Question

Consider the 4th harmonic (standing wave with n = 4) on a string of length L with fixed ends, mass density μ and tension T

.a) On a standing wave, the nodes are the points that are not moving, and the antinodes the ones that move with the biggest amplitude. How many nodes and antinodes are on the 4th harmonic? Count them and make a graph of the function clearly showing where all the nodes and antinodes are located.

b) Write down an expression for the instantaneous velocity of each point of the string (i.e. for each x ) as it moves up and down.

c) Using your answer to part b) write the maximum velocity vmax for each point along the x axis as it moves up and down.

Answer 1

Solution: anti node node a) On the 4th harmonic, it's clear from the figure that there are 5 nodes and 4 antinodes. (b) The instantaneous velocity of a string depends on the position of the point. The velocity frequency relation is given by nv 2L 2L (c) At the maximum point, the instantaneous velocity is zero. At the extreme point, the string is instantly at rest.