Ratio of the damping time and period of the torsional wave versus the longitudinal wave number when the plasma-β is equal to zero (black lines) and equal to 0.5 (red lines). The solid lines are for when the tube is rotating ℛ = 0.2 without twist, while the dashed lines are for when the tube is twisted 𝒦 = 0.2 without rotation. The plots have been obtained using the analytic expression represented by Eq. (20).