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Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions

Abstract

We consider an acoustic waveguide (the Neumann problem for the Helmholtz equation) shaped like a periodic family of identical beads on a thin cylinder rod. Under minor restrictions on the bead and rod geometry, we use asymptotic analysis to establish the opening of spectral gaps and find their geometric characteristics. The main technical difficulties lie in the justification of asymptotic formulas for the eigenvalues of the model problem on the periodicity cell due to its arbitrary shape.

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Correspondence to F. L. Bakharev.

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Original Russian Text Copyright © 2015 Bakharev F.L. and Nazarov S.A.

The authors were supported by St. Petersburg State University (Project 0.38.237.2014) and the Russian Foundation for Basic Research (Grant 15–01–02175).

St. Petersburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 732–751, July–August, 2015; DOI: 10.17377/smzh.2015.56.402.

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Bakharev, F.L., Nazarov, S.A. Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions. Sib Math J 56, 575–592 (2015). https://doi.org/10.1134/S0037446615040023

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Keywords

  • Neumann problem
  • junction of domains with different limiting dimensions
  • periodic waveguide
  • spectral gaps
  • asymptotics