text: Interference of two sinusoidal waves with slightly different wavelengths, but the same amplitude and propagation direction, results in a beat pattern, called a wave group. As can be seen in the animation, the group moves with a group velocity c different from the phase velocity c, due to frequency dispersion.
  The group velocity is depicted by the red lines (marked B) in the two figures above.
  In shallow water, the group velocity is equal to the shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: c = ½ c.
  The group velocity also turns out to be the energy transport velocity. This is the velocity with which the mean wave energy is transported horizontally in a narrow-band wave field.
  In the case of a group velocity different from the phase velocity, a consequence is that the number of waves counted in a wave group is different when counted from a snapshot in space at a certain moment, from when counted in time from the measured surface elevation at a fixed position. Consider a wave group of length Λ and group duration of τ. The group velocity is:
  The number of waves in a wave group, measured in space at a certain moment is: Λ / λ. While measured at a fixed location in time, the number of waves in a group is: τ / T. So the ratio of the number of waves measured in space to those measured in time is:
  So in deep water, with c = ½ c, a wave group has twice as many waves in time as it has in space.
  The water surface elevation η(x,t), as a function of horizontal position x and time t, for a bichromatic wave group of full modulation can be mathematically formulated as:
  with:
  * a the wave amplitude of each frequency component in metres,
  * k and k the wave number of each wave component, in radians per metre, and
  * ω and ω the angular frequency of each wave component, in radians per second.
  Both ω and k, as well as ω and k, have to satisfy the dispersion relation:
  :   and   
  Using trigonometric identities, the surface elevation is written as:
  The part between square brackets is the slowly varying amplitude of the group, with group wave number ½ ( k − k ) and group angular frequency ½ ( ω − ω ). As a result, the group velocity is, for the limit k → k :
  Wave groups can only be discerned in case of a narrow-banded signal, with the wave-number difference k − k small compared to the mean wave number ½ (k + k).