  The acoustic modes of a rotating fluid-filled cavity can be used to determine
the effective rotation rate of a fluid (since the resonant frequencies are
modified by the flows). To be accurate, this method requires a prior knowledge
of the acoustic modes in rotating fluids. Contrary to the Coriolis force,
centrifugal gravity has received much less attention in the experimental
context. Motivated by on-going experiments in rotating ellipsoids, we study how
global rotation and buoyancy modify the acoustic modes of fluid-filled
ellipsoids in isothermal (or isentropic) hydrostatic equilibrium. We go beyond
the standard acoustic equation, which neglects solid-body rotation and gravity,
by deriving an exact wave equation for the acoustic velocity. We then solve the
wave problem using a polynomial spectral method in ellipsoids, which is
compared with finite-element solutions of the primitive fluid-dynamic
equations. We show that the centrifugal acceleration has measurable effects on
the acoustic frequencies when $M_\Omega \gtrsim 0.3$, where $M_\Omega$ is the
rotational Mach number defined as the ratio of the sonic and rotational time
scales. Such a regime can be reached with experiments rotating at a few tens of
Hz, by replacing air with a highly compressible gas (e.g. SF$_6$ or
C$_4$F$_8$).
