In the derivation of velocity of a wave on a string, a/c to HCV book, it is assumed that the - Physics - Waves

Question

In the derivation of velocity of a wave on a string, a/c to HCV
book, it is assumed that the particles move in an approximately
circular path but actually the particles only move up and down a
string and not sideways How is this possible? Also the weight force
is not taken into account Please explain

Sanjay Singh · Accepted Answer

Dear student
While deriving this formula he assumed that this equation is made by the person who is moving in  the direction of wave velocity and with the same velocity as thevelocity of the wave 
So he will always observe different particles of the string instead of the same particle which ofcourse move up and down but when the motion of  different particles are observed in sequence as is possible in the moving frame of the observer the combined motion of these particles constitute circular motion
Further the string is assumed to be very light
regards