Abstract
We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter e > 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 Λ k of the spectral parameter λ in continuous spectrum. The greatest attention is paid to analyzing the range λ e = Λ k + e2μ2 of the spectral parameter with μ ≤ μ0, which generates scattering coefficients depending on μ > 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 μ → +∞) and approaching it (as μ → +0) characterized by simpler asymptotics.
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