Application of the Euler-Maclaurin Sum Formula to Obtain a Closed form Analytical Solution for Acoustical and Heat Diffusion in a Partially Bounded Region

Abstract

Green’s function in the Laplace transform domain for the sound wave field in an infinite area between two parallel reflecting planes may be represented as an infinite series of images caused by the planes. Based on a comparison of the wave equation and the diffusion equation, the Green’s function representation as an infinite series for diffusion in the same region is directly obtained. A closed form expression in space time for the diffusion problem is obtained by applying the Euler-Maclaurin sum formula to a modified diffusion form of the series in the Laplace transform domain and then inverting to the time domain. Using several sets of numerical values for system parameters applicable to acoustic diffusion in the region, numerical comparisons of the infinite series vs Euler-Maclaurin closed form representation of the Green function is presented. Comparison of the infinite series vs Euler-Maclaurin transient response to an exponential and constant input are presented for cases of acoustical noise diffusion and heat diffusion respectively. Transfer function comparisons are given for the diffusion models along with the use of the closed form representation in a model-based control scheme.