On the validity and convergence of the Helmholtz equation least squares solutions

Abstract

The problem of reconstruction of acoustic radiation from arbitrary shaped vibrating bodies is often encountered in engineering applications. The Helmholtz equation least-squares (HELS) method proposed by Wu (2000) is an assumed-form solution method in which the reconstructed acoustic pressure is found as a superposition of spherical waves. The unknown coefficients are determined by matching the assumed-form solution to the measured acoustic pressures at a number of points around the source and their errors are minimized by the least-squares method. To further minimize the errors of reconstruction, constraints are used on the surface of the object. In this paper a two-dimensional, noncircular object is considered and the two-dimensional Helmholtz equation is solved. The locations of singularities in the analytic continuation of the solution across the source surface are found using the Schwarz function of the source surface. It will be shown that the HELS solution remains bounded and close to the true pressures inside the minimum circle enclosing the object as well as outside. Other types of modifications of the HELS method for reconstruction of acoustic radiation in an exterior two-dimensional region will be considered. [Work supported by NSF.]