  We revisit the theoretical analysis of an expanding ring-shaped Bose-Einstein
condensate. Starting from the action and integrating over dimensions orthogonal
to the phonon's direction of travel, we derive an effective one-dimensional
wave equation for azimuthally-travelling phonons. This wave equation shows that
expansion redshifts the phonon frequency at a rate determined by the effective
azimuthal sound speed, and damps the amplitude of the phonons at a rate given
by $\dot{\cal V}/{\cal V}$, where $\cal{V}$ is the volume of the background
condensate. This behavior is analogous to the redshifting and "Hubble friction"
for quantum fields in the expanding universe and, given the scalings with
radius determined by the shape of the ring potential, is consistent with recent
experimental and theoretical results. The action-based dimensional reduction
methods used here should be applicable in a variety of settings, and are well
suited for systematic perturbation expansions.
