Period, damping time, and ratio of the damping time to period versus the horizontal wavenumber for the long (solid) and short (dashed) period Alfvén waves in a PIP with and k_z = 1.57 × 10^-5 m^-1. The location of the critical wavenumbers is defined by the presence of peaks and bifurcations in the plots. The background flow speed is 15 km s^-1 and the shaded region corresponds to an interval of horizontal wavelengths coincident with observed wavelengths in prominence oscillations.