In the standard simplified derivation of group velocity (which can be found here) we use two waves y1=Asin(K1x−ω1t) y2=Asin(K2x−ω2t) In the proof we then get Vg=ΔωΔk But I do not understand the step where this is then turned into Vg=dωdk why do we assume that Δω and Δk are small? The derivation is valid in the case where they are not small, which means that Vg=dωdk does not hold in this case and therefore does not hold in general.
Consider this example
Let K1=3 and k2=1 and let us say we have relationship ω=k3 using my first fromula we get Vg=13 but using the second (with ˉk=2) we get Vg=12, theses are diferent.
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