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Transformation Elastodynamics and Active Exterior Acoustic Cloaking

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Acoustic Metamaterials

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

Abstract

This chapter consists of three parts. In the first part we recall the elastodynamic equations under coordinate transformations. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a mass-spring network model. The transformed networks can be realized with “torque springs”, which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. In the second and third parts we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. In the second part, the exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of functions. An explicit solution is given that allows cloaking of larger objects at a fixed distance from the cloaking device, compared to previous explicit solutions. In the third part we consider the active exterior cloaking problem for the Helmholtz equation in 3D. Our method uses the Green’s formula and an addition theorem for spherical outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green’s formula.

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Acknowledgements

We are thankful to Michael Bentley for noticing an error in an early version of this chapter. GWM is grateful for support from the University of Toulon-Var. GWM and DO are grateful to the National Science Foundation for support through grant DMS-0707978. FGV is grateful to the National Science Foundation for support through grant DMS-0934664. FGV, GWM and DO are grateful to the Mathematical Sciences Research Institute where parts of this manuscript were completed. The computations of the device and scattered fields in Sect. 12.3 were facilitated by the freely available spherical harmonics library SHTOOLS by Mark Wieczorek, available at http://www.ipgp.fr/~wieczor/SHTOOLS/SHTOOLS.html.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Fernando Guevara Vasquez, Graeme W. Milton & Daniel Onofrei

  2. Institut de Mathématiques de Toulon, Université de Toulon et du Var, BP 132-83957, La Garde Cedex, France

    Pierre Seppecher

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  1. Fernando Guevara Vasquez
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  2. Graeme W. Milton
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  4. Pierre Seppecher
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Correspondence to Graeme W. Milton .

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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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Guevara Vasquez, F., Milton, G.W., Onofrei, D., Seppecher, P. (2013). Transformation Elastodynamics and Active Exterior Acoustic Cloaking. 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_12

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