(987) |

(Batchelor 2000). Here, is the

(988) |

which reduces to

(989) |

This boundary condition can be combined with the solution (943), in the deep water limit , to give the modified deep water dispersion relation (see Exercise 21)

Hence, the phase velocity of the waves takes the form

(991) |

and the ratio of the group velocity to the phase velocity can be shown to be

(992) |

We conclude that the phase velocity of surface water waves attains a minimum value of when , which corresponds to . The group velocity equals the phase velocity at this wavelength. For long wavelength waves (i.e., ), gravity dominates surface tension, the phase velocity scales as , and the group velocity is half the phase velocity. As we have already mentioned, this type of wave is known as a gravity wave. On the other hand, for short wavelength waves (i.e., ), surface tension dominates gravity, the phase velocity scales as , and the group velocity is times the phase velocity. This type of wave is known as a