Guntersville Yacht club

Jeff Horton

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The Heart of Dixie
We have had our share of bad weather lately. Tornadoes, bad thunderstorms. Then we had a rare phenomenon called a Gravity wave hit us. You can Google it but it meant sustained 50 mph winds and gusts up to near 70 for about an hour. Not part of a storm either, just strong winds. After all the rain the down town area was a mess. Lots of trees down and leaves everywhere!

I heard that that Yacht club was a real mess and that I needed to drive over and see it. What I saw was a major surprise!

Drive up I see this. Now you shouldn't see this here.

yacht_club1.jpg


I get up closed I just had to pull off the road, I couldn't believe it.

yacht_club2.jpg


yacht_club3.jpg


The winds broke two(?) floating sections loose from their anchors and down onto the causeway. The far end is wedged on the bank and the causeway. Bascially wrapped around a few boats.

If you look and see the boats in the distance. That is where the Yachtclub used to be. About a 1/4 mile away.

yacht_club4.jpg
 
This should help explain gravity waves:

The phase speed c of a linear gravity wave with wavenumber k is given by the formula
688728eab7320dac8739fc8936be629b.png

where g is the acceleration due to gravity. When surface tension is important, this is modified to
01c7b6ff0e87eb7f24dab8afc251d83f.png

where g is the acceleration due to gravity, σ is the surface tension coefficient, ρ is the density, and k is the wavenumber of the disturbance.

The gravity wave represents a perturbation around a stationary state, in which there is no velocity. Thus, the perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,
15ee7be13ac831c107e67ec7db4eea22.png
Because the fluid is assumed incompressible, this velocity field has the streamfunction representation
cf42f626d583e0582d61088750cdab2c.png
where the subscripts indicate partial derivatives. In this derivation it suffices to work in two dimensions
3ce2f0cb99c39ed36cb8d40df5e9ad09.png
, where gravity points in the negative z-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence
83354f45003eb7f2ef38af5420a99238.png
. In the streamfunction representation,
bb71720d4cbd8569fc85874d630e393c.png
Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz
64d2556360e8e20602f11c85d0508e2f.png
where k is a spatial wavenumber. Thus, the problem reduces to solving the equation
9b718191df619e45d49b0eee6d43a8a9.png
We work in a sea of infinite depth, so the boundary condition is at z = − ∞. The undisturbed surface is at z = 0, and the disturbed or wavy surface is at z = η, where η is small in magnigude. If no fluid is to leak out of the bottom, we must have the condition
3530eda8185960174f9c5b5f3343ffa4.png
. Hence, Ψ = Aekz on
388d68809dff4bfbcc3929b944930fee.png
, where A and the wave speed c are constants to be determined from conditions at the interface.
The free-surface condition: At the free surface
8fe6a1057a521d0703f82017570a4149.png
, the kinematic condition holds:
d664c0226dab5249b83108d46f77fb7c.png
Linearizing, this is simply
24046ee15e5ad03a505eed1a252f7579.png
where the velocity
228101242da73ec872c216934e205f8b.png
is linearized on to the surface
ce2ccc0e0ac17ccc26345c70c3cbf45f.png
. Using the normal-mode and streamfunction representations, this condition is
7996c9a1d934c67f1bf8ef3c4cddb017.png
, the second interfacial condition.
Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at z = η is given by the Young–Laplace equation:
bfd2c43da0d7056ce66756c2245cf206.png
where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is
01d5d9dfd9c626522a640d3d6a6a3290.png
Thus,
5a303cfdc1cefe95e6e1295ca19d6bdb.png
However, this condition refers to the total pressure (base+perturbed), thus
b10cb4ab3c97dde04aa46bdb514d8c2d.png
(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form P = − ρgz + Const.,
this becomes
1a7f84ace63953f506ecfdc4f9fd2f66.png
The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations,
4b457b1e3b7dfe1719344cee54912537.png
to yield p' = ρcDΨ.
Putting this last equation and the jump condition together,
9b129291e01e6b6602b01b87f578bfb6.png
Substituting the second interfacial condition
7996c9a1d934c67f1bf8ef3c4cddb017.png
and using the normal-mode representation, this relation becomes cDΨ = gΨρ + σk2Ψ.
Using the solution Ψ = ekz, this gives
35b2dd7a90147e9c7bd3192f0c6695e6.png


Since c = ω / k is the phase speed in terms of the frequency ω and the wavenumber, the gravity wave frequency can be expressed as
5c8bcd8ab3f883a9647e1874e21f280f.png

The group velocity of a wave (that is, the speed at which a wave packet travels) is given by
67d466f0dddc2fb8e7a5e06e97608fe4.png

and thus for a gravity wave,
604d3f895ccb0340fcf96c055d65d825.png

The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.


Any questions?
 
C'mon, Frank...everybody knows that already. :rolleyes: I even remember the rhyme we learned in school to keep the theory straight:


If
688728eab7320dac8739fc8936be629b.png
today,

Then
35b2dd7a90147e9c7bd3192f0c6695e6.png
for me.



Jeff, that looks like one heck of a wind. :eek:
 
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