# Multiple resonance system

In the first half of this section, we will think of the situation where the both ends of a rope are fixed. Let us think that you grab the left end of the 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. In this section, we assume 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.

The next video shows when you repeatedly send pulses, with its amplitude of 1, at the timings when each pulse travels back from the right end (fixed-end) and is just reflected at the left end (fixed-end). You can observe the height of the resulting pulse grows.

Previously, pulses were sent every T1 [s], which is the time duration of the pulse to reciprocate and come back to the starting point. The next video shows when pulses are sent every 2 T1, or the time duration of the pulse travels two reciprocations. Before the next pulse is sent, the previous pulse is damped at every reflection. However, the new pulse is sent with its amplitude of 1 at every certain period, the height of the resulting pulse gradually grows anyway.

In the second half of this section, we will think of the situation where the left and right ends of the rope are fixed and free, respectively. Let us think that you grab the left end of the rope and move it for a half period of a simple harmonic motion, again. As a result, a pulse travels from left to right. When the first pulse travels from left to right, it still travels from right to left after the first reflection at the right end (free-end), and it continues to travel from left to right after the second reflection at the left end (fixed-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 end but not the free end. We still assume 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.

The next video shows when you repeatedly send pulses every 2 T1 [s], with its amplitude of 1, at the timings when each pulse travels two reciprocations. You can observe the height of the resulting pulse grows.

Previously, pulses were sent every 2 T1, which is the time duration of the pulse to travel two reciprocations. The next video shows when pulses are sent every 4 T1, or the time duration of the pulse travels four reciprocations. Before the next pulse is sent, the previous pulse is damped at every reflection. However, the new pulse is sent with its amplitude of 1 at every certain period, the height of the resulting pulse gradually grows anyway.