On the propagation of second-sound in linear and nonlinear media: Results from Green–Naghdi theory

We investigate thermal wave propagation in one-dimensional media according to Green–Naghdi's heat conduction theory. Under the linearized theory, the dynamic propagation of a Heaviside input signal in a half-space is examined. Exact analytical solutions are derived for the three cases (i.e., types I–III) of this theory. We then numerically compare the evolution of the linear and nonlinear type-II temperature profiles, and track the finite-time blow-up of the latter's temperature rate wave, in the setting of an initial-boundary value problem involving a sudden sinusoidal input signal. Lastly, an exact traveling wave solution of a lossless, nonlinear equation, which arises under type-II theory, is determined and analyzed.