  We use matched asymptotics to derive analytical formulae for the acoustic
impedance of a subwavelength orifice consisting of a cylindrical perforation in
a rigid plate. In the inviscid case, an end correction to the length of the
orifice due to Rayleigh is shown to constitute an exponentially accurate
approximation in the limit where the aspect ratio of the orifice is large; in
the opposite limit, we derive an algebraically accurate correction, depending
upon the logarithm of the aspect ratio, to the impedance of a circular aperture
in a zero-thickness screen. Viscous effects are considered in the limit of thin
Stokes boundary layers, where a boundary layer analysis in conjunction with a
reciprocity argument provides the perturbation to the impedance as a quadrature
of the basic inviscid flow. We show that for large aspect ratios the latter
perturbation can be captured with exponential accuracy by introducing a second
end correction whose value is calculated to be in between two guesses commonly
used in the literature; we also derive an algebraically accurate approximation
in the small-aspect-ratio limit. The viscous theory reveals that the resistance
exhibits a minimum as a function of aspect ratio, with the orifice radius held
fixed. It is evident that the resistance grows in the long-aspect-ratio limit;
in the opposite limit, resistance is amplified owing to the large velocities
close to the sharp edge of the orifice. The latter amplification arrests only
when the the plate is as thin as the Stokes boundary layer. The analytical
approximations derived in this paper could be used to improve circuit modelling
of resonating acoustic devices.
