A temperature pulse propagating in a solid is studied by simple wave theory. The aim is to determine the shape change of the pulse, which initially is given by a Gaussian function, using a generalized nonlinear Cattaneo model proposed, in the framework of extended thermodynamics, by Ruggeri and co-workers. We prove that the characteristic temperature \ensuremath{\theta}\ifmmode \tilde{}\else \~{}\fi{}, already pointed out in previous papers, plays an essential role in the shape change of the propagating second sound wave. In fact, three different shape changes can occur: (i) The back branch of the wave becomes steeper than the front one, (ii) vice versa, the steepening is forwards, or (iii) a double shock effect arises. These possibilities depend on the relation of the unperturbed temperature of the body to the characteristic temperature and, in some cases, on the wave amplitude. We discuss both the usual experimental case of a hot wave and the case of a cold wave, proving that this last process is not symmetric with respect to the previous one. Nevertheless, our calculations show that the effects described here are not so evident in the usual experimental conditions. Finally, for NaF and Bi crystals we discuss the best choice of some parameters in order to see clearly the changes of the wave profile and, in particular, the double shock effect. \textcopyright{} 1996 The American Physical Society.