Acoustics of a fluid-loaded plate with attached oscillators. Part I. Feynman rules

Abstract

The “fuzzy structure” paradigm developed by Soize and others provides a framework within which the acoustic properties of systems with complex internal structure can be explored. In this two part paper, the acoustic properties of one of the simplest such systems, a fluid-loaded plate with attached internal oscillators, are explored. In Part I, the theoretical tools needed in this investigation, the representation of the Neumann series in terms of Feynman diagrams, is presented. In most ways, the formal aspects of this development are identical to similar results from other disciplines, but the presence of damping in the system, either internal or due to radiation into the fluid, requires one to make some modifications to existing theories. The rules for evaluating averaged quantities such as G⋯GG * ⋯G * involving higher-order products of the Green’s function are given. These quantities are necessary to determine a number of relevant properties of the system including the scattering cross section and the size of typical fluctuations. The Feynman rules, detailed mathematical expressions which enable averaged quantities to be evaluated for particular systems, are explicitly given for the case in which the internals are simple oscillators distributed uniformly and independently in both space and frequency. In Part II of this paper, the theoretical approach will be applied to predict a number of acoustic properties of the system.