0.002

0.0024

0.0028

0.0032

|Z4|

α=0.5α=1.0

Fig. 17 Variation of amplitude ratio |Z4| w.r.t. Frequency

1 2 3 4 5

Frequency

0.1704

0.1708

0.1712

0.1716

0.172

0.1724

0.1728

|Z5|

α=0.5α=1.0

Fig. 18 Variation of amplitude ratio |Z5| w.r.t. Frequency (ω)

6.1 Incident P-wave

It is clear from Fig. 2 that for both fractional orders α = 0.5 and 1.0, the values of energy ratio E1 decrease withthe increase in angle of incidence θ0 from 0 to 74◦ and then increase as θ0 increases further. Figure 3 shows thatfor fractional order α = 0.5 the values of energy ratio E2 increase with the increase of the angle of incidenceθ0 from 0 ≤ θ0 ≤ 7◦ and decrease for 0 ≤ θ◦ ≤ 13◦ and again slightly increase for 13◦ ≤ θ0 ≤ 75◦ anddecreases suddenly as θ0 approaches 90◦. For α = 1.0, the values of E2 increase slightly for 0 ≤ θ0 ≤ 35◦ anddecrease in the range 35◦ ≤ θ0 ≤ 40◦ and again increase for 40◦ ≤ θ◦ ≤ 75◦ and then suddenly decrease asθ0 approaches 90◦. Figure 4 indicates that for α = 0.5, E11 oscillates for 0 ≤ θ0 ≤ 10◦ and slightly decreasesfor 10◦ ≤ θ0 ≤ 90◦. On the other hand, for α = 1.0, it increases in 0◦ ≤ θ0 ≤ 37◦and than slightly decreasesfor 37◦ ≤ θ0 ≤ 90◦. Figure 5 depicts that for α = 0.5, the values of E22 oscillate in the range 0 ≤ θ0 ≤ 10◦,increase for 10◦ ≤ θ0 ≤ 70◦ and suddenly decrease for 70◦ ≤ θ0 ≤ 90◦. On the other hand for α = 1.0,E22 has constant value of 1 for 0 ≤ θ0 ≤ 80◦ and decrease as θ0 approaches 90◦. Figure 6 shows that for


Fig. 19 Variation of Amplitude ratio |Z1| w.r.t. Frequency (ω)

1 2 3 4 5

Frequency

0.617

0.618

0.619

0.62

0.621

0.622

|Z2|

α=0.5α=1.0


α = 1.0, E33 increases slightly 0 ≤ θ0 ≤ 35◦, decreases 35◦ ≤ θ0 ≤ 40◦ and again increases 40◦ ≤ θ0 ≤ 75◦and then decreases rapidly 75◦ ≤ θ0 ≤ 90◦. On the other hand for α = 0.5, the value of ERR oscillates for0 ≤ θ0 ≤ 12◦, again slightly decreases for 12◦ ≤ θ0 ≤ 75◦ and increases rapidly as θ0 approaches 90◦. Andfor α = 1.0, the energy ratio ERR decreases slightly for 0 ≤ θ0 ≤ 37◦ and increases for 37◦ ≤ θ0 ≤ 40◦ andagain decreases for 40◦ ≤ θ0 ≤ 75◦ and rapidly increases as θ0 approaches 90◦.It is noticed that the sum of thevalues of energy ratios E1, E2, E11, E22, E33 and ERR is found to be exactly unity at each value of θ0 whichproves the law of conservation of energy at the interface.

Figure 14 shows that the values of |Z1| increase smoothly for α = 0.5 whereas for α = 1.0, the valuesdecrease for small values of ω and become constant as ω increases. Figure 15 shows that for α = 1.0, thevalues of |Z2| increase slowly for small values of ω and then become constant whereas for α = 0.5, |Z2|decreases for all values of ω. The values of |Z2| are higher in comparison with |Z1|. From Fig. 16 it is evident


1 2 3 4 5

Frequency

0.01

0.012

0.014

0.016

0.018

|Z3|

α=0.5α=1.0


1 2 3 4 5

Frequency

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

|Z4|

α=0.5α=1.0


that for α = 0.5, the values of |Z3| increase sharply whereas for α = 1.0, values of |Z3| decrease initiallyand become constant as ω increases. Figure 17 depicts that the value of |Z4| increases sharply for α = 0.5,whereas for α = 1.0, it shows a constant behavior. Figure 18 indicates that |Z5| shows the same behavior andvariation as |Z2| but with different magnitude values. The values of |Z2| are higher in comparison with |Z5|.

6.2 Incident SV-wave

From Fig. 8, it is evident that there is a rapid increase in the values of energy ratio E1 for 0 ≤ θ0 ≤ 36◦, butfor 36◦ ≤ θ0 ≤ 90◦, values of energy ratio E1 become negligibly small for both fractional orders. Figure 9depicts that values of energy ratio E1 initially fluctuate for 0 ≤ θ0 ≤ 40◦, but finally reach to nearly unity asθ0 approaches 90◦. Figure 10 indicates that for α = 1.0, values of energy ratio E11 increase for 0 ≤ θ0 ≤ 20◦and decrease rapidly to 0 at θ0 = 22◦. On the other hand for α = 0.5, E11 increases for 0 ≤ θ0 ≤ 37◦ and then


1 2 3 4 5

Frequency

0.304

0.3044

0.3048

0.3052

0.3056

|Z5|

α=0.5α=1.0


decreases onward in the range 37◦ ≤ θ0 ≤ 90◦. Figure 11 shows that values of energy ratio E22 increase for0 ≤ θ0 ≤ 40◦ and slightly decrease for 40◦ ≤ θ0 ≤ 90◦ for both fractional orders. But value of E22 is higherin case of α = 0.5. Figure 12 indicates that values of energy ratio E33 decrease for 0 ≤ θ0 ≤ 37◦ and rapidlyincrease for the range 37◦ ≤ θ0 ≤ 55◦ and slightly decrease for 55◦ ≤ θ0 ≤ 90◦ for both fractional orders.We notice from Fig. 13 that firstly, the value of energy ratio ERR shows fluctuating behavior for 0 ≤ θ0 ≤ 40◦and increases continuously for 40◦ ≤ θ0 ≤ 90◦. Like in case of incident P-wave, the sum of all energy ratiosis also found to be unity in case of incident SV-wave.

From Fig. 19, it is evident that the values of |Z1| increase sharply for α = 0.5, whereas for α = 1.0the values decrease initially and then become constant. Figure 20 shows that for α = 0.5, the values of |Z2|decrease for all values of ω whereas for α = 1.0, there is small decrease in values initially and then becomesconstant. Figures 21 and 23 indicate that behavior and variation of |Z3| and |Z5| are same as that of |Z1| butwith different magnitude values. Figure 22 shows that value of |Z4| increases sharply for α = 0.5 whereas forα = 1.0, it attains the constant value.

7 Conclusions

In the present article, the phenomenon of reflection and transmission of obliquely incident elastic wave at theinterface between an elastic solid half-space and a fractional thermoelastic solid half-space has been studied.The three waves in fractional thermoelastic solid medium are identified and explained through different waveequations in terms of displacement potentials. The energy ratios of different reflected and refracted waves tothat of incident wave are computed numerically and presented graphically with respect to the angle of incidencefor fractional order α = 0.5 and α = 1.0. For P-wave, values of E1 E22 are higher and E2, E33, ERR are lowerfor α = 0.5 than for α = 1.0. For α = 0.5 the value of E11 initially has higher value but after a certain stagebecomes lower than α = 1.0. For SV-wave, values of E1, E22, E33 are higher and E2, E11, ERR are lower forα = 0.5 than for α = 1.0.

From numerical results, we conclude that the effect of angle of incidence and fractional order on amplitudeand the energy ratios of the reflected and refracted waves is significant. The sum of all energy ratios of thereflected waves, refracted waves and interference between refracted waves is verified to be always unity whichensures the law of conservation of incident energy at the interface.


References

1. Achenbach, J.D.: Wave propagation in elastic solids. Elsevier Science Publishers B. V., North –Holland, Amsterdam (1973)2. Biswas, R.K., Sen, S.: Fractional optimal control problems: a pseudo state-space approach. J. Vib. Control 17,

1034–1041 (2011)3. Borcherdt, R.D.: Reflection-refraction of general P and type-I S waves in elastic and anelastic solids. Geophys. J. R. Astron.

Soc. 70, 621–638 (1982)4. Borejko, P.: Reflection and refraction coefficients for three-dimensional plane waves in elastic media. Wave Motion 24,

371–393 (1996)5. Bullen, K.E.: An Introduction to the Theory of Seismology. Cambridge University Press, England (1963)6. Caputo, M.: Linear model of dissipation whose Q is always frequency independent-II. Geophys. J. R. Astron. Soc. 13,

529–539 (1967)7. Caputo, M.: Vibrations on an infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56, 897–904 (1974)8. Ezzat, M.A.: Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Physica

B 406, 30–35 (2011a)9. Ezzat, M.A.: Theory of fractional order in generalized thermoelectric MHD. Appl. Math. Model. 35, 4965–4978 (2011b)

10. Jiang, X., Xu, M.: The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and thecylindrical coordinate systems. Physica A 389, 3368–3374 (2010)

11. Kumar, R., Sarthi, P.: Reflection and refraction of thermoelastic plane waves at an interface of two thermoelastic mediawithout energy dissipation. Arch. Mech. 58, 155–185 (2006)

12. Miller, K.S., Ross, B.: An Introduction to the Fractional Integrals and Derivatives, Theory and Applications. Wiley,New York (1993)

13. Povstenko, Y.Z.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)14. Povstenko, Y.Z.: Thermoelasticity that uses fractional heat conduction equation. J. Math. Stress. 162, 296–305 (2009)15. Povstenko, Y.Z.: Fractional Radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal

stresses. Mech. Res. Commun. 37, 436–440 (2010)16. Povstenko, Y.Z.: Fractional Catteneo-type equations and generalized thermoelasticity. J. Therm. Stress. 34, 97–114 (2011)17. Sharma, M.D., Gogna, M.L.: Reflection and refraction of plane harmonic waves at an interface between elastic solid and

porous solid saturated by viscous liquid. PAGEOPH 138, 249–266 (1992)18. Sherief, H.H., Saleh, H.A.: A half space problem in the theory of thermoelastic diffusion. Int. J. Solid Struct. 42,

4484–4493 (2005)19. Sinha, S.B., Elsibai, K.A.: Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with

two relaxation times. J. Therm. Stress. 20, 129–145 (1997)20. Tomar, S., Arora, A.: Reflection and refraction of elastic waves at an elastic/porous solid saturated by immiscible fluids.

International Journal of Solids and Structures 43, (2006)21. Wu, C.M., Lundberg, B.: Reflection and refraction of the energy of harmonic elastic waves in a bent bar. J. Sound

Vib. 190, 645–659 (1996)22. Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 1–7 (2010)

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Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space