Analysis of the radiated information in spinning sound fields

Abstract

The information content of a spinning sound field is analyzed using a combination of exact and asymptotic results, in order to set limits on how accurately source identification can be carried out. Using a transformation of the circular source to an exactly equivalent set of line source modes, given by Chebyshev polynomials, it is found that the line source modes of order greater than the source wavenumber generate exponentially small fields. Asymptotic analysis shows that the remaining, lower order, modes radiate efficiently only into a region around the source plane, with this region shrinking as the mode order is increased. The results explain the ill-conditioning of source identification methods; the successful use of low order models in active noise control; and the low radiation efficiency of subsonic jets.

The radiating part of circular sources

“…The analysis of the nature of the sound field from an arbitrary disk source is based on a transformation of the disk to an exactly equivalent line source, an approach which has been used to study transient radiation from pistons [3,4], rotor noise [9,10] and source identification methods [23][24][25]. The transformation to a line source is shown in Figure 2, which shows the new coordinate system (r 2 , θ 2 , z)…”

“…This can be attributed to physical causes, and is not merely an artifact of the methods used. Recent analysis [23][24][25] has given a framework for the study of this ill-conditioning by quantifying the source information which is radiated into the acoustic near and far fields. As described below, it has been found that the source can be decomposed into orthogonal modes based on Chebyshev polynomials, only a limited number of which radiate a detectable acoustic field, with the limit being fixed by the source frequency.…”

“…The results to be presented arise from two different exact theories for radiation from circular sources [24,25,34] which are combined to give a formulation for the information in the acoustic field in terms of radiation functions and weighted integrals of the source term. The implications of the results are discussed in terms of the information content of the acoustic field and with regard to some of the measurement methods used to study noise sources.…”

Jet noise: sound generation by disc and cylinder sources

“…Under this transformation (Carley 2009), which gathers terms of constant phase to reduce the double integral to a set of single integrals: (Carley 2010) and can be written as where U q (s) is a Chebyshev polynomial of the second kind, s = s 2 − r and the coefficients u n q (r) are functions of r but not of z. Inserting (2.4) into (2.2) gives…”

“…Our analysis will demonstrate that this ill conditioning is an unavoidable result of the nature of the source and that there is a limit on the information available for source identification. It is well-known that owing to the arbitrary nature of source terms (Ffowcs Williams 1984), and the filtering effect of propagation, which removes source terms with subsonic phase velocity (Carley 2010), there is no unambiguous relationship that characterizes a source using only field information.…”