The equations of thermoelasticity are briefly derived. A non-linear constitutive equation for the heat flux vector is proposed as a more accurate description of heat flow at low temperatures. In the linear theory, it is shown that an exact equation, governing dilatational wave propagation, must be used for certain real materials. The corresponding approximate equation deduced by Lord and Shulman  predicts unstable lower order waves. The strong dilatational shock equations are completed for the extended heat conduction law. Dilatational constant profile waves are studied and used to discuss the shock structure of the dilatational non-heat conducting shock. Two methods for predicting the breakdown of the constant profile wave solution are compared. Both methods coincide in their estimation of the strain that would cause the breakdown. Some conclusions are drawn for real materials. © 1973 Springer-Verlag.