Difference between revisions of "Wave Energy Density and Flux"
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<center><math> \Phi = \mathbf{Re} \{ \frac{igA}{\omega} e^{KZ-iKX+i\omega t} \} </math></center> | <center><math> \Phi = \mathbf{Re} \{ \frac{igA}{\omega} e^{KZ-iKX+i\omega t} \} </math></center> | ||
+ | |||
+ | <center><math> \Phi_X = \mathbf{Re} \{ \frac{igA}{\omega} (-iK) e^{KZ-iKX+i\omega t} \} </math></center> <br> | ||
+ | <center><math> = A \mathbf{Re} \{ \omega e^{KZ-iKX+i\omega t} \} </math></center> | ||
+ | |||
+ | <center><math> \Phi_Z = \mathbf{Re} \{ \frac{iSA}{\omega} K e^{KZ-iKX+i\omega t} \} </math></center> <br> | ||
+ | <center><math> = A \mathbf{Re} \{ i \omega e^{KZ-iKX+i\omega t} \} </math></center> | ||
+ | |||
+ | <u>Lemma</u> | ||
+ | |||
+ | Let: |
Revision as of 09:23, 26 January 2007
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
Consider now as a special case plane progressive waves defined by the velocity potential in deep water (for simplicity):
Lemma
Let: