An anonymous person writes:
>> Group velocity is the speed at which the envelope moves. Phase
>> velocity is the speed at which the peaks move, and is constrained to be
>> the speed of light in the medium. Group velocity is the speed at which
>> energy is transported, but signalling cannot be done at greater than
>> the phase velocity.
> No, you've got it backwards. Phase velocity is irrelevant, group
> velocity is the speed at which signals travel. See the sci.physics
> FAQ at http://math.ucr.edu/home/baez/physics/FTL.html#5:
Signals cannot propagate faster than the speed of light, so this is true
only in the case where group velocity is less than phase velocity. One
has to be careful when identifying energy transport with signalling,
because an envelope can move by interference of different energy at each
point at which it appears, and tweeking it at one point will not
necessarily propagate in the obvious manner.
If I set up a mile long row of flashbulbs, and set them off to produce
what appears to be a bright thing moving at twice the speed of light to an
observer viewing it from the side, I have luminosity moving around in ways
that signals can't. Envelopes can do the same sort of thing, in that the
point where maximum constructive interference occurs may move at arbitrary
speed, but actual mass/energy is not exceeding the speed of light.
> "The paradox is resolved by distinguishing this velocity which is known as
> the phase velocity vph from another velocity known as the group velocity
> vgr which is given by,
> "vgr = c / vph
So if the phase velocity is c, the group velocity is one? :/
> "If a wave solution has a frequency dispersion it will take the form of
> a wave packet which travels at the group velocity which is less than
> c. Only its wave trains travel at the phase velocity. It is only possible
> to send information with such a wave equation at the group velocity so
> the phase velocity is yet another example of a speed faster than light
> which cannot carry a message."
Does anyone else think that this is just plain wrong? Phase velocity
is the speed at which the peaks move within the medium, according to the
wave equation, and is what we conventionally identify with the speed of
light, sound, or whatever.
> The actual resolution of the paradox in the NY Times article is that the
> pulse changes shape but its leading edge does not advance. The peak
> of the pulse arrives sooner but that its because the peak is shifted
> forward within the pulse itself. The energized medium is acting like
> an amplifier, raising the early trailing edge of the pulse up to the
> level of the peak:
Given that the peak is identified with the envelope, this would seem
to be the very definition of how the envelope can appear to move in
strange ways without anything actually going faster than the speed of
light.
--
Eric Michael Cordian 0+
O:.T:.O:. Mathematical Munitions Division
"Do What Thou Wilt Shall Be The Whole Of The Law"
