Wave Energy Density and Flux

Energy Density, Energy Flux and Momentum Flux of Surface Waves

[math]\displaystyle{ \varepsilon(t) = \ \mbox{Energy in control volume} \ \gamma(t) }[/math] :

[math]\displaystyle{ \varepsilon (t) = \rho \iiint_V \left( \frac{1}{2} V^2 + gZ \right) dV }[/math]

Mean energy over unit horizongtal surface area [math]\displaystyle{ S \, }[/math] :

[math]\displaystyle{ \overline{\varepsilon} = \overline{\frac{\varepsilon(t)}{S}} = \rho \overline{ \int_{-H}^{\zeta(t)} \left( \frac{1}{2} V^2 + gZ \right) dZ} = \frac{1}{2} \rho \overline{ \int_{-H}^{\zeta(t)} V^2 dZ} + \overline{ \frac{1}{2} \rho g ( \zeta^2 - H^2 ) } }[/math]

where [math]\displaystyle{ \zeta(t) \, }[/math] is free surface elevation.

Ignore term [math]\displaystyle{ -\frac{1}{2} \rho g H^2 \, }[/math] which represents the potential energy of the ocean at rest.

The remaining perturbation component is the sum of the kinetic and potential energy components

[math]\displaystyle{ \overline{\varepsilon} = \overline{\varepsilon_KIN} + \overline{\varepsilon_POT} }[/math]

[math]\displaystyle{ \overline{\varepsilon_KIN} = \frac{1}{2} \rho \overline{\int_{-H}^{\zeta(t) V^2 dZ}, \ V^2 = \nabla\Phi \cdot \nabla \Phi = \Phi_X^2 + \Phi_Z^2 }[/math]

[math]\displaystyle{ \overline{\varepsilon_POT} = \overline{\frac{1}{2} \rho g \zeta^2 (t)} }[/math]