phase and group velocity

[neelakash]

1. The problem statement, all variables and given/known data

I do not know if it is a homework problem or not.The moderators will determine that.However,I read in acoustics books that it is a standard practice to use superposition of harmonic functions to denote a finite wave train, which cannot be given by a sine/cosine functions(because they are infinitely extended in time and space).This wave tran is known as wave group.
The impression I got is that a wave group is a mathematical concept and have no physical reality.Yet,its velocity, i.e. group velocity is different fro the phase velocity(w/k) of the wave.
So,where we are?It seems that phase velocity (w/k) is the physical reality but the group velocity is a mathematical concept only.However,phase velocity may be >c,whereas group velocity always<c.Please make the misconception clear.


2. Relevant equations



3. The attempt at a solution

[mjsd]

group velocity is the speed of actual information propagation and that's why it must be bounded by c.
if you have just one wave, you can't transmit information, but if you have say two waves then, you can transmit information by adjusting the phase between the two waves accordingly. In that case, the phase difference "encodes" your information, and it travels at the group velocity, not the phase velocity (which is the velocity of the individual wave...determined by the frequency and wave length). same principle works when you have many waves put together. mathematically, group velocity is given by dw/dk (the derivative clear indicating the "difference" between two things)