  We demonstrate the use of a ring-shaped Bose-Einstein condensate as a
rotation sensor by measuring the interference between two counter-propagating
phonon modes imprinted azimuthally around the ring. We observe rapid decay of
the excitations, quantified by quality factors of at most $Q \approx 27$. We
numerically model our experiment using the c-field methodology, allowing us to
estimate the parameters that maximise the performance of our sensor. We explore
the damping mechanisms underlying the observed phonon decay, and identify two
distinct Landau scattering processes that each dominate at different driving
amplitudes and temperatures. Our simulations reveal that $Q$ is limited by
strong damping of phonons even in the zero temperature limit. We perform an
experimental proof-of-principle rotation measurement using persistent currents
imprinted around the ring. We demonstrate a rotation sensitivity of up to
$\Delta \Omega \approx 0.3$ rad/s from a single image, with a theoretically
achievable value of $\Delta \Omega \approx 0.04$ rad/s in the atomic shot-noise
limit. This is a significant improvement over the shot-noise-limited $\Delta
\Omega \approx 1$ rad/s sensitivity obtained by Marti et al. [Phys. Rev. A 91,
013602 (2015)] for a similar setup.
