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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!