  The polytropic hydrodynamic vortex describes an effective $(2+1)$-dimensional
acoustic spacetime with an inner reflecting boundary at $r=r_{\text{c}}$. This
physical system, like the spinning Kerr black hole, possesses an ergoregion of
radius $r_{\text{e}}$ and an inner non-pointlike curvature singularity of
radius $r_{\text{s}}$. Interestingly, the fundamental ratio
$r_{\text{e}}/r_{\text{s}}$ which characterizes the effective geometry is
determined solely by the dimensionless polytropic index $N_{\text{p}}$ of the
circulating fluid. It has recently been proved that, in the $N_{\text{p}}=0$
case, the effective acoustic spacetime is characterized by an {\it infinite}
countable set of reflecting surface radii,
$\{r_{\text{c}}(N_{\text{p}};n)\}^{n=\infty}_{n=1}$, that can support static
(marginally-stable) sound modes. In the present paper we use {\it analytical}
techniques in order to explore the physical properties of the polytropic
hydrodynamic vortex in the $N_{\text{p}}>0$ regime. In particular, we prove
that in this physical regime, the effective acoustic spacetime is characterized
by a {\it finite} discrete set of reflecting surface radii,
$\{r_{\text{c}}(N_{\text{p}},m;n)\}^{n=N_{\text{max}}}_{n=1}$, that can support
the marginally-stable static sound modes (here $m$ is the azimuthal harmonic
index of the acoustic perturbation field). Interestingly, it is proved
analytically that the dimensionless outermost supporting radius
$r^{\text{max}}_{\text{c}}/r_{\text{e}}$, which marks the onset of superradiant
instabilities in the polytropic hydrodynamic vortex, increases monotonically
with increasing values of the integer harmonic index $m$ and decreasing values
of the dimensionless polytropic index $N_{\text{p}}$.
