In the upper panel, simulations of the fraction of H in H i are plotted for a periodic step function light-curve with Δt = 1.6 × 10⁸ s (shown in the lower panel). The hydrogen number density of the thick black line was chosen to give a recombination timescale equal to half the period of the lightcurve. The red dashed line has a timescale that is 4 times shorter, and the green line has a timescale that is 4 times longer. The dotted black lines are analytic solutions for a flux change f = 1 (lower line) and f = 0.5 (upper line).