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.