On the Propagation of Second-Sound in Nonlinear Media: Shock, Acceleration and Traveling Wave Results

Abstract

An exhaustive study of thermal shock, acceleration (or temperature-rate) and traveling waves in media with temperature-dependent thermal conductivity, and within which the flux is described by the Maxwell–Catteneo law, is presented. The resulting quasilinear, hyperbolic system of equations predicts a variety of interesting phenomena in such media including, but not limited to, dynamic thermal shock waves with jump-dependent wave speed, finite-time temperature-rate wave blow-up, and temperature-rate wave formation as traveling waves reach a critical speed. Both analytical (integral transforms, singular surface theory, solution of transcendental equations using special functions) and numerical approaches are used; the former are benchmarked against the latter where appropriate. Parallels are drawn between the nonlinear heat waves discussed in the present work and nonlinear wave phenomena in related “classical” continuum theories such as acoustics.