Image conditions for polar and cylindrical coordinate separation-of-variables acoustic multiple scattering models with perfectly reflecting flat boundaries

Ho-Chul Shin

doi:10.1093/qjmam/hby005

Abstract

This article addresses efficient implementation of the method of images for acoustic multiple scattering models (MSM) with perfectly reflecting flat boundaries. Time-harmonic problems are first solved in the polar coordinate system for circular scatterers; then the models are extended to the cylindrical coordinate system with (semi-)infinitely long circular cylinders. The MSM in this article is based on the method of separation of variables and Graf’s addition theorem. Derivations are provided for ‘image conditions’ which relate the unknown coefficients of outgoing waves from image scatterers with those of real counterparts. The method of images is applied to wedge-shaped domains with apex angles of π/n rad for a positive integer n. Image conditions make the MSM numerically more efficient: the system of linear equations for unknown coefficients is formulated 2n times faster; its memory requirements are reduced by 4n2 times for direct solvers. The proposed model is applied to a benchmark wedge in ocean environment with n=64. Good agreement is observed between the MSM with image conditions and the boundary element method. Furthermore, half- and quarter-space measurements in an anechoic chamber are in accordance with the correct use of image conditions. Incorrect image conditions reported elsewhere for polar coordinates are also discussed.

Highlights

Multiple scattering (1) of waves has been one of the classical problems which has captured the imagination of research communities: acoustics and beyond
Image conditions proposed in this article enable efficient implementation of the method of images for polar and cylindrical coordinate multiple scattering models (MSM), based on the method of separation of variables approach and Graf’s addition theorem
Image conditions were derived for a single flat boundary of either pressure release or rigid nature

Summary

Introduction

Multiple scattering (1) of waves has been one of the classical problems which has captured the imagination of research communities: acoustics and beyond. There are several numerical modalities such as the boundary element method and finite element method for analysis These are very versatile in terms of scatterer shapes, boundary types, material properties, and so on. When all the scatterers are circles in 2D domains (equivalent to infinitely long circular cylinders in 3D) or can be approximated as such, scattering problems can be addressed much more efficiently by using the method of separation of variables and addition theorems (2–7). These combinations are the constituents of the multiple scattering model addressed in this article, which is equivalent to the T-matrix method for circular scatterers (1).

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Conclusion