  We (re)cosider the propagation of small disturbances (sound waves) in the
presence of a pinned irrotational vortex in a superfluid with the help of the
formalism of acoustic spacetimes. We give closed formulas for the scattering
angle for sound rays, formulate the sound-propagation problem in the
Hamiltonian form, and discuss the form of boundary conditions at the core of
the vortex, where the Hamiltonian has a singular point. The wave equation is
simplified to a single ordinary differential equation of Mathieu type. We give
an extensive discussion of perturbations localized close to the core, which are
similar to what is known as the Kelvin waves. The spectra of modes depend
strongly on the type of boundary condition employed close to the vortex core.
The existence of the gapless Kelvin mode, which is one of the modes with
angular number -1, is usually discussed in the context of unpinned vortices in
superfluid helium or rotating Bose-Einstein condensates. We prove that this
particular mode is absent if the vortex is pinned, and consequently one must
discuss the full family of modes in this case. The question of whether or not
the acoustic spacetime admits an ergoregion turns out to have a decisive
(qualitative) influence on many aspects of sound-propagation phenomena.
