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In this answer a possible derivation of the group velocity is provided.

It is, anyway, based on the assumption that there will always be a point where all the cosines will sum with the same phase:

The peak will still be where the phases of the component waves are the same.

But if the waves travel at different velocities, the existence of such a peak is unlikely. So, what are the particular conditions (hypotheses) when the above statement is valid and the computation in the linked answer is acceptable?

What are the particular cases when all the waves within a certain $k$-range will always sum with the same phase in some points, during their propagation along $x$?

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  • $\begingroup$ This one feels like it's asking a bit too much, to be honest. I'm not sure any such result will actually exist. $\endgroup$ – Emilio Pisanty Feb 20 '17 at 15:06
  • $\begingroup$ I would start from a more physical basis and consider what the kinematic and dynamical significance of the group velocity actually is. The geometric properties are not necessarily as fundamental, imo. $\endgroup$ – Nick P Feb 28 '17 at 20:29
  • $\begingroup$ @EmilioPisanty If I were asking for some mathematical condition, I would definitely agree with you. But my question was simpler: the linked answer gives some hypotheses, and under them there is actually a point in the envelope which moves at the group velocity. Somehow, in the linked example the waves can sum in that way. I would simply like to know what are the physical conditions to be satisfied (should the $k$ range be as small as possible? Should the evolution of the waves be considered only in a limited time?): this is the point of view proposed by Nick P in the other comment. $\endgroup$ – BowPark Mar 9 '17 at 16:54
  • $\begingroup$ @NickP Yes, exactly, that is the right point of view. $\endgroup$ – BowPark Mar 9 '17 at 16:56
  • $\begingroup$ @BowPark The thing is, what happens if there are multiple different nonequivalent situations that lead to the result you want? That is: it's perfectly easy to give one possible sufficient condition, but you're asking for either all possible sufficient conditions (you're asking too much) or some subset of some possible sufficient conditions (a vaguely-posed list question). $\endgroup$ – Emilio Pisanty Mar 9 '17 at 18:58

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