Nonuniqueness of the solution of the sound field reproduction problem
Filippo M. Fazi, and Philip A. Nelson
The sound field reproduction problem is formulated as an inverse problem, in which the reproduction of a target sound field is attempted, in the interior of a given control region, with an array of loudspeakers (referred to as a secondary source distribution). The determination of the loudspeaker gains represents an ill-posed problem. This paper studies under what circumstances the said inverse problem allows for a unique solution.The general solution of the problem is derived, and it is shown that nonuniqueness arises when the wave number is one of the Dirichlet eigenvalues of the control region. It is shown that, when this is not the case, the solution of the problem is unique. Numerical simulations illustrate the effect of nonuniqueness of the solution for the case of spherical secondary source distribution and control region. The case is also studied of the wave number being one of the Dirichlet eigenvalues of the region bounded by the secondary source distribution.