We propose an analytical framework to study the propagation of transition waves in one-dimensional free-standing lattices comprising concentrated masses and bistable springs. Starting from the discrete model, we derive the partial differential equation that describes the non-linear dynamic response of the system and obtain its exact closed-form solution. Such solution enables us to uncover the effect of the system’s parameters on the characteristics of the supported transition waves as well as to characterize the rarefaction front that precedes the pulse. As such, this work provides new opportunities to elucidate how transition waves propagate in systems in which the phase transition results in macroscopic volumetric changes.