Near-discontinuities in the density of a star, such as that encountered in the transition between a chromosphere and a corona, reflect acoustic waves and thereby partially contain them within the star. These reflections help to establish a resonant acoustic cavity with discrete characteristic frequencies, even for modes with frequencies above the acoustic cutoff frequency. Reflection is not complete, however, and energy propagates away from the star, contributing to the decay of the mode. Nevertheless, for coronae with temperatures of order 10 to the 6th K, the contribution that such decay is likely to make to the line width in a power spectrum of the oscillations is substantially less than the frequency separation between modes of adjacent order and like degree. Therefore, in that case, characteristic acoustic frequencies exist. A physical discussion of the phenomena is presented, and is illustrated by a simple analytical model. The results are compared with numerical stability calculations of high-order radial modes of a model of the sun. 33 refs.