Water waves vs. other wave types

[zacman2400]

I know that as water waves get closer to shore, amplitude increases via energy conservation, and from what I have read shallow water behaves analogously to an increasingly dense medium (or at least higher index of refraction). If I try and extend this analogy to sound wave/wave on string/earthquakes in a more dense medium, I find that the amplitude should decrease. Consider the sound wave first: Intensity is proportional to square root of the bulk modulus of the material and density, it is also proportional to the square of the displacement amplitude. Assume minimal reflection between the two media. If the density increases, the amplitude should decrease somewhat..however this reasoning does not feel correct. I just need some idea of how or where my reasoning is incorrect.

 

[Studiot]

The equations for gravity surface waves in water contain a dispersion term, not present in simpler wave forms such as the acoustic waves you mention.