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