Adams, S. D. M., Cherednichenko, Kirill, Craster, R. V. and Guenneau, S.
2010.
Highfrequency spectral analysis of thin periodic acoustic strips: theory and numerics.
European Journal of Applied Mathematics
21
(6)
, pp. 557590.
10.1017/S0956792510000215

Abstract
This paper is devoted to the study of the asymptotic behaviour of the highfrequency spectrum of the wave equation with periodic coefficients in a ‘thin’ elastic strip Ση=(0, 1)×(−η/2, η/2), η > 0. The main geometric assumption is that the structure period is of the order of magnitude of the strip thickness η and is chosen in such a way that η−1 is a positive large integer. On the boundary ∂Ση, we set Dirichlet (clamped) or Neumann (tractionfree) boundary conditions. Aiming to describe sequences of eigenvalues of order η−2 in the above problem, which correspond to oscillations of high frequencies of order η−1, we study an appropriately rescaled limit of the spectrum. Using a suitable notion of twoscale convergence for bounded operators acting on twoscale spaces, we show that the limiting spectrum consists of two parts: the Bloch (or band) spectrum and the ‘boundary’ spectrum. The latter corresponds to sequences of eigenvectors concentrating on the vertical boundaries of Ση, and is characterised by a problem set in a semiinfinite periodic strip with either clamped or stressfree boundary conditions. Based on the observation that some of the related eigenvalues can be found by solving an appropriate periodiccell problem, we use modal methods to investigate finitethickness semiinfinite waveguides. We compare our results with those for finitethickness infinite waveguides given in Adams et al. (Proc. R. Soc. Lond. A, vol. 464, 2008, pp. 2669–2692). We also study infinitethickness semiinfinite waveguides in order to gain insight into the finiteheight analogue. We develop an asymptotic algorithm making use of the unimodular property of the modal method to demonstrate that in the weak contrast limit, and when wavenumber across the guide is fixed, there is at most one surface wave per gap in the spectrum. Using the monomode property of the waveguide we can consider the gap structure for the nth mode, when doing so, for tractionfree boundaries, we find exactly one surface wave in each nband gap.
Item Type: 
Article

Date Type: 
Publication 
Status: 
Published 
Schools: 
Mathematics 
Subjects: 
Q Science > QA Mathematics 
Additional Information: 
Pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/09567925/ (accessed 25/02/2014). 
Publisher: 
Cambridge University Press 
ISSN: 
09567925 
Funders: 
EPSRC 
Date of First Compliant Deposit: 
30 March 2016 
Last Modified: 
20 Oct 2017 09:52 
URI: 
http://orca.cf.ac.uk/id/eprint/13209 
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