  The hydrodynamic vortex, an effective spacetime geometry for propagating
sound waves, is studied analytically. In contrast with the familiar Kerr
black-hole spacetime, the hydrodynamic vortex model is described by an
effective acoustic geometry which has no horizons. However, this acoustic
spacetime possesses an ergoregion, a property which it shares with the rotating
Kerr spacetime. It has recently been shown numerically that this physical
system is linearly unstable due to the superradiant scattering of sound waves
in the ergoregion of the effective spacetime. In the present study we use
analytical tools in order to explore the onset of these superradiant
instabilities which characterize the effective spacetime geometry. In
particular, we derive a simple analytical formula which describes the physical
properties of the hydrodynamic vortex system in its critical
(marginally-stable) state, the state which marks the boundary between stable
and unstable fluid configurations. The analytically derived formula is shown to
agree with the recently published numerical data for the hydrodynamic vortex
system.
