deletions | additions       diff --git a/Theory.tex b/Theory.tex  index caec486..76e4164 100644  --- a/Theory.tex  +++ b/Theory.tex 
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\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 $\vec{V}_{gr}=\frac{\mathrm\partial\omega}{\mathrm \partial\vec{k}}$ and the phase velocity is given by $V_{ph}=\frac{\omega}{k}$ $V_{ph}=\frac{\omega}{k}$.   The group velocity can be rewritten as $V_{group}=\frac{c}{\frac{d}{df}\left\etaf\right}