Oscillations and Waves > Wave propagation > Problem 3.7.8

Statement

A string consisting of two parts with linear densities and is stretched by longitudinal forces . At the point where the parts are connected, the string is pulled by a transverse force . How does the shape of the string change with time?

Solution

The speeds of propagation of the waveform to the left and the right are and , respectively. Consequently, the mass of the portion of the rope that has transverse momentum increases linearly with time, and its transverse speed must be constant, because the transverse force is constant. After time , transverse momentum of the waveform is

Note that even though the lengths of the stretched parts of the rope are and , they used to have lengths and , so they must have masses and , respectively.

As a result,

Therefore,

Answer

See the solution.