In this study, propagation of transient dilatational waves in a viscoelastic cylinder contained in an elastic case is investigated. The viscoelastic material of the cylinder is modelled as standard linear solid. The encased viscoelastic cylinder is subjected to uniform time dependent dynamic inputs at its inner and outer boundaries. Method of characteristics is employed to obtain the solutions. Solutions for the case of an ablating inner surface are obtained as well. The numerical results are displayed in curves denoting the variations of circumferential and radial stresses with time at different locations. The curves reveal clearly the effects of reflections and refractions at the inner and outer surfaces and the interface between the viscoelastic cylinder and the elastic case. Furthermore, the effects of damping, geometric dispersion and ablating inner boundary are clearly seen in the curves.