Nonuniqueness of solutions to variationally formulated acoustic radiation problems

Abstract

Determination of surface acoustic pressure when the normal velocity is prescribed over a vibrating body's surface can be formulated in various ways, but for some such formulations the solution is not unique at certain discrete frequencies. The question arises as to whether uniqueness problems are present for variational formulations of this problem. The answer is ordinarily “yes,” but proper selection of the variational approach can help to circumvent such difficulties. The variational formulation based on the normal derivative of the Kirchhoff‐Helmholtz integral has a unique solution for vibrating disks and platelike bodies. For bodies of finite volume, but for which each surface point is vibrating in phase, calculated total radiated acoustic power is always unique, even though the pressure may not be. Use of the Gerjuoy‐Ran‐Spruch technique and judicious choice of what physical quantity is stationary under variations leads to a formulation such that the surface pressure is unique for any specified frequency. Similarities with the Burton‐Miller techniques are discussed; an appropriate variational formulation provides a method for choosing the weighting factor for the linear combination of the two surface relations (an integral equation and an integrodifferential equation). [Work supported by ONR.]