# Physics of tsunami: the relationship between wavelength, sea depth and the height of the water

If I understand correctly, when an earthquake occurs, energy will be transferred to the water, resulting in water waves. As the waves reach seashore, because the sea depth is getting shallower and wavelength is getting shorter, the height of the wave gets push up, resulting in tsunami. In other words in deep sea, water won't get pushed up as high as the water in shallow seashore.

Is my understanding correct? Is there a quantitative way to express the physics behind all this?

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Here is something on Tao's blog. terrytao.wordpress.com/2011/03/13/… – MBN Mar 16 '11 at 4:04
@MBN, I would appreciate if you could post this as an answer; I think this comment is good enough to be an answer. – Graviton Mar 16 '11 at 4:05
@Graviton It's a comment, of course. Usually people post a link as a comment if they don't have a lot of time. If they want to go in more depth, they post an answer with their link and then summarize its contents. – Mark Eichenlaub Mar 16 '11 at 4:25
@Mark: Should I delete the answer? – MBN Mar 16 '11 at 4:29
@MBN It's your answer; I don't know. I don't think this is some official thing - just my observations. You could start a meta thread if you want to clarify the difference between comments and answers. – Mark Eichenlaub Mar 16 '11 at 4:45

A tsunami is basically a shallow-water wave, even in deep seas. This means its velocity is $v=\sqrt{gH}$, where $H$ is the water depth and $g$ is the gravitational acceleration.

The energy of the tsunami scales as the square of its amplitude $A$, and thus the energy flux $S$ goes as $S\sim A^2 \sqrt{H}$. Conservation of energy then implies that the wave amplitude depends on the sea depth as $$A \sim H^{-1/4}$$ a result known as Green's law.

For example, Green's law predicts that a tsunami with amplitude $A=1$m at $H=5000$m will run up to $A=4$m if the depth becomes $H=20$m.

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I think it is just the gravity wave solution. We have a periododic function (sine wave) in the horizontal direction, and an exponential in the vertical direction. I think the wave number and the exponential decay rate are the same number, but hopefully someone in the know can fill in the details. In any case for short waves and deep water you need only consider the exponential term which decays with depth. But for tsunami's the wavelength is greater than the depth, so you have to use both types. Not sure what the boundary conditions are (at the water surface, and at the sea bottom are), but satisfying them would give you the allowable form for the wave at a given wavelength and depth. But in any case, for the tsunami, considerable motion is seen throughout the water column. If the wave is not reflected going into shallower water (I think this means the depth doesn't change much within a horizontal wavelength) then conservation of energy & momentum means the wave amplitude grows.

You can get a hydrolic jump (moving wall of water), because the wavespeed is higher in deeper water, so the higher portion of the wave can catch up with the slower moving portions ahead of it. If that happens, instead of a gradually increasing sealevel, you can get one or several stepfunction type waves coming in.

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