We investigate the scalar-wave resonances of systems composed of identical
Neumann-type inclusions arranged periodically around a circular ring. Drawing
on natural similarities with the undamped Rayleigh-Bloch waves supported by
infinite linear arrays, we deduce asymptotically the exponentially small
radiative damping in the limit where the ring radius is large relative to the
periodicity. In our asymptotic approach, locally linear Rayleigh-Bloch waves
that attenuate exponentially away from the ring are matched to a ring-scale
WKB-type wave field. The latter provides a descriptive physical picture of how
the mode energy is transferred via tunnelling to a circular
evanescent-to-propagating transition region a finite distance away from the
ring, from where radiative grazing rays emanate to the far field. Excluding the
zeroth-order standing-wave modes, the position of the transition circle
bifurcates with respect to clockwise and anti-clockwise contributions,
resulting in striking spiral wavefronts.