  The precision of compact inertial sensing schemes using trapped- and
guided-atom interferometers has been limited by uncontrolled phase errors
caused by trapping potentials and interactions. Here, we propose an acoustic
interferometer that uses sound waves in a toroidal Bose-Einstein condensate to
measure rotation, and we demonstrate experimentally several key aspects of this
type of interferometer. We use spatially patterned light beams to excite
counter-propagating sound waves within the condensate and use \emph{in situ}
absorption imaging to characterize their evolution. We present an analysis
technique by which we extract separately the oscillation frequencies of the
standing-wave acoustic modes, the frequency splitting caused by static
imperfections in the trapping potential, and the characteristic precession of
the standing-wave pattern due to rotation. Supported by analytic and numerical
calculations, we interpret the noise in our measurements, which is dominated by
atom shot noise, in terms of rotation noise. While the noise of our acoustic
interferometric sensor, at the level of $\sim \mbox{rad}\,
\mbox{s}^{-1}/\sqrt{\mbox{Hz}}$, is high owing to rapid acoustic damping and
the small radius of the trap, the proof-of-concept device does operate at $10^4
- 10^6$ times higher density and in a volume $10^9$ times smaller than
free-falling atom interferometers.
