Wave Interaction with Subwavelength Resonators

Ammari, Habib; Davies, Bryn; Hiltunen, Erik Orvehed; Lee, Hyundae; Yu, Sanghyeon

doi:10.1007/978-3-031-05321-4_3

Cited by 8 publications

(10 citation statements)

References 31 publications

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“…It turns out that the eigenvalues and eigenvectors of the generalized capacitance matrix determine the subwavelength resonant frequencies and subwavelength resonant modes of the system of resonators, at leading order. This is made precise by the following result, that was proved in Lemma 3.2.3 and Theorem 3.2.5 of [17].…”

Section: Asymptotic Methodsmentioning

confidence: 88%

“…Then, non-zero 𝛿 > 0 and 𝜔 can be studied as an asymptotic perturbation of this limiting case. In particular, we can prove the existence of 𝑁 subwavelength resonant frequencies, see Theorem 3.2.4 of [17].…”

Section: Asymptotic Methodsmentioning

confidence: 99%

\lambda ^{(1)},\dots,\lambda ^{(N)}

and

\mathbf {v}^{(1)},\dots,\mathbf {v}^{(N)}

be the eigenvalues and associated eigenvectors of the generalized capacitance matrix

\mathcal {C}

, defined in Equation ().…”

Section: Setting and Preliminariesmentioning

confidence: 98%

“…In particular, it has been shown that the resonant frequencies and associated resonant modes of these systems are given, at leading order, by the eigenvalues and eigenvectors of the generalized capacitance matrix. 16,17 This is a generalization of the capacitance matrix that often appears in electrostatics, having been introduced by Maxwell 18 to model the relationship between the distributions of potential and charge in a many-body system of conductors. The application of capacitance coefficients to the high-contrast, low-frequency setting considered here is qualitatively similar, in the sense that the generalized capacitance matrix can be thought of as capturing the strength of the coupling interactions between each of the inclusions.…”

Section: Introductionmentioning

confidence: 99%

“…The subwavelength resonance of systems of high‐contrast inclusions can be characterized using asymptotic analysis. In particular, it has been shown that the resonant frequencies and associated resonant modes of these systems are given, at leading order, by the eigenvalues and eigenvectors of the generalized capacitance matrix 16,17 . This is a generalization of the capacitance matrix that often appears in electrostatics, having been introduced by Maxwell 18 to model the relationship between the distributions of potential and charge in a many‐body system of conductors.…”

Section: Introductionmentioning

confidence: 99%

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Landscape of wave focusing and localization at low frequencies

Davies,

Lou

2023

Stud Appl Math

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High‐contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high‐contrast systems that resonate in a subwavelength regime. Our from‐first‐principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three‐dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three‐dimensional wave focusing and localization in random and non‐periodic systems of subwavelength resonators.

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Section: Asymptotic Methodsmentioning

confidence: 88%

Section: Asymptotic Methodsmentioning

confidence: 99%

\lambda ^{(1)},\dots,\lambda ^{(N)}

and

\mathbf {v}^{(1)},\dots,\mathbf {v}^{(N)}

be the eigenvalues and associated eigenvectors of the generalized capacitance matrix

\mathcal {C}

, defined in Equation ().…”

Section: Setting and Preliminariesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Landscape of wave focusing and localization at low frequencies

Davies,

Lou

2023

Stud Appl Math

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Subwavelength resonant acoustic scattering in fast time-modulated media

Feppon,

Ammari

2024

Journal de Mathématiques Pures et Appliquées

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Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

Feppon

Ammari

2023

ESAIM: M2AN

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We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s, randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O(1), we obtain that the effective medium is governed by a dissipative Lippmann-Schwinger equation which approximates the total field with a relative mean-square error of order O(max((sN)^2N^(-1/3),N^(-1/2))). We retrieve the critical size s ∼ 1/N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes (si (δ))1≤i≤K and is governed by a Lippmann-Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies (ωi (δ))1≤i≤K . These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o(N^(-2)), while the considered frequency is “not too close” to the resonance, i.e. N δ 2 = O(|1 − s/si (δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime N δ 2 |1 − s/si (δ)|−1 → +∞.

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Wave Interaction with Subwavelength Resonators

Cited by 8 publications

References 31 publications

Landscape of wave focusing and localization at low frequencies

Landscape of wave focusing and localization at low frequencies

Subwavelength resonant acoustic scattering in fast time-modulated media

Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

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