deletions | additions       diff --git a/Theory.tex b/Theory.tex  index ee15e7d..1c59da7 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 $V_{ph}=\frac{\omega}{k}$. $\vec{V_{ph}}=\frac{\omega}{k}$.  The group velocity can be rewritten as $V_{group}=\frac{c}{\frac{d}{df}(\eta f)}$, which for whistler waves becomes approximately $V_{group}=2 c \frac{\sqrt{f}}{f_{pe}f_{ce}\cos{2\theta}}(f_{ce}\cos{\theta}-f)^{\frac{3}{2}}$ where $\theta$ is the angle between the group velocity and the magnetic field. It can then be shown that this angle is represented as \[\theta_{group}=a\tan\left(\frac{\sin\theta\left(\cos\theta-2\frac{f}{f_{ce}}\right)}{1+\cos\theta\left(\cos\theta-2\frac{f}{f_{ce}}\right)}\right)\]