What is the difference between particle velocity and wave velocity of a simple harmonic progressive wave?

Consider a string tied at one end and you are moving the other end in your hand. If you carefully observe, the wave is traveling through the rope to the tied end. But, the particles, which constitute the rope move only in up and down direction. They do not move parallel to the rope at all. If that was the case, after a bit of time, you would have had a rope that was denser on the tied end. So particle velocity is perpendicular to the wave velocity in a transverse wave. It is equal to (omega) squared times the wave velocity.