1. ## Guntersville Yacht club

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.

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

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.

2. This should help explain gravity waves:

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

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

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, Because the fluid is assumed incompressible, this velocity field has the streamfunction representation
where the subscripts indicate partial derivatives. In this derivation it suffices to work in two dimensions , 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 . In the streamfunction representation, Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz
where k is a spatial wavenumber. Thus, the problem reduces to solving the equation
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
. Hence, Ψ = Aekz on , 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 , the kinematic condition holds:
Linearizing, this is simply
where the velocity is linearized on to the surface . Using the normal-mode and streamfunction representations, this condition is , 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:
where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is
Thus,
However, this condition refers to the total pressure (base+perturbed), thus
(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form P = − ρgz + Const.,
this becomes
The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations,
to yield p' = ρcDΨ.
Putting this last equation and the jump condition together,
Substituting the second interfacial condition and using the normal-mode representation, this relation becomes cDΨ = gΨρ + σk2Ψ.
Using the solution Ψ = ekz, this gives

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

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

and thus for a gravity wave,

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

Any questions?

3. Member
Join Date
Feb 2009
Posts
2,028
Thanks Frank, now I understand perfectly.

4. Happy to help Alan!

5. Member
Join Date
Oct 2008
Location
Kea'au Hawaii. Just down the road from Hilo town!
Posts
1,357
Geez, Frank your fast on the keys!! I was just going to say the same thing@

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

If today,

Then for me.

Jeff, that looks like one heck of a wind.

7. Thanks Royall and Vaughn.
You know me, a big show-off!
breaks me up every time!

8. i'm just waiting for larry to chime in on this one...

9. Just wait a gosh darn minute,,,,, I was told there would be no maths

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