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Bands in the spectrum of a periodic elastic waveguide

Abstract

We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order \( h >0\). The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as \(h \rightarrow 0\).

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Correspondence to J. Taskinen.

Additional information

F. L. Bakharev was supported by the St. Petersburg State University Grant 6.38.64.2012, by the Chebyshev Laboratory—RF Government Grant 11.G34.31.0026, and by JSC “Gazprom Neft”. J. Taskinen was supported by Grants from the Magnus Ehrnroot Foundation and the Väisälä Foundation of the Finnish Academy of Sciences and Letters.

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Bakharev, F.L., Taskinen, J. Bands in the spectrum of a periodic elastic waveguide. Z. Angew. Math. Phys. 68, 102 (2017). https://doi.org/10.1007/s00033-017-0846-0

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Mathematics Subject Classification

  • Primary 35J57
  • Secondary 35P99
  • 35Q74
  • 74J05

Keywords

  • Elliptic system
  • Linearized elasticity problem
  • Essential spectrum
  • Spectral band
  • Spectral gap
  • Asymptotic analysis
  • Floquet–Bloch theory