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Sibirsk. Mat. Zh., 2018, Volume 59, Number 1, Pages 110–129 (Mi smj2958)  

Transmission of waves through a small aperture in the cross-wall in an acoustic waveguide

S. A. Nazarovab

a St. Petersburg State University, St. Petersburg, Russia
b Institute of Problems of Mechanical Engineering, St. Petersburg, Russia

Abstract: We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter $\varepsilon>0$. We describe the effects of almost complete reflection or transmission of waves related to the classical Vainstein anomaly and the presence of almost standing waves for the threshold value $\Lambda_k$ of the spectral parameter $\lambda$ in continuous spectrum. The greatest attention is paid to analyzing the range $\lambda^\varepsilon=\Lambda_k+\varepsilon^2\mu^2$ of the spectral parameter with $\mu\leq\mu_0$, which generates scattering coefficients depending on $\mu>0$ and presents the greatest difficulties in constructing and justifying the asymptotics. Almost complete reflection and transmission correspond to the cases of going away from the threshold (as $\mu\to+\infty$) and approaching it (as $\mu\to+0$) characterized by simpler asymptotics.

Keywords: acoustic waveguide, cross-wall, small aperture, scattering, almost complete transmission and reflection of waves.

Funding Agency Grant Number
Russian Science Foundation 17-11-01003
The author was supported by the Russian Science Foundation (Grant 17-11-01003).


DOI: https://doi.org/10.17377/smzh.2018.59.110

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English version:
Siberian Mathematical Journal, 2018, 59:1, 85–101

Bibliographic databases:

UDC: 517.956.8+517.958:535.4
Received: 22.05.2017

Citation: S. A. Nazarov, “Transmission of waves through a small aperture in the cross-wall in an acoustic waveguide”, Sibirsk. Mat. Zh., 59:1 (2018), 110–129; Siberian Math. J., 59:1 (2018), 85–101

Citation in format AMSBIB
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\paper Transmission of waves through a~small aperture in the cross-wall in an acoustic waveguide
\jour Sibirsk. Mat. Zh.
\yr 2018
\vol 59
\issue 1
\pages 110--129
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\crossref{https://doi.org/10.17377/smzh.2018.59.110}
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\jour Siberian Math. J.
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\vol 59
\issue 1
\pages 85--101
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