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

Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity