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We examine second-sound phenomena in a class of rigid, thermally conducting,solids that are described by a special case of the Maxwell--Catteneo flux law.Employing both analytical and numerical methods, we examine bothtemperature-rate waves and thermal traveling waves in this class of thermalconductor, which have recently been termed Graffi--Franchi--Straughan typeconductors. In the present study, the temperature-dependent nature of thethermal relaxation time, which is the distinguishing feature of this class ofconductors, gives rise to a variety of nonlinear effects; in particular,finite-time temperature-rate wave blow-up and temperature traveling waveformswhich exhibit a "tongue". The presentation concludes with a discussion ofpossible follow-on studies.

On the propagation of temperature-rate waves and traveling waves in rigid conductors of the Graffi--Franchi--Straughan type