Normalized velocities of: a) the fundamental mode (first-kind) for ϵ₂ = 0.01 Mm^-2 (solid line), 0.1 Mm^-2 (dotted line), −0.01 Mm^-2 (dashed line), and −0.1 Mm^-2 (dot-dashed line); b) the first overtone (second-kind) for different expansion parameters. The half-length of the tube is set to L = 100 Mm, the thread half-length l = 5 Mm, the radius of curvature R = 60 Mm, the coronal sound speed c_sc = 200 km s^-1 , the contrast χ = 100, and ϵ₂ = 0.005 Mm^-2. The shaded area represents the position of the cool prominence plasma.