7.3 Phase Velocity & Group Velocity

As we have seen, the speed you need to keep up with a point of constant phase along the wave is given by

v_{\phi}=f\lambda=\frac{\omega}{k}.

(7.37)

This is known as the phase velocity. The dependence of $\omega$ on $k$ (or vice-versa) is called the dispersion relation. If the relationship is linear, i.e. if $v_{\phi}$ is constant, the wave is said to be dispersionless. Otherwise, the wave will undergo dispersion as different frequencies will travel at difference speeds.

The group velocity is defined as

v_{g}=\frac{\operatorname{\mathrm{d}}\!\omega}{\operatorname{\mathrm{d}}\!k}.

(7.38)

So if a wave is dispersionless, the phase velocity and group velocity will be the same. In the case where $v_{g}\neq v_{\phi}$ , the group velocity is the speed that the wave envelope propagates.