Asymptotic and computational analyses of a well-posed initial-boundary-value problem are used to describe the time history of co-existing acoustic and rotational velocity disturbances in a long, narrow cylinder with uniform steady sidewall mass injection. Transient planar pressure disturbances prescribed on the open exit plane of the cylinder are the source of acoustic disturbances in the axisymmetric flow. Both the asymptotic and numerical solutions describe the nonlinear aspects of the flow interactions. The full computational results are compared favorably with those of the asymptotic study to show that; (1) transient vorticity is generated near the injection surface and is transported into the cylinder by the radial velocity component of the flow field, (2) at any sufficiently small value of time, a well defined front separates the fluid containing transient vorticity from a flow field in the interior of the cylinder containing a much smaller amplitude vorticity and, (3) at sufficiently large values of time, vorticity is present throughout the cylinder. In addition, the analytically derived acoustic solution obtained from the asymptotic analysis is used to show that the present numerical solution and all earlier studies of similar problems are missing travelling waves (eigenfunctions) which should be present in a complete mathematical solution of the defined initial-boundary-value problem.