This has always puzzled me and nobody (professors, textbooks) ever seemed to elaborate on it (maybe it is very simple and I merely didn't catch on). Anyway, I understand that when a wave passes through a medium, the individual particles of the medium are briefly displaced from their equilibrium positions and then brought back to equilibrium by a restoring force (tension, gravity) unique to whatever the medium is (a string, the surface of a lake). Therefore the medium itself does not change position and the wave transfers only energy as it passes by. So, my question is, is the speed at which the individual particles are displaced the same as the speed of the wave as it passes that particle (the particle moves a very small distance over a very small time, which turns out to be the same as the wave speed when calculated)? Or is it a totally different speed that is too difficult or too meaningless to bother calculating?
They are related, but not the same. The position of a particle on the surface is governed by an equation of harmonic motion, such as y=A*sin(w*t) where A is the amplitude and w is the angular velocity (2*pi*frequency) of the wave. The velocity is the derivative of that equation, v=A*w*cos(w*t). I think the relationship might be more complicated for real water waves due to other forces at work though.