A numerical solution technique is presented for determining the dynamic response of a thin, elastic, circular, cylindrical shell of constant wall thickness and density, in a potential fluid. The shell may be excited by any radial forcing function with a specified time history and spatial distribution. In addition, a pressure history may be specified over a segment of the fluid outer boundary. Any of the natural shell end conditions may be prescribed. The numerical results are compared to experimental results for a 1/12-scale model of a nuclear-reactor core-support barrel. Natural frequencies and modes are determined for this model in air, water, and oil. The computed frequencies are within 15 percent of experimental results. A sample application compares the numerical technique to an analytical solution for shell beam modes. The comparison resolves an uncertainty concerning the proper effective mass to use in the analytical technique.