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I still would like to thank those who participated to my previous thread about group velocity and dispersion. Now there is a (maybe) simpler question.

A sinusoidal, electro-magnetic plane wave in the vacuum propagates in a certain direction with the following wavenumber, which is supposed to equal the propagation constant:

[itex]\beta = k = \omega \sqrt{\mu_0 \epsilon_0}[/itex]

so

[itex]\omega = \displaystyle \frac{k}{\sqrt{\mu_0 \epsilon_0}}[/itex]

[itex]\displaystyle \frac{d \omega}{dk}= \displaystyle \frac{1}{\sqrt{\mu_0 \epsilon_0}} = c = v_g[/itex]

and the medium is non dispersive, because [itex]v_g[/itex] is always the same for all frequencies.

But even in this case, the wavenumber is frequency-dependent. Maybe this question is not strictly related to dispersion itself, but to the waves in general: why should the wavenumber linearly increase with frequency? I know that, being [itex]v_g = d\omega / dk[/itex], if [itex]\omega[/itex] were not dependent on [itex]k[/itex], group velocity would be always zero.

But what is the physical meaning of an increasing wavenumber with frequency? Why do waves with higher frequencies propagate with greater wavenumbers?