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Sbornik: Mathematics, 2015, Volume 206, Issue 6, Pages 782–813
DOI: https://doi.org/10.1070/SM2015v206n06ABEH004479 (Mi sm8381) 		 
This article is cited in 19 scientific papers (total in 19 papers)

Scattering anomalies in a resonator above the thresholds of the continuous spectrum

S. A. Nazarovabc

a Saint-Petersburg State Polytechnical University
b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
c St. Petersburg State University, Department of Mathematics and Mechanics
English version PDF (733 kB) Citations (19)
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DOI: https://doi.org/10.1070/SM2015v206n06ABEH004479
Abstract: We consider the Dirichlet spectral problem for the Laplace operator in a multi-dimensional domain with a cylindrical outlet to infinity, a Helmholtz resonator. Using asymptotic analysis of the scattering matrix we demonstrate different types of reflection of high-amplitude near-threshold waves. One scattering type or another, unstable or stable with respect to variations of the resonator shapes, is determined by the presence or absence of stabilizing solutions at the threshold frequency, respectively. In a waveguide with two cylindrical outlets to infinity, we discover the effect of almost complete passage of the wave under ‘fine tuning’ of the resonator.
Bibliography: 26 titles.
Keywords: Helmholtz resonator, scattering problem, thresholds of continuous spectrum, waves at near-threshold frequencies, almost complete reflection and passage.
Funding agency Grant number
Russian Foundation for Basic Research 15-01-02175
This research was supported by the Russian Foundation for Basic Research (grant no. 15-01-02175).
Received: 28.04.2014
Russian version:
Matematicheskii Sbornik, 2015, Volume 206, Number 6, Pages 15–48
DOI: https://doi.org/10.4213/sm8381
Bibliographic databases:
Document Type: Article
UDC: 517.958:535.4+517.956.8
MSC: Primary 35P25; Secondary 35B20, 35B34, 35B40, 35J05
Language: English
Original paper language: Russian
Citation: S. A. Nazarov, “Scattering anomalies in a resonator above the thresholds of the continuous spectrum”, Mat. Sb., 206:6 (2015), 15–48; Sb. Math., 206:6 (2015), 782–813
Citation in format AMSBIB
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    • This publication is cited in the following 19 articles:
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