Steam of low wetness fraction (<20%) is a dispersive medium and two limiting sound speeds can be identified corresponding to complete equilibrium and fully frozen flow respectively. A stationary normal shock wave in wet steam can exhibit either a fully dispersed or a partially dispersed structure. A theoretical analysis of the structure of such shocks demonstrates the role of the three different time scales associated with the relaxation processes occurring within the wave. In contrast, the internal structure of a moving shock wave can change with time. An unsteady time-marching computational method has been developed and has been applied to solve the classical piston and cylinder problem for wet steam. The results have a direct bearing on the periodically unsteady flow of a condensing medium in a converging-diverging nozzle and the non-equilibrium two-phase flow in low-pressure wet -steam turbines.