  The axisymmetric acoustic perturbations in the velocity potential of a
Bose-Einstein condensate in the presence of a single vortex behave like
minimally coupled massless scalar fields propagating in a curved (1+1)
dimensional Lorentzian space-time, governed by the Klein-Gordon wave equation.
Thus far, the amplified scattering of these perturbations from the vortex, as a
manifestation of the acoustic superradiance, has been investigated with a
constant background density. This paper goes beyond by employing a
self-consistent condensate density profile that is obtained by solving the
Gross-Pitaevskii equation for an unbound BEC. Consequently, the loci of the
event horizon and the ergosphere of the acoustic black hole are modified
according to the radially varying speed of sound. The superradiance is
investigated both for transient features in the time-domain and for spectral
features in the frequency domain. In particular, an effective energy-potential
function defined in the spectral formulation correlates with the existence and
the frequency dependence of the acoustic superradiance. The numerical results
indicate that the constant background density approximation underestimates the
maximum superradiance and the frequency at which this maximum occurs.
