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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on April 3, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(3):221-233; doi:10.1093/qjmam/hbp007
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© The author 2009. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

On the existence of waves guided by a cavity in an elastic film

K. D. Cherednichenko{dagger}

( School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG )

F. J. Sabina{ddagger}

( Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, Delegación de Alvaro Obregón, 01000 México Distrito Federal, México )

{dagger} <cherednichenkokd{at}cardiff.ac.uk>

{ddagger} <fjs{at}mym.iimas.unam.mx>

Received 29 December 2007. Revise 21 June 2008. Revise 27 January 2009. Accepted 25 February 2009.


  Abstract

In the regime where linearised elasticity is a suitable approximation to the behaviour of an elastic body, the existence of a wave guided along the cavity in a film of a homogeneous and isotropic elastic material is proved, at least for a certain range of frequencies. Using the theory of linear self-adjoint operators, it is shown that the associated eigenvalue problem has a nontrivial solution in an appropriate Sobolev class, with the corresponding eigenvalue lying below the continuous spectrum. We study the existence of localised modes of this kind in two particular cases: under the assumption that the cavity is sufficiently narrow in the direction transverse to the film and for a rectangular cavity of arbitrary size.


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