Phase vs group vs signal velocities

[nomadreid]

www.mathpages.com/home/kmath210/kmath210.htm outlines the distinction between phase and group velocities, and why the group velocity of electromagnetic waves going faster than the speed of light c does not mean that information is going faster than c, because the phase velocity will always be less than or equal to c. The arguments seem to be clear.

However, although the site http://www.physique.usherbrooke.ca/grosdidier/phq210/phasegroup.pdf
seems to agree with the previous site's mathematical definitions of phase and group velocities, it interprets them differently to say that neither one of them represents the signal velocity, namely the velocity of information transfer. It also defends relativity, but in saying that both phase and group velocities can exceed c, but not the signal velocity. Its arguments are a bit more involved, but also seem clear. However, one of them is not completely correct, since the first one has phase velocity always less than or equal to c, and the second one has phase velocities greater than c.

I am missing something here. Please help. Thanks.

 

[netheril96]

In general,the phase velocity can definitely exceed c.
Suppose a long line of computer-controlled spring oscillators.The computers are programmed to release the oscillator at a certain,scheduled time.In this way,one can generate a wave with very long wavelength,and its phase velocity can easily exceed c.
This phase velocity is definitely not the signal velocity.In fact,nothing is propagated except the "phase".The coordination is made in advance,not a result of propagation.

But in terms of electromagnetic waves,things may be different.Maybe the Maxwell equations prohibit a electromagnetic wave with phase velocity greater than c.
I don't know more about it.

 

[nomadreid]

So, if the second link mentioned in my first post is correct, this brings me to the basis of its arguments, which is that if a wavelength with wavelength $$\lambda$$ is observed obliquely, the wavelength the observer will measure is $$\lambda$$/cos$$\theta$$, where $$\theta$$ is the angle of the line of the observer to the line of the pulse. However, applying this to electromagnetic pulses seems to me to be using geometry which may not be valid at relativistic speeds. It seems to be assuming what we want to prove: that you can have an observer traveling at superluminal speeds. Something is rotten in the state of Denmark...

 

[bcrowell]

www.mathpages.com/home/kmath210/kmath210.htm outlines the distinction between phase and group velocities, and why the group velocity of electromagnetic waves going faster than the speed of light c does not mean that information is going faster than c, because the phase velocity will always be less than or equal to c. The arguments seem to be clear.

In typical cases, the group velocity is the velocity at which the information travels, and the group velocity is less than c, even if the phase velocity is greater than c. The link you gave talks about some unusual situations where the group velocity is not the velocity at which the information travels, but taking the usual case, in the quote above you've basically got "phase" and "group" turned around.

However, although the site http://www.physique.usherbrooke.ca/grosdidier/phq210/phasegroup.pdf
seems to agree with the previous site's mathematical definitions of phase and group velocities, it interprets them differently to say that neither one of them represents the signal velocity, namely the velocity of information transfer.

Actually both pages agree that neither one necessarily corresponds to the signal velocity.

 

[nomadreid]

First, thanks. Secondly, I am red-faced in switching "group" and "phase" in my question, and in fact not having read the first article closely enough. My apologies.
Thirdly: the arguments of the first article are much simpler than the arguments of the second article; are they also just as valid?