Tsunami and wave dispersion

I would like to understand a little more about the low waters and high waters observed during a tsunami.

My first understanding was that the bottom of the ocean collapsed by a few meters in a 500 km² region in front of Sumatra.
I could then understand low waters on the coasts.
However, I could not understand the 10 m high waters that followed.
Possible explanations are:
• shaking of the bottom of the ocean instead of a simple collapse
• wave breaking near the coasts
• wave dispersion
The well-known law for deep-water wave propagation tells us that the phase velocity depends on the wave vector k :

$$v_{ph} = \sqrt{gk}$$​
The wave dispersion will distord an initial disturbance.
An initially negative disturbance (low water) can even change its sign and show positive portions (high water).
It is quite easy to check that numerically by a simple Fourier analysis.

But now the question is: what is the main reason for the high waters?
Is it dispersion?
Is it shaking?
Is it a coastal effect?
Is it still something else?

Any idea ? Any reference ?

Thanks

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HallsofIvy
Homework Helper
It's ENERGY (which is initially due to the "shaking"). A tsunami in mid ocean would be a wave that is not very high but has a lot of energy. As that wave approaches the coast the water becomes shallower (that's the "coastal effect") that energy becomes "focused" in less area so the water builds up.

HallsofIvy

I have two questions now!

- Do you mean that a dip in surface caused by earthquake would remain a dip until it reaches shallow waters? Or do you think the 'overshoot' is due to another reason? If it remains a dip (negative displacement) you hint that it will increase when the wave reaches the coast?

- What do you think is the expression that represent the energy of a wave ?

Thank,

Michel

/SQRT gk is dimensionally incorrect--should be /SQRT gd.Can anybody here give a derivation for the phase velocity?

HallsofIvy
Homework Helper
It isn't a matter of a "dip" that remains a dip. The instantaneous dip becomes a wave: with both troughs and heights.

gptejms:

Sorry for the typing error, the phase velocity for deep water waves is given by:

$$v_{ph} = \sqrt{\frac{g}{k}}$$​

You can find the derivation on this site for example:

http://www.ocean.washington.edu/people/faculty/parsons/OCEAN549B/lwt-lect.pdf [Broken]​

I found an interresting discussion about tsunamis there:

You can read there that tsunami waves have typically a wavelength of 100km.
Therefore, the shallow water phase velocity applies instead of the deep water limit.

HallsofIvy:

I agree with your remark. I come to realise that the shaking of the ocean floor produces troughs and heights.
However, are the amplitudes of these wave troughs and heights essentially the image of the floor motions? (more or less periodic motions then)
Or does the dispersion of the wave play a non negligible role ?
Indeed, since the phase velocity depends on the wavelength, and since the excitation is certainly not harmonic, the wavepacket could change its shape quite a lot. It can even develop heights when the initial condition contains troughs only.

In addition, I have the belief that the floor of the ocean does not move up and down but that it repeatedly moves downward.
If that was true, then I don't know how wave heigths could be explained without dispersion.
Obviously, I would need better information on the boundary condition for typical tsunamis.

Thank a lot

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i was wondering how fast a Tsunami travels ..it was all over the news that Tsunami's travel at hundreds of miles an hour , i dont agree with that.

As the wavelength of Tsunamis on open sea is larger then the depth of the water they can be thought of as 'shallow water' waves. The speed of such a wave is:

$$v=\sqrt{gd}$$

With g the gravitational acceleration and d the depth of the water. E.g. for a depth of 5km thi syields a speed of about 800km/h.

then how come it took so long to hit Africa ?
it seems like it took till the next day to hit the east coast of Africa..

rcgldr
Homework Helper
From what I understand, the waves weren't that high. Hawaii has 30 foot waves every now and then. Mavericks, near the bay area near San Fransisco has similar waves on good days. That's a 60+ foot wave face that the surfers are negotiating.

The issue with the 20ft or so Tsunami waves is their length, not their height. The larger amount of energy results in a wave that affects larger masses of water as it passes through. As the waves reach a shoreline, their exteme length causes them to continue flowing inwards for quite a ways.

Bigger waves move at faster speeds. At Mavericks, the really big waves require the surfers be towed into the wave front with a jet ski pulling the surfer.

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rcgldr
Homework Helper
Some big wave pics here:

http://billabongxxl05.com [Broken]

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LURCH