First- and second-kind solutions for the asymmetric tube. In a) the thick lines correspond to first-kind solutions with positive expansion parameters ϵ₁ = 0.01 Mm^-1 (solid line), ϵ₁ = 0.1 Mm^-1 (dashed line), and ϵ₁ = 1 Mm^-1 (dot-dashed line). The thin lines correspond to negative expansion parameters ϵ₁ = −0.01 Mm^-1 (solid line), ϵ₁ = −0.1 Mm^-1 (dashed line), and ϵ₁ = −1 Mm^-1 (dot-dashed line). In this plot it is clear that the solutions with ϵ₁ positive or negative are identical under the inversion of the coordinate, s to −s. Similarly, in b) the second-kind solutions are plotted. For simplicity, in b) only positive expansion parameters solutions are plotted.