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My book here states that for a medium where the index of refraction n increases with increasing frequency (or wavenumber), "the group velocity is less than the phase velocity". This is stated for a wave which is the sum of two waves with equal amplitude and differing frequency.

Mathematically there is no leak to me as u

_{g}=dω/dk=u(1-k*dn/(n*dk)).

However let's say the medium is as stated; as for definition of n=c/u, when n increases, u (the wave velocity) decreases. Let's consider a modulated plane wave travelling in vacuum and reaching the dispersive medium. If I consider the modulation envelope as a wave, having wave number smaller than the one associated to the modulated wave, I am brought to the conclusion that the envelope will travel in the dispersive medium with a velocity greater than the phase velocity, since the medium "slows" the modulated wave(with greater k) more than it does with the envelope wave.

Is it therefore incorrect to consider (for the sake of intuitivity) the envelope as a separate wave and deduce it's relative velocity from it's wavenumber?

Thank you