Theory of adiabatic fountain resonance

The theory of "Adiabatic Fountain Resonance" with superfluid $^4$He is clarified. In this geometry a film region between two silicon wafers bonded at their outer edge opens up to a central region with a free surface. We find that the resonance in this system is not a Helmholtz resonance as claimed by Gasparini and co-workers, but in fact is a fourth sound resonance. We postulate that it occurs at relatively low frequency because the thin silicon wafers flex appreciably from the pressure oscillations of the sound wave.