Group velocity and Dispersion Relation

[Xian]

Hi there


So I was looking into group velocity and related matters and found myself quite confused. So now I have a few questions which I feel I need to understand (primarily the first one). Any help with these would be awesome and I would be very grateful...

1) Why is the group velocity defined as v_group = \frac{\partial \omega}{\partial k}?
What does this physically mean?
2) For what kinds of functions is this meaningful/valid and why?
3) How is group velocity related to signal velocity and the transfer of energy
4) For that matter what is the explicit definition of signal velocity
5) In general, for what functions can we define ω and k?

I'm looking for some pretty rigorous derivation for the first one, as I've seen some heuristics but am not convinced by their generalization. Anyways, thanks in advance!

 

[genneth]

For 1): http://en.wikipedia.org/wiki/Stationary_phase_approximation

 

[Xian]

For 1): http://en.wikipedia.org/wiki/Stationary_phase_approximation

Awesome, looks very promising but unfortunately in its current presentation might be a little to dense for me.

I do notice that it mentions the group velocity definition very briefly as a sufficient condition for the phase being "stationary". I'm having a hard time understanding the implications of stationary phases in the integral the mention. It seems to have something to do with the Fourier components of F(w) but I'm not sure. Would you be able to enlighten me on the implications of "stationary" phase for this kind of integral?

 

[DrDu]

Usually you start from a spacially very broad wavepacket and cosider its Fourier transform, which is very peaked at some k value k' and frequency omega(k). Then you consider how the maximum (or mean) does move in an infinitesimal instant of time. As the Fourier transform is very peaked, you may expand omega(k) into a series around k=k'.

 

[Cybertib]

Hello Xian,

Concerning 2), I found this interesting abstract. http://prola.aps.org/abstract/PR/v104/i6/p1760_1
Concerning 3), to my knowledge, it is similar.

For most of the points, I found a good book which could give you some deep answers:
HM Nussezveig, Causality and dispersion relations. http://books.google.com/books?id=QD...1&cd=12&source=gbs_gdata#v=onepage&q&f=false"

Sincerly yours,
Thibault

 

[chrisbaird]

1) Why is the group velocity defined as v_group = \frac{\partial \omega}{\partial k}?
What does this physically mean?

It means the velocity of a wavepacket. A wavepacket can be represented spatially as a sine wave times an envelope function, such as a Gaussian. The velocity of the sine wave is the phase velocity and the velocity of the envelope is the group velocity. You can quantify this by tracking the peak of the envelope and measuring the distance it travels in a given amount of time.

2) For what kinds of functions is this meaningful/valid and why?

The group velocity only has meaning if the wave packet retains its general shape as it travels through the medium, so that there is still a dominant central peak of the envelope that you can track. Materials with high enough dispersion will jumble up the wave packet enough that there is no peak to track. If you try to calculate the group velocity of such a case, you will get unphysical values, such as imaginary-valued or infinite.

3) How is group velocity related to signal velocity and the transfer of energy

Signal velocity is the same as group velocity. It is the the velocity at which information (and energy) is traveling. In the special case where the front of a wave packet is amplified and the back is destroyed, the group velocity may seem to exceed the speed of light while the signal velocity has not. However, I would argue this is such a case where the group velocity looses physical meaning.

4) For that matter what is the explicit definition of signal velocity

It is the same as group velocity.

5) In general, for what functions can we define ω and k?

ω and k are traditionally defined to mean the angular frequency and angular wavenumber of the wave traveling through the medium. The response of the material to the traversing wave links ω and k in what we call a dispersion relation.