  Localised wave oscillations in an open system that do not decay or grow in
time, despite their frequency lying within a continuous spectrum of radiation
modes carrying energy to or from infinity, are known as bound states in the
continuum (BIC). Small perturbations from the typically delicate conditions for
BIC 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. We
study the asymptotic nature of this weak coupling in the case of acoustic waves
interacting with a rigid substrate featuring a partially partitioned slit -- a
setup that supports quasi-BIC that exponentially approach BIC as the slit is
made increasingly narrow. In that limit, we use the method of matched
asymptotic expansions in conjunction with reciprocal relations to study those
quasi-BIC and their resonant excitation. In particular, we derive a leading
approximation for the exponentially small imaginary part of each wavenumber
eigenvalue (inversely proportional to quality factor), which is beyond all
orders of the expansion for the wavenumber eigenvalue itself. Furthermore, we
derive a leading approximation for the exponentially large amplitudes of the
states in the case where they are resonantly excited by a plane wave at oblique
incidence. These resonances occur in exponentially narrow wavenumber intervals
and are physically manifested in cylindrical-dipolar waves emanating from the
slit aperture and exponentially large field enhancements inside the slit. The
asymptotic approximations are validated against numerical calculations.
