An analysis is made of the canonical problem of flow at very high Reynolds number past a circular aperture in a thin rigid wall. The motion is incompressible, and the shear layer over the aperture is treated as a vortex sheet separating two parallel flows of unequal mean velocities. Viscosity is neglected except for its role in shedding vorticity from the upstream semicircular edge of the aperture. Nominally steady flow is unstable, and often accompanied by large amplitude self–sustaining oscillations at certain discrete frequencies, whose values are governed by a mechanism involving the periodic shedding of vorticity from the leading edge of the aperture and feedback of pressure disturbances produced by interaction of the vorticity with the downstream edge. Admissible frequencies are identified with the real parts of complex characteristic frequencies of the linearized equation of motion of the vortex sheet. These eigenfrequencies are also poles of the Rayleigh conductivity of the aperture, and their dependence on the mean velocity ratio across the aperture is calculated for the first four ‘operating stages’ of the motion. Results are presented in both graphical and tabular forms to facilitate their ready incorporation into numerical models of more complicated flow problems. The investigation completes the linearized study of this problem initiated by Scott (1995), which dealt with forced, time–harmonic oscillations of the shear layer.