Viliam_Bur comments on Configurations and Amplitude - Less Wrong
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I think you are fighting a strawman here. By using the word "wave" physicists are not suggesting that something moves along a sinusoidal path. That was your intepretation, an incorrect one, and you have successfully disproved it, which is great. Just please don't assume that physicists are doing the same mistake.
When you look at the waves on the water, yes, there is a kind of sinusoidal shape. But the word "wave" in physics means that at some places the density of something is higher, and at other places, the density is lower, and the map of the densities looks and moves... well, like the waves on the water.
The waves on the water make a sinusoidal shape, because there is an air above the water, so the higher density of water creates a "wave" in a layman's meaning of the word. But imagine an explosion deep below the ocean surface, and how the density of the surrounding water changes in time. This is also called "waves" in physics, but there is nothing moving along the sinusoidal path. Similarly, sound makes "waves" (areas of different density) in the air. And a photon also makes some kind of a "wave" (some its properties measured in space and time show the same kind of pattern).