RESONANCE

You can demonstrate harmonics if you have a piece of rope about 10 m long. Anchor one end of the rope to an immovable object such as a fence post or a hook in a wall. Be sure that the rope is tied securely so that it won’t shake loose. Hold the other end, and back off until the rope is tight. Then start pumping, slowly at first and then gradually faster. At a certain pumping speed, the rope will get into the rhythm and will seem to move up and down with a will of its own. This is a condition of resonance. Get it going this way for awhile. Then double the rate of pumping. If you keep at it, you’ll get a full wave cycle to appear along the rope. The wave will reverse itself in phase each time you pump, and its curvature will attain a familiar shape: the sinusoid. Keep on pumping at this rate for awhile. Then, if you can, double the pumping speed once more. This experiment requires some conditioning and coordination, but eventually you’ll get two complete wave cycles to appear along the length of the rope. You’re at the second harmonic of the previous oscillation, and resonance occurs again.
If you’re strong and fast enough, and if you have enough endurance, you might double the frequency again, getting four complete standing waves to appear along the rope (the fourth harmonic). If you’re a professional athlete, maybe you can double it yet another time, getting eight standing waves (the eighth harmonic). Theoretically, there is no limit to how many cycles can appear between the shaker and the anchor. In the real world, of course, the diameter and elasticity of the rope impose a limit. When you pump a rope, the wave impulses have longitudinal motion; they travel lengthwise along the rope. The individual molecules in the rope undergo transverse motion; they move from side to side (or up and down).
The waves along the rope resemble swells on the surface of the ocean. Stop shaking the rope and let it come to rest. Then give it a quick, hard, single pump. A lone wave shoots from your hand toward the far end of the rope and then reflects from the anchor and travels back toward you. As the pulse travels, its amplitude decays. Hold your hand steady as the pulse returns. The pulse energy is partially absorbed by your arm and partially reflected from your hand, heading down toward the far end again. After several reflections, the wave dies down. Some of its energy has been dissipated in the rope. Some has been imparted to the object to which the far end of the rope is anchored. Some has been absorbed by your body. Even the air has taken up a little of the original wave energy.