It is helpful to introduce the capillary length,

(11.123) |

(See Section 3.4.) The capillary length of an air/water interface at s.t.p. is (Batchelor 2000). The associated

(11.124) |

This critical velocity takes the value for an air/water interface at s.t.p. (Batchelor 2000). It follows from Equation (11.122) that the phase velocity, , of a surface water wave can be written

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

In the long wavelength limit (i.e., ), we obtain

and

(11.128) |

We can identify this type of wave as the deep water gravity wave discussed in Section 11.3.

In the short wavelength limit (i.e., ), we get

and

(11.130) |

This corresponds to a completely new type of wave known as a

(11.131) |

which demonstrates that gravity plays no role in the propagation of a capillary wave. In fact, its place is taken by surface tension. Finally, it is easily seen that the phase velocity (11.125) attains the minimum value when (i.e., when ). Moreover, from Equation (11.126), at this wavelength. It follows that the phase velocity of a surface wave propagating over a body of water can never be less than the critical value, .