Because water is essentially incompressible, its equations of motion are

where is the (uniform) mass density, the (uniform) viscosity, and the (uniform) acceleration due to gravity. (See Section 1.14.) Let us write

where is atmospheric pressure, and the pressure perturbation due to the wave. Of course, in the absence of the wave, the water pressure a depth below the surface is . (See Chapter 2.) Substitution into Equation (11.2) yields

(11.4) |

where we have neglected terms that are second order in small quantities (i.e., terms of order ).

Let us also neglect viscosity, which is a good approximation provided that the wavelength is not ridiculously small. [For instance, for gravity waves in water, viscosity is negligible as long as .] It follows that

Taking the curl of this equation, we obtain

(11.6) |

where

where is the velocity potential. (See Section 4.15.) However, from Equation (11.1), the velocity field is also divergence free. It follows that the velocity potential satisfies Laplace's equation,

Finally, Equations (11.5) and (11.7) yield

We now need to derive the physical constraints that must be satisfied at the water's upper and lower boundaries. It is assumed that the water is bounded from below by a solid surface located at . Because the water must always remain in contact with this surface, the appropriate physical constraint at the lower boundary is (i.e., the normal velocity is zero at the lower boundary), or

The water's upper boundary is a little more complicated, because it is a free surface. Let represent the vertical displacement of this surface due to the wave. It follows that

The appropriate physical constraint at the upper boundary is that the water pressure there must equal atmospheric pressure, because there cannot be a pressure discontinuity across a free surface (in the absence of surface tension--see Section 11.11). Accordingly, from Equation (11.3), we obtain

or

(11.13) |

which implies that

(11.14) |

where use has been made of Equation (11.11). The previous expression can be combined with Equation (11.9) to give the boundary condition

Let us search for a wave-like solution of Equation (11.8) of the form

(11.16) |

This solution actually corresponds to a propagating plane wave of wave vector , angular frequency , and amplitude (Fitzpatrick 2013). Substitution into Equation (11.8) yields

(11.17) |

whose independent solutions are and . Hence, a general solution to Equation (11.8) takes the form

(11.18) |

where and are arbitrary constants. The boundary condition (11.10) is satisfied provided that , giving

The boundary condition (11.15) then yields

(11.20) |

which reduces to the dispersion relation

The type of wave described in this section is known as a