Reflection of inhomogeneous plane waves at the surface of an unsaturated porothermoelastic media

A generalized porothermoelasticity theory developed by Wang et al. (IEEE Trans Geosci Remote Sens 60:1–14, 2021) for plane wave propagation in an unsaturated porothermoelastic media is employed in this study. This theory incorporates LS (Lord–Shulman) and GL (Green–Lindsay) theories. The generalized equations of motions are solved using the potential functions approach and predict the four dilatational waves and one shear wave. The incidence of the P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1$$\end{document} (or SV) wave generates the five reflected waves. The suitable potentials for the incident and reflected waves are considered, which meet the necessary boundary constraints with the aid of Snell’s law. The reflection coefficients are computed as a system of five linear non-homogeneous equations based on the permeable and impermeable boundary constrains. The partitioning of incident energy is then computed using these reflection coefficients. The mathematical derivations introduced in this study can investigate the impact of subsurface features (liquid saturation, porosity, surface pores characteristics, thermal expansion coefficients, wave frequency) on the propagation characteristics (propagation and attenuation directions, phase shift, energy ratio) of reflected waves. In addition, the propagation characteristics for LS (Lord–Shulman) and GL (Green–Lindsay) theories, and permeable and impermeable boundary constraints are compared numerically. Moreover, energy conservation is also verified at the stress-free surface of unsaturated porothermoelastic media.