Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces

Shin, Ho-Chul (2017). Image Conditions for Spherical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly-Reflecting Flat Surfaces. The Quarterly Journal of Mechanics and Applied Mathematics, 70(4) pp. 419–453.

DOI: https://doi.org/10.1093/qjmam/hbx018

Abstract

Efficient implementation of the method of images is addressed for 3D multiple scattering models (MSM) for spheres with perfectly-reflecting flat surfaces. The Helmholtz equation is solved by the spherical-coordinate separation-of-variables approach and addition theorems for spherical wavefunctions. The unknown coefficients of outgoing waves from image objects are related with those of real counterparts through ‘image conditions’ which are derived in this article. The method of images is applied to concave part of wedge-shaped acoustic domains with apex angles of π/n rad for a positive integer n, which includes half-space and corners. Image conditions make the MSM numerically more efficient: memory space is reduced by 4n2 times; matrix is populated 2n- times faster for infinite or 2D wedges. Savings are 16n2 times in memory and 4n times in speed for semi-infinite or 3D wedges. Image conditions are valid regardless of the type of scatterers as long as they are spheres and submerged in acoustic domains; they are also suitable for the modified Helmholtz equation and radiation problems. However, specific formulae of image conditions depend on definitions of the spherical harmonics. Image conditions for rigid flat surfaces are verified by measurements of 13 balls in an anechoic chamber for configurations of half-space, 2D & 3D corners and 3D wedge with n=3. Image conditions for pressure-release flat interfaces are validated by the boundary element method (BEM) for the pulsation mode of an underwater air sphere in half-space and for scattering by a sphere in an ocean environment with the wedge angle of 1.2° by n=150. Agreement is very good between the MSM and measurements and is impeccable between the MSM and BEM.

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