 Solution

For a medium with constant coefficients, $\bar{\omega }=\omega (\bar{k})$ on the selected branch. The spatial chain rule gives ${\bar{\omega }}_x=(d\omega /dk{)}_{\bar{k}}{\bar{k}}_x=c_g(\bar{k}){\bar{k}}_x$. Thus wave conservation becomesAlong a characteristic curve with $dx/dt=c_g(\bar{k})$, the total derivative $d\bar{k}/dt$ vanishes. A slowly varying wavenumber feature, and hence the corresponding modulation of a narrow-band packet, travels with the group velocity rather than the carrier's phase velocity. This interpretation assumes the modulation remains smooth on the scale required in part (e).