= Solution
A fixed <wave phase> satisfies $kx-\omega t=$ a constant. Differentiating gives the <phase velocity>, called the <wave speed> here:
$$
\boxed{c=\frac\omega k.}
$$
It is the speed of an individual crest or other fixed-phase point. The <group velocity> is
$$
\boxed{c_g=\frac{d\omega}{dk},}
$$
evaluated on the relevant <dispersion relation>. To see its meaning, superpose two equal-<amplitude> <waves> with nearby $(k,\omega)$:
$$
\cos(k_1x-\omega_1t)+\cos(k_2x-\omega_2t)
=2\cos\frac{\Delta k\,x-\Delta\omega\,t}{2}\cos(\bar kx-\bar\omega t).
$$
The slowly varying <wave envelope> moves with speed $\Delta\omega/\Delta k$, tending to $d\omega/dk$ as the two components approach one another. Thus <group velocity> describes a narrow <wave packet> and, for conservative linear <surface gravity waves> in still water, their <energy> transport. A single <time-harmonic wave> has no distinguished packet envelope. When $c=c(k)$, the equivalent relation is $c_g=c+k\,dc/dk$.
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