Negative bending mode curvature via Robin boundary conditions

Samuel D.M. Adams; Richard V. Craster; Sébastien Guenneau

doi:10.1016/j.crhy.2009.03.009

Negative bending mode curvature via Robin boundary conditions
[Conditions de Robin pour modes de flexion à dispersion négative]

Samuel D.M. Adams ¹ ; Richard V. Craster ² ; Sébastien Guenneau ³

¹ Department of Mathematics, Imperial College London, London SW7-2AZ, UK
² Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1
³ Department of Mathematical Sciences, Liverpool University, Liverpool L69-3BX, UK

Comptes Rendus. Physique, Volume 10 (2009) no. 5, pp. 437-446.

Résumé
Abstract

Nous étudions le spectre de bande associé aux modes de Floquet–Bloch dans des guides d'ondes planaires acoustiques périodiques. Nous imposons des conditions d'impédance homogènes (conditions de Robin) sur les bords du guide d'épaisseur finie, et observons dans certains cas une courbure négative de la première bande de dispersion (mode de flexion). Cette trouvaille est pour le moins inattendue, car une telle anomalie n'a pas été reportée à ce jour pour d'autres types de conditions limites dans nombre de problèmes physiques. Encore plus surprenante est l'existence de modes de flexions dans un segment de la zone de Brillouin ainsi que des vitesses de groupe extrêmes. Finalement, nous suggérons certaines pistes pour obtenir de telles conditions d'impédances, telle que la voie classique de l'homogénéisation, même si cela reste pour l'heure un problème ouvert.

We examine the band spectrum, and associated Floquet–Bloch eigensolutions, arising in straight walled acoustic waveguides that have periodic structure along the guide. Homogeneous impedance (Robin) conditions are imposed along the guide walls and we find that in certain circumstances, negative curvature of the lowest (bending) mode can be achieved. This is unexpected, and has not been observed in a variety of physical situations examined by other authors. Further unexpected properties include the existence of the bending mode only on a subset of the Brillouin zone, as well as permitting otherwise unobtainable velocities of energy transmission. We conclude with a discussion of how such boundary conditions might be physically reproduced using effective conditions and homogenization theory, although the methodology to achieve these effective conditions is an open problem.

Publié le : 2009-05-09

DOI : 10.1016/j.crhy.2009.03.009

Keywords: Robin conditions, Negative refraction, Sub-wavelength imaging
Mot clés : Conditions de Robin, Réfraction négative, Imagerie haute résolution

Affiliations des auteurs :

Samuel D.M. Adams ¹ ; Richard V. Craster ² ; Sébastien Guenneau ³

¹ Department of Mathematics, Imperial College London, London SW7-2AZ, UK
² Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1
³ Department of Mathematical Sciences, Liverpool University, Liverpool L69-3BX, UK

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     author = {Samuel D.M. Adams and Richard V. Craster and S\'ebastien Guenneau},
     title = {Negative bending mode curvature via {Robin} boundary conditions},
     journal = {Comptes Rendus. Physique},
     pages = {437--446},
     publisher = {Elsevier},
     volume = {10},
     number = {5},
     year = {2009},
     doi = {10.1016/j.crhy.2009.03.009},
     language = {en},
}

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Samuel D.M. Adams; Richard V. Craster; Sébastien Guenneau. Negative bending mode curvature via Robin boundary conditions. Comptes Rendus. Physique, Volume 10 (2009) no. 5, pp. 437-446. doi : 10.1016/j.crhy.2009.03.009. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2009.03.009/

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