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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  $\vec{V}_{gr}=\frac{\mathrm\partial\omega}{\mathrm \partial\vec{k}}$ \begin{$\vec{V}_{gr}=\frac{\mathrm\partial\omega}{\mathrm \partial\vec{k}}$}