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Mat. Sb., 2015, Volume 206, Number 6, Pages 15–48 (Mi msb8381)  

This article is cited in 11 scientific papers (total in 11 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

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).


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English version:
Sbornik: Mathematics, 2015, 206:6, 782–813

Bibliographic databases:

UDC: 517.958:535.4+517.956.8
MSC: Primary 35P25; Secondary 35B20, 35B34, 35B40, 35J05
Received: 28.04.2014

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
\by S.~A.~Nazarov
\paper Scattering anomalies in a~resonator above~the~thresholds of the continuous spectrum
\jour Mat. Sb.
\yr 2015
\vol 206
\issue 6
\pages 15--48
\jour Sb. Math.
\yr 2015
\vol 206
\issue 6
\pages 782--813

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    This publication is cited in the following articles:
    1. Bruneau V., Miranda P., Parra D., Popoff N., “Eigenvalue and Resonance Asymptotics in Perturbed Periodically Twisted Tubes: Twisting Versus Bending”, Ann. Henri Poincare  crossref  mathscinet  isi
    2. A. I. Korolkov, S. A. Nazarov, A. V. Shanin, “Stabilizing solutions at thresholds of the continuous spectrum and anomalous transmission of waves”, ZAMM Z. Angew. Math. Mech., 96:10 (2016), 1245–1260  crossref  mathscinet  isi  scopus
    3. S. A. Nazarov, K. Ruotsalainen, P. Uusitalo, “Multifarious transmission conditions in the graph models of carbon nano-structures”, Mater. Phys. Mech., 29:2 (2016), 107–115  isi
    4. S. A. Nazarov, “The asymptotic behaviour of the scattering matrix in a neighbourhood of the endpoints of a spectral gap”, Sb. Math., 208:1 (2017), 103–156  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. A. V. Shanin, A. I. Korolkov, “Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide”, Wave Motion, 68 (2017), 218–241  crossref  mathscinet  isi  scopus
    6. A.-S. Bonnet-Ben Dhia, L. Chesnel, S. A. Nazarov, “Perfect transmission invisibility for waveguides with sound hard walls”, J. Math. Pures Appl., 111 (2018), 79–105  crossref  mathscinet  zmath  isi  scopus
    7. L. Chesnel, S. A. Nazarov, V. Pagneux, “Invisibility and perfect reflectivity in waveguides with finite length branches”, SIAM J. Appl. Math., 78:4 (2018), 2176–2199  crossref  mathscinet  zmath  isi  scopus
    8. S. A. Nazarov, “Transmission of waves through a small aperture in the cross-wall in an acoustic waveguide”, Siberian Math. J., 59:1 (2018), 85–101  mathnet  crossref  crossref  isi  elib
    9. S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides”, Comput. Math. Math. Phys., 58:11 (2018), 1838–1855  mathnet  crossref  crossref  isi  elib
    10. S. A. Nazarov, “Rasseyanie uprugikh voln na malykh chastotakh v beskonechnoi plastine Kirkhgofa”, Matematicheskie voprosy teorii rasprostraneniya voln. 49, Zap. nauchn. sem. POMI, 483, POMI, SPb., 2019, 142–177  mathnet
    11. S. A. Nazarov, “Waveguide with double threshold resonance at a simple threshold”, Sb. Math., 211:8 (2020), 1080–1126  mathnet  crossref  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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