First- and second-kind solutions of the symmetric tube as functions of the expansion parameter ϵ₂. In a) a large range of ϵ₂ values is considered. The dotted lines correspond to the asymptotic values of the frequency for ϵ₂ → ∞, Ω₊, given by Eq. (31). The dot-dashed lines correspond to the asymptotic values of the frequencies for ϵ₂ → −∞, Ω₋. In b) only the range of realistic ϵ₂ values is considered. The half-length of the tube is set to L = 100 Mm, the thread half-length is l = 5 Mm, the radius of curvature is R = 60 Mm, the coronal sound speed c_sc = 200 km s^-1, and the temperature contrast χ = 100.