Integral Equation Method for Radiation from Vibrating Bodies

Abstract

Physically realizable distributions of pressure and normal velocity over a vibrating surface submerged in an unbounded acoustic medium can not be independently prescribed, but must satisfy some acoustic relationship. The surface pressure and normal velocity can be obtained by solving an acoustic relationship simultaneously with an equilibrium relationship expressing dynamic equilibrium of the structure whose surface is vibrating. The present method uses as the acoustic relationship a regular integral equation derived from the classical Helmholtz integral. A numerical technique successful in solving the integral equation and economical of computational effort is described. It represents the solution functions in terms of their values at a set of mesh points but employs another, much larger, set of points for evaluating the integrals. The pressure at any field point is obtained by numerical quadrature in the Helmholtz integral of the calculated surface pressure and normal velocity. Results are presented of calculations for a spheroid and for finite capped cylinders, all with prescribed axisymmetric velocity distributions. The surface pressures calculated for the spheroid agree closely with an exact solution, while those for the cylinders are consistent with understood acoustical phenomena. The far-field directivities for the cylinders agree reasonably with an independent approximate solution.