We extend our original analysis begun in Tarkenton et al. [Phys. Rev. B 49, 11 794 (1994)] to include dissipative effects that are important in real cryogenic systems where nonlinear second sound exists. We present results concerning arrival times of thermal pulses propagated in cryogenic crystals, namely the behavior of the arrival times as a function of pulse amplitude. These arrival times show some surprising effects due to competing nonlinear terms: after decreasing with increasing amplitude, as one would expect, the arrival times start to lengthen due to nonlinear effects and finally saturate at a level slightly above the shortest arrival times. All these surprising effects arise from competing nonlinear terms in the expression for the wave speed. We finally relate these results to the experiment we proposed in our original paper.