k -Space Formulation of the Acoustic Scattering Problem

Abstract

The acoustic scattering problem is solved by means of a k-space formulation of the field equations, thereby replacing the conventional integral equation formulation by a set of two simultaneous algebraic equations in two unknowns in two spaces (the constitutive boundary condition being an algebraic equation in x space). These equations are solved by an iterative method with the aid of the fast Fourier transform (FFT) algorithm connecting the two spaces, requiring trivial initial approximations. Since algebraic and FFT equations are used, the number of arithmetic multiply-add operations and storage allocations required for a numerical solution are reduced from the order of N2 and N2, respectively (for solving the matrix equations resulting from the conventional integral equations), to the order of N log2N and N, respectively (where N is the number of data points required for the specification of the problem). The advantage gained in speed and storage is thus of the order of N2/log2N and N, respectively. This method is thus considerably more efficient, and permits exact numerical solutions for much larger problems than possible with the conventional integral equation-matrix inversion method. The details and some numerical results of the application of this method to two- and three-dimensional acoustic scattering are presented.