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

S. A. Nazarov

Sbornik. Mathematics, Tome 206 (2015) no. 6, pp. 782-813

Résumé

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.

Détail

Keywords: Helmholtz resonator, scattering problem, thresholds of continuous spectrum, waves at near-threshold frequencies, almost complete reflection and passage.

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