# Multiple reflections and resonance (fixed-fixed ends)

Let us think that you grab the left end of a rope and move it for a half period of a simple harmonic motion. As a result, a pulse travels from left to right. When the both ends are fixed, the pulse travels from left to right, it still travels from right to left after the first reflection at the right end, and it continues to travel from left to right after the second reflection at the left end, etc. Thus, the multiple reflections will occur repeatedly. Please note that the direction of the displacement of each pulse becomes inverted at the fixed ends.

Ideally, it will continue forever. However, in a reality, there is “damping”. If there is no damping at a reflection, it is called “perfect reflection.” The next video shows multiple reflections by assuming there is a small damping at each reflection, resulting the amplitude becomes 90% of the one before each reflection. Please note the original amplitude is 1.

What is going to happen, if you send the second pulse, with its amplitude of 1, when the first pulse travels back from the right end (fixed-end) and is just reflected at the left end (fixed-end). The first and the second pulses are superimposed and becomes a single pulse with its amplitude more than 1. Let us call this timing T1 [s], which is the time duration of the pulse to reciprocate and come back to the starting point. The next video clip shows when you repeatedly send pulses at the same timings (T1, 2T1, 3T1, etc.), resulting the amplitude of the pulse is growing.

Previously, pulses were sent every T1. What is going to happen, if pulses are sent every timing less than T1. The next video clip shows when pulses are sent every 3/4 of T1. Then, the pulses are not growing due to bad timings.

Is it always true that pulses do not grow when the pulses are sent every timing less than T1? Let us take a look an the next video of the case, when pulses are sent every a half period of T1. Indeed, the pulses are growing in this case. Furthermore, we can observe the nodes at the both ends and at the center.

What is going to happen, when pulses are sent every timing of 1/3 of T1? In this case, we can also observe that the pulses are growing and there are 4 nodes at the both ends as well as the one-third point (the x-coordinate of 16 or 17) and the two-third point (around the x-coordinate of 33).

In summary, when the periodic external force applies in proper timings, displacements can grow, and it is the phenomenon called “resonance.” In this fixed-fixed case, we can observe resonance when pulses are sent every 1/n of T1, where n is a natural number. In the frequency domain, it is n F1, where F1 = 1/T1.