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

In the case that the both ends of a rope are fixed, we can observe resonance when pulses are sent every 1/n of T1, where T1 is the duration of a single pulse traveling its reciprocation 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 the both ends are nodes and the center is the antinode.

Next, when the simple harmonic motion of the left-most red dot is twice as fast as the previous example that was the fundamental/first mode, a created sinusoidal wave with the frequency of F2 = 2 F1 will resonate (the second mode). In this case, you can observe three nodes at the both ends and the center. The antinodes are located in the middle of each consecutive pair of two nodes.

Finally, when the simple harmonic motion of the left-most red dot is three times as fast as the first mode, a created sinusoidal wave with the frequency of F3 = 3 F1 will also resonate (the third mode). In this case, you can observe four nodes at the both ends, the one-third point (around the x-coordinate of 16 and 17) and the two-third point (around the x-coordinate of 33). Again, the antinodes are located in the middle of each consecutive pair of two nodes.

Thus, in case of the both ends fixed of a rope with the 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 n F1, where n is a natural number and F1 = c / 2L (c is the wave speed). In the first (fundamental) mode, the length of the rope, L, equals to a half of the wavelength, it is also called a half-wavelength resonance.