The reconstruction of a spatially incoherent two-dimensional source in an acoustically rigid rectangular cavity.

This paper applies Green's function modal expansion techniques to inverse source problems within an acoustically rigid rectangular cavity. In particular, solutions are developed for the reconstruction of a random spatially incoherent distributed source and a deterministic point source by using pressure-field measurements on adjacent walls. Such problems have seemingly not been addressed previously. Several example problems are studied in this paper. As is characteristic of inverse problems in general, numerical instabilities are encountered in the example problems addressed in this paper. These are linked to the notion of compactness in the forward operator, and dealt with using magnitude regularization and projections onto convex sets. The behavior of the example problems studied in this paper greatly deteriorates at frequencies near cavity resonances, particularly low frequency resonances (i.e., with ka < or = 2pi, where a is the smallest cavity dimension). Other results concerning the example problems are also given. A proof of compactness of an infinite rank version of the type of operator used in this paper is given in the Appendix. This proof implies that the convergence of the Green's function modal expansion in the kernel of the forward problem is related to the instability of the inverse problem. This implies that, if it suffices to know an integral of the source profile rather than the actual profile, the problem becomes more stable.