Waveguide with double threshold resonance at a simple threshold

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.
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