Waves at the imperfect boundary of elastic and bio-thermoelastic diffusive media

Abstract

The present investigation deals with the reflection and transmission phenomena in isotropic elastic half-space (ES) and bio-thermoelastic diffusive half-space (BDS) for the study of plane harmonic waves. The two half-spaces are separated by an imperfect interface and the governing equations for BDS are modified by introducing phase lag parameters in Fourier’s law of heat conduction and Fick’s law of mass diffusion. After expressing the equations in two-dimension, the potential functions are used to simplify the equations for further calculation. The plane P or SV wave is incident from ES which results in two reflected waves in ES and four transmitted waves in BDS. The amplitude ratios are obtained for these reflected and transmitted waves concerening the incident P or SV wave for stiffness forces. The obtained amplitude ratios are further used to drive the energy ratios of various reflected and transmitted waves. These ratios are impacted by the characteristics of the wave and material properties of the media. The resulting energy ratios are computed numerically by developing a numerical algorithm and displayed in the form of graphs to show the effects of blood perfusion rate, lagging times, and stiffness parameters.

Appendices

Appendix A

$$\begin{aligned} A_1&=CH+EC, \ A_2=HB+GC-DF+\omega ^2CH+AH-AF, \ A_3=GB+\omega ^2(HB+GC-DF)+AG, \ A_4=\omega ^2GB,\\ A&=-i\omega \tau _{q2}R_1,\ B=-\omega \tau _{q2}-\tau _{q2}R_3,\ C=-\tau _{T2},\ D=-\omega \tau _{q2}R_2,\ E=\tau _{P2q_1^*},\\ F&=\tau _{P2}q_2^*,\ G=-i\omega \tau _{\eta 2}q_4^*,\ H=-\tau _{P2}q_3^*,\ q_1^*=D\beta _2,\ q_2^*=\frac{Da\rho c_1^2}{\beta _1},\\ q_2^*&=\frac{Db\rho c_1^2}{\beta _2},\ q_4^*=\frac{\rho c_1^4}{\omega _1^* \beta _2},\ \tau _{T2}=1-i\omega \tau _T,\ \tau _{q2}=1-i\omega \tau _q+(-i\omega )^2\frac{\tau _q^2}{2},\ \tau _{P2}=1-i\omega \tau _P,\ \tau _{\eta 2}=1-i\omega \tau _{\eta }+(-i\omega )^2\frac{\tau _{\eta }^2}{2},\\ n_i&=\frac{-(AH-ED)\omega ^4+AH\omega ^2V_i^2}{CH\omega ^4-\omega ^2V_i^2(CG+HB-DF)+BGV_i^4},\\ k_i&=\frac{EC\omega ^6-(EB-AF)\omega ^4V_i^2}{CH\omega ^4-\omega ^2V_i^2(CG+HB-DF)+BGV_i^4}. \end{aligned}$$

Appendix B

$$\begin{aligned} d_{11}&=2\mu ^e\bigg (\frac{\xi _r}{\omega }\bigg )^2-\rho ^ec_1^2,\ d_{12}=2\mu ^e \frac{\xi _r}{\omega }\frac{dV_{\beta }}{\omega },\\ d_{1j}&=\lambda \bigg (\frac{\xi _r}{\omega }\bigg )^2+\frac{\rho c_1^2}{\omega ^2}\bigg [dV_j^2+n_j+k_j\bigg ],\ d_{16}=2\mu \frac{\xi _r}{\omega }\frac{dV_4}{\omega },\\ d_{21}&=2\mu ^e\frac{\xi _r}{\omega }\frac{dV_{\alpha }}{\omega },\ d_{22}=\mu ^e\bigg [\bigg (\frac{dV_{\beta }}{\omega }\bigg )^2+\bigg (\frac{\xi _r}{\omega }\bigg )^2\bigg ],\\ d_{2j}&=2\mu \frac{\xi _r}{\omega }\frac{dV_j}{\omega },\ d_{26}=\mu \bigg [\bigg (\frac{\xi _r}{\omega }\bigg )^2-\bigg (\frac{dV_4}{\omega }\bigg )^2\bigg ],\\ d_{31}&=k_t\frac{\xi _r}{\omega },\ d_{32}=k_t\frac{dV_{\beta }}{\omega },\\ d_{3j}&=2\mu \frac{\xi _r dV_j}{i\omega }-\xi _r\frac{k_t}{\omega }, \ d_{36}=\mu \frac{\xi _r^2-dV_4^2}{i\omega }+k_t\frac{dV_4}{\omega },\\ d_{41}&=-k_n\frac{dV_{\alpha }}{\omega },\ d_{42}=k_n\frac{\xi _r}{\omega },\\ d_{4j}&=-k_n\frac{dV_i}{\omega }+(\rho c_1^2-2\mu )\frac{\xi _r^2}{i\omega }+\frac{\rho c_1^2}{i\omega }(dV_j^ 2+n_j+k_j), \\ d_{46}&=-k_n\frac{\xi _r}{\omega }+2\mu \xi _r\frac{dV_4}{i\omega },\\ d_{51}&=d_{52}=d_{56}=0,\\ d_{5j}&=in_j\frac{dV_j}{\omega },\ d_{61}=d_{62}=d_{66}=0,\\ d_{6j}&=ik_j\frac{dV_j}{\omega },\ j=1,2,3,\\ \frac{dV_{\alpha }}{\omega }&=\bigg (\frac{1}{V_P^2}-\bigg (\frac{\xi _r}{\omega }\bigg )^2\bigg )^{\frac{1}{2}}=\bigg (\frac{1}{V_P^2}-\frac{\sin ^2\theta _0}{V_0^2}\bigg )^{\frac{1}{2}}, \frac{dV_{\beta }}{\omega }=\bigg (\frac{1}{V_S^2}-\frac{\sin ^2\theta _0}{V_0^2}\bigg )^{\frac{1}{2}},\\ \frac{dV_j}{\omega }&=p.v.\bigg (\frac{1}{V_j^2}-\frac{\sin ^2\theta _0}{V_0^2}\bigg )^{\frac{1}{2}}, \ j=1,2,3,4. \end{aligned}$$

Appendix C

$$\begin{aligned} \langle P_{ij}^* \rangle&=-\frac{\omega ^4}{2}Re\bigg [\bigg (2\mu \frac{dV_i}{\omega }\frac{\xi _r}{\omega }\frac{\bar{\xi _r}}{\omega }\bigg )+\bigg (\lambda \bigg (\frac{\xi _r}{\omega }\bigg )^2\frac{\bar{dV_j}}{\omega }+\rho c_1^2\bigg (\frac{dV_i}{\omega }\bigg )^2+\frac{\rho c_1^2 n_i}{\omega ^2}+\frac{\rho c_1^2 k_i}{\omega ^2}\bigg )Z_{i+1}\bar{Z}_{j+2}\bigg ],\\ \langle P_{i4}^* \rangle&=-\frac{\omega ^4}{2}Re \bigg [\bigg (-2\mu \frac{dV_i}{\omega }\frac{dV_4}{\omega }\frac{\xi _r}{\omega }\bigg )Z_{i+2}\bar{Z}_{6}\bigg ],\\ \langle P_{4j}^* \rangle&=-\frac{\omega ^4}{2}Re \bigg [\bigg (\mu \bigg (\frac{\xi _r}{\omega }-\frac{dV_4}{\omega }\bigg )\frac{\bar{\xi _r}}{\omega }+2\mu \frac{\xi _r}{\omega }\frac{dV_4}{\omega }\frac{dV_j}{\omega }\bigg )Z_6\bar{Z}_{j+2}\bigg ],\\ \langle P_{44}^* \rangle&=-\frac{\omega ^4}{2}\bigg [\bigg ( \mu \bigg (\bigg (\frac{\xi _r}{\omega }\bigg )^2-\bigg (\frac{dV_4}{\omega }\bigg )^2\bigg )\frac{\bar{dV}_4}{\omega }+2\mu \frac{\xi _r}{\omega }\frac{\bar{\xi _r}}{\omega }\frac{dV_4}{\omega }\bigg ) Z_6\bar{Z}_6 \bigg ]. \end{aligned}$$