Reflection and Refraction of P Wave at the Interface Between Thermoelastic and Porous Thermoelastic Medium

Abstract

Reflection and refraction phenomenon due to an obliquely incident longitudinal wave at a plane interface between an isotropic, homogeneous, thermoelastic medium and a porous thermoelastic medium is studied. Firstly, the generalized thermoelastic theory (G-TE) of wave propagation in a saturated porous thermoelastic medium is developed in the context of the coupled thermoelastic theories of Biot, the theory of generalized thermoelasticity of Lord–Shulman (with one relaxation time) and that of Green–Lindsay (with two relaxation times). Then, the expressions of amplitude ratios of various reflected and refracted waves to that of incident P wave are derived. Numerical results are obtained and used to analyze the difference of reflection amplitude of four kinds of reflection waves, and refraction amplitude of three kinds of refraction waves among the G-TE theory, the Lord–Shulman theory and the Green–Lindsay theory in the case of porous thermoelasticity. The effect of relaxation time $\tau _{_1}$ and thermal expansion coefficient $a_\mathrm{s}$ on the reflection and refraction of the incident P-wave is also discussed.

Appendix

$$\begin{aligned} d_{11}= & {} \left[ {\lambda +\alpha ^{2}M+\alpha M\zeta _1 +2G\cos ^{2}\theta _1 } \right] l_\mathrm{p1}^2 +\left( {a_\mathrm{c} \alpha M+\lambda ^{{\prime }}\left( {1\hbox {+i}\omega \tau _\mathrm{1} } \right) } \right) \eta _1 \\ d_{12}= & {} \left[ {\lambda +\alpha ^{2}M+\alpha M\zeta _2 +2G\cos ^{2}\theta _2 } \right] l_\mathrm{p2}^2 +\left( {a_\mathrm{c} \alpha M+\lambda ^{{\prime }}\left( {1\hbox {+i}\omega \tau _\mathrm{1} } \right) } \right) \eta _2 \\ d_{13}= & {} \left[ {\lambda +\alpha ^{2}M+\alpha M\zeta _3 +2G\cos ^{2}\theta _3 } \right] l_\mathrm{T}^2 +\left( {a_\mathrm{c} \alpha M+\lambda ^{{\prime }}\left( {1\hbox {+i}\omega \tau _\mathrm{1} } \right) } \right) \eta _3 \\ d_{14}= & {} -G\sin 2\theta _4 l_\mathrm{s}^2 \\ d_{15}= & {} -\left\{ {\left[ {\bar{{\lambda }}+2\bar{{G}}\cos ^{2}\theta _5 } \right] \bar{{l}}_\mathrm{p}^2 +\bar{{\lambda }}^{{\prime }}\left( {1\hbox {+i}\omega \tau _\mathrm{1} } \right) \beta _1 } \right\} \\ d_{16}= & {} -\left\{ {\left[ {\bar{{\lambda }}+2\bar{{G}}\cos ^{2}\theta _6 } \right] \bar{{l}}_\mathrm{T}^2 +\bar{{\lambda }}^{{\prime }}\left( {1\hbox {+i}\omega \tau _\mathrm{1} } \right) \beta _2 } \right\} \\ d_{17}= & {} \bar{{G}}\sin 2\theta _7 \bar{{l}}_\mathrm{s}^2 \\ d_{21}= & {} \hbox {G}l_\mathrm{p1}^2 \sin 2\theta _1 , d_{22} =\hbox {G}l_\mathrm{p2}^2 \sin 2\theta _2 , d_{23} =\hbox {G}l_\mathrm{T}^2 \sin 2\theta _3 , d_{24} =\hbox {G}\cos 2\theta _4 l_\mathrm{s}^2 \\ d_{25}= & {} \bar{{G}}\bar{{l}}_\mathrm{p}^2 \sin 2\theta _5 , d_{26} =\bar{{G}}\bar{{l}}_\mathrm{T}^2 \sin 2\theta _6 , d_{27} =-\bar{{G}}\cos 2\theta _7 \bar{{l}}_\mathrm{s}^2 \\ d_{31}= & {} l_\mathrm{p1} \cos \theta _1 , d_{32} =l_\mathrm{p2} \cos \theta _2 , d_{33} =l_\mathrm{T} \cos \theta _3 , d_{34} =-l_\mathrm{s} \sin \theta _4 \\ d_{35}= & {} \bar{{l}}_\mathrm{p} \cos \theta _5 , d_{36} =\bar{{l}}_\mathrm{T} \cos \theta _6 , d_{37} =\bar{{l}}_\mathrm{s} \sin \theta _7 \\ d_{41}= & {} l_\mathrm{p1} \sin \theta _1 , d_{42} =l_\mathrm{p2} \sin \theta _2 , d_{43} =l_\mathrm{T} \sin \theta _3 , d_{44} =l_\mathrm{s} \cos \theta _4 \\ d_{45}= & {} -\bar{{l}}_\mathrm{p} \sin \theta _5 , d_{46} =-\bar{{l}}_\mathrm{T} \sin \theta _6 , d_{47} =\bar{{l}}_\mathrm{s} \cos \theta _7 \\ d_{51}= & {} \eta _1 , {d_{52} =\eta _2}, {d_{53} =\eta _3 }, {d_{54} \hbox {=0}} \\ d_{55}= & {} -\beta _1, {d_{56} =-\beta _2 }, {d_{57} \hbox {=0}} \\ d_{61}= & {} \left( {\alpha +\zeta _1 } \right) l_\mathrm{p1}^2 +a_\mathrm{c} \eta _1 , d_{62} =\left( {\alpha +\zeta _2 } \right) l_\mathrm{p2}^2 +a_\mathrm{c} \eta _2 , d_{63} =\left( {\alpha +\zeta _3 } \right) l_\mathrm{T}^2 +a_\mathrm{c} \eta _3 \\ d_{64}= & {} 0, {d_{65} =0}, {d_{65} =0}, d_{66} =0, {d_{67} =0}, \\ d_{71}= & {} l_\mathrm{p1} \eta _1 \cos \theta _1 , d_{72} =l_\mathrm{p2} \eta _2 \cos \theta _2 , d_{73} =l_\mathrm{T} \eta _3 \cos \theta _3 \\ d_{74}= & {} 0, d_{75} =\bar{{l}}_\mathrm{p} \beta _1 \cos \theta _5 , d_{76} =\bar{{l}}_\mathrm{T} \beta _2 \cos \theta _6 , {d_{77} =0,} \\ b_1= & {} -\left[ {\lambda +\alpha ^{2}M+\alpha M\zeta _0 +2G\cos ^{2}\theta _0 } \right] l_\mathrm{p1}^2 -\left( {a_\mathrm{c} \alpha M+\lambda ^{{\prime }}\left( {1\hbox {+i}\omega \tau _\mathrm{1} } \right) } \right) \eta _0\\ b_2= & {} \hbox {G}l_\mathrm{p1}^2 \sin 2\theta _0 , b_3 =l_\mathrm{p1} \cos \theta _0 , b_4 =-l_\mathrm{p1} \sin \theta _0 , b_5 =-\eta _0\\ b_6= & {} -\left[ {\left( {\alpha +\zeta _0 } \right) l_\mathrm{p1}^2 +a_\mathrm{c} \eta _0 } \right] , b_7 =l_\mathrm{p1} \eta _0 \cos \theta _0 \end{aligned}$$