Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1128 https://internationalpubls.com Reflection at the Free Surface of Generalized Thermoelastic Medium with Voids Under Three Phase Lag Effect Neelam Hooda1 and Renu Yadav* (Corresponding Author) 1Assistant Professor of Mathematics Govt. College for Women, Lakhan Majra, Rohtak, Haryana * Assistant Professor of Mathematics F. G. M. Govt. College Adampur, Hisar, Haryana Email: renumath87@gmail.com Article History: Received: 02-10-2024 Revised: 25-11-2024 Accepted: 26-12-2024 Abstract: In this paper, propagation of plane waves in an isotropic, homogeneous, generalized magneto-thermo-viscoelastic semi-infinite medium (having stress- free insulated boundary) containing a distribution of vacuous pores (voids) has been investigated. Basic governing equations are based on the linear theory of thermoelasticity, namely, three-phase-lag (3PL) thermoelasticity to account for finite velocity of the temperature. It is found that three compressional waves and a shear vertical (SV) wave can exist in the medium adopted. Reflection phenomena of compressional and shear waves from the free surface of the medium is considered. The formulae for reflection coefficients of various reflected waves have been obtained in the closed form. The numerical values of the modulus of reflection coefficients are presented graphically. Keywords: Three-phase-lag thermoelasticity, Reflection, Voids. Introduction Concept of generalized thermoelasticity has drawn the attention of many researchers during the last decades. Series of generalized theories of thermoelasticity has been developed to overcome the shortcomings inherent in the classical coupled dynamical theory of elasticity. These theories are characterized by the finite speed of propagation of thermal disturbance. Generalized thermoelastic model known as dual-phase-lag model was developed by Tzou (1995) by considering micro-structural effects into the delayed response in time at the macroscopic level by taking into account that the increase in the lattice temperature is delayed due to phonon-electron interactions on the macroscopic level. Tzou introduced two different phase lags, one for the heat flux vector,πœπ‘ž and the other for the temperature gradient,πœπ‘‡. For this model, classical Fourier’s law is thus modified to οΏ½βƒ—οΏ½(𝑝, 𝑑 + πœπ‘ž) = βˆ’πΎ[�⃗⃗�𝑇(𝑝, 𝑑 + πœπ‘‡)]. The delay time πœπ‘ž is interpreted as the relaxation time due to fast transient effects of thermal inertia. The other delay time πœπ‘‡is interpreted as that caused by the microstructural interactions. Stability of the dual-phase-lag heat conduction is discussed by Quintanilla and Racke (2006). Sixth and the most recent development in generalized thermoelasticity is three-phase-lag thermoelastic model. Roychoudhuri (2007) established this model by modifying heat conduction law to the form οΏ½βƒ—οΏ½(𝑝, 𝑑 + πœπ‘ž) = βˆ’[𝐾�⃗⃗�𝑇(𝑝, 𝑑 + πœπ‘‡) + πΎβˆ—οΏ½βƒ—βƒ—οΏ½π‘£(𝑝, 𝑑 + 𝜏𝜐)]. Here𝜏𝜐, the delay time in thermal displacement gradient is also introduced in addition to πœπ‘ž and πœπ‘‡. Stability of three-phase-lag heat conduction equation and the relations among the three material Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1129 https://internationalpubls.com parameters are discussed by Quintanilla and Racke (2008). Mukhopdhyay and Kumar (2010) analysed the effects of phase lags on wave propagation in a thick plate under axisymmetric temperature distribution. A magneto-thermoelastic problem in an unbounded, perfectly conducting medium with three phase lags has been discussed by Das and Kanoria (2012). Wave propagation in a porous media has its significance in diversified fields of science and engineering. The general non-linear theory of elastic materials with voids has been formulated by Nunziato and Cowin (1979). Linearized version of the said theory is developed by Cowin and Nunziato (1983), where void volume has been included as an additional kinematic variable. This theory reduces to the classical theory of elasticity in the limiting case when the void volume vanishes. Iesan (1986) established a theory of linear thermoelastic materials with voids Singh and Tomar (2007) investigated the propagation of plane waves in a thermoelastic material with voids. Many problems of waves and vibrations are studied by several researchers in an elastic material with voids. Some of them are Abo-Dahab et al. (2013) , Sharma and Kumar (2013), Malik et. al.(2022), and Kundu et al. (2022). The two-dimensional deformation of a generalized thermoelastic half-space with voids and microtemperatures under the action of a mechanical force was investigated by L. Rani (2023). In this paper, we have studied reflection of plane waves at the free surface of generalized thermoelastic medium with voids. The study is in the context of three phase lag thermoelasticity. It is found that there exist three sets of coupled dilatational waves and one set of coupled transversal waves. The reflection coefficients of various reflected waves are computed numerically for a specific model and their variations with angle of incidence are presented graphically. The effect of viscosity on reflection coefficients is demonstrated graphically. Nomenclature πœπ‘–π‘— Components of stress tensor πœ†, πœ‡ Lame’s constants 𝛽 (3πœ† + 2πœ‡)𝛼𝑑 𝛼𝑑 Coefficient of linear thermal expansion 𝑐𝑒 Specific heat at constant strain 𝐾 Thermal conductivity πΎβˆ— 𝑐𝑒(πœ†+2πœ‡) 4 , material constant 𝑇 Absolute temperature 𝑇0 Reference temperature 𝛩 Temperature deviation from the reference temperature 𝛩 = 𝑇 βˆ’ 𝑇0, | 𝛩 𝑇0 | β‰ͺ 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1130 https://internationalpubls.com 𝑒𝑖 Components of displacement vector 𝜌 Density of the medium 𝑒𝑖𝑗 Components of strain tensor π‘’π‘˜π‘˜ e, cubical dilatation πœ™ Change in volume fraction field 𝛿𝑖𝑗 Kronecker delta function Components of equilibrated stress vector Components of heat flux vector 𝛼, 𝑏, πœ‰1 Void material parameters π‘š Thermo-void coefficient πœ’ Equilibrated inertia t Time variable Basic equations and Problem Formulation In this section, a resumΓ© of the basic equations has been presented for the analysis of wave propagation in a generalized thermo-viscoelastic medium containing voids. The constitutive relation is πœπ‘–π‘— = πœ†π›Ώπ‘–π‘—π‘’π‘˜,π‘˜ + πœ‡(𝑒𝑖,𝑗 + 𝑒𝑗,𝑖) + π‘πœ™π›Ώπ‘–π‘— βˆ’ 𝛽𝛩𝛿𝑖𝑗, (1) The balance of linear momentum in the absence of body forces may be written as πœŒοΏ½ΜˆοΏ½π‘– = πœπ‘—π‘–,𝑗 (2) Again, the volume fraction field πœ™ satisfies the following equation [Iesan (1986)] 𝛼𝛻2πœ™ βˆ’ 𝑏(𝛻 β‹… οΏ½βƒ—βƒ—οΏ½) βˆ’ πœ‰1πœ™ + π‘šπ›© = πœŒπœ’οΏ½ΜˆοΏ½ (3) The heat equation corresponding to generalized thermoelasticity theory with three phase lags [Roychoudhuri (2008)] is [πΎβˆ— (1 + 𝜏𝜐 πœ• πœ•π‘‘ ) + 𝐾 πœ• πœ•π‘‘ (1 + πœπ‘‡ πœ• πœ•π‘‘ )] 𝛻2𝛩 = (1 + πœπ‘ž + πœπ‘ž 2 2 πœ•2 πœ•π‘‘2) (πœŒπ‘π‘’οΏ½ΜˆοΏ½ + 𝛽𝑇0�̈� + π‘šπ‘‡0�̈�) (4) In the above, the usual summation convention on repeated indices has been followed. Indices following comma indicate partial derivative with respect to those indices. We consider the propagation of plane waves in a homogeneous, isotropic, generalized semi-infinite solid occupying the region 𝑧 β‰₯ 0. Solid is assumed to be composed of material possessing voids. The surface 𝑧 = 0 is assumed to be unstressed, unstrained, thermally ih iq Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1131 https://internationalpubls.com insulated and initially at uniform temperature T0. Rectangular Cartesian coordinate system has been chosen with origin on the surface 𝑧 = 0. The z-axis is pointing vertically downward into the medium. Assuming xz-plane as the plane of incidence, the wave motion will be the same in every plane parallel to xz-plane which in turn implies that all physical quantities will be independent of y- coordinate. Thus the displacement vector (οΏ½βƒ—βƒ—οΏ½ = (𝑒, 0, 𝑀)) components, change in void volume fraction πœ™ and thermal parameter 𝛩 are 𝑒 = 𝑒(π‘₯, 𝑧, 𝑑), 𝑣 = 0, 𝑀 = 𝑀(π‘₯, 𝑧, 𝑑) , πœ™ = πœ™(π‘₯, 𝑧, 𝑑) and 𝛩 = 𝛩(π‘₯, 𝑧, 𝑑). Hence the dynamical equations for xz-plane may be obtained as follows 𝜌 πœ•2𝑒 πœ•π‘‘2 = (πœ† + πœ‡) πœ•π‘’ πœ•π‘₯ + πœ‡π›»2𝑒 βˆ’ 𝛽 πœ•π›© πœ•π‘₯ + 𝑏 πœ•πœ™ πœ•π‘₯ , (5) 𝜌 πœ•2𝑀 πœ•π‘‘2 = (πœ† + πœ‡) πœ•π‘’ πœ•π‘§ + πœ‡π›»2𝑀 βˆ’ 𝛽 πœ•π›© πœ•π‘§ + 𝑏 πœ•πœ™ πœ•π‘§ . (6) For convenience, we will make use of the following non-dimensional quantities (π‘₯β€², 𝑧′) = οΏ½Μ„οΏ½ 𝑐1 (π‘₯, 𝑧), (𝑑′, πœπ‘ž β€² , 𝜏𝜐 β€² , πœπ‘‡ β€² , 𝛽′) = οΏ½Μ„οΏ½(𝑑, πœπ‘ž, 𝜏𝜐, πœπ‘‡, 𝛽) , (𝑒′, 𝑀′) = πœŒοΏ½Μ„οΏ½π‘1 𝛽𝑇0 (𝑒, 𝑀),  𝛩′ = 𝛩 𝑇0 , β€„πœ™β€² = οΏ½Μ„οΏ½2πœ’ 𝑐1 2 πœ™, (πœπ‘§π‘₯ β€² , πœπ‘§π‘§ β€² ) = 1 𝛽𝑇0 (πœπ‘§π‘₯, πœπ‘§π‘§). (7) where οΏ½Μ„οΏ½ = πœŒπ‘π‘’π‘1 2 𝐾 , 𝑐1 = √ πœ†+2πœ‡ 𝜌 are the characteristic frequency and longitudinal wave velocity in the medium respectively. For investigation of plane waves, the potentials πœ“1(π‘₯, 𝑧, 𝑑),πœ“2(π‘₯, 𝑧, 𝑑) are introduced. They are related to displacement components u and w by the relation 𝑒 = πœ•πœ“1 πœ•π‘₯ βˆ’ πœ•πœ“2 πœ•π‘§ , 𝑀 = πœ•πœ“1 πœ•π‘§ + πœ•πœ“2 πœ•π‘₯ . (8) Plugging the above potentials from Eq.(8) into non-dimensional form of the Equations (1)-(4), we get (1 + 𝛼1 πœ• πœ•π‘‘ ) 𝛻2πœ“2 = π‘Ž1 πœ•2πœ“2 πœ•π‘‘2 , (9) [ πœ†π‘’ πœ‡π‘’ + 2] 𝛻2πœ“1 βˆ’ π‘Ž1 πœ•2πœ“1 πœ•π‘‘2 βˆ’ (1 + 𝛽 πœ• πœ•π‘‘ ) 𝛾2𝛩 βˆ’ π‘Ž2πœ™ = 0, (10) 𝛻2πœ™ βˆ’ π‘Ž3(𝛻2πœ“1) βˆ’ π‘Ž4πœ™ + π‘Ž5𝛩 βˆ’ π‘Ž6�̈� = 0, (11) [π‘Ž7 (1 + 𝜏𝜐 πœ• πœ•π‘‘ ) + πœ• πœ•π‘‘ (1 + πœπ‘‡ πœ• πœ•π‘‘ )] 𝛻2𝛩 = (1 + πœπ‘ž + πœπ‘ž 2 2 πœ•2 πœ•π‘‘2) (�̈� + π‘Ž8(1 + 𝛽 πœ• πœ•π‘‘ )�̈�1 + π‘Ž9�̈�). (12) where 𝛾2 = πœŒπ‘1 2 πœ‡π‘’ , π‘Ž1 = 𝛾2, π‘Ž2 = πœŒπ‘π‘1 4 𝛽𝑇0πœ‡οΏ½Μ„οΏ½2πœ’ , π‘Ž3 = π‘πœ’π›½π‘‡0 π›ΌπœŒπ‘1 2 , π‘Ž4 = πœ‰1𝑐1 2 οΏ½Μ„οΏ½2𝛼 ,β€„π‘Ž5 = π‘šπ‘‡0πœ’ 𝛼 , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1132 https://internationalpubls.com π‘Ž6 = πœŒπ‘1 2πœ’ 𝛼 , π‘Ž7 = πΎβˆ— 𝐾�̄� ,β€„π‘Ž8 = 𝛽2𝑇0 πΎπœŒοΏ½Μ„οΏ½ , β€„π‘Ž9 = π‘šπ‘1 4 πΎπœ’οΏ½Μ„οΏ½3 . Eq. (9) is uncoupled while equations (10)-(12) are coupled in πœ“1, 𝛩 and πœ™. To suit the actual situation of the problem, we seek solutions of differential equations (9)-(12) in the following forms: [πœ“1, πœ“2, 𝛩, πœ™](π‘₯, 𝑧, 𝑑) = [οΏ½Μ„οΏ½1, οΏ½Μ„οΏ½2, οΏ½Μ„οΏ½, οΏ½Μ„οΏ½] 𝑒π‘₯𝑝{ πœ„π‘˜(π‘₯ 𝑠𝑖𝑛 πœƒ βˆ’ 𝑧 π‘π‘œπ‘  πœƒ) βˆ’ πœ„πœ”π‘‘}, (13) where k is the wave number and πœ” is angular frequency connected by the relation πœ” = π‘˜π‘‰, V being the phase velocity and (𝑠𝑖𝑛 πœƒ , βˆ’ π‘π‘œπ‘  πœƒ) denotes the projection of wave normal of incident wave onto the xz-plane. Barred quantities are the amplitudes of the field quantities. Injecting Eq.(13) into Eqs.(10)-(12), we get respectively (π‘Ž1πœ”2𝑉2 + πœ„πœ”3π‘Ž10)οΏ½Μ„οΏ½1 + (πœ„πœ”π‘‰2𝛾2𝛽00)οΏ½Μ„οΏ½ + π‘Ž2𝑉2οΏ½Μ„οΏ½ = 0, (14) πœ”2π‘Ž3οΏ½Μ„οΏ½1 + π‘Ž5𝑉2οΏ½Μ„οΏ½ + (π‘Ž6πœ”2𝑉2 βˆ’ πœ”2 βˆ’ π‘Ž4𝑉2)οΏ½Μ„οΏ½ = 0, (15) and πœ„πœ”3𝛽00π‘Ž8π‘Ž11οΏ½Μ„οΏ½1 + (πœ”2πœπ‘‡0 + πœ„πœ”π‘Ž7𝜏𝜐0 + 𝑉2π‘Ž11)οΏ½Μ„οΏ½ + π‘Ž9π‘Ž11𝑉2οΏ½Μ„οΏ½ = 0, (16) where 𝛽00 = 𝛽 + πœ„ πœ” , 𝜏𝜐0 = 𝜏𝜐 + πœ„ πœ” , πœπ‘‡0 = πœπ‘‡ + πœ„ πœ” , π‘Ž10 = ( πœ†+2πœ‡ πœ‡ ), π‘Ž11 = 1 βˆ’ πœ„πœ”πœπ‘ž βˆ’ πœπ‘ž 2πœ”2 2 . The condition for the existence of non-trivial solution of the system of equations (14)-(16) provides us 𝑉6 + 𝐴𝑉4 + 𝐡𝑉2 + 𝐢 = 0 (17) where 𝐴 = 𝐴′ 𝐹 , 𝐡 = 𝐡′ 𝐹 , 𝐢 = 𝐢′ 𝐹 , 𝐹 = π‘Ž1π‘Ž6π‘Ž11πœ”4 βˆ’ π‘Ž1π‘Ž4π‘Ž11πœ”2 βˆ’ π‘Ž1π‘Ž5π‘Ž9π‘Ž11πœ”2, 𝐴′ = π‘Ž1π‘Ž6πœπ‘‡0πœ”6 + (πœ„π‘Ž1π‘Ž6π‘Ž7𝜏𝜐0 + πœ„π‘Ž6π‘Ž10π‘Ž11)πœ”5 βˆ’ (π‘Ž1π‘Ž4πœπ‘‡0 + π‘Ž1π‘Ž11)πœ”4 βˆ’(πœ„π‘Ž1π‘Ž4π‘Ž7𝜏𝜐0 + πœ„π‘Ž4π‘Ž10π‘Ž11 βˆ’ πœ„π‘Ž3π‘Ž9π‘Ž11𝛽00𝛾2 + πœ„π‘Ž5π‘Ž9π‘Ž10π‘Ž11)πœ”3 βˆ’ π‘Ž2π‘Ž3π‘Ž11πœ”2 , 𝐡′ = πœ„π‘Ž6π‘Ž10πœπ‘‡0πœ”7 βˆ’ (π‘Ž1πœπ‘‡0 + π‘Ž6π‘Ž7π‘Ž10𝜏𝜐0)πœ”6 βˆ’ (πœ„π‘Ž4π‘Ž10πœπ‘‡0 + πœ„π‘Ž1π‘Ž7𝜏𝜐0 + πœ„π‘Ž10π‘Ž11)πœ”5 βˆ’(π‘Ž2π‘Ž3πœπ‘‡0 βˆ’ π‘Ž4π‘Ž7π‘Ž10𝜏𝜐0)πœ”4 βˆ’ πœ„π‘Ž2π‘Ž3π‘Ž7𝜏𝜐0πœ”3, 𝐢′ = π‘Ž7π‘Ž10𝜏𝜐0πœ”6 + (πœ„π‘Ž8π‘Ž11𝛽00 βˆ’ πœ„π‘Ž10πœπ‘‡0)πœ”3. Using the transformation 𝑉2 = π‘Œ in Eq. (17), we obtain π‘Œ3 + π΄π‘Œ2 + π΅π‘Œ + 𝐢 = 0. (18) Eq. (18) is a cubic in 𝑉2, which implies that there shall be three dilatational waves travelling with three different velocities. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1133 https://internationalpubls.com Using Cardan’s method in Eq.(18), we get 𝑍3 + 3𝐻𝑍 + 𝐺 = 0, (19) where 𝑍 = π‘Œ + 𝐴 3 ,  𝐻 = 𝐡 3 βˆ’ 𝐴2 9 and 𝐺 = 2𝐴3 27 βˆ’ 𝐴𝐡 3 + 𝐢. Since the quantities A, B and C are complex, therefore the coefficients H and G are complex. Thus the roots of Eq.(19) and hence of Eq.(18) are given by 𝑉1 2 = 𝑆 βˆ’ 𝐴 3 , 𝑉2 2 = βˆ’ 1 2 𝑆 + πœ„ √3 2 𝑇 βˆ’ 𝐴 3 , 𝑉3 2 = βˆ’ 1 2 𝑆 βˆ’ πœ„ √3 2 𝑇 βˆ’ 𝐴 3 (20) where 𝑆 = π‘ˆ + 𝑉, 𝑇 = π‘ˆ βˆ’ 𝑉, π‘ˆ3 = 1 2 [βˆ’πΊ + √𝐺2 + 4𝐻3] and 𝑉 = βˆ’π» π‘ˆ . 𝑉1,2,3 are the speeds of propagation of three coupled dilatational waves namely longitudinal displacement wave (𝑃1), thermal wave (𝑃2) and longitudinal void volume fraction wave (𝑃3). It can be easily observed that speeds of all the coupled longitudinal waves are influenced by three phase lags (πœπ‘ž,𝜏𝜐andπœπ‘‡) and void parameters. Eq. (9) corresponds to the uncoupled transverse displacement wave (SV) whose velocity is given by 𝑉4 = √ 1 π‘Ž1 . (21) Reflection at the free surface Here, we shall consider the problem of incidence of a coupled longitudinal wave on the free and thermally insulated boundary of a thermoelastic half-space with voids. Fig. 1 Schematic diagram for the problem We assume that a set of coupled longitudinal waves of amplitude A0 propagating with the phase velocity V1 becomes incident obliquely at the free plane surface, making an angle ΞΈ0 with the normal. In order to satisfy the boundary conditions, this incident coupled longitudinal wave gives rise to the following reflected waves: (i) Three coupled longitudinal waves with amplitudes A1, A2 and A3 propagating with speeds V1,2,3 and making angles ΞΈ1,2,3 respectively with the normal. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1134 https://internationalpubls.com (ii) A transverse wave of amplitude B1 propagating with speed V4 making an angle ΞΈ4 with the normal. Full structure of the wave field consisting of the incident and reflected waves can be written as: πœ“1 = 𝐴0 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ 𝑠𝑖𝑛 πœƒ0 βˆ’ 𝑧 π‘π‘œπ‘  πœƒ0) βˆ’ πœ„πœ”π‘‘} + βˆ‘ 𝐴𝑖 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ 𝑠𝑖𝑛 πœƒπ‘– + 𝑧 π‘π‘œπ‘  πœƒπ‘–) βˆ’ πœ„πœ”π‘‘}3 𝑖=1 , (22) 𝛩 = π‘Ž1 βˆ—π΄0 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ 𝑠𝑖𝑛 πœƒ0 βˆ’ 𝑧 π‘π‘œπ‘  πœƒ0) βˆ’ πœ„πœ”π‘‘} + βˆ‘ π‘Žπ‘– βˆ—π΄π‘– 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ 𝑠𝑖𝑛 πœƒπ‘– + 𝑧 π‘π‘œπ‘  πœƒπ‘–) βˆ’ πœ„πœ”π‘‘}3 𝑖=1 , (23) πœ™ = 𝑏1 βˆ—π΄0 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ 𝑠𝑖𝑛 πœƒ0 βˆ’ 𝑧 π‘π‘œπ‘  πœƒ0) βˆ’ πœ„πœ”π‘‘} + βˆ‘ 𝑏𝑖 βˆ—π΄π‘– 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ 𝑠𝑖𝑛 πœƒπ‘– + 𝑧 π‘π‘œπ‘  πœƒπ‘–) βˆ’ πœ„πœ”π‘‘}3 𝑖=1 , (24) πœ“2 = 𝐡1 𝑒π‘₯𝑝{πœ„π‘˜4(π‘₯ 𝑠𝑖𝑛 πœƒ4 + 𝑧 π‘π‘œπ‘  πœƒ4) βˆ’ πœ„πœ”π‘‘}, (25) where π‘Žπ‘– βˆ— and 𝑏𝑖 βˆ— (𝑖 = 1,2,3) are the coupling parameters. Their expressions are given by π‘Žπ‘– βˆ— = (π‘Ž1π‘Ž6πœ”4βˆ’π‘Ž1π‘Ž4πœ”2)𝑉𝑖 4+(πœ„π‘Ž6π‘Ž10πœ”5βˆ’π‘Ž1πœ”4βˆ’πœ„π‘Ž4π‘Ž10πœ”3βˆ’π‘Ž2π‘Ž3πœ”2)𝑉𝑖 2βˆ’πœ„π‘Ž10πœ”5 (𝑔1𝑉𝑖 4+𝑔2𝑉𝑖 2) , 𝑏𝑖 βˆ— = (βˆ’π‘Ž1π‘Ž5πœ”2)𝑉𝑖 4+πœ„πœ”3(π‘Ž3𝛽00𝛾2βˆ’π‘Ž5π‘Ž10)𝑉𝑖 2 (𝑔1𝑉𝑖 4+𝑔2𝑉𝑖 2) , where 𝑔1 = πœ„πœ”π‘Ž4𝛽00𝛾2 βˆ’ πœ„πœ”3π‘Ž6𝛽00𝛾2 + π‘Ž2π‘Ž5 , 𝑔2 = πœ„πœ”3𝛽00𝛾2. The amplitudes 𝐴1, 𝐴2, 𝐴3 and 𝐡1 can be determined from the boundary conditions at 𝑧 = 0. The surface 𝑧 = 0 is assumed to be traction free and thermally insulated so that there is no variation of temperature and volume fraction field on it. Therefore, the boundary conditions are written as πœπ‘§π‘§ = 0, πœπ‘§π‘₯ = 0, πœ•π›© πœ•π‘§ = 0, πœ•πœ™ πœ•π‘§ = 0 at 𝑧 = 0 (26) The above boundary conditions are identically satisfied if and only if π‘˜1 𝑠𝑖𝑛 πœƒ1 = π‘˜2 𝑠𝑖𝑛 πœƒ2 = π‘˜3 𝑠𝑖𝑛 πœƒ3 = π‘˜4 𝑠𝑖𝑛 πœƒ4 and π‘˜1𝑉1 = π‘˜2𝑉2 = π‘˜3𝑉3 = π‘˜4𝑉4 . (27) Now, substituting the values of potentials πœ“1, 𝛩, πœ™ and πœ“2 from Eqs.(22)-(25) into the above boundary conditions one can obtain the following system of four simultaneous equations βˆ‘ 𝐴𝑖𝑗𝑍𝑗 = 𝐢𝑖, (𝑖, 𝑗 = 1,2,3,4), (28) where 𝐴1𝑗 = [π‘Ž12 βˆ’ 𝛿4 + π‘Ž13 π‘π‘œπ‘ 2 πœƒπ‘— + 𝑏1 𝑏𝑗 βˆ— π‘˜π‘— 2 + π‘Ž14 π‘Žπ‘— βˆ— π‘˜π‘— 2] π‘˜π‘— 2 π‘˜1 2, 𝐴2𝑗 = 𝑠𝑖𝑛 2 πœƒπ‘— π‘˜π‘— 2 π‘˜1 2, 𝐴3𝑗 = π‘Žπ‘— βˆ— π‘π‘œπ‘  πœƒπ‘— π‘˜π‘— π‘˜π‘— 2 π‘˜1 2, 𝐴4𝑗 = 𝑏𝑗 βˆ— π‘π‘œπ‘  πœƒπ‘— π‘˜π‘— π‘˜π‘— 2 π‘˜1 2. (𝑗 = 1,2,3). 𝐴14 = π‘Ž13 𝑠𝑖𝑛 πœƒ4 π‘π‘œπ‘  πœƒ4 π‘˜4 2 π‘˜1 2, 𝐴24 = βˆ’ π‘π‘œπ‘  2 πœƒ4 π‘˜4 2 π‘˜1 2, 𝐴34 = 0, 𝐴44 = 0, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1135 https://internationalpubls.com π‘Ž12 = βˆ’πœ”π›Ώ2, π‘Ž13 = βˆ’2𝛿3, π‘Ž14 = πœ„πœ”π›½00, 𝐢1 = βˆ’π΄11, 𝐢2 = 𝐴21, 𝐢3 = 𝐴31, 𝐢4 = 𝐴41, 𝑍1 = 𝐴1 𝐴0 , 𝑍2 = 𝐴2 𝐴0 , 𝑍3 = 𝐴3 𝐴0 , 𝑍4 = 𝐡1 𝐴0 . Here, 𝑍𝑖 (𝑖 = 1,2,3,4) are the reflection coefficients for the incidence of coupled longitudinal wave travelling with speed V1 . Numerical results and discussion Following Dhaliwal and Singh (1980), we consider an example where magnesium crystal like material is modeled as an isotropic thermoelastic solid with voids for numerical computations. For this test material, elastic and thermal constants are πœ† = 2.17 Γ— 1010π‘π‘šβˆ’2, β€„πœ‡ = 3.278 Γ— 1010π‘π‘šβˆ’2, 𝛽 = 2.68 Γ— 106 π‘π‘šβˆ’2 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1   ,   𝑇0 = 298𝐾, 𝐾 = 1.7 Γ— 102π‘Šπ‘šβˆ’1 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1, 𝑐𝑒 = 1.04 Γ— 103π½πΎπ‘”βˆ’1 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1, β€„πœŒ = 1.74 Γ— 103πΎπ‘”π‘šβˆ’3. Void parameters are given by 𝛼 = 3.688 Γ— 10βˆ’5𝑁,β€„β€„β€„πœ‰1 = 1.475 Γ— 1010π‘π‘šβˆ’2, β€„β€„πœ’ = 1.753 Γ— 10βˆ’15π‘š2, 𝑏 = 1.13849 Γ— 1010π‘π‘šβˆ’2, β€„β€„β€„π‘š = 2 Γ— 106π‘π‘šβˆ’2 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1. Other constants involved in the problem are: 𝜏𝜐 = 0.1,β€„πœπ‘ž = 0.2,β€„πœπ‘‡ = 0.15, β€„πœ” = 3.5 Γ— 1012. To discuss the nature of dependence of reflection coefficients on the angle of incidence, we have computed their expressions using Matlab software. All the relative amplitudes are found to be complex valued. All the figures have been taken in the context of thermoelastic theory based on: (i) Three-phase-lag model with voids (3PLV) (ii) GN-III model with voids (GN3V) (iii) Three-phase-lag model without voids (3PLWV). Figures 2-5 are drawn for incidence of a P-wave of speed 𝑉1. Considered range for angle of incidence is 0∘ ≀ πœƒ ≀ 90∘. Figure 2 shows the variations in the absolute values of reflection coefficient 𝑍1. It is clear that |𝑍1|has value almost equal to unity during the whole range of incidence for all the three models considered. Despite of this, difference in trends of variations of |𝑍1| for 3PLV, GN3V and 3PLWV models is easily noticeable from the graph. Presence of voids increases the values of |𝑍1|in the entire range of angle of incidence. Absence of three phase lags increases the values till πœƒ = 45∘and after that causes a decrement in the values. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1136 https://internationalpubls.com In Figure 3, the reflection coefficients |𝑍2| decrease with increasing angle of incidence, which are affected due to void parameters and relaxation times. In the absence of voids and three relaxation times πœπ‘ž,𝜏𝜐andπœπ‘‡, the values of |𝑍2| rise at each angle of incidence. Variations in the values of |𝑍3| against the angle of incidence are depicted in the figure 4. Magnitude of 𝑍3 is very small for all the three cases. Also pattern of variations in the graph is similar to that for |𝑍2|. Figure 5 characterizes the behaviour of |𝑍4|with increasing angle of incidence. Its trend of variations is different from the remaining reflection coefficients. |𝑍4| starts from zero value, increases till it attains its maximum value at πœƒ = 45∘and then decreases smoothly to zero. Absence of relaxation times ( three phase lags) causes an increment in the numerical values of 𝑍4 while absence of voids decreases the numerical values. Concluding remarks The facts extracted from our study can be concluded as: (i) It can be checked out from the calculatory part that amplitude ratios are independent of the wavelength of the incident wave but depend only upon angle of incidence and wave numbers of the incident wave. (ii) At grazing incidence (πœƒ = 90∘) of longitudinal wave of speed 𝑉1, no other reflected wave appears except the longitudinal wave of the same amplitude as that of incident wave. In case of normal incidence (πœƒ = 0∘), the reflected longitudinal wave having maximum amplitude is the wave of amplitude equal to that associated with incident P-wave. Thus at normal and grazing incidence, it appears that incident P-wave is reflected as a P-wave. (iv) Voids and three phase lag parameters are having a pronounced effect on reflection coefficients. References 1. Abo-Dahab, S.M., Abd-Alla, A.M. and Mahmoud, S.R., Effects of voids and rotation on plane waves in generalized thermoelasticity, J. Mech. Sci. Tech., Vol. 27, pp. 3607- 3614, 2013 2. Cowin, S.C. and Nunziato, J.W., Linear theory of elastic materials with voids, J. Elasticity, Vol. 13, pp. 125-147, 1983 3. Das, P. and Kanoria, M., Magneto-thermo-elastic response in a perfectly conducting medium with three-phase-lag effect, Acta Mech., Vol. 223, pp. 811-828, 2012. 4. Dhaliwal, R.S. and Singh, A., Dynamic Coupled Thermoelasticity, Hindustan Publ. Corp., New Delhi. p. 726, 1980. 5. Iesan, D., A theory of thermoelastic materials with voids, Acta Mech., Vol. 60, pp. 67- 89, 1986. 6. Kundu, S., Kalkal, K.K., Sangwan, M. and Sheoran, D. , Two-dimensional deformations in an initially stressed nonlocal micropolar thermoelastic porous medium subjected to a moving thermal load, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33 No. 3, pp. 1116-1143.(2023). https://www.emerald.com/insight/search?q=Sandeep%20Kundu https://www.emerald.com/insight/search?q=Kapil%20Kumar%20Kalkal https://www.emerald.com/insight/search?q=Monika%20Sangwan https://www.emerald.com/insight/search?q=Devender%20Sheoran https://www.emerald.com/insight/publication/issn/0961-5539 https://www.emerald.com/insight/publication/issn/0961-5539 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1137 https://internationalpubls.com 7. Malik, S., Gupta, D., Kumar, K., Sharma, R.K., Plane wave propagation and fundamental solution in functionally graded couple stress micropolar thermoelastic solid with diffusion and voids, Wave Rand. Compl. Media. DOI: 10.1080/17455030.2022.2155331, 2022. 8. Mukhopdhyay, S. and Kumar, R., Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution, Acta Mech., Vol. 210, pp. 331-344, 2010. 9. Nunziato, J.W. and Cowin, S.C., A non-linear theory of elastic materials with voids, Arch. Ration. Mech. Anal., Vol. 12, pp. 175-201, 1979. 10. Quintanilla, R. and Racke, R., A note on stability in dual-phase-lag heat conduction, Int. J. Heat Mass Transf., Vol. 49, pp. 1209-1213, 2006. 11. Roychoudhuri, S.K., On a thermoelastic three-phase-lag model, J. Therm. Stress., Vol. 30, pp. 231-238, 2007. 12. Sharma, K. and Kumar, P., Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, J. Therm. Stress., Vol. 36, pp. 94-111, 2013. 13. Singh, J. and Tomar, S.K., Plane waves in thermoelastic materials with voids, Mech. Mate., Vol. 39, pp. 932-940, 2007. 14. Tzou, D.Y., A unified field approach for heat conduction from macro to micro scales, ASMEJ. Heat. Transf., Vol. 117, pp. 8-16, 1995. Figure Captions: Fig. 2 Variation of the modulus of reflection coefficient Z1 with the angle of incidence of coupled longitudinal wave with speed V1 Fig. 3 Variation of the modulus of reflection coefficient Z2 with the angle of incidence of coupled longitudinal wave with speed V1. Fig. 4 Variation of the modulus of reflection coefficient Z3 with the angle of incidence of coupled longitudinal wave with speed V1. Fig. 5 Variation of the modulus of reflection coefficient Z4 with the angle of incidence of coupled longitudinal wave with speed V1. https://doi.org/10.1080/17455030.2022.2155331 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1138 https://internationalpubls.com