The simplest form of longitudinal constitutive equation for a compressible solid which exhibits second order convective and dissipative effects is σ = Am+Bm2+ηδm/δt. A constant profile waveform, analogous to that discovered independently both by Taylor and by Rayleigh in 1910 for a compressible fluid, can propagate in such a solid. It is shown that any monotonic waveform ultimately adopts this profile; the width and time of formation of the profile are found. The constant profile waveform appropriate to transverse vibration in an incompressible solid is investigated. Both waveforms propagate at the velocities given by the Rankine-Hugoniot equations in the absence of dissipation.