S. A. Nazarov, “Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide”, Mat. Zametki, 87:5 (2010), 764–786; Math. Notes, 87:5 (2010), 738

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Matematicheskie Zametki, 2010, Volume 87, Issue 5, Pages 764–786
DOI: https://doi.org/10.4213/mzm8719 (Mi mzm8719)

This article is cited in 40 scientific papers (total in 40 papers)

Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Full-text PDF (702 kB) Citations (40)

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DOI: https://doi.org/10.4213/mzm8719

Abstract: It is established that a small periodic singular or regular perturbation of the boundary of a cylindrical three-dimensional waveguide can open up a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator in the resulting periodic waveguide. A singular perturbation results in the formation of a periodic family of small cavities while a regular one leads to a gentle periodic bending of the boundary. If the period is short, there is no gap, while if it is long, a gap appears immediately after the first segment of the continuous spectrum. The result is obtained by asymptotic analysis of the eigenvalues of an auxiliary problem on the perturbed cell of periodicity.

Keywords: cylindrical waveguide, gap in a continuous spectrum, Laplace operator, Dirichlet problem, Helmholtz equation, cell of periodicity, Sobolev space.

Received: 18.08.2008

English version:
Mathematical Notes, 2010, Volume 87, Issue 5, Pages 738–756
DOI: https://doi.org/10.1134/S0001434610050123

Bibliographic databases:

Document Type: Article

UDC: 517.956.227:517.958

Language: Russian

Citation: S. A. Nazarov, “Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide”, Mat. Zametki, 87:5 (2010), 764–786; Math. Notes, 87:5 (2010), 738–756

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Linking options:

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https://doi.org/10.4213/mzm8719

https://www.mathnet.ru/eng/mzm/v87/i5/p764

This publication is cited in the following 40 articles:

S. A. Nazarov, “Gaps in the Spectrum of Thin Waveguides with Periodically Locally Deformed Walls”, Comput. Math. and Math. Phys., 64:1 (2024), 99
S. A. Nazarov, “Lakuny v spektre tonkikh volnovodov s periodicheski raspolozhennymi lokalnymi deformatsiyami stenok”, Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, 64:1 (2024)
Delfina Gómez, Sergei A. Nazarov, Rafael Orive-Illera, María-Eugenia Pérez-Martínez, “Spectral gaps in a double-periodic perforated Neumann waveguide”, ASY, 131:3-4 (2023), 385
Richard Craster, Sébastien Guenneau, Muamer Kadic, Martin Wegener, “Mechanical metamaterials”, Rep. Prog. Phys., 86:9 (2023), 094501
Nazarov S.A., Taskinen J., “Band-Gap Structure of the Spectrum of the Water-Wave Problem in a Shallow Canal With a Periodic Family of Deep Pools”, Rev. Mat. Complut., 2022
D'Elia L., Nazarov S.A., “Gaps in the Spectrum of Two-Dimensional Square Packing of Stiff Disks”, Appl. Anal., 2022
Rosler F., “A Strange Vertex Condition Coming From Nowhere”, SIAM J. Math. Anal., 53:3 (2021), 3098–3122
Nazarov S.A., Chesnel L., “Transmission and Trapping of Waves in An Acoustic Waveguide With Perforated Cross-Walls”, Fluid Dyn., 56:8 (2021), 1070–1093
D. Gómez, S. A. Nazarov, R. Orive-Illera, M.-E. Pérez-Martínez, “Remark on Justification of Asymptotics of Spectra of Cylindrical Waveguides with Periodic Singular Perturbations of Boundary and Coefficients”, J Math Sci, 257:5 (2021), 597
S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
Nazarov S.A., “Anomalies of Acoustic Wave Scattering Near the Cut-Off Points of Continuous Spectrum (a Review)”, Acoust. Phys., 66:5 (2020), 477–494
Nazarov S.A. Orive-Illera R. Perez-Martinez M.-E., “Asymptotic Structure of the Spectrum in a Dirichlet-Strip With Double Periodic Perforations”, Netw. Heterog. Media, 14:4 (2019), 733–757
Bakharev F.L. Eugenia Perez M., “Spectral Gaps For the Dirichlet-Laplacian in a 3-D Waveguide Periodically Perturbed By a Family of Concentrated Masses”, Math. Nachr., 291:4 (2018), 556–575
Nazarov S.A. Taskinen J., “Essential Spectrum of a Periodic Waveguide With Non-Periodic Perturbation”, J. Math. Anal. Appl., 463:2 (2018), 922–933
Bakharev F.L., Exner P., “Geometrically Induced Spectral Effects in Tubes With a Mixed Dirichlet-Neumann Boundary”, Rep. Math. Phys., 81:2 (2018), 213–231
Piat V.Ch., Nazarov S.A., Ruotsalainen K.M., “Spectral Gaps and Non-Bragg Resonances in a Water Channel”, Rend. Lincei-Mat. Appl., 29:2 (2018), 321–342
S. A. Nazarov, “Breakdown of cycles and the possibility of opening spectral gaps in a square lattice of thin acoustic waveguides”, Izv. Math., 82:6 (2018), 1148–1195
S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, J. Math. Sci. (N. Y.), 243:5 (2019), 746–773
Nazarov S.A., “Wave Scattering in the Joint of a Straight and a Periodic Waveguide”, Pmm-J. Appl. Math. Mech., 81:2 (2017), 129–147
Cardone G. Khrabustovskyi A., “Spectrum of a Singularly Perturbed Periodic Thin Waveguide”, J. Math. Anal. Appl., 454:2 (2017), 673–694

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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