Year: 2024
Author: Wangtao Lu, Wei Wang, Jiaxin Zhou
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 2 : pp. 234–263
Abstract
This paper presents a Fourier matching method to rigorously study resonances in a sound-hard slab with a finite number of narrow cylindrical holes. The cross sections of the holes, of diameters $\mathcal{O}(h)$ for $h≪1,$ can be arbitrarily shaped. Outside the slab, a sound field can be represented in terms of its normal derivatives on the apertures of the holes. Inside each hole, the field can be represented in terms of a countable Fourier basis due to the zero Neumann boundary condition on the side surface. The countably infinite Fourier coefficients for all the holes constitute the unknowns. Matching the two field representatives leads to a countable-dimensional linear system governing the unknowns. Due to the invertibility of a principal submatrix of the infinite-dimensional coefficient matrix, we reduce the linear system to a finite-dimensional one. Resonances are those when the finite-dimensional linear system becomes singular. We derive asymptotic formulae for the resonances in the subwavelength structure for $h≪1.$ They reveal that a sound field with its real frequency close to a resonance frequency can be enhanced by a magnitude $\mathcal{O}(h^{−2}).$ Numerical experiments are carried out to validate the proposed resonance formulae.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2023-0019
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 2 : pp. 234–263
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: Acoustic scattering problem resonance frequency subwavelength structure Helmholtz equation field enhancement.