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High-frequency spectral analysis of thin periodic acoustic strips: Theory and numerics

Published online by Cambridge University Press:  27 July 2010

S. D. M. ADAMS ,
K. D. CHEREDNICHENKO ,
R. V. CRASTER  and
S. GUENNEAU
S. D. M. ADAMS
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
K. D. CHEREDNICHENKO
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK
R. V. CRASTER
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, CanadaT6G 2G1 email: [email protected]
S. GUENNEAU
Affiliation:
Department of Mathematical Sciences, Liverpool University, Liverpool L69 3BX, UK
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Abstract

This paper is devoted to the study of the asymptotic behaviour of the high-frequency 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 (traction-free) 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 two-scale convergence for bounded operators acting on two-scale 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 semi-infinite periodic strip with either clamped or stress-free boundary conditions. Based on the observation that some of the related eigenvalues can be found by solving an appropriate periodic-cell problem, we use modal methods to investigate finite-thickness semi-infinite waveguides. We compare our results with those for finite-thickness infinite waveguides given in Adams et al. (Proc. R. Soc. Lond. A, vol. 464, 2008, pp. 2669–2692). We also study infinite-thickness semi-infinite waveguides in order to gain insight into the finite-height 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 traction-free boundaries, we find exactly one surface wave in each n-band gap.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 6 , December 2010 , pp. 557 - 590
Copyright
Copyright © Cambridge University Press 2010

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