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?
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.
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.
The ewuatiin for water wave energy is
Energy = ¼ × density × gravity × wavelength × wave span × amplitude²
Which simplifies to
Energy ≈ 2,452.5 × wavelength × wave span × amplitude²
Wavelength Vs speed relation is
Wavelength² = period² / (gravity × wavelength × tanh(2π × depth / wavelength) / 2π)
Which simplifies to
Wavelength² ≈ period² / (1.56 × wavelength × tanh(6.28 × depth / wavelength))
If we assume that the ration if wavelength to depth exceeds 20, this equation further simplifies to
Wavelength² ≈ period² / (9.81 × depth)
This can also be used down to wavelengths three times the depth where it will overestimate the speed by 1.5×.
Hence amplitude is
Amplitude⁴ ≈ 613,089 × depth × energy² / (period × wave span)²
