In a plasma for instance, the dispersion relation is [itex]k^2=\omega ^2 \epsilon_0 \mu_0 -\epsilon_0\mu_0 \omega_p^2[/itex], where I have made the approximation that the permitivity and permeability in a plasma are about those in empty space. Let's take the case where k is real (which happens as soon as [itex]\omega >\omega_p[/itex]). From this, I can calculate the phase velocity:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] v_{\phi}=\frac{\omega}{k}=\sqrt{c^2+\frac{\omega_p^2}{k^2}}>c[/tex]

On the other hand, the group velocity turn out to be c, or a little less than c if we disregard the approximations.

But this is for amonochromaticplane wave. It's not like we have a wave packet where the modulation envelope moves with the group velocity. The wavedoesspread at the phase velocity, transporting with it an energy density. So energy is carried at a speed greater than c. :grumpy:

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# Phase velocity > c

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