Wave equation for sound in fluids with vorticity

We use Clebsch potentials and an action principle to derive a complete closed system of gauge-invariant equations for sound superposed on a general background flow. Our system reduces to the Unruh [Phys. Rev. Lett. 46 (1981) 1351] and Pierce [J. Acoust. Soc. Am. 87 (1990) 2292] wave equations when the flow is irrotational, or slowly varying. We illustrate our formalism by applying it to waves propagating in a uniformly rotating fluid where the sound modes hybridize with inertial waves.