Variational methods in acoustics: An assessment.

Abstract

A predecessor of modern variational principles is Rayleigh’s stationary expression for the natural frequency, which is such that insertion of a ‘‘good guess’’ for the mode shape gives an ‘‘excellent’’ approximation for the frequency. It is now known how to construct principles such that the stationary expression corresponds to any physical magnitude of interest such as for example, the acoustic power radiated by a body in a given state of vibration. Application typically involves the exercise of some insight into the generic form of the trial functions, with a trade-off between simplicity and generality. Intermediate results tend to refine the insight and to determine what effects are of principal importance. Criticisms are that one usually has no firm knowledge of the accuracy and that the insight used in selection of the generic trial functions can be construed as subjective. The application often presents complicated multiple integrals with singular integrands, so it does not necessarily require less computational effort. The technique is, however, suitable for those with strong ‘‘physical’’ understanding of experimental and analytical results of analogous problems. For example, theoretical considerations predict specific singular behavior at corners and edges. Trial functions that incorporate such behavior will yield better approximations. From a broader viewpoint, variational principles can be used as a framework for the derivation of finite element idealizations, and from this viewpoint, there are no limitations on the achievable accuracy. Variational techniques should have an enduring role in the conceptualization of complex systems by simpler lumped-parameter and lumped-function models. One may legitimately question, however, whether they are needed for routine numerical predictions. [Work supported by the Office of Naval Research and by the William E. Leonhard endowment to Pennsylvania State University.]