PROPAGATION OF WAVES AT AN IMPERFECTLY

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1 Journal of Theoretical and Applied Mechanics, Sofia, 2011, vol. 41, No. 3, pp PROPAGATION OF WAVES AT AN IMPERFECTLY BONDED INTERFACE BETWEEN TWO MONOCLINIC THERMOELASTIC HALF-SPACES Joginder Singh Department of Mathematics, Indo Global College of Engineering, Abhipur, Mohali, India Baljeet Singh Department of Mathematics, Post Graduate Government College, Sector-11, Chandigarh , India Praveen Ailawalia Department of Mathematics, RIMT Institute of Engineering and Technology, Mandi Govindgarh, Punjab, India [Received 09 February Accepted 14 November 2011] Abstract. An imperfectly bonded interface between two monoclinic thermoelastic half-spaces is chosen to study the reflection and transmission of plane waves in context of generalized thermoelasticity. Six relations between amplitudes of incident, reflected and transmitted quasi-p (qp) waves, quasi thermal (qt) waves and quasi-sv (qsv ) waves are obtained. Some particular cases are obtained which agree with earlier well established results. A procedure for computing reflected and refracted angles is derived for a given incident wave, where the angles of reflection are found not equal to the angles of incident waves in a monoclinic thermoelastic medium. Key words: generalized thermoelasticity, imperfect boundary, plane waves, reflection and transmission, monoclinic anisotropic. 1. Introduction Keith and Crampin [1] investigated that three types of body waves with mutually orthogonal particle motion can propagate in an anisotropic elastic solid medium. In general, the particle motion is neither purely longitudinal nor * Corresponding author bsinghgc11@gmail.com

2 78 Joginder Singh, Baljeet Singh, Praveen Ailawalia purely transverse. Due to this, the three types of body waves in an anisotropic medium are referred to as qp, qsv and qsh, rather than as P, SV and SH, the symbols used for propagation in an isotropic medium. A monoclinic medium has one plane of elastic symmetry. SH motion is decoupled from the P SV motion for wave propagation in the plane of symmetry. It is neither purely longitudinal nor purely transverse motion in the case of P SV waves, while the particle motion of SH waves is purely transverse. Chattopadhyay and Choudhury [2] discussed the reflection of qp waves at the plane free boundary of amonoclinic half-space. Inasubsequentpaper, Chattopadhyaet al [3] studied the reflection of qsv waves at a plane free boundary of a monoclinic half-space. Inthese papers, they assumedthatqp waves are purelylongitudinal andqsv waves are purely transverse. Singh [4] pointed out the errors in these papers and SinghandKhurana [5] restudiedthe reflection ofp and SV waves at the free surface of a monoclinic half-space. Biot [6] developed the coupled theory of thermoelasticity, where heat equation being parabolic predicts an infinite speed of propagation for thermal wave, which is not physically relevant. Lord and Shulman [7] introduced the theory of generalized thermoelasticity with one relaxation time by postulating a new law of heat conduction to replace the classical Fourier s law. This law contains the heat flux vector as well as its time derivative. In addition, it contains a new constant which acts as a relaxation time. The heat equation of this theory is of wave type, ensuring finite speeds of propagation for thermal and elastic waves. The remaining governing equations for this theory, namely the equation of motion and the constitutive relation remain the same as those for the coupled and uncoupledtheories. Dhaliwal and Sherief [8] extended the Lord and Shulman (L-S) theory for an anisotropic media. The second generalization was developed by Green and Lindsay [9] and is known as G-L theory. This theory contains two constants that act as relaxation times. Many authors contribute towards the wave propagation in isotropic and anisotropic medium with the application of generalized thermoelasticity, for example, Sinha and Sinha [10], Sharma [11], Kumar and Singh [12] and Singh [13 15]. The boundary between the solids may behave as slip, perfect or neither, depending on the properties of the layers, as demonstrated by Rokhlin and Marom [16], and its state significantly affects elastic wave reflection phenomena. Imperfect bonding means that the stress components, heat flux are continuous and the small displacement field and temperature are not continuous. The small vector difference in the displacement is assumed to depend linearly on the traction vector and in temperature it is assumed to depend linearly on the heat flux. In the present paper, the reflection and transmis-

3 Propagation of waves at an imperfectly bonded interface sion of plane waves is studied at an imperfectly bonded interface between two dissimilar thermoelastic half-spaces of monoclinic type in the context of gerneralized thermoelasticity. Relations between amplitudes of incident, reflected and transmitted quasi-p (qp) waves, quasi thermal (qt) waves and quasi- SV (qsv ) waves are obtained, which are reduced to earlier well established relations for some limiting cases. A procedure for computing reflected and refracted angles is also derived for a given incident angle of a striking wave. 2. Nomenclature c phase velocity c ij isothermal elasticities C e specific heat at constant strain e ij components of strain tensor k wave number t time τ 0,τ 1 relaxation times K 2,K 3 thermal conductivities T absolute temperature T 0 the initial uniform temperature α 2,α 3 coefficients of linear thermal expansion ρ density σ ij components of stress tensor ω circular frequency Θ = T T 0, Θ T 1 β 2 = (c 12 + c 22 )α 2 + c 23 α 3, β 3 = 2c 23 α 2 + c 33 α Formulation of the problem and solution Considerahomogeneous, anisotropic, generalized thermoalstic medium of monoclinic type at a uniform temperature T 0. The origin is taken on the thermally insulated and stress-free plane surface and z-axis is directed normally into the half-space, which is representedby z >0. Following Lord andshulman [7] and Green ( and Lindsay ) [9] theories, the governing field equations in the y z plane x 0 for generalized monoclinic thermoelasticity, in absence of body forces and heat sources, are: 2 v (1) c 22 y 2 + c 2 v 44 z 2 + c 2 w 24 y 2 + c 2 w 34 z 2 + 2c 2 v 24 y z

4 80 Joginder Singh, Baljeet Singh, Praveen Ailawalia ( ) + (c 23 + c 44 ) 2 w y z β Θ 2 Θ + τ 1 = ρ 2 v y t t 2, 2 v c 24 y 2 + c 2 v 34 z 2 + c 2 w 44 y 2 + c 2 w 33 z 2 + 2c 2 w 34 y z (2) + (c 23 + c 44 ) 2 v y z β 3 z ( ) Θ Θ + τ 1 = ρ 2 w t t 2, 2 Θ K 2 y 2 + K 2 [ ( Θ 2 ) 3 z 2 T v 0 β 2 y t + τ 0Ω 3 v y t 2 ( 2 )] ( w (3) +β 3 z t + τ 0Ω 3 w Θ z t 2 = ρc e t + τ 2 ) Θ 0 t 2. The use of symbol Ω and τ 1 in equation (3) makes these equations possible for L-S and G-L theories of generalized thermoelasticity. For the L-S theory, τ 1 = 0, Ω = 1 and for the G-L theory τ 1 > 0, Ω = 0. The thermal relations τ 0 and τ 1 satisfy the inequality τ 1 > τ 0 > 0 for the G-L theory only. For plane waves of circular frequency ω, wave number k and phase velocity c, incident at the free boundary z = 0 at an angle θ with the z-axis, we assume: (4) (v, w, Θ) = (A, B, C)exp (ιp 1 ), where A, B, C are the amplitude factors and: (5) P 1 = ωt k (y sin θ z cos θ), is the phase factor. For waves reflected at z = 0, we assume: (6) (v, w, Θ) = (A, B, C)exp (ιp 2 ), where: (7) P 2 = ωt k (y sin θ + z cos θ). (8) Making use of Eq. (4) or (6) in Eqs. (1) to (3), we obtain: ( D1 ρc 2) ( ι A + L 1 B τ k) 1 β 2 sin θ C = 0,

5 Propagation of waves at an imperfectly bonded interface (9) L 1 A + ( D 2 ρc 2) ( ι B ± τ k) 1 β 3 cos θ C = 0, ( ι ) (D3 (10) τt 0 c 2 β 2 sin θ A τt 0 c 2 β 3 cos θ B ρc 2 ) τ C e C = 0, k Here, D 1 (θ) = c 22 sin 2 θ + c 44 cos 2 θ 2c 24 sinθ cos θ, D 2 (θ) = c 44 sin 2 θ + c 33 cos 2 θ 2c 34 sinθ cos θ, D 3 (θ) = K 3 cos 2 θ + K 2 sin 2 θ, L 1 (θ) = c 24 sin 2 θ + c 34 cos 2 θ (c 23 + c 44 ) sinθ cos θ, ( ι ( ι τ = τ 0, τ = τ 0 Ω, τ ω) ω) 1 = (1 + ιωτ 1 ), where, the upper sign corresponds to the incident waves and the lower sign corresponds to the reflected waves. Equations (8) (10) in A, B, C admit a non-trivial solution only if the determinant of their coefficients vanishes, i.e.: (11) A 0 ζ 3 + A 1 ζ 2 + A 2 ζ + A 3 = 0 where A 0 = τ, A 1 = [ D 1 τ + D 2 τ + D 4 + εη ( β2 cos 2 θ + sin 2 θ )], A 2 = ( D 1 D 2 L 2 ) 1 τ + D 1 D 4 + D 2 D 4 + εη ( β2 D 1 cos 2 θ + D 2 sin 2 θ ) ± 2L 1 εη β sin θ cos θ, ( A 3 = D 4 D1 D 2 L 2 1), ( ) ( ζ = ρc 2 D3 β 2, D 4 =, ε = 2 T 0 C e ρc e v1 2 ), η = ττ 1 v 2 1, v 2 1 = c 22 ρ, β = ( β3 Following Singh [13], the cubic equation (11) with complex coefficients has three roots ζ 1, ζ 2, ζ 3 corresponding to phase velocities of three plane waves, namely quasi-p (qp), quasi-thermal (qt) and quasi-sv (qsv ) waves, respectively. If we write c 1 j = Vj 1 iω 1 q j, j = 1,2,3, then V j and q j are the speeds of propagation and the attenuation coefficients of qp, qt and qsv waves, respectively. β 2 ).

6 82 Joginder Singh, Baljeet Singh, Praveen Ailawalia 4. Reflection and transmission Consider a homogeneous, monoclinic generalized thermoelastic halfspace occupying the region z > 0 in imperfectly bonded contact with another homogeneous, monoclinic generalized thermoelastic half-space occupying the region z < 0. Incident qp or qsv or qt waves at the interface will generate reflected qp,qsv,qt waves inthe half-space z >0and transmitted qp,qsv,qt waves in the half-space z < 0 as shown in Fig. 1. Fig. 1. Geometry of the problem showing incident, reflected and transmitted waves at an interface between two dissimilar thermoelastic half-spaces of monoclinic type The required boundary conditions at an imperfectly bonded interface z = 0 are (Lavrentyev and Rokhlin [17]): (i) σ 33 = K n [w w ], (ii) σ 23 = K t [v v ], (12) (iii) K 3 T z = K c [T T ], (iv) σ 33 = σ 33, (v) σ 23 = σ 23, (vi) K 3 T z = K T 3 z,

7 Propagation of waves at an imperfectly bonded interface where, K n, K t are the normal and transverse stiffness coefficients. The symbols with primes correspond to the upper half-space z < 0. The appropriate components of the displacement vector and the temperature field which satisfy the boundary condition (12) at z = 0 are as follows: For half-space z > 0, (13a) v = 6 A j e ip j, w = j=1 For half-space z < 0, (13b) v = 9 A j e ip j, w = j=7 6 B j e ip j, T = j=1 9 B j e ip j, T = j=7 6 C j e ip j. j=1 9 C j e ip j, j=7 where [ P j = ω t (y sin e ] j z cos e j ), (j = 1,2,3,7,8,9); c j [ (14) P j = ω t (y sine ] j + z cos e j ), (j = 4,5,6), c j (15) A j = F j B j, C j = F j B j, For j = 1, 2, 3, 7, 8, 9, ( ) D 2j ρc 2 j β 2 sin θ + L 1j β 3 cos θ F j = ( ), D 1j ρc 2 j β 3 cos θ + L 1j β 2 sin θ F j = ( i k )( ) L 2 1j (D 1j ρc 2 j D 2j ρc 2 j ) [ ( ) ], τ L 1j β 2 sin θ + D 1j ρc 2 j β 3 cos θ where, D 2j = c 44 sin 2 e j + c 33 cos 2 e j 2c 34 sine j cos e j, D 1j = c 22 sin 2 e j + c 44 cos 2 e j 2c 24 sine j cos e j, L 1j = c 24 sin 2 e j + c 34 cos 2 e j (c 23 + c 44 ) sine j cos e j.

8 84 Joginder Singh, Baljeet Singh, Praveen Ailawalia For j = 4, 5, 6, ( ) D 2j ρc 2 j β 2 sin θ L 1j β 3 cos θ F j = ( ), D 1j ρc 2 j β 3 cos θ L 1j β 2 sin θ ( )( ) D 1j ρc 2 Fj = j D 2j ρc 2 j L 2 1j ) [( ) ], τ D 1j ρc 2 j β 3 cos θ β 2 sin θ L 1j where, ( i k D 2j = c 44 sin 2 e j + c 33 cos 2 e j + 2c 34 sine j cos e j, D 1j = c 22 sin 2 e j + c 44 cos 2 e j + 2c 24 sine j cos e j, L 1j = c 24 sin 2 e j + c 34 cos 2 e j + (c 23 + c 44 ) sine j cos e j. Here, the subscriptj =1 corresponds to incident qp waves, j =2 corresponds to incident qt waves, j = 3 corresponds to incident qsv waves, j = 4 corresponds to reflected qp waves, j = 5 corresponds to reflected qt waves, j = 6 corresponds to reflected qsv waves, j = 7 corresponds to transmitted qp waves, j = 8 corresponds to transmitted qt waves and j = 9 corresponds to transmitted qsv. The boundary conditions at z = 0 given by equation (12) must be satisfied for all values of y, then we have: (16) P 1 (y, 0, t) = P 2 (y, 0, t) = P 3 (y, 0, t) = P 4 (y, 0, t) = P 5 (y, 0, t) which can be written as: (17) = P 6 (y, 0, t) = P 7 (y, 0, t) = P 8 (y, 0, t) = P 9 (y, 0, t) sin e 1 c 1 (e 1 ) = sin e 2 c 2 (e 2 ) = sin e 3 c 3 (e 3 ) = sin e 4 c 4 (e 4 ) = sin e 5 c 5 (e 5 ) From equations (12) to (17), we have = sine 6 c 6 (e 6 ) = sin e 7 c 7 (e 7 ) = sin e 8 c 8 (e 8 ) = sin e 9 c 9 (e 9 ) (18) a 1 B 1 + a 2 B 2 + a 3 B 3 + a 4 B 4 + a 5 B 5 + a 6 B 6 + a 7 B 7 + a 8 B 8 + a 9 B 9 = 0, b 1 B 1 + b 2 B 2 + b 3 B 3 + b 4 B 4 + b 5 B 5 + b 6 B 6 + b 7 B 7 + b 8 B 8 + b 9 B 9 = 0, d 1 B 1 + d 2 B 2 + d 3 B 3 + d 4 B 4 + d 5 B 5 + d 6 B 6 + d 7 B 7 + d 8 B 8 + d 9 B 9 = 0, f 1 B 1 + f 2 B 2 + f 3 B 3 + f 4 B 4 + f 5 B 5 + f 6 B 6 + f 7 B 7 + f 8 B 8 + f 9 B 9 = 0, g 1 B 1 + g 2 B 2 + g 3 B 3 + g 4 B 4 + g 5 B 5 + g 6 B 6 + g 7 B 7 + g 8 B 8 + g 9 B 9 = 0, h 1 B 1 + h 2 B 2 + h 3 B 3 + h 4 B 4 + h 5 B 5 + h 6 B 6 + h 7 B 7 + h 8 B 8 + h 9 B 9 = 0,

9 Propagation of waves at an imperfectly bonded interface where the expressions for a l, b l, d l, f l, g l and h l are given in Appendix. The system of equations (18) can be reduced for different boundaries by taking: (i) (ii) K n 0, K t, K c, for Normal stiffness boundary, K n, K t 0, K c, for Transverse stiffness boundary, (iii) K n, K t, K c 0, for Thermal contact conductance, (iv) K n, K t 0, K c, for Slip boundary, (v) K n, K t, K c, for Welded contact. 5. Particular cases (a) If we take, c 24 = c 34 = c 24 = c 34 = 0, the system of equations (18) reduces for imperfectly bounded interface between two dissimilar orthotropic thermoelastic half-spaces as discussed by Kumar and Singh [12], where we have found minor but significant error in the coefficients of their equations (28) to (33). (b) If we take, K 1 = K 3 = K 1 = K 3 = 0, β 1 = β 3 = β 1 = β 3 = 0, the system of equations (18) reduces to for imperfectly bounded interface between two dissimilar monoclinic elastic half-spaces. If furthermore, we take K n, K t, the above results agree with Khurana and Singh [18]. (c) If we take, c 23 = c 33 2c 44 c 23 = c 33 2c 44, c 24 = c 34 = c 24 = c 34 = 0, the system of equations (18) reduces for imperfectly bonded interface between two dissimilar transversely isotropic thermoelastic half-spaces. (d) If we take, c 22 = c 33 = λ + 2µ, c 13 = c 23 = c 12 = λ, c 44 = c 55 = c 66 = µ, c 14 = c 24 = c 34 = c 56 = 0, β 2 = β 3 = β = (3λ + 2µ)α t, α 2 = α 3 = α t, K 2 = K 3 = K, c 22 = c 23 = λ + 2µ, c 13 = c 23 = c 12 = λ, c 44 = c 55 = c 66 = µ, c 14 = c 24 = c 34 = c 56 = 0, β 2 = β 3 = β = (3λ + 2µ )α t, α 2 = α 3 = α t, K 2 = K 3 = K, the system of equation (18) reduces for imperfectly bondedinterface between two dissimilar isotropic thermoelastic half-spaces. If we take further K n, K t, K c the above case agrees with Sinha and Elsibai [19, 20]. 6. Procedure for computing reflected and refracted angles The reflection coefficients depend on the velocities c i (e i ), i = 1, 2, 3, 4, 5, 6, which are the functions of reflected and refracted angles. They are

10 86 Joginder Singh, Baljeet Singh, Praveen Ailawalia supposed to be known for an incident qp wave, e 1 and, therefore c 1 (e 1 ). One has to compute e 4, e 5 and e 6 for given e 1. The velocities c 4 (e 4 ), c 5 (e 5 ) and c 6 (e 6 ) can then be computed from explicit algebraic formulae. The procedure is given below for computing e 4, e 5 and e 6 for given e 1 in the case of incident qp wave, for given e 2 in the case of incident qt wave, and for given e 3 in the case of incident qsv wave. Equation (11) can be written as (19) A 0 c 6 + A 1 c 4 + A 2 c 2 + A 3 = 0. Here, A 0 = ρ 3 τ, A 1 = ρ 2 [Uτ + Zτ + W C e + εη ( β2 p p2 2) ], [ (UZ A 2 = ρ V 2 ) τ + UW C e A 3 = W C e ( UZ V 2 ), where, + ZW ] + εη ( β2 Up 2 3 C + ) Zp2 2 2V η βp2 p 3, e U = c 22 p c 44p c 24p 2 p 3, Z = c 44 p c 33p c 34p 2 p 3, V = c 44 p c 34p (c 23 + c 24 )p 2 p 3, W = K 3 p K 2p 2 2. We define dimensionless apparent velocity c by the relation: (20) c = c a /β = c/p 2 β, where, β = (c 44 /ρ) 1 2, ρc 2 = p 2 2c 44 c 2. With the help of relation (20), the equation (19) becomes [ (21) c 6 (Ū + Z)τ + W ] c ( β 2 p ) τ + εη C e c 2 44 [ + (Ū Z W V 2 )τ + (Ū + Z) + εη C e c 2 ( β 2 Ūp 2 + Z) 2εη β V 44 c 2 p 44 ] c 2 τ W C e (Ū Z V 2 ) 1 τ = 0.

11 Propagation of waves at an imperfectly bonded interface Here, (22) Ū = c 22 + p c 24 p, Z = 1 + c 33 p c 34 p, V = c 24 + c 34 p 2 + ( c )p, W = K3 p 2 + K 2, where, c ij = c ij c 44, K3 = K 3 c 44, K2 = K 2 c 44, p = p 3 p 2. For incident qp waves, p = cot e 1 ; for incident qt waves, p = cot e 2 ; for incident qsv waves, p = cot e 3 ; for reflected qp waves, p = cot e 4 ; for reflected qt waves, p = cot e 5 ; for reflected qsv waves, p = cot e 6. For a given p, equation (21) can be solved for c 3, the three roots corresponding to qp, qt and qsv waves. However, for a given c, equation (21) is a six degree equation in p, corresponding to incident qp, qt and qsv, reflected qp, qt and qsv. The positive roots corresponding to the reflected waves and the negative roots corresponding to the incident waves. On inserting the expressions for Ū, Z, V and W from equation (22) into equation (21), we obtain the six degree equation in p as: (23) G 0 p 6 + G 1 p 5 + G 2 p 4 + G 3 p 3 + G 4 p 2 + G 5 p + G 6 = 0. The expressions for G 0, G 1, G 2, G 3, G 4, G 5 and G 6 are given in Appendix. If we neglect thermal effects, the equation (23) reduces to the equation (48) obtained by Singh and Khurana [5]. Now, we define q = 1, The equation p (23) transforms into (24) G 6 q 6 + G 5 q 5 + G 4 q 4 + G 3 q 3 + G 2 q 2 + G 1 q + G 0 = 0. Equation will possess three positive roots for angles of incidence, for which all three reflected qp, qsv and qt waves exist, the smaller positive root (say q 6 ) corresponding to reflected qt and the root (q 5 ) corresponding to reflected qsv and the larger positive root (q 4 ) corresponding to reflected qp. Further, e 4 = tan 1 (q 4 ), e 5 = tan 1 (q 5 ), e 6 = tan 1 (q 6 ). For an isotropic thermoelastic medium, if we take: c 14 = c 24 = c 34 = 0, c 11 = c 22 = c 33 = λ + 2µ, c 44 = c 55 = c 66 = µ, c 12 = c 13 = c 23 = λ, K2 = K 3 = K, β 2 = β 3 = β

12 88 Joginder Singh, Baljeet Singh, Praveen Ailawalia Then, G 0 = K C e τ γ, G 1 = 0, { G 2 = γ + K εη (1 + γ) + C e τ µτ { G 4 = 1 + γ + K C e τ + εη µτ G 5 = 0, } c 2 + } c 4 + ( G 6 = c γ + K C e τ + εη ) ( µτ c 4 + γ + K C e τ ( 3γ), G 3 = 0, { 2γ + 2 K 2εη C e τ (1 + γ) + µτ K ) εη C e τ (1 + γ) + µτ c 2 } c 2 3 K C e τ γ, K C e τ γ, and, the equation (23) reduces to (25) sγ ( p 2 c ) ( p 2 α 1 c 2 γ + 1 )(p 2 β 1 c 2 ) γ + 1 = 0, where α 1 = r + r 2 4sγ 2s, β 1 = r r 2 4sγ, r = γ + K 2s In this case, the Snell s law becomes: C e τ + εη K µτ, s = C e τ. (26) sin e 1 v qp = sin e 2 v qsv = sin e 3 v qt. Therefore, the roots of equation (25) are given by p 2 = c 2 1 = cot 2 e 3 corresponding to qsv waves, p 2 = c2 γ α 1 1 = cot 2 e 1, corresponding to qp wave and p 2 = c2 γ β 1 1 = cot 2 e 2 corresponding to qt wave. Thus, we may choose q = 1 p : (27) q 1 = tan e 1, q 2 = tan e 2, q 3 = tan e 3, q 4 = tan e 1, q 5 = tan e 2, q 6 = tan e 3,... as the six roots of the equation (24) for an isotropic medium. This choice will act as a guiding factor in computing the angles of reflection of qp, qt and qsv

13 Propagation of waves at an imperfectly bonded interface waves in a monoclinic thermoelastic medium. For an orthotropic medium, it can be shown that g 1 = g 3 = g 5 = 0 in equation (24). Therefore, equation (24) reduces to a cubic equation in q 2. Thus we may choose q 1 = q 4, q 2 = q 5, q 3 = q 6, so that the angle of reflection of qp, qt and qsv waves is equal to the angle of incidence of qp, qt and qsv waves, respectively. This is not true for a monoclinic thermoelastic medium. A similar analysis can be derived to compute the angles of refraction of qp, qsv and qt waves in the upper monoclinic half-space for given incident angle. 7. Conclusion A problem of reflection and transmission is considered at an imperfectly bonded interface between two dissimilar thermoelastic solid half-spaces of monoclinic type, where we have obtained the six relations between amplitudes of incident, reflected and transmitted waves. These relations are verified theoretically for some limiting cases. A procedure for computing the reflected and refracted angles for a given incident angle of a striking wave is explained in detail. It is concluded that the angle of particular reflected wave is not equal to the angle of incidence of that wave in a monoclinic medium. Numerical verification of present theoretical results can be attempted by following the procedure given in the previous section. REFERENCES [1] Keith, C. M., S. Crampin. Seismic Body Waves in Anisotropic Media: Reflection and Refraction at a Plane Interface. Geophys. J. R. Astr. Soc, 49 (1977), [2] Chattopadhyay, A., S. Choudhary. The Reflection Phenomenon of P- waves in a Medium of Monoclinic Type. Int. J. Engg. Sci, 33 (1995), [3] Chattopadhyay, A., S. Saha, M. Chakraborty. The Reflection of SVwaves in a Monoclinic Medium. Indian J. pure Appl. Math., 27 (1996), [4] Singh, S. J. Comments on The Reflection Phenomenon of P-waves in a Medium ofmonoclinic Type by ChattopadhyayandChoudhary. (Int. J. Engg. Sci., 33 (1995), ), Int. J. Engg. Sci., 37 (1999),

14 90 Joginder Singh, Baljeet Singh, Praveen Ailawalia [5] Singh, S., S. Khurana. Reflection of P and SV Waves at the Free Surface of a Monoclinic Elastic Half- Space, Proc. Indian Acad. Sci. (Earth Planet. Sci.), 111, 2002, [6] ] Biot, M. Thermoelasticity and Irreversible Thermodynamics. J. Appl. Phys., 27 (1956), [7] Lord, H. W., Y. Shulman. A Generalized Dynamical Theory of Thermoelasticity. J. Mech. Phys. Solids, 15 (1967), [8] Dhaliwal, R. S., H. H. Sherief. Generalized Thermoelasticity for Anisotropic Media. Quart. Appl. Math., 33 (1980) 1 8. [9] Green, A. E., K. A. Lindsay. Thermoelasticity. J. Elasticity, 2 (1972), 1 7. [10] Sinha, A. N., S. B. Sinha. Reflection of Thermoelastic Waves at a Solid Halfspace With Thermal Relaxation. J. Phys. Earth, 22 (1974), [11] Sharma, J. N. Reflection of Thermoelastic Waves from the Stress-free Insulated Boundary of an Anisotropic Half-space. Indian J. pure Appl. Math., 19 (1988), [12] Kumar, R., M. Singh. Reflection/ Transmission of Plane Waves at an Imperfectly Bonded Interface of Two Orthotropic Generalized Thermoelastic Half- Spaces. Material Sci. Engg. A, 472 (2008), [13] Singh, B. Dispersion Relations in a Generalized Monoclinic Thermoelastic Solid half-space. Journal of Technical Physics, 47 (2006), [14] Singh, B. Reflection of Plane Waves at the Free Surface of a Monoclinic Thermoelastic Solid Half-space. European J. Mech. A/Solids, 29 (2010), [15] Singh, B. Wave Propagation in an Initially Stressed Transversely Isotropic Thermoelastic Solid Half-space. Applied Mathematics and Computation, 217 (2010), [16] Rokhlin, S.I., D. Marom. Study of Adhesive Bonds using Low-frequency Obliquely Incident Ultrasonic Waves. J. Acoust. Soc. Am., 80 (1986), [17] Lavrentyev, A.I., S.I. Rokhlin. Ultrasonic Spectroscopy of Imperfect ContactInterfaces Betweena Layerand TwoSolids. J. Acoust. Soc. Am., 103 (1998), [18] Singh, S. J., S. Khurana. Reflection and Transmission of P and SV Waves atthe Interface BetweenTwo Monoclinic Elastic Half-spaces, Proc. Natl. Acad. Sci. India 71(A), 2001, [19] Sinha, S. B., K. A. Elsibai. Reflection of Thermoelastic Waves at a Solid Half-space with two Relaxation Times. J. Thermal Stresses, 19 (1996), [20] Sinha, S. B., K. A. Elsibai. Reflection and Refraction of Thermoelastic Waves atan Interface of Two Semi-infinite Media with two RelaxationTimes. J. Thermal Stresses, 20 (1997),

15 Propagation of waves at an imperfectly bonded interface Appendix The expressions for a l, b l, d l, f l, g l and h l are a l = ik n, (l = 1,...,6) [ ] a l = ik n c ω sine l 23F l + c ω cose l ω cose l 33 + c 34 F l c ω sine l 34 + iβ c l c l c l c 3τ 1F l, l b l = ik t F l, (l = 1,...,6) [ ] b l = ik t F l c ω sine l 24 F l + c ω cose l 34 + c ω cose l 44 F l c ω sine l 44, c l c l c l c l [ d l = ik c Fl, (l = 1,...,6), d l = ik c Fl + K 3 ω cos e l c l ω sine l ω cose l f l = c 23 F l + c 33 c l c l ω sine l ω cose l f l = c 23 F l c 33 c l c l F l ω cose l ω sine l + c 34 F l c 34 + iβ 3 τ 1 Fl c l c, l ω cose l ω sine l c 34 F l c 34 + iβ 3 τ 1 Fl, c l c l (l = 7,...,9) (l = 7,...,9) ], (l = 7,...,9) (l = 1,...,3) (l = 4,...,6) f l = c ω sine l 23 F l c ω cose l 33 c ω cose l 34 F l + c ω sine l 34 iβ 3 c l c l c l c τ 1 F l, l (l = 7,...,9) ω sine l ω cose l g l = c 24 F l + c 34 c l c l ω sine l ω cose l g l = c 24 F l c 34 c l c l ω cose l ω sine l + c 44 F l c 44, (l = 1,...,3) c l c l ω cose l ω sine l c 44 F l c 44, (l = 4,...,6) c l c l g l = c ω sine l 24 F l c ω cose l 34 c ω cose l 44 F l + c ω sine l 44, (l = 7,...,9) c l c l c l c l h l = K 3 F l ω cose l, (l = 1,...,3), h l = K 3 F ω cose l l, (l = 4,...,6) c l c l h l = K 3F l ω cos e l c l, (l = 7,...,9)

16 92 Joginder Singh, Baljeet Singh, Praveen Ailawalia τ 1 = (1 + iωt 1), τ 1 = (1 + iωt 1 ) The expressions for G 0, G 1, G 2, G 3, G 4, G 5 and G 6 are G 0 = K 3 ) 2 ( c C e τ 34 c 33, G1 = 2 K 3 C e τ ( c 34 c 23 c 24 c 33 ) ( G 2 = c 33 c K 3 C e τ (1 + c 33) + εη ) c 44 τ β 2 c 2 K 3 ( c22 c C e τ 33 2 c 24 c 34 c c ) K2 23 ( c33 C e τ c 2 ) 34 [{ G 3 = 2 c 24 c 33 c 34 c 23 + K } 2 3 C e τ ( c εη β 24 + c 34 ) + c 44 τ c εη β 24 c 44 τ c 34 K ] 3 C e τ ( c 22 c 34 c 24 c 23 ) K 2 C e τ ( c 24 c 33 c 34 c 23 ) { G 4 = (1 + c 33 ) + K 3 C e τ + εη } c 44 τ β 2 G 5 = 2 { c 4 + c 22 c c 24 c 34 (1 + c 23 ) 2 + K 3 C e τ ( c 22+1) + K 2 2 C e τ (1+ c εη β 33)+ c 44 τ + c 22 2 εη β c 44 τ (1+ c 23) + εη } c 44 τ c 33 c 2 + K 3 ) ( ) 2 K2 ( c C e τ 24 c 22 + C e τ (1 + c 23 ) 2 c 22 c c 24 c 34 [ { ( c 24 + c 34 ) c 4 + c 22 c 34 c 24 c 23 + K 2 + K 2 G 6 = c 6 ] C e τ ( c 24 c 23 c 22 c 34 ) { (1 + c 22 ) + K 2 C e τ + εη c 44 τ + K 2 C e τ ( c2 24 c 22) C e τ ( c 24 + c 34 ) + εη c 2 } c 44 τ ( c 34 c 24 β) } { c 4 + ( c 22 c 2 24)+ K 2 C e τ ( c )+ εη } c 44 τ c 2 c 2