Over the last few days I have been looking at a derivation of group velocity. The derivation is the one shown in this question Deriving group velocity. I have seen this derivation in many places, and in all they without comment say do the following.
In the derivation we get
$$v_g=\frac{\Delta \omega}{\Delta k}$$ And letting $\Delta k \rightarrow 0$ we then get: $$v_g=\frac{d \omega}{d k}$$ They then say that this holds in general. The only way that I can see this to be true is if $\Delta k$ is always small, no matter what wave we use. So is there some ristriction on how large $\Delta k$ can be so that we can always go from my first expression to my second.
If there is no restriction, then I could chose $\Delta k$ to be large my first expression would hold and my second would not.
Answer
The group velocity depends on the relationship between $\omega$ and $k$ which is not, in general, constant. Thus, the group velocity will be a function of $\omega$. But for a given value of $\omega$ there is a corresponding value of $k$.
If you permit the initial wave packet to contain a wide range of frequencies, you get a phenomenon called dispersion in which different frequencies travel at different velocities, and in general will cause the packet to spread. That makes it more difficult to talk about group velocity - is it the front of the packet, the middle, or the back that you are looking at to determine the velocity?
It's not a rigorous explanation but I hope it helps - as the frequency content increases, group velocity becomes a more poorly defined concept.
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