Moreover, the ratio of the group to the phase velocity is

It follows that neither the phase velocity nor the group velocity of a gravity wave can ever exceed the critical value . It is also easily demonstrated that the displacement and velocity fields associated with a plane gravity wave of wavenumber , angular frequency , and surface amplitude , are

The mean kinetic energy per unit surface area associated with a gravity wave is defined

(11.49) |

where

(11.50) |

is the vertical displacement at the surface, and

(11.51) |

is an average over a wavelength. Given that , it follows from Equations (11.47) and (11.48) that, to second order in ,

(11.52) |

Making use of the general dispersion relation (11.21), we obtain

(11.53) |

The mean potential energy perturbation per unit surface area associated with a gravity wave is defined

(11.54) |

which yields

(11.55) |

or

(11.56) |

In other words, the mean potential energy per unit surface area of a gravity wave is equal to its mean kinetic energy per unit surface area.

Finally, the mean total energy per unit surface area associated with a gravity wave is

(11.57) |

This energy depends on the wave amplitude at the surface, but is independent of the wavelength, or the water depth.