Uniqueness of Solutions to Variationally Formulated Acoustic Radiation Problems

Abstract

Determination of the surface acoustic pressure given the surface velocity of a vibrating body can be formulated in various ways. However, for some such formulations such as the surface Helmholtz integral equation, solutions are not unique at certain discrete frequencies. Such uniqueness problems can also be present for variational formulations of the problem, but the variational formulation based on the normal derivative of the Kirchhoff integral theorem has unique solutions for vibrating disks and plate-like bodies. For bodies of finite volume, but for which each surface point is vibrating in phase, the total radiated acoustic power is always unique, even though the pressure may not be. The latter conclusion is supported by numerical calculations based on the Rayleigh-Ritz technique for the case of a finite cylinder vibrating as a rigid body in the axial direction.