phase velocity

The phase velocity is a term describing the velocity of a point of constant (phase) in a pure sine wave through or across a medium.

The phase speed v, the frequency f, and the wavelength λ are related as follows:

v = λf

The phase speed can vary depending on the medium through which it's travelling, and what mode of propagation. For example, in electromagnetic waves travelling through space the phase speed v is related as:

byv² = 1/ε₀μ₀

where ε₀ is the electric constant (absolute permittivity of free space ... 8.854 x 10^-12 Fm^-1), and μ₀ is the magnetic constant (permeability of free space ... 4π x 10^-7 Hm^-1).

See group velocity - there's a difference =)

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 packets, 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 guides! 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 velocity

v_F=lim k→∞(ω(k)/k)

And again, relativity is saved!


Source: R. U. Sexl and H. K. Urbantke, Relativität, Gruppen, Teilchen, chap. 2, 24, 3rd edn., Springer, Wien (1992)