Sound power emitted by a pure-tone source in a reverberation room

Abstract

Energy considerations are of enormous practical importance in acoustics. In “energy acoustics,” sources of noise are described in terms of the sound power they emit, the underlying assumption being that this property is independent of the particular environment where the sources are placed. However, it is well known that the sound power output of a source emitting a pure tone or a narrow band of noise actually varies significantly with its position in a reverberation room at low frequencies, and even larger variations occur between different rooms. The resulting substantial uncertainty in measurements of sound power as well as in predictions based on knowledge of sound power is one of the fundamental limitations of energy acoustics. The existing theory for this phenomenon is fairly complicated and has only been validated rather indirectly. This paper describes a far simpler theory and demonstrates that it gives predictions in excellent agreement with the established theory. The results are confirmed by experimental results as well as finite element calculations.

Statistical properties of kinetic and total energy densities in reverberant spaces

“…14 The reason is that the radiation impedance is affected by the random reverberant part of the sound field, which moreover is increased at the source position because of coherent backscattering or "weak Anderson localization" as predicted by Weaver and Burkhardt. 17 Besides, one can no longer expect the same statistics with respect to position as would be found in an ensemble of rooms.…”

“…It is statistically independent of the other quantities because the variations in the sound power depends only on the reverberant part of the sound pressure at the source position. 14 The relative ensemble variance of the mean-square particle velocity component in an arbitrary direction now becomes…”

“…17 Besides, one can no longer expect the same statistics with respect to position as would be found in an ensemble of rooms. 14,15 The resulting relative ensemble variance of the sound power emitted by a monopole has been found, based on Eq. ͑1͒, to be 2…”

“…5 Each set of random amplitudes and wave number vectors corresponds to an outcome of a stochastic process, and above the Schroeder frequency there is no difference between the statistics with respect to position and the full ensemble statistics. 5,14 It is easy to show that the mean-square values of the pressure and each component of the particle velocity can be expressed as a sum of two independent squared Gaussian variables ͑random sums͒ with zero mean. 5,13 Thus, the mean-square pressure as well as the mean-square value of any individual component of the particle velocity have a chi-square distribution with two degrees of freedom ͑also known as the exponential distribution͒, 13,16 from which it follows that their relative ͑normalized͒ ensemble variance is 1.…”

Measurement with verification of stationary signals and noise in extremely quiet environments: Measuring below the noise floor

“…Applying the methods to investigate very low level, stationary in frequency, pure tone target measurements has not been investigated here. Consideration of frequency or room position dependencies in room sound pressure level which might be related to room size, shape, and nodal behavior as discussed by Jacobsen and Molares (2009) is beyond the scope of this paper but it is possible the methods described here could be useful to study the effects of these parameters.…”