General statements concerning phase and group velocities of stress waves propagating in elastic media where at least one dimension is infinite are valuable for separating the distinct characteristics of the various operating phenomena. The problem of impact of a finite body, however, is so greatly complicated by the inherent reflections, reinforcements, and cancellations of the stress pulses that little use can be made of these general statements in describing the overall transient response of such a body. Presented here is a finite difference numerical method for integrating over space and time the appropriate equations of motion and boundary conditions for determining the transient stress state in a finite elastic body of revolution resulting from longitudinal impact against a rigid wall. Also presented here are the results obtained from programming this numerical procedure for the Rice University digital computer and using this program to evaluate the transient stress states in elastic bodies of nine distinct boundary configurations.