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Ultimately, it turns into heat, so that isn't wrong. But If it were immediately converted to heat as it crashed into shore, the shorelines would be boiling. Most of the "heat" is in the form of moving air in the atmosphere as the wave passes below it, and most of that heat radiates into space.
Most of the energy that crashes onto the beach ends up going back into the sea as a reflected wave.
This is an old lecture on the properties of waves. Pay attention to the types of reflection.
Polynesian "Wayfinding" relied heavily on understanding how waves reflect off shorelines, enabling navigators to locate islands beyond the horizon.
Yeah. I felt like the spirit of the question was about how moving water can rapidly stop without breaking energy conservation, so I focused the answer there. I did mention reflection back into the sea at the end.
How do you mean? Waves carry significant energy, but moving water carries even more significant cooling capacity.
Dam spillways are designed to dissipate the full power capacity of the dam with splashing, and there are modifications made to some shorelines to increase the amount of absorption of incoming waves, specifically.
My point is that the moving water doesn't "rapidly stop" when it hits the beach. The beach does not absorb very much of the energy of the wave at all.
As it arrives at the beach, the water flows uphill, against gravity, converting kinetic energy into potential energy, not heat, at least not in any significant quantities. Now the water is high up on the beach. Gravity drags it right back downhill. That energy travels back out to sea. Virtually all of the energy of the wave is reflected back into the sea. The proportion of the wave energy that is converted to heat/noise at the beach is a tiny fraction of the total wave energy.
The video I linked discusses mechanical wave energy absorbed by a "dashpot". The dashpot analogizes conversion to heat. If we were to model a beach, we would need to use an infinitesimally small dashpot.
Sure. I don't think we're disagreeing on much of any substance here.
(You'll find that the shallow water is much, much warmer than the deep water at the beach, but I always blamed the sun for that.)
That is true.
That temperature difference is also true in the middle of the ocean, far away from any beach for a wave to crash.
Blame for the temperature delta you have identified rests with insolation.
Maybe there's still a delta, but when the water is a metre or two deep it's barely noticeable, whereas it's really really stark between a foot and two inches depth.