Joseph, Phillip Frederick
(1990)
Active control of high frequency enclosed sound fields.
*University of Southampton, Doctoral Thesis*.

## Abstract

This thesis is concerned with the active control of random sound fields. Initially, the thesis considers applying active control to sound fields which exhibit random behaviour in time. The causally constrained minimum sound power output of field sources radiating plane waves into an infinite duct is determined in terms of the sound power spectral densities from each of the sources. This time domain analysis is then later extended to investigate the effects of reflections where the causally constrained minimum sound power output of two white noise sources is derived situated close to a single reflecting surface of known reflection coefficient. The remainder of this thesis is given to applying active control to harmonically varying, three dimensional sound fields driven above the Schroder frequency where the sound field is regarded as `diffuse'. Chapter 3 considers the possibilities for global control where the minimum sound power output of two closely point sources radiating into a diffuse environment is deduced. The various moments of the minimum sound power are derived together with those of the secondary source strength requirements. Recognising that a point primary source distribution is unrealistic, the maximum secondary source diffuse field sound power absorption is derived which is shown to correspond to a cross sectional area of absorption approximately equal to λ2/π which is also the area of absorption afforded by a Helmholtz resonator in a diffuse field at resonance. For frequencies greater than about the Schroder frequency therefore, this control scheme is ineffective. The equivalence between an optimally absorbing secondary source and an equivalent element of passive absorption is highlighted. In chapter 4, the more realistic objective of imposing localised quiet is investigated by considering the active cancellation of the diffuse field pressure at a point using a remote secondary source. Expressions are derived for the spatial extent of the diffuse field quiet zone together with an analysis of the optimal secondary source strength statistics and the resulting increase in the total potential energy radiated by the source pair. This analysis proves that this control strategy is poorly conditioned for which the outcome for any single source configuration is highly unpredictable inasmuch that the various moments of the various field variables are no longer defined. The results of chapter 4 provide the motivation for the work presented in chapters 5 and 6. Chapter 5 deals with constrained active control in diffuse fields, in particular, looking at the effects of `hard limiting' where the secondary source strength is prevented from exceeding some upper bound value. Various multi-channel control schemes are also investigated. The last chapter concerns the use of a secondary source to cancel the pressure at point situated in its near field. This control strategy is shown to be well behaved in that comparatively large quiet zones are possible, causing increases in the sound pressure level well away from the control point of less than only 1 dB. Experimental results validating this theory are also presented.

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