Vertical stratification of the isothermal cut-off frequency at the axis of the sunspot models with B_u = 1500 G (top panel), B_u = 2500 G (middle panel), and B_u = 3500 G (bottom panel), where z is height and z = 0 is defined at the quiet-Sun photosphere. The colored lines correspond to the Wilson depression of the model: z_WD = 250 km (black), z_WD = 350 km (blue), z_WD = 450 km (green), and z_WD = 550 km (red). The vertical dotted lines illustrate the height where β = 1 for the model with the Wilson depression associated to the corresponding color. The horizontal thin dashed lines mark some selected frequencies (3.25, 4.25, 5.25 mHz). The thick black dashed line illustrates the cut-off frequency at the quiet-Sun.