Resonance scattering in a waveguide with identical thick perforated barriers

We consider wave propagation across an infinite waveguide of an arbitrary bounded cross-section, whose interior is blocked by two identical thick perforated barriers with holes. When the holes are small, the waves over a broad range of frequencies are almost fully reflected. However, we show the existence of a resonance frequency at which the wave is almost fully transmitted, even for very small holes. Counter-intuitively, this resonance effect occurs for barriers of arbitrary thickness. We also discuss another asymptotic limit, in which the thickness of barriers grows to infinity but the fixed diameter of the holes can be large and even arbitrarily close to the diameter of the waveguide. The resonance scattering, which is known as tunneling effect in quantum mechanics, is demonstrated in a constructive way by rather elementary tools such as separation of variables and matching of the resulting series, in contrast to commonly used abstract methods such as searching for complex-valued poles of the scattering matrix or non-stationary scattering theory. In particular, we derived an explicit equation that determines the resonance frequency. The employed basic tools make the paper accessible to non-experts and educationally appealing.