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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References
Khrabustovskyi A., “Periodic elliptic operators with asymptotically preassigned spectrum,” Asymptot. Anal., 82, No. 1–2, 1–37 (2013).
Khrabustovskyi A. and Khruslov E., “Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps,” Math. Methods Appl. Sci., DOI: 10.1002/ mma.3046 (2014).
Arsen’ev A. A., “The existence of resonance poles and scattering resonances in the case of boundary conditions of the second and third kind,” USSR Comput. Math. Math. Phys., 16, No. 3, 171–177 (1976).
Beale J. T., “Scattering frequencies of resonators,” Comm. Pure Appl. Math., 26, No. 4, 549–563 (1973).
Gadyl’shin R. R., “On eigenfrequencies of bodies with thin branches. II. Asymptotics,” Math. Notes, 55, No. 1, 14–23 (1994).
Kozlov V. A., Maz’ya V. G., and Movchan A. B., “Asymptotic analysis of a mixed boundary value problem in a multistructure,” Asymptot. Anal., 8, 105–143 (1994).
Nazarov S. A., “Junctions of singularly degenerating domains with different limit dimensions. II,” J. Math. Sci., 97, No. 3, 4085–4108 (1999).
Nazarov S. A., “Asymptotic analysis and modeling of the jointing of a massive body with thin rods,” J. Math. Sci., 127, No. 127, 2192–2262 (2005).
Gadyl’shin R. R., “On the eigenvalues of a ‘dumb-bell with a thin handle’,” Izv. Math., 69, No. 2, 265–329 (2005).
Nazarov S. A., “Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler,” Siberian Math. J., 53, No. 2, 274–290 (2012).
Gelfand I. M., “Expansion in eigenfunctions of an equation with periodic coefficients,” Dokl. Akad. Nauk SSSR, 73, 1117–1120 (1950).
Skriganov M. M., “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Proc. Steklov Inst. Math., 171, 1–121 (1987).
Kuchment P. A., “Floquet theory for partial differential equations,” Russian Math. Surveys, 37, No. 4, 1–60 (1982).
Ladyzhenskaya O. A., The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York etc. (1985).
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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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DOI: https://doi.org/10.1134/S0037446615040023