
The eigenvalue problem stated by Equation (20) supplemented by the boundary conditions can be solved analytically (Nakariakov and Roberts, 1995b; Cooper et al., 2003). The eigenfunctions describing kink and sausage modes are respectively given by
where is the amplitude. Here is given by with being the phase speed, which is determined by the dispersion relations and where is the Alfvén velocity at the centre of the profile, .http://www.livingreviews.org/lrsp20053 
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