Fazi, Filippo Maria
(2010)
Sound field reproduction.
*University of Southampton, Institute of Sound and Vibration Research, Doctoral Thesis*, 312pp.

## Abstract

This thesis is concerned with the problem of reproducing a desired sound field with an array of loudspeakers. A theory based on functional analysis and the theory of

integral equations is developed for the study of this problem. An attempt is made to develop a mathematical framework that can be adopted as a generalized theory of sound field reproduction. The reproduction problem is formulated as an acoustical inverse problem, in which the target sound field is given on the boundary of a control volume located in the interior of the loudspeaker array, while the loudspeaker signals required for the reproduction of the desired field are to be determined. The loudspeaker array is initially modeled as a continuous distribution of secondary sources, mathematically

represented by a single layer potential, whose density is to be determined. The singular value decomposition of the integral operator involved is proposed as a method for solving the inverse problem. Closed form expressions are derived for the singular system for the

cases of secondary sources arranged on a sphere and on a circle. An attempt is also made to extend the calculation to unbounded geometries, such as an infinite line and a

plane. The inverse problem under consideration is in general ill-posed, and the existence and uniqueness of its solution are studied in relation to sound fields of practical interest. It is shown that an exact and unique solution exists for a large family of sound fields.

Strategies are proposed for overcoming the problem of nonexistence and nonuniqueness of the solution, arising in cases such as the reproduction of focused sources or when the operating frequency corresponds to one of the Dirichlet eigenvalues of the control region.

An important analogy is also drawn between the problem of sound field reproduction and the theory of acoustic scattering. In a later part of this work, the assumptions of a continuous layer of secondary sources and of a single operating frequency are removed, and the resulting consequences are analyzed. The experimental validation of some of the theoretical results is described in the final part of the thesis. A large spherical loudspeaker

array is used in an attempt to reproduce the sound field generated by a single virtual source, located in the exterior of the array. Experimental results are in good agreement with the theoretical results over a wide range of frequencies.

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