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.     

Second-sound propagation in rigid,nonlinear conductors

The main part of this work is devoted to investigating second-sound in metals. The analysis is carried out in the context of shock and traveling wave phenomena. First, it is shown that the thermal transport equation exactly linearizes under a simple, algebraic transformation. This feature is then used, following energy and traveling wave analyses, to obtain both singular surface-based shock results and the exact solution to a signaling-type problem involving the Heaviside boundary condition (BC). Finally, the question of second-sound in dielectric solids is briefly considered and it is shown how the approach used for metals can be applied to the former.