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Group velocity and Dispersion Relation 
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#1
Apr1510, 01:00 AM

P: 25

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} = [tex]\frac{\partial \omega}{\partial k}[/tex]? 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! 


#2
Apr1510, 07:52 AM

P: 980



#3
Apr1610, 10:48 AM

P: 25

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? 


#4
Apr1910, 02:43 AM

Sci Advisor
P: 3,628

Group velocity and Dispersion Relation
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'.



#5
Aug2811, 03:17 AM

P: 4

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. Here is the link to google book mostly readable. Sincerly yours, Thibault 


#6
Aug2911, 10:24 AM

P: 617




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