"Edward Green" <spamspamspam3@netzero.com> wrote in message
news:eca320d0.0406120352.70d64571@posting.google.com...

I've been misapprending something about the "group" of group

velocity

for all these years (not surprising since its been on my list of
"things which won't be too much trouble to figure out when I have a
moment" list for ... uh ... never mind). I pictured a traveling
vaguely Gaussian hump, and had imagined the method for deriving the
"group velocity" would be to track the maximum of this hump. But
trying to understand the hint of the actual derivation provided by
Timo Nieminen, I now see a problem with this simple picture!

The condition for the velocity of the group locus is not that the
components of the packet remain at constant phase to first order
(which would just return the phase velocity) but that the components
remain _in_ phase. This means that if the phase of the center
frequency changes by pi, as we move along a trjaectory defined by

the

group velocity, the phase of the a nearby component also changes by
pi, to first order. This means the hump is not traveling like a

bulge

in a carpet, but like an _oscillating_ bulge in a carpet: an

observer

co-moving at group velocity sees things waving up and down in front

of

his face, unlike the comoving observer at phase velocity, who sees
stasis.

Am I on the right trajectory?

Not really. The easiest way to visualise the nature of the group
velocity is to consider an amplitude modulated wave . The group
velocity is the velocity with which the *envelope* propagates.

The easiest way to visualise the nature of the group
velocity is to consider an amplitude modulated wave . The group
velocity is the velocity with which the *envelope* propagates.

In article <eca320d0.0406120352.70d64571@posting.google.com>,

I've been misapprending something about the "group" of group velocity
for all these years (not surprising since its been on my list of
"things which won't be too much trouble to figure out when I have a
moment" list for ... uh ... never mind). I pictured a traveling
vaguely Gaussian hump, and had imagined the method for deriving the
"group velocity" would be to track the maximum of this hump. But
trying to understand the hint of the actual derivation provided by
Timo Nieminen, I now see a problem with this simple picture!

The condition for the velocity of the group locus is not that the
components of the packet remain at constant phase to first order
(which would just return the phase velocity) but that the components
remain _in_ phase. This means that if the phase of the center
frequency changes by pi, as we move along a trjaectory defined by the
group velocity, the phase of the a nearby component also changes by
pi, to first order. This means the hump is not traveling like a bulge
in a carpet, but like an _oscillating_ bulge in a carpet: an observer
co-moving at group velocity sees things waving up and down in front of
his face, unlike the comoving observer at phase velocity, who sees
stasis.

Am I on the right trajectory?

Yes, pretty much so. Note that the observer travelling at phase
velocity only sees stasis when the wave is monochromatic, or in
vacuum.

Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"