What Causes a Particle on a String to Return to the Origin in Wave Motion?

[Darren93]

In my wave course at the moment we are looking at waves on a string and their applications. At one point we consider the transverse force at a particular point on the string due to a uniform tension throughout a string. Something about this has confused me, the force isn't at all in the form I expected. Specifically when you look at a particle at peak amplitude, here velocity=0 as is the point the particle changes direction. However the transverse force due to tension=0. What force causes that particle to return back to the origin? I'm interpreting it as that particular particle acting as if their was no wave going through the string as tension is in same direction and magnitude as when string was at rest.

 

[jbriggs444]

The tension in the string is approximately uniform in magnitude. But it is not uniform in direction. Any curvature in the string will result in a net force toward the center of the curve -- the tensions on either side of an infinitesimal element will not quite align and the resultant will be non-zero.