Sound Propagation from an Arbitrarily Oriented Multi-Pole Placed Near a Plane, Finite Impedance Surface

Abstract

Theories that have been developed for the purpose of predicting sound propagation over plane, finite impedance surfaces have usually been expressed in terms of sound radiation from point monopoles, or at least in terms of sources that generate cylindrically symmetric sound fields. However, not all practical noise sources are monopolar in character, nor do they necessarily generate cylindrically symmetric sound fields. In this paper, a two-dimensional finite Henkel transform technique is described that makes it possible to predict sound propagation from sources that generate non-cylindrically symmetric sound fields: e.g., arbitrarily oriented dipoles and quadrupoles. As a result, the proposed technique may prove useful for predicting sound propagation from aerodynamic noise sources placed near plane outdoor surfaces. The prediction procedure is based on representing the direct field of a source as a two-dimensional wavenumber spectrum in which the radial wavenumber and azimuth are the transform variables. The direct wavenumber spectrum is then combined with the plane wave reflection coefficient of the impedance plane to yield the wavenumber spectrum of the reflected field. The latter component is added to the direct wavenumber spectrum and the result is inversely transformed to give the sound pressure as a function of radius and azimuth angle. Owing to the finite nature of the transforms that are used in the discrete implementation of this procedure, the predictions are valid only to a finite distance from the source. Thus the transform procedure must be supplemented by an asymptotic theory if predictions of the whole sound field are to be made: an appropriate asymptotic theory is summarized here. The use of the transform procedure is illustrated by using a number of example calculations, and general conclusions are drawn regarding ways of choosing the transform parameters to ensure accurate predictions. The results of the transform procedure have also been found to be in excellent agreement with measurements of sound propagation over a finite impedance surface from a small, unbaffled loudspeaker.