  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 both
temperature-rate waves and thermal traveling waves in this class of thermal
conductor, which have recently been termed Graffi--Franchi--Straughan type
conductors. In the present study, the temperature-dependent nature of the
thermal relaxation time, which is the distinguishing feature of this class of
conductors, gives rise to a variety of nonlinear effects; in particular,
finite-time temperature-rate wave blow-up and temperature traveling waveforms
which exhibit a "tongue". The presentation concludes with a discussion of
possible follow-on studies.
