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\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 $V_{ph}=\frac{\omega}{k}$.   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 $\theta_{group}=(a\tan\left({\frac{\sin\theta}(\cos\theta-2\frac{f}{f_{ce}}){1+\cos\theta((\cos\theta-2\frac{f}{f_{ce}))}\}}\right)$ $\theta_{group}=(a\tan\left({\frac{\sin\theta(\cos\theta-2\frac{f}{f_{ce}}){1+\cos\theta((\cos\theta-2\frac{f}{f_{ce}))}\}}\right)$