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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 $\vec{V_{ph}}=\frac{\omega}{k}$.   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 \[\theta_{group}=a\tan\left(\frac{\sin\theta\left(\cos\theta-2\frac{f}{f_{ce}}\right)}{1+\cos\theta\left(\cos\theta-2\frac{f}{f_{ce}}\right)}\right)\]  Consider the case when the wave propagationangle is small with respect to the magnetic field and there are no collisions collisions,  then appletons equation reduces to \[\eta=\sqrt{1-\frac{\omega_{pe}^2}{\omega(\omega-\omega_{ce}\cos\theta)}}\]