Representation of the Field of an Acoustic Source as a Series of Multipole Fields

Abstract

A representation of the field of a sound source of arbitrary shape and size as a series of multipole fields of increasing order is derived. This series is convergent for sources of any size if the surface is sufficiently regular, but its main advantage is for sources small compared to the wavelength. In this case the source has, in general, the directivity pattern of a multipole, the order and strength of which can be determined from the formula. Although the representation is not unique, certain coefficients have physical significance. The presently used definition of “strength of a sound source” is discussed and a modified definition suggested.

The Multipole Expansion: A New Look

“…di!erent values (since pairs of Q L GHI 2 whose indexes di!er by a permutation are equal to each other). It can be shown, following the procedure of Oestreicher [3], that in general only the lowest order non-zero Q L is independent of the choice of the reference point. If Q is taken as the right-hand side of the d'Alembert equation (1) (or of the Poisson equation), the above result yields directly the same multipole expansion of the source distribution Q that would be obtained by the more lengthy procedure, described earlier, of recovering the multipole strengths from the expansion of the "eld of Q, the strengths of the 2L-pole components being given by the corresponding Q L GHI 2 .…”

Fast computation of time-dependent acoustic fields

“…For single frequency fields, a multipole expansion is the classical approach and can be derived for the field generated by sources of arbitrary shape, 1 based on the work of Oestreicher. 2 Similar methods exist in more specialized contexts, such as a recently developed technique for the far-field noise of a propeller, using data from a small number of nearfield points to generate an equivalent expansion, 3 or the approach developed for quadrupole terms arising from flow noise. 4 When a transient signal is required, however, methods for equivalent expansions or replacement sources are less common.…”