The interaction of a light pulse with reflective and either a passive, lossy medium or an active medium with population inversion gives rise to elastic waves, already as a result of the change in the momentum carried by the incident light. We derived a 1D analytic displacement field that quantitatively predicts the shape and amplitude of such waves in semi-infinite and finite elastic rods in a half-space and infinite layer. The results are compatible with the conservation of momentum and energy of the light-matter system. They can be used as a signature for direct measurements of the radiation-pressure-induced elastic waves and to clarify the Abraham-Minkowski momentum dilemma.