The propagation of small amplitude stress discontinuities in linear viscoelastic media with differential stress‐strain laws is considered. The theory of weak solutions is used to determine the curves of discontinuity and the jumps in the stress and the velocity and their successive derivatives on the discontinuity. A Taylor series expansion is then used to approximate the stress and the velocity behind and sufficiently near the discontinuity. These results are applied to study the propagation of longitudinal impact waves in semi‐infinite and finite viscoelastic rods.