Past exam of the mathematical methods II course of the University of Oxford / 2014 / 4 / a / Solution

For a branch of the dispersion relation $\omega =\omega (k)$ with nonzero $k$, the carrier phase velocity and narrow-band group velocity areThe first follows the constant-phase points of a monochromatic wave; the second follows a slowly varying envelope under the usual weak-distortion wavepacket assumptions.