  Waves in thin fluid layers are important in various stellar and planetary
problems. Due to rapid rotation such systems will become oblate, with a
latitudinal variation in the gravitational acceleration across the surface of
the object. In the case of accreting neutron stars, rapid rotation could lead
to a polar radius smaller than the equatorial radius by a factor $\sim 0.8$. We
investigate how the oblateness and a changing gravitational acceleration affect
different hydrodynamic modes that exist in such fluid layers through analytic
approximations and numerical calculations. The wave vectors of $g$-modes and
Yanai modes increase for more oblate systems compared to spherical
counterparts, although the impact of variations in the changing gravitational
acceleration is effectively negligible. We find that for increased oblateness,
Kelvin modes show less equatorial confinement and little change in their wave
vector. For $r$-modes, we find that for more oblate systems the wave vector
decreases. The exact manner of these changes for the $r$-modes depends on the
model for the gravitational acceleration across the surface.
