I am working thru a derivation of the group velocity formula and I get to this stage: $$y=2A\cos(x\frac{\Delta K}{2} -t\frac{\Delta \omega}{2})\sin( \bar k x-\bar \omega t)$$ Then all the derivations I have seen say that $\frac{\Delta \omega}{\Delta K} $ is the group velocity. I know mathematically why this is a velocity but what I don't get is why do we know that this is the group velocity rather then the phase velocity and that $\frac{\bar \omega}{\bar k}$ is the phase velocity and not the group velocity?
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