Relative motion of a particle of a moving wave with respect to the wave itself

When deriving the equation for the velocity of a transverse wave in a string of tension $T$ and linear mass density $d$, we assume that a small part of the wave is undergoing circular motion with respect to the medium of the wave.

However if i take a particle on the string, other particles on the string should also exhibit SHM motion. So in which reference frame does the small part of the string exhibits circular motion?