Steady Flow over a Corrugated Bottom

We expect the velocity potential, perturbed pressure, and vertical displacement of the water to be of the form (11.91), (11.92), and (11.93), respectively, with , because we are looking for a stationary (i.e., non-propagating) perturbation driven by the static corrugations in the bottom. The boundary condition at the bottom is

(11.104) |

which yields

At the free surface, we have

(11.106) |

which gives

In addition, pressure balance across the free surface yields

(11.108) |

which leads to

Hence, from Equations (11.105), (11.107), and (11.109),

(11.110) |

or

(11.111) |

where is the phase velocity of a gravity wave of wave number . [See Equation (11.21).] It follows that the peaks and troughs of the free surface coincide with those of the bottom when , and the troughs coincide with the peaks, and vice versa, when . If then the ratio becomes infinite, implying that the oscillations driven by the corrugations are not of small amplitude, and, therefore, cannot be described by linear theory.