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

Shin, Ho-Chul

doi:10.1093/qjmam/hbx018

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…Acoustic fields in the concave part of a corner or acute-angled wedges can be analysed. However, there are certain restrictions in realising a wedgeshaped acoustic domain through the method of images (17,26). Having a finite number of image scatterers requires a wedge angle of β = π/Q where Q is the number of total wedges in a given halfplane; hence, 2Q wedges in total.…”

Section: Methods Of Imagesmentioning

confidence: 99%

“…Especially, it is not surprising that spherical coordinate image conditions should not be used in place of polar coordinate image conditions due to the difference in their respective wavefunctions. More details of spherical coordinate image conditions can be found in (17).…”

Section: Incorrect Image Conditionsmentioning

confidence: 99%

“…The second procedure is to use 'image conditions' to merge these addresses for image scatterers with those of the corresponding real scatterers in the matrix. This terminology of 'image condition' is named such because it is a necessary condition to make one scatterer an image of the other (17). Therefore, such use of image conditions reduces memory requirements and results in faster computation compared to the former approach.…”

Section: Introductionmentioning

confidence: 99%

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

Shin

2018

The Quarterly Journal of Mechanics and Applied Mathematics

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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 wedgeshaped 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 4n 2 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.

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Section: Methods Of Imagesmentioning

confidence: 99%

Section: Incorrect Image Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Shin

2018

The Quarterly Journal of Mechanics and Applied Mathematics

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Image conditions and addition theorems for prolate and oblate spheroidal-coordinate separation-of-variables acoustic multiple scattering models with perfectly-reflecting flat surfaces

Shin

2019

IMA Journal of Applied Mathematics

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Three-dimensional time-harmonic acoustic multiple scattering problems are considered for a finite number of prolate and oblate spheroidal objects adjacent to flat surfaces. Wave propagation by spheroids is modelled by the method of separation of variables equipped with the addition theorems in the spheroidal coordinates. The effect of flat surfaces is accounted for by using the method of images; hence, the flat surfaces are of (semi-)infinite extent and perfectly reflecting: either rigid or pressure release. Wedge-shaped acoustic domains are constructed including half-space and right-angled corners with the wedge angle of $\pi /n$ rad with positive integer $n$. First, Euler angles are implemented to rotate image spheroids to realize the mirror reflection. Then, the ‘image conditions’ are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image-scattered fields in terms of real counterparts. Use of image conditions to 2D wedges, therefore, leads to the $4n^2$-fold reduction in the size of a matrix for direct solvers and $2n$-times faster computation than the approach without using them; for 3D wedges, the savings are $16n^2$-fold and $4n$-times, respectively. Multiple scattering models (MSMs) are also formulated for fluid, rigid and pressure-release spheroids under either plane- or spherical-wave incidence; novel addition theorems are also derived for spheroidal wavefunctions by using two rotations of spherical wavefunctions and a $z$-axis translation in-between, which is shown numerically more efficient than other addition theorems based on an arbitrary-direction translation and a single rotation. Finally, MSMs using image conditions are numerically validated by the boundary element method for a configuration populated with both prolate and oblate spheroids.

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Image Conditions for Elliptical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly Reflecting Flat Boundaries: Application to in Situ Tunable Noise Barriers

Shin

2020

The Quarterly Journal of Mechanics and Applied Mathematics

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Summary Two-dimensional time-harmonic multiple scattering problems are addressed for a finite number of elliptical objects placed in wedge-shaped acoustic domains including half-plane and right-angled corners. The method of separation of variables in conjunction with the addition theorems is employed in the elliptical coordinates. The wavefunctions are represented in terms of radial and angular Mathieu functions. The method of images is applied to consider the effect of the infinitely long flat boundaries which are perfectly reflecting: either rigid or pressure release. The wedge angle is $\pi/n$ rad with integer $n$; image ellipses must be appropriately rotated to realise the mirror reflection. Then, the ‘image conditions’ are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image scattered fields in terms of real counterparts. Use of image conditions, therefore, leads to the $4n^2$-fold reduction in the size of a matrix for direct solvers and $2n$-times faster computation in building the system of linear equations than the approach without using them. Multiple scattering models using image conditions are formulated for rigid, pressure release and fluid ellipses under either plane- or cylindrical-wave incidence, and are numerically validated by the boundary element method. Furthermore, potential applications are presented: arrays of elliptically shaped scatterers make in situ tunable noise barriers by rotating scatterers. Finally, polar-coordinate image conditions (for circular objects) are also discussed when coordinates local to circles are also rotated. In Appendix, analytic formulae are provided, which permits the elliptical-coordinate addition theorems used in this article to be calculated by summation instead of numerical integration.

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

Cited by 3 publications

References 18 publications

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

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

Image conditions and addition theorems for prolate and oblate spheroidal-coordinate separation-of-variables acoustic multiple scattering models with perfectly-reflecting flat surfaces

Image Conditions for Elliptical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly Reflecting Flat Boundaries: Application to in Situ Tunable Noise Barriers

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