Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T01:08:53.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of near-cloaking

Published online by Cambridge University Press:  16 July 2020

J. R. OCKENDON ,
H. OCKENDON ,
B. D. SLEEMAN  and
R. H. TEW Open the ORCID record for R. H. TEW [Opens in a new window]
J. R. OCKENDON
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK emails: [email protected]; [email protected]
H. OCKENDON
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK emails: [email protected]; [email protected]
B. D. SLEEMAN
Affiliation:
School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK email: [email protected]
R. H. TEW
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper describes how asymptotic analysis can be used to gain new insights into the theory of cloaking of spherical and cylindrical targets within the context of acoustic waves in a class of linear elastic materials. In certain cases, these configurations allow solutions to be written down in terms of eigenfunction expansions from which high-frequency asymptotics can be extracted systematically. These asymptotics are compared with the predictions of ray theory and are used to describe the scattering that occurs when perfect cloaking models are regularised.

Keywords

Cloakingray theoryelastic wavesasymptotics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 6: Special issue featuring papers on Professor Sam Howison , December 2021 , pp. 1106 - 1126
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Colton, D. & Kress, R. (2013) Inverse Acoustic and Electromagnetic Scattering Theory. Springer-Verlag New York, New York, USA.Google Scholar
Crosskey, M. M., Nixon, A. T., Schick, L. M. & Kovacic, G. (2011) Invisibility cloaking via non-smooth transformation optics and ray tracing. Phys. Lett. A. 375, 1903–1911.CrossRefGoogle Scholar
Cummer, S. A., Popa, B.-I., Schurig, D., Smith, D. R., Pendry, J., Rahm, M. & Starr, A. (2008) Scattering theory derivation of a 3D acoustic. Shell. Phys. Rev. Lett. 100, 024301.CrossRefGoogle ScholarPubMed
Dassios, G. & Kleinman, R. E. (2000) Low-Frequency Scattering. Oxford University Press, Oxford, UK.Google Scholar
Gao, P. & Wu, L. (2016) Non–singular acoustic cloak derived by the ray tracing method with rotationally symmetric transformations. Proc. Roy. Soc. Lond. (A) 472, 20150348. http://dx.doi.org/10.1098/rspa.2015.0348 Google Scholar
Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. (2009) Cloaking devices, electromagnetic wormholes and transformation optics. SIAM Rev. 51, 3–33.CrossRefGoogle Scholar
Hu, C.-Y. & Lin, C.-H. (2015) Reverse ray tracing for transformation optics. Opt. Exp. 23, 23017622.CrossRefGoogle ScholarPubMed
Hu, G. & Liu, H. (2015) Nearly cloaking the elastic wave fields. J. Math. Pure Appl. 104, 1045–1074.CrossRefGoogle Scholar
Jones, D. S. (1986) Acoustic and Electromagnetic Waves, Oxford University Press, Oxford, UK.Google Scholar
Kohn, R. V., Onofrei, D., Vogelius, M. S. & Weinstein, M. I. (2010) Cloaking via change of variables in the Helmholtz equation. Comm. Pure Appl. Math. 63, 973–1016.CrossRefGoogle Scholar
Kohn, R. V., Shen, H., Vogelius, M. S. & Weinstein, M. I. (2008) Cloaking via change of variables in electric impedance tomography. Inverse Probl. 24, 15–16.CrossRefGoogle Scholar
Milton, G. W., Briane, M. & Willis, J. R. (2006) On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, Article 248CrossRefGoogle Scholar
Norris, A. N. (2008) Acoustic cloaking in 2D and 3D using finite mass. arxiv.org/abs/0802.0701.Google Scholar
Norris, A. N. (2008) Acoustic cloaking theory. Proc. R. Soc. A 464, 2411–2434.CrossRefGoogle Scholar
Norris, A. N. & Shuvalov, A. (2011) Elastic cloaking theory. Wave Motion 48, 525–538.CrossRefGoogle Scholar
Nussenszveig, H. M. (1965) High-frequency scattering by an impenetrable sphere. Ann. Phys. 34, 23–95.CrossRefGoogle Scholar
Ockendon, J., Howison, S., Lacey, A. & Movchan, A. (2003) Applied Partial Differential Equations. Oxford University Press, Oxford, UK.Google Scholar
Ockendon, J. R. & Tew, R. H. (2012) Thin layer solutions of the Helmholtz and related equations. SIAM Rev. 54, 3–51.CrossRefGoogle Scholar
Olver, F. W. J., Lozier, D. W. & Bolsvert, R. F. (eds.) (2010) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, UK.Google Scholar
Spence, E. A. (2014) The Watson transformation revisited. http:people.bath.ac.uk/eas25/Sp13b.pdfGoogle Scholar
Tanaka, T. & Matoba, O. (2017) Evolution and design of a large-scale cloaking device by the Hamilton-based ray-tracing method. J. Opt. Soc. Am. 34, 1052–1059.CrossRefGoogle Scholar
Urzhumov, Y., Ghezzo, F., Hunt, J. & Smith, D. R. (2010) Acoustic cloaking transformations from attainable properties. New J. Phys. 12(7), 073014.CrossRefGoogle Scholar