# Resonance (fixed-end/free-end reflections)

In the case that the left and right ends of a rope are fixed and free, respectively, we can observe resonance when pulses are sent every 1/n of T1, where T1 is the duration of a single pulse traveling its two reciprocations and n is a natural number. Let us take a look at the fundamental vibration when the left-most red dot is vibrates in simple harmonic motion with its period of T1. The frequency of a sinusoidal wave created by this simple harmonic motion is F1, which is the reciprocal of T1. In the next video, you can observe that the left (fixed) end is node and the right (free) end is antinode.

Next, when the simple harmonic motion of the left-most red dot is three times as fast as the previous example that was the fundamental/first mode, a created sinusoidal wave with the frequency of F2 = 3 F1 will resonate (the second mode). In this case, you can observe two nodes at the left (fixed) end and at the two-third point (around the x-coordinate of 33), and two antinodes at the right (free) end and the one-third point (around the x-coordinate of 16 and 17).

Finally, when the simple harmonic motion of the left-most red dot is five times as fast as the first mode, a created sinusoidal wave with the frequency of F3 = 5 F1 will also resonate (the third mode). In this case, you can observe three nodes at the left (fixed) end, the two-fifth point (around the x-coordinate of 20) and the four-fifth point (around the x-coordinate of 40). The antinodes are located in the middle of each consecutive pair of two nodes and at the right (free) end.

Thus, in case that a rope has the fixed and free ends with its length of L [m], it is summarized as follows. Resonance will occur when an external force in simple harmonic motion is applied with the frequency of (2n-1) F1, where n is a natural number and F1 = c / 4L (c is the wave speed). In the first (fundamental) mode, the length of the rope, L, equals to a quarter of the wavelength, it is also called a quarter-wavelength resonance.

The next video shows a multiple spring-mass system producing the longitudinal wave in the fundamental vibration. In this case, because the left end is fixed and the right end is free, they become node and antinode, respectively.

The next is the case of the second mode. You can observe node, antinode, node, and antinode, from left to right.