Group Velocity

It is the velocity at which the envelope of a wave’s amplitudes propagates through space.

The group velocity $v_g$ is defined as:

$v_g=\frac{\partial \omega }{\partial k}$ where $\omega$ is the wave’s angular frequency and $k$ is it’s wavenumber.

The phase velocity is $v_p=\omega /k$.

If $\omega$ is directly proportional to $k$, then the group velocity is equal to the phase velocity. Otherwise they differ by the partial derivative slope.

Relation to refractive index

Phase velocity: $n=c/v_p=ck/\omega$ Group velocity: $v_g=\frac{\partial \omega }{\partial k}=\frac{c}{n+\omega \frac{\partial n}{\partial \omega }}$

The group velocity is only equal to the phase velocity only if the refractive index is independent of the frequency ($\frac{\partial n}{\partial \omega }=0$).

Examples of distinctions

• Deep water gravity waves
• Surface gravity waves