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.
\Bibitem{Naz15}
\by S.~A.~Nazarov
\paper Scattering anomalies in a~resonator above~the~thresholds of the continuous spectrum
\jour Sb. Math.
\yr 2015
\vol 206
\issue 6
\pages 782--813
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https://doi.org/10.1070/SM2015v206n06ABEH004479
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This publication is cited in the following 19 articles:
S. A. Nazarov, L. Chesnel, “Almost complete transmission of waves through perforated cross-walls in a waveguide with Dirichlet boundary condition”, Siberian Math. J., 62:2 (2021), 272–291
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
S. A. Nazarov, L. Chesnel, “Anomalies of Acoustic Wave Propagation in Two Semi-Infinite Cylinders Connected by a Flattened Ligament”, Comput. Math. and Math. Phys., 61:4 (2021), 646
S. A. Nazarov, “Scattering of Low-Frequency Elastic Waves in An Infinite Kirchhoff Plate”, J Math Sci, 252:5 (2021), 664
Bruneau V., Miranda P., Parra D., Popoff N., “Eigenvalue and Resonance Asymptotics in Perturbed Periodically Twisted Tubes: Twisting Versus Bending”, Ann. Henri Poincare, 21:2 (2020), 377–403
S. A. Nazarov, “Waveguide with double threshold resonance at a simple threshold”, Sb. Math., 211:8 (2020), 1080–1126
S. A. Nazarov, “Anomalies of acoustic wave scattering near the cut-off points of continuous spectrum (a review)”, Acoust. Phys., 66:5 (2020), 477–494
S. A. Nazarov, “Scattering matrix at small frequencies in a junction of cylindrical acoustic waveguides”, Mech. Sol., 55:8 (2020), 1340–1350
S. A. Nazarov, “Abnormal Behavior of Eigenvalues of Mixed Boundary Value Problems for the Laplace Operator in Truncated, but Long Cylinders”, J Math Sci, 250:2 (2020), 351
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
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
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
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
S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides”, Comput. Math. Math. Phys., 58:11 (2018), 1838–1855
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
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
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
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
S. A. Nazarov, “Transmission Conditions in One-Dimensional Model of a Rectangular Lattice of Thin Quantum Waveguides”, J Math Sci, 219:6 (2016), 994
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