One thing to keep in mind about the phase velocity is that it is an entirely mathematical construct. Pure sine waves do not exist, as a monochromatic wave train is infinitely long. They are merely a tool to construct wave packet
s, which have a group velocity
, and that is what we are measuring in experiments.
In fact, it may very well be that the phase velocity comes out as higher than c
, eg in wave guide
s! This puzzles people, and some use that fact to claim that the theory of relativity
is wrong. However, even if you had a pure sine wave, you couldn't use it to transmit any information
, because it is unmodulated, so there is no contradiction.
But it turns out that even the group velocity may be higher than c
, namely in the case of anomalous dispersion
. Now how do we get around this? Well, this kind of dispersion is so bad that the definition of our wave packet loses its meaning because it just disintegrates, and again we cannot use it to transmit information. The only way would be to switch the signal
on and off - these discontinuities propagate with the wavefront
And again, relativity is saved!
Source: R. U. Sexl and H. K. Urbantke, Relativität, Gruppen, Teilchen, chap. 2, 24, 3rd edn., Springer, Wien (1992)