Radial vibration of a thick-shell hollow piezoceramic sphere

A thick-shell hollow piezoceramic sphere electroded completely on both inside and outside surfaces and radially polarized can be used as a source and receiver of underwater sound. Restricting the analysis to moderately thick shells the electromechanical equations of displacement are solved to generate the frequency equation for the pure radial modes. The frequency equation is specialized to reflect the behavior of a barium-titanate shell and numerical values are tabulated for the first three frequencies for a broad range of shell thicknesses.