= 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 becomes
$$
\boxed{\bar k_t+c_g(\bar k)\bar k_x=0.}
$$
Along 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).
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