Singular surface analysis and properties of solutions for a theoretical thermoporoacoustic model

Within the framework of theoretical thermoporoacoustics, some recent results are proposed that were obtained by investigating the linearized version of the so-called Eringen-Cattaneo-Christov-Straughan model; it arises from a well-established theory of swelling porous elasticity developed by A.C. Eringen (1994) involving, however, the Cattaneo-Christov approach for the heat flux (B. Straughan, 2020). Wave propagation in a rigid, stationary porous matrix saturated by a perfect gas is studied, allowing for temperature variations; in this regard, the classical Fourier law is abandoned and second sound—i.e., the phenomenon whereby heat travels as a thermal wave—is contemplated. A singular surface-based analysis is performed to examine the evolution of coupled thermal and acoustic pressure wavefronts arising in a signaling problem, that is, considering a Heaviside temperature input source. The Laplace transform and its properties are applied for this purpose, and numerical results are presented to support the analytical findings. In detail, the amplitudes and speeds of the emerging shock waves are derived and analyzed, several special/limiting cases are identified, and the effects of variations in the Eringen coefficient are determined. Finally, for a suitably constructed initial-boundary value problem, some results on the qualitative properties of the solution are provided.