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

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.