# Multiple reflections and resonance (fixed-free 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 left end is fixed and the right end is free, 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 stays the same at the free end but becomes inverted at the fixed end.

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 (free-end) and is just reflected at the left end (fixed-end). The first and the second pulses are canceled out because the directions of the two pulses are opposite. You can observe that resonance does not occur in this case.

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

Let us take a look an the next video of the case, when pulses are sent every timing of 1/3 of T1. Indeed, the pulses are growing in this case. Furthermore, we can observe the nodes at the left (fixed) end and at the two-third point (around the x-coordinate of 33).

Next, let us send pulses every timing of 1/4 of T1. You can observe that the pulses do not grow.

Finally, pulses are sent every timing of 1/5 of T1. Then, you can observe the three nodes at the left (fixed) end, at the two-fifth point (around the x-coordinate of 20), and at the four-fifth point (around the x-coordinate of 40).

Thus, in case of a rope with one end fixed and the other end free, it is summarized as follows. When the periodic external force applies in proper timings, displacements can grow, and it is the phenomenon called “resonance.” In this fixed-free case, we can observe resonance when pulses are sent every 1/(2n-1) of T1, where n is a natural number and T1 is the duration of a pulse traveling two reciprocations. In the frequency domain, it is (2n-1) F1, where F1 = 1/T1.