Gravity Waves

Water is essentially incompressible (i.e., its bulk modulus is very large) (Batchelor 2000). Thus, any (low speed) wave disturbance in water is constrained to preserve the volume of a co-moving volume element. Equivalently, the inflow rate of water into a stationary volume element must match the outflow rate. Consider a stationary cubic volume element lying between and , and , and and . The element has two faces, of area , perpendicular to the -axis, located at and . Water flows into the element through the former face at the rate (i.e., the product of the area of the face and the normal velocity), and out of the element though the latter face at the rate . The element also has two faces perpendicular to the -axis, but there is no flow through these faces, because . Finally, the element has two faces, of area , perpendicular to the -axis, located at and . Water flows into the element through the former face at the rate , and out of the element though the latter face at the rate . Thus, the net rate at which water flows into the element is . Likewise, the net rate at which water flows out of the element is . If the water is to remain incompressible then the inflow and outflow rates must match. In other words,

(920) |

or

(921) |

Hence, the incompressibility constraint reduces to

which is consistent with Equation (667) in the incompressible limit .

Consider the equation of motion of a small volume element of water lying between and , and , and and . The mass of this element is , where is the uniform mass density of water. Suppose that is the pressure in the water, which is assumed to be isotropic (Batchelor 2000). The net horizontal force on the element is (because force is pressure times area, and the external pressure forces acting on the element are directed inward normal to its surface). Hence, the element's horizontal equation of motion is

(923) |

which reduces to

[See Equation (665).] The vertical equation of motion is similar, except that the element is subject to a downward acceleration, , due to gravity. Hence, we obtain

[See Equation (666).]

We can write

where is atmospheric pressure (i.e., the air pressure just above the surface of the water), and is the pressure perturbation due to the wave. In the absence of the wave, the water pressure at a depth below the surface is (Batchelor 2000). Substitution into Equations (924) and (925) yields

It follows that

(929) |

which implies that

(930) |

or

(Actually, this quantity could be non-zero and constant in time, but this is not consistent with an oscillating wave-like solution.)

Equation (931) is automatically satisfied by writing the fluid velocity in terms of a velocity potential: that is,

(932) | ||

(933) |

(See Section 8.10.) Equation (922) then gives

Finally, Equations (927) and (928) yield

As we have just demonstrated, surface waves in water are governed by Equation (934),
which is known as *Laplace's equation*. We now need to derive the physical
constraints that must be satisfied by the solution to this equation at
the water's upper and lower boundaries. The water is bounded from below by a
solid surface located at
. Assuming that the water always remains in contact
with this surface, the appropriate physical constraint at the lower boundary is
(i.e., there is no vertical motion of the water at the lower boundary), or

The physical constraint at the water's upper boundary is a little more complicated, because this boundary is a free surface. Let represent the vertical displacement of the water's surface. It follows that

The physical constraint at the surface is that the water pressure is equal to atmospheric pressure, because there cannot be a pressure discontinuity across a free surface (in the absence of surface tension). Thus, it follows from Equation (926) that

Finally, differentiating with respect to , and making use of Equations (935) and (937), we obtain

Hence, the problem boils down to solving Laplace's equation, (934), subject to the physical constraints (936) and (939).

Let us search for a propagating wave-like solution of Equation (934) of the form

(940) |

Substitution into Equation (934) yields

(941) |

whose independent solutions are and . Hence, the most general wave-like solution to Laplace's equation is

(942) |

where and are arbitrary constants. The boundary condition (936) is satisfied provided that , giving

The boundary condition (939) yields

(944) |

which reduces to the dispersion relation

The type of wave just described is generally known as a

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

Fianlly, the velocity fields associated with a gravity wave of surface amplitude are

In *shallow water* (i.e.,
), Equation (945) reduces to
the linear dispersion relation

(950) |

Here, use has been made of the small argument expansion for . (See Appendix B.) It follows that gravity waves in shallow water are non-dispersive in nature, and propagate at the phase velocity . On the other hand, in

(951) |

Here, use has been made of the large argument expansion for . We conclude that gravity waves in deep water are dispersive in nature. The phase velocity of the waves is , whereas the group velocity is . In other words, the group velocity is half the phase velocity, and is largest for long wavelength (i.e., small ) waves.

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

(952) |

where

(953) |

is the vertical displacement at the surface, and

(954) |

represents an average over a wavelength. Given that , it follows from Equations (948) and (949) that, to second order in ,

(955) |

Making use of the general dispersion relation (945), we obtain

(956) |

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

(957) |

which yields

(958) |

or

(959) |

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

(960) |

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