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:

\(\large v_g = \dfrac{\partial{\omega}}{\partial{k}}\) where \(\large \omega\) is the wave's angular frequency and \(k\) is it's wavenumber.

The phase velocity is \(\large v_p = \omega/k\).

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

Relation to refractive index

Phase velocity: \(\large n = c/v_p = ck/\omega\)
Group velocity: \(\large v_g= \dfrac{\partial \omega}{\partial k} = \dfrac{c}{n + \omega \dfrac{\partial n}{\partial \omega}}\)

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

Examples of distinctions

• Deep water gravity waves
• Surface gravity waves