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Fundamentals of Acoustic Metamaterials

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Acoustic Metamaterials

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 166))

Abstract

This chapter introduces the field of acoustic metamaterials in light of correspondences with related phenomena in electromagnetics. The semantic frontier between phononic/photonic crystals (PCs) and metamaterials is underpinned by low-frequency high-contrast and high-frequency homogenization models for periodic structures, the former being well suited for metamaterials, while the latter unveils the band structure and associated anomalous dispersion of PCs. We find it therefore worthwhile to outline the corresponding asymptotic models for waves propagating in such structured media. The mathematics behind the physical scene are illustrated by numerical simulations including cloaking, lensing and confinement effects via artificial anisotropy (motivated by transformational optics and acoustics), negative refraction and slow waves.

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Notes

  1. 1.

    At this stage, the meaning of the word effectively seems to be rather vague, but its definition will be made more precise with specific physical contexts and mathematical models as we proceed.

  2. 2.

    Kohn and Shipman were the first mathematicians to retrieve Pendry’s formula for artificial magnetism via two-scale homogenization of split ring resonators in 2008 [25]. However, Bouchitté and Schweizer obtained a more general form for the tensor of effective permeability in the case of a cubic array of toroidal SRRs with a thin slit of high contrast material in 2010 [3]. In the two-dimensional case, Farhat et al. retrieved Pendry’s result in 2009, using a three-scale homogenization approach for SRRs with thin slits with Neumann data which model infinite conducting boundaries for transverse electromagnetic waves as in Fig. 1.2 or rigid boundaries in the context of acoustics [15].

  3. 3.

    One can also assume that Ω=ℝ2, in which case the displacement goes to zero at infinity if it is assumed of finite energy and the surface term naturally vanishes.

  4. 4.

    Negative refraction can also be seen as some kind of space folding, see for instance [21].

  5. 5.

    If one assumes Dirichlet (clamped) conditions hold on the upper and lower edges of the thin-domain, this kills the field oscillations in the thin bridge, which is of no physical interest.

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Authors and Affiliations

  1. Institut Fresnel, UMR CNRS 7249, Aix-Marseille Université, Marseille, France

    Sébastien Guenneau

  2. Imperial College London, London, UK

    Richard V. Craster

Authors
  1. Sébastien Guenneau
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  2. Richard V. Craster
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Correspondence to Sébastien Guenneau or Richard V. Craster .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College, South Kensington Campus, London, SW7 2BZ, United Kingdom

    Richard V. Craster

  2. Institut Fresnel CNRS, Rue du Coteau 71, Marseille, 13770, France

    Sébastien Guenneau

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Guenneau, S., Craster, R.V. (2013). Fundamentals of Acoustic Metamaterials. In: Craster, R., Guenneau, S. (eds) Acoustic Metamaterials. Springer Series in Materials Science, vol 166. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4813-2_1

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