# Group velocity

From AMS Glossary

## group velocity

- The velocity of the envelope of a group of waves of nearly equal frequencies.

From the dispersion relation*k*(ω) the group velocity is defined as*d*ω/*dk*, as distinguished from the phase velocity (or phase speed) ω/*k*. The origin of this term is made clearer by considering the superposition of two equal-amplitude plane harmonic waves with wavenumbers*k*± Δ*k*and frequencies ω ± Δω:Because , this composite wave may be looked upon as a high-frequency wave, with phase velocity ω/*k*, modulated by a wave of much lower frequency Δω. The envelope of the high- frequency wave is a low-frequency wave propagating with the group velocity Δω/Δ*k*. Note the similarity with beating. Indeed, a group of waves may be looked upon as a moving beat.

In deep water, on the basis of linear water wave theory, it can be shown to be equal to one half the phase velocity.