  For axially symmetric accretion maintained in hydrostatic equilibrium along
the vertical direction, we investigate how the characteristic features of the
embedded acoustic geometry depends on the background Kerr metric, and how such
dependence is governed by three different expressions of the thickness of the
matter flow. We first obtain the location of the sonic points and stationary
shock between the sonic points. We then linearly perturb the flow to obtain the
corresponding metric elements of the acoustic space-time. We thus construct the
causal structure to establish that the sonic points and the shocks are actually
the analogue black hole type and white hole type horizons, respectively. We
finally compute the value of the acoustic surface gravity as a function of the
spin angular momentum of the rotating black hole for three different flow
thicknesses considered in the present work. We find that for some flow models,
the intrinsic acoustic geometry, although in principle may be extended up to
the outer gravitational horizon of the astrophysical black hole, cannot be
constructed beyond a certain truncation radius as imposed by the expressions of
the thickness function of the corresponding flow.
