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lanebeale added Theory.tex
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\section{theory}
The study of whistler waves in ducting will still be determined by the dispersion relation given by the approximation of Appleton's Equation
\begin{tiny}{\[\eta^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{(\omega_{ce}^2\sin^2\theta)^2}{4(1-\frac{\omega_{pe}^2}{\omega^2})}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}\]}
\end{tiny}
One of the remarkable aspects of ducted whistler waves is that the phase velocity travels at a separate angle to the group velocity and direction of propogation of the wave. The group velocity is given by
$\vec{V}_{gr}=\frac{\mathrm\partial\omega}{\mathrm \partial\vec{k}}
