Transverse wave mode solutions for χ = 0.1. Top: dispersion plot, showing the solutions of Eq. (14) for χ = 0.1 as the normalised phase speed v_ph/v_Ai versus the normalised wave number ka, for the first eleven transverse harmonics. Solid (dashed) curves indicate trapped (leaky) modes. The normalised internal and external Alfvén speeds are indicated by dashed black lines. Bottom: normalised transverse displacement profile P_n(tanh(x/a)) sech^α(x/a) as a function of x/a for three modes whose phase speed and wave number are indicated as coloured circles above. The solid (dashed) curves are the real (imaginary) part.