Eigenfunction v = v(s) solutions of Eq. (4) normalised to its maximum value for (a) circular, (b) elliptical, and (c) sinusoidal tube models (see Fig. 1). The thick black line corresponds to the mode with the minimum radius of curvature for each model (see Sect. 2.3), and the thick gray line corresponds to R₀ = 1000 Mm. The thin lines correspond to eigenfunctions with intermediate R₀ values, smoothly transitioning between both extreme functions. The two red dashed lines in (a) are the normal modes for both maxima and minima of R₀ for the uniform-gravity case.