A mathematical model for the transmitting response and electrical impedance of a free-flooded, thickness-polarized piezoelectric cylinder transducer has been developed. The elasticity of the cylinder is described using membrane equations which have been modified to account for piezoelectric effects. An additional consequence of the piezoelectricity of the shell is a significant alteration in the boundary conditions for the ends of the cylinder. A particular solution to the membrane equations which satisfies these modified boundary conditions is obtained. The complete solution for the vibration of the cylinder is then found in the form of a sum of the particular solution and the normal modes appropriate to a nonpiezoelectric cylinder with free ends. The acoustic field loads the solution, couples the normal modes, further couples the particular solution and the normal modes, and changes the boundary conditions. The acoustic influence coefficients needed to incorporate these effects are calculated using the ship computer program. The water-loaded normal surface velocities are determined and the surface pressures and farfield radiation characteristics are calculated using ship. The model fully accounts for such effects as the cavity resonance, the velocity of the ends of the cylinder, distribution of velocity along the length of the cylinder, and the difference between the velocity on the inner and outer surfaces of the cylinder. Comparisons between theoretical and experimental results are presented.
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Peter H. Rogers (1986) studied this question.