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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}\left\etaf\right} $V_{group}=\frac{c}{\frac{d}{df}\left\eta f \right}