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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{7484fig1.eps} \end{figure}$ Figure 1: Results from normal mode theory in which $\Sigma =19.75$ and q=3/2 and $\Omega _0 = 1$ . a) The exact eigenfunctions for $\hat u_{\alpha\beta}$ depicted for the first three overtones each at two values of $\alpha$ . The eigensolutions developed via the asymptotic theory lie exactly on the eigenfunctions for $\alpha = 0.02$ where $A_0 = 1/2\delta$ . b) The frequencies $\omega$ as a function of $\alpha$ for the first three overtones are shown (open circles). The predicted frequencies using the asymptotic theory are shown with solid lines. The values corresponding to the eigenfunctions depicted in the left panel are designated with diamonds. Note that only the positive overtones (i.e., n>0) are shown here - the negative overtones are reflection symmetric about x=1/2 as discussed in the text.
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In the text