Answer
No, the medium in which a wave travels does not move along with a wave.
Work Step by Step
Picture a wave traveling down a rope that is shaken at one end, or a wave moving along the surface of a swimming pool. If you could label a particle of the rope or the water, you would see that, over time, the medium oscillates around some equilibrium position, but does not actually go anywhere.
This is discussed on page 360.