 Solution

A fixed wave phase satisfies $kx-\omega t=$ a constant. Differentiating gives the phase velocity, called the wave speed here:It is the speed of an individual crest or other fixed-phase point. The group velocity isevaluated on the relevant dispersion relation. To see its meaning, superpose two equal-amplitude waves with nearby $(k,\omega )$: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 $c_g=c+kdc/dk$.