The aim is to determine the displacement and the stress fields for a cylindrical cavity under a transient pressure which is an arbitrary function of time. A fundamental solution, given by Kadioglu and Ataoglu, has been revised to be used in reciprocal theorem for the solutions of axially symmetric transient problems of elastodynamics. An integral equation, whose unknown is the radial displacement on the boundary, has been written, using this fundamental solution in reciprocal identity. The boundary values of two sample problems, have been determined by solving this integral equation. To calculate the propagation of the displacement field in the region for any axially symmetric problem, a second elastodynamic state has also been derived. This new elastodynamic state has been used in reciprocity theorem to compute the time-variations of the displacement and the stresses at any point in the region for both sample problems. Whole singularities arising in every stage of the formulation have been eliminated. The most interesting result of the presented solution is the representation of a propagating wave by two different two-dimensional integrals at any interior point. Each of these integrals is valid for a different time interval and these intervals are defined by the position of the mentioned interior point.