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Localised wave oscillations in an open system that do not decay or grow intime, despite their frequency lying within a continuous spectrum of radiationmodes carrying energy to or from infinity, are known as bound states in thecontinuum (BIC). Small perturbations from the typically delicate conditions forBIC almost always result in the waves weakly coupling with the radiation modes,leading to leaky states called quasi-BIC that have a large quality factor. Westudy the asymptotic nature of this weak coupling in the case of acoustic wavesinteracting with a rigid substrate featuring a partially partitioned slit -- asetup that supports quasi-BIC that exponentially approach BIC as the slit ismade increasingly narrow. In that limit, we use the method of matchedasymptotic expansions in conjunction with reciprocal relations to study thosequasi-BIC and their resonant excitation. In particular, we derive a leadingapproximation for the exponentially small imaginary part of each wavenumbereigenvalue (inversely proportional to quality factor), which is beyond allorders of the expansion for the wavenumber eigenvalue itself. Furthermore, wederive a leading approximation for the exponentially large amplitudes of thestates in the case where they are resonantly excited by a plane wave at obliqueincidence. These resonances occur in exponentially narrow wavenumber intervalsand are physically manifested in cylindrical-dipolar waves emanating from theslit aperture and exponentially large field enhancements inside the slit. Theasymptotic approximations are validated against numerical calculations.

Acoustics of a partially partitioned narrow slit connected to a  half-plane: case study for exponential quasi-bound states in the continuum  and their resonant excitation