Difference between revisions of "Wave Energy Density and Flux"
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<center><math> U = \frac{\rho \ {\overline{\int_{-\infty}^0 \frac{\partial\Phi}{\partial t} \frac{\partial\Phi}{\partial x} dZ}}^t}{{\overline{\int_{-\infty}^0 \left( \frac{P}{\rho} + \frac{\partial\Phi}{\partial t} \right) dZ}}^t} </math></center> | <center><math> U = \frac{\rho \ {\overline{\int_{-\infty}^0 \frac{\partial\Phi}{\partial t} \frac{\partial\Phi}{\partial x} dZ}}^t}{{\overline{\int_{-\infty}^0 \left( \frac{P}{\rho} + \frac{\partial\Phi}{\partial t} \right) dZ}}^t} </math></center> | ||
+ | |||
+ | Upon substitution of the plane progressive wave velocity potential and definition of pressure from Bernoulli's equation we obtain: | ||
+ | |||
+ | <center><math> U \equiv V_g = \frac{1}{2} \frac{g}{\omega} = \frac{1}{2} V_P </math></center> | ||
+ | |||
+ | Note that <math> U \equiv V_g </math> by definition. If the above exercise is repeated in water of finite depth the solution for <math> U </math> after some algebra is: | ||
+ | |||
+ | <center><math> U = V_g = \left( \frac{1}{2} + \frac{KH}{\sinh 2KH} \right) V_P </math></center> | ||
+ | |||
+ | with | ||
+ | |||
+ | <center><math> \omega^2 = gK \tanh KH </math></center> | ||
+ | |||
+ | It may be shown that the group velocity <math> V_g </math> is given in terms of <math> \omega \ne k \, </math> by the relation | ||
+ | |||
+ | <center> V_g = \frac{d\omega}{d K} </math></center> | ||
+ | |||
+ | This relation follows from the very elegant "device" due to rayleigh which applies to any wave form: |
Revision as of 23:42, 15 February 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:
[math]\displaystyle{ \overline{\varepsilon_{pot}} = \frac{1}{2} \rho g {\overline{\zeta(t)}}^2 = \frac{1}{4} \rho g A^2 }[/math]
Hence:
Energy flux = rate of change of energy density [math]\displaystyle{ \varepsilon(t) }[/math]
Transport theorem where [math]\displaystyle{ U_n }[/math] is normal velocity of surface [math]\displaystyle{ S(t) }[/math] outwards of the enclosed volume [math]\displaystyle{ V }[/math].
Invoking the scalar form of Gauss's theorem in the frist term, we obtain:
An alternative form for the energy flux [math]\displaystyle{ P(t) \, }[/math] crossing the closed control surface [math]\displaystyle{ S(t) \, }[/math] is obtained by invoking Bernoulli's equation in the second term. Recall that:
At any point in the fluid domain and on boundarie.
Here we did allow [math]\displaystyle{ \ P_a \equiv \mbox{Atmospheric pressure} \ }[/math] to be non-zero for the sake of physical clarity. Upon substitution in [math]\displaystyle{ P(t) }[/math] we obtain the alternate form:
So the energy flux across [math]\displaystyle{ S(t)\, }[/math] is given by the terms under the interral sign. They can be collected in the more compact form:
Note that [math]\displaystyle{ P(t) \, }[/math] measures the energy flux into the volume [math]\displaystyle{ V(t) \, }[/math] or the rate of growth of the energy density [math]\displaystyle{ \varepsilon(t)\, }[/math].
We are ready now to apply the above formulae to the surface wave propagation problem.
Break [math]\displaystyle{ S(t) \, }[/math] into tis components and derive specialized forms of [math]\displaystyle{ P(t) \, }[/math] pertinent to each.
Therefore over [math]\displaystyle{ S_F; \ P(t) \equiv 0 }[/math] as expected. No energy can flow into the atmosphere!
This case will be of interest later in the course when we consider ships moving with constant velocity [math]\displaystyle{ U }[/math].
The formulae derived above are very general for potential flows with a free surface and solid boundaries. We are now ready to apply them to plane progressive waves.
Energy flux across a vertical fluid boundary fixed in space.
Mean energy flux for a plane progressive wave follows upon substitution of the regular wave velocity potential and taking mean values:
or
It follows from this exercise that the mean energy flux of a plane progressive wave is the product of its mean energy density times a velocity which equals [math]\displaystyle{ \frac{1}{2} }[/math]. The phase velocity in deep water we call this the group velocity of deep water waves and it is defined as:
A more formal proof that this is the velocity with which the energy flux of plane progressive waves propagates is to ask the following question:
[math]\displaystyle{ \longrightarrow }[/math] What needs to be the horizontal velocity [math]\displaystyle{ U_n \equiv U }[/math] of a fluid boundary so that the mean energy flux across it vanishes?
This can be found from the solution of the following equation:
Where terms of [math]\displaystyle{ O(A^3) }[/math] have been neglected. Note that within linear theory, energy density and energy flux are quantities of [math]\displaystyle{ O(A^2) }[/math]. If higher-order terms are kept then we need to consider the treatment of second-order surface wave theory, at least.
Solving the above equation for [math]\displaystyle{ U }[/math] we obtain:
Upon substitution of the plane progressive wave velocity potential and definition of pressure from Bernoulli's equation we obtain:
Note that [math]\displaystyle{ U \equiv V_g }[/math] by definition. If the above exercise is repeated in water of finite depth the solution for [math]\displaystyle{ U }[/math] after some algebra is:
with
It may be shown that the group velocity [math]\displaystyle{ V_g }[/math] is given in terms of [math]\displaystyle{ \omega \ne k \, }[/math] by the relation
This relation follows from the very elegant "device" due to rayleigh which applies to any wave form: