Oscillations and Waves > Sound. Acoustic resonators > Problem 3.9.9

Statement

Two external longitudinal forces act in two sections of an infinite rod. The force on the left side changes according to the law , and the force on the right side changes according to the law . At what distance between the sources of the force will the traveling wave propagate from left to right only or from right to left only?

Solution

Let and be the positions of the left and the right sources, respectively. Then, the superposition of the waves generated by both sources is given by

where is the wave number and is the speed of the wave. Clearly, a standing wave is formed for . In order that left traveling wave for vanishes and only right traveling wave for remains, we must have

where is an integer. Thus, , where .

On the other hand, in order that right traveling wave for vanishes and only left traveling wave for remains, we must have

Thus, .

Answer

Only right traveling wave remains when , where .

Only left traveling wave remains when .