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Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays

We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness beco...

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Autores principales: Smith, Michael J. A., Abrahams, I. David
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9215215/
https://www.ncbi.nlm.nih.gov/pubmed/35756874
http://dx.doi.org/10.1098/rspa.2022.0125
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author Smith, Michael J. A.
Abrahams, I. David
author_facet Smith, Michael J. A.
Abrahams, I. David
author_sort Smith, Michael J. A.
collection PubMed
description We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the extremely thick-walled limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that extremely thick-walled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications.
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spelling pubmed-92152152022-06-24 Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays Smith, Michael J. A. Abrahams, I. David Proc Math Phys Eng Sci Research Articles We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the extremely thick-walled limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that extremely thick-walled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications. The Royal Society 2022-06 2022-06-22 /pmc/articles/PMC9215215/ /pubmed/35756874 http://dx.doi.org/10.1098/rspa.2022.0125 Text en © 2022 The Authors. https://creativecommons.org/licenses/by/4.0/Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) , which permits unrestricted use, provided the original author and source are credited.
spellingShingle Research Articles
Smith, Michael J. A.
Abrahams, I. David
Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays
title Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays
title_full Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays
title_fullStr Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays
title_full_unstemmed Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays
title_short Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays
title_sort tailored acoustic metamaterials. part ii. extremely thick-walled helmholtz resonator arrays
topic Research Articles
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9215215/
https://www.ncbi.nlm.nih.gov/pubmed/35756874
http://dx.doi.org/10.1098/rspa.2022.0125
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