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# Energy of Gravity Waves

It is easily demonstrated, from the analysis contained in the previous sections, that a gravity wave of arbitrary wavenumber , propagating horizontally through water of depth , has a phase velocity

 (11.43)

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

 (11.44)

It follows that neither the phase velocity nor the group velocity of a gravity wave can ever exceed the critical value . It is also easily demonstrated that the displacement and velocity fields associated with a plane gravity wave of wavenumber , angular frequency , and surface amplitude , are

 (11.45) (11.46) (11.47) (11.48)

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

 (11.49)

where

 (11.50)

is the vertical displacement at the surface, and

 (11.51)

is an average over a wavelength. Given that , it follows from Equations (11.47) and (11.48) that, to second order in ,

 (11.52)

Making use of the general dispersion relation (11.21), we obtain

 (11.53)

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

 (11.54)

which yields

 (11.55)

or

 (11.56)

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

 (11.57)

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

Next: Wave Drag on Ships Up: Waves in Incompressible Fluids Previous: Gravity Waves in Shallow
Richard Fitzpatrick 2016-03-31