We (re)consider the propagation of small disturbances (sound waves) in the presence of an irrotational vortex in a superfluid pinned/wound on a stiff central wire, 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 for vortices that are bendable (not pinned). The spectra of modes depend strongly on the type of boundary condition employed close to the vortex core. The gapless mode with the angular number −1, the Kelvin mode usually discussed in the context of unpinned vortices in superfluid helium or rotating Bose–Einstein condensates, turns out either not to exist or to have a completely different dispersion relation if the vortex is pinned. 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.