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Waveguide with double threshold resonance at a simple threshold. (English. Russian original) Zbl 1454.35063

Sb. Math. 211, No. 8, 1080-1126 (2020); translation from Mat. Sb. 211, No. 8, 20-67 (2020).
Summary: A threshold resonance generated by an almost standing wave occurring at a threshold – a solution of the problem that do not decay at infinity, but rather stabilizes there – brings about various anomalies in the diffraction pattern at near-threshold frequencies. Examples when a simple threshold resonance occurs or does not occur are trivial. For the first time an acoustic waveguide (the Neumann spectral problem for the Laplace operator) of a special shape is constructed in which there is a maximum possible number (namely two) of linearly independent almost standing waves at a threshold (equal to a simple eigenvalue of the model problem on the cross-section of the cylindrical outlets to infinity). Effects in the scattering problem for acoustic waves, which are caused by these standing waves are discussed.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
35P25 Scattering theory for PDEs
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