An energy explanation for why waves slow down

 

It occurred to me that as a wave approaches the coast, it starts to get taller. I think that this is because the water is shallower so there is les space beneath the surface for the wave motion. Now the fact that the water has been pushed up higher means that it has more gravitational potential energy. This energy has to come from somewhere. The total energy of the wave will remain the same if it is a closed system. This means that the energy must come from the wave's kinetic energy. With less kinetic energy, the wave slows down.