The Helmholtz equation least-squares method and Rayleigh hypothesis in near-field acoustical holography

Abstract

In this paper we present a numerical investigation of reconstructing time-harmonic acoustic pressure field in two dimensional space by using a series expansion—the so-called Helmholtz equation least-squares (HELS) method. Series expansion methods (or the Rayleigh methods) have been widely used in predicting the scattered acoustic pressure. With regularization, they can also be applied to reconstruction of acoustic pressure on the source surface from the measurements taken in the field, and HELS is the first such attempt for these problems. In this paper, we establish HELS in the framework of the Rayleigh methods and reveal its interrelationship with the Rayleigh hypothesis. In particular, to regularize a reconstruction problem, we use the method of quasisolutions, i.e., a Tikhonov regularization with an a posteriori choice of the regularization parameter. It is shown that without regularization HELS can still yield a satisfactory reconstruction of acoustic radiation from an arbitrary object when enough measurements are taken at sufficiently close range to the source. With regularization the number of measurements can be reduced and reconstruction accuracy be enhanced. It is concluded that HELS can be used to reconstruct acoustic radiation from a convex arbitrarily shaped vibrating object regardless of the validity of the Rayleigh hypothesis, although in practice the results will depend on the rate of convergence of the approximating sequence.