Nonuniqueness of the Solution of the Sound Field Reproduction Problem with Boundary Pressure Control

This paper studies the circumstances under which the problem of reproducing a desired sound field using the boundary pressure control approach has a unique solution. The sound field reproduction problem is formulated as an inverse problem, in which the reproduction of the target sound field is attempted in the interior of a bounded control region surrounded by a continuous distribution of secondary sources. The determination of the secondary source strength is an ill-posed problem. A general formula for the solution is derived (assuming its existence) and it is shown that nonuniqueness arises when the wavenumber 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. Some strategies are presented that enable the nonuniqueness to be overcome. The case is also studied of the wavenumber being one of the Dirichlet eigenvalues of the region bounded by the secondary source distribution, which contains but generally does not coincide with the control region. The results derived are illustrated for a two dimensional problem with a finite number of secondary sources.