Symmetric Instability

Do at least two of the following:
☺ Use Mathematica to derive(14) from (9)-(13).
☺ Make a plot of the ratio of the two terms on the right hand side
 of (51).
☺ Make plots that show how superimposing two modes of the same
frequency ω, but different β, allows for satisfaction of the rigid boundary conditions.
☺ After equation (63) I make the bold claim that "It turns out (though it will not be shown here) that the minimum of (63) occurs as α->0"....  I think I might be bluffing.  I am not quite sure how to prove this, and I am not sure that Mathematica can help.  Can you prove it?