The method of the characteristics is used to study the influence of thermomechanical coupling on the magnitudes of discontinuities at the wavefronts of dilatational waves. A set of unified equations is employed, which is applicable to plane, cylindrical and spherical waves. The thermal conductivity is taken as a function of the space coordinate. For cylindrical and spherical symmetry the discontinuity at the wavefront of a diverging wave is subjected to both material damping and decay due to geometry. The sharp wavefront is attenuated over a very short distance. For converging waves the material damping is counteracted by an increase in magnitude due to geometry. It is shown that material damping is predominant, except at extremely small distances from the center which fall outside the realm of continuum theory.