Airy Wave Theory - Second-order Wave Properties - Wave Energy Density

Wave Energy Density

Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains. As can be seen above, many wave quantities like surface elevation and orbital velocity are oscillatory in nature with zero mean (within the framework of linear theory). In water waves, the most used energy measure is the mean wave energy density per unit horizontal area. It is the sum of the kinetic and potential energy density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area Epot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:

E_\text{pot}\, =\, \overline{\int_{-h}^{\eta} \rho\,g\,z\;\text{d}z}\, -\, \int_{-h}^0 \rho\,g\,z\; \text{d}z\, =\, \overline{\frac12\,\rho\,g\,\eta^2}\, =\, \frac14\, \rho\,g\,a^2,

with an overbar denoting the mean value (which in the present case of periodic waves can be taken either as a time average or an average over one wavelength in space).

The mean kinetic energy density per unit horizontal area Ekin of the wave motion is similarly found to be:

E_\text{kin}\, =\, \overline{\int_{-h}^0 \frac12\, \rho\, \left\; \text{d}z}\, -\, \int_{-h}^0 \frac12\, \rho\, \left| \boldsymbol{U} \right|^2\; \text{d}z\, =\, \frac14\, \rho\, \frac{\sigma^2}{k\, \tanh\, (k\, h)}\,a^2,

with σ the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for surface gravity waves is:

As can be seen, the mean kinetic and potential energy densities are equal. This is a general property of energy densities of progressive linear waves in a conservative system. Adding potential and kinetic contributions, Epot and Ekin, the mean energy density per unit horizontal area E of the wave motion is:

In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving

E_\text{pot}\, =\, E_\text{kin}\, =\, \frac14\, \left( \rho\, g\, +\, \gamma\, k^2 \right)\, a^2, \qquad \text{so} \qquad E\, =\, E_\text{pot}\, +\, E_\text{kin}\, =\, \frac12\, \left( \rho\, g\, +\, \gamma\, k^2 \right)\, a^2,

with γ the surface tension.

Read more about this topic:  Airy Wave Theory, Second-order Wave Properties

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