Water waves vs. other wave types

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Water waves increase in amplitude as they approach shore due to energy conservation, behaving similarly to waves in a denser medium. In contrast, sound waves and other wave types, like those on a string or seismic waves, typically experience a decrease in amplitude when moving into a denser medium. This is because intensity is linked to both the bulk modulus and density, while amplitude is related to the square of displacement. The presence of a dispersion term in water wave equations differentiates them from simpler wave forms, complicating direct analogies. Understanding these distinctions is crucial for accurately comparing wave behaviors across different mediums.
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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.
 
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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.
 
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