Chemical reactions and phase transformations couple in solid systems with the stress fields. When reaction rates become of the same order as the rate of relaxation of mechanical perturbations, the reactions can stabilize elastic waves in the system. Density varies with temperature and conversion and since the elastic waves are driven by the gradients in temperature and conversion, these source terms travel with the same speed as the reaction front. A model is presented for elastic waves in solid media, driven by a moving source term that is well approximated by a delta function. The propagation velocity of this source is constant, but it can propagate either subsonically or supersonically. The effects of precompressing the sample, applying a force at one boundary, and varying the strength of the source term with time are included in the discussion. The model is useful to study transformation reactions under shock-compression, ultrafast deflagrations, and detonations in the solid phase.