Application of a Neumann‐series method to two problems in acoustic radiation theory that are formulated in terms of Green’s functions

Abstract

Two problems in radiation theory are analyzed. In the first problem, two acoustic sources in a fluid interact via their radiated pressure field. Each source may have an arbitrary shape and an arbitrary nonuniform surface vibration. The two sources may be placed in the fluid in an arbitrary fashion, so that there is no preferred orientation of one with respect to the other. A Neumann‐series method is used to simplify that term in the integral equation expressing the pressure field which describes the interactions taking place between the sources. By use of this method, the pressure field can be expressed in terms of the Green’s functions of the individual sources, rather than in terms of an abstract ad hoc Green’s function defined on the disjoint surface that comprises the surfaces of both sources. This field formulation is used to express the self and mutual radiation impedances of the sources. A physical description of the interactions taking place between the sources is given. This description is suggested by the form of the equation describing the pressure field that results when the Neumann‐series method is used. In the second problem, the exterior Neumann and Dirichlet Green’s functions are analytically determined for a radiating acoustic source with an arbitrary shape and with an arbitrary nonuniform surface vibration. These Green’s functions are constructed by using a Neumann‐series method to simplify that term in the integral equation expressing the pressure field which describes the interaction of the source with its own radiation. In principle, the exterior Green’s functions, which are expressed in terms of the geometry of the source and the free‐space Green’s function for the Helmholtz equation, are calculable for any source by performing a series of elementary mathematical operations.