Analytic solutions to the acoustic source reconstruction problem

Abstract

In this paper, we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed-form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far-field or intermediate regions. Exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain. A stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field. It is found that the degree of ill-posedness of the inverse problem is significantly reduced, when considering a hemi-circular observation arc with respect to a linear array of sensors, by a factor scaling on the small angular aperture subtended by the observation line. Estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace.