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

Other articles related to "energy density, waves, wave, energy, wave energy density":

Wave Power - Physical Concepts - Wave Energy and Wave-energy Flux
... In a sea state, the average energy density per unit area of gravity waves on the water surface is proportional to the wave height squared, according to linear wave ... The potential energy density is equal to the kinetic energy, both contributing half to the wave energy density E, as can be expected from the equipartition theorem ... In ocean waves, surface tension effects are negligible for wavelengths above a few decimetres ...
Airy Wave Theory - Second-order Wave Properties - Wave Action, Wave Energy Flux and Radiation Stress
... In general, there can be an energy transfer between the wave motion and the mean fluid motion ... This means, that the wave energy density is not in all cases a conserved quantity (neglecting dissipative effects), but the total energy density – the sum of the energy density ... However, there is for slowly varying wave trains, propagating in slowly varying bathymetry and mean-flow fields, a similar and conserved wave quantity, the wave action with ...

Famous quotes containing the words wave and/or energy:

    Now I stand as one upon a rock,
    Environed with a wilderness of sea,
    Who marks the waxing tide grow wave by wave,
    Expecting ever when some envious surge
    Will in his brinish bowels swallow him.
    William Shakespeare (1564–1616)

    I rather think the cinema will die. Look at the energy being exerted to revive it—yesterday it was color, today three dimensions. I don’t give it forty years more. Witness the decline of conversation. Only the Irish have remained incomparable conversationalists, maybe because technical progress has passed them by.
    Orson Welles (1915–1984)