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}, \qquad 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]
Consider now as a special case plane progressive waves defined by the velocity potential in deep water (for simplicity):
[math]\displaystyle{ \Phi = \mathbf{Re} \{ \frac{igA}{\omega} e^{KZ-iKX+i\omega t} \} }[/math]
[math]\displaystyle{ \Phi_X = \mathbf{Re} \{ \frac{igA}{\omega} (-iK) e^{KZ-iKX+i\omega t} \} }[/math]
[math]\displaystyle{ = A \mathbf{Re} \{ \omega e^{KZ-iKX+i\omega t} \} }[/math]
[math]\displaystyle{ \Phi_Z = \mathbf{Re} \{ \frac{iSA}{\omega} K e^{KZ-iKX+i\omega t} \} }[/math]
[math]\displaystyle{ = A \mathbf{Re} \{ i \omega e^{KZ-iKX+i\omega t} \} }[/math]
Lemma
Let:
[math]\displaystyle{ \mathbf{Re} \{ A e^{i\omega t} \} = A(t) }[/math]
[math]\displaystyle{ \mathbf{Re} \{ B e^{i\omega t} \} = B(t) }[/math]
[math]\displaystyle{ \overline{A(t)B(t)} = \frac{1}{2} \mathbf{Re} \{ A B^* \} }[/math]
[math]\displaystyle{ \overline{\epsilon_{kin}} = \frac{1}{2} \rho \overline{ ( \int_{-\infty}^0 + \int_0^\zeta ) \left( \Phi_X^2 + \Phi_Z^2 \right) } dZ }[/math]
