Dimensionless frequency, ωτ_Ap, versus time in units of the internal Alfvén travel time, τ_Ap = L/v_Ap, for a) L_p/L =  0.05, 0.1, 0.2 with v₀/v_Ap =  0.05, and b) v₀/v_Ap =  0.03, 0.05, 0.1 with L_p/L = 0.1. In all cases, z₀/L =  −0.25. The solid lines are the results obtained by numerically solving Eq. (14), whereas the symbols correspond to the analytical approximation given by Eq. (16). The vertical dotted line in panel a) denotes the minimum of the curves, i.e., the time when the prominence thread is centered within the flux tube.