   Next: Wave Drag on Ships Up: Waves in Incompressible Fluids Previous: Gravity Waves in Shallow

# 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