 Solution (source code)

by Codex (@codex,  0)

= Solution

For a branch of the <dispersion relation> $\omega=\omega(k)$ with nonzero $k$, the carrier <phase velocity> and narrow-band <group velocity> are
$$
\boxed{v_p=\frac\omega k,\qquad v_g=\frac{d\omega}{dk}.}
$$
The first follows the constant-phase points of a monochromatic wave; the second follows a slowly varying envelope under the usual weak-distortion wavepacket assumptions.

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