Combined Helmholtz Integral Equation - Fourier series formulation of acoustical radiation and scattering problems

Abstract

The Combined Helmholtz Integral Equation – Fourier series Formulation (CHIEFF) is based on representation of a velocity potential in terms of Fourier series and finding the Fourier coefficients of this expansion. The solution could be substantially simplified if the Green function and its normal derivative are represented by Fourier series. Unfortunately the direct expansion of the Green function and its normal derivative are impossible because these functions do not satisfy the Dirichlet’s theorem due to the singularities. To take advantage of the Fourier methods it is necessary to reformulate the original Helmholtz integral equation so that the modified Green function does not contain any singularities. The corresponding revision of the problem is proposed in the present paper. The Green function is modified so to satisfy the Dirichlet’s theorem. The tradeoff is that the original Helmholtz integral equation contains new double singular integrals which could be calculated numerically by an adaptive procedure or by means of quadrature formulae. Fourier coefficients of the modified Green functions are calculated using a discrete Fourier transform, in particular case by FFT. Using orthogonality of the sine and cosine functions the original problem is reduced to an over determined system of linear algebraic equations to obtain the unknown coefficients of the Fourier series expansion. The CHIEFF method is applicable to a broad range of acoustical problems of radiation and scattering. It is especially effective for calculation of near acoustic fields of large-scale structures.