Harmonics and Resonance Help (page 2)
Seiche and Harmonics
Any child who lives in a house with a bathtub knows about seiche (pronounced “saysh”). Any enclosed or semienclosed body of water can be made to slosh back and forth at a rate that depends on the size and shape of the container. In a bathtub, this sloshing can be set up with a period of 1 or 2 seconds. Give the water a little push, and then another, and then another. Keep this up at a certain regular repetitive rate, and soon there is water all over the bathroom. The same thing can happen in a swimming pool during an earthquake, although the period is longer. When waves moving in opposite directions collide, the peaks and troughs are exaggerated (Fig. 17-2).
Fig. 17-2 . When waves collide, the effects are magnified.
Harmonics are familiar to anyone who plays a musical instrument such as a clarinet, flute, trumpet, or trombone. If you can blast out a note with certain keys pressed or with the slide at a given position, then if you tighten your lips enough, you can sound a note one octave higher. The higher note is the second harmonic of the first note. The chamber of the instrument contains twice as many wave peaks and valleys at the higher note as compared with the lower note. If you’re a virtuoso, you might get the instrument to toot at three times the original, or fundamental, frequency. This is the third harmonic . Mathematically, there is no limit to how far this can go (Fig. 17-3). When the frequency of one wave is a harmonic of the frequency of another wave, the two waves are said to be harmonically related .
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.
Start shaking the rope rhythmically once again. Set up waves along its length just as you did before. Send sine waves down the rope. At certain shaking frequencies, the impulses reflect back and forth between your hand and the anchor so that their effects add together: Each point on the rope experiences a force upward, then downward, and then up again, then down again. The reflected impulses reinforce; the sideways motion of the rope is exaggerated. Standing waves appear.
Standing waves get their name from the fact that they do not, in themselves, travel anywhere. But they can acquire tremendous power. Some points along the rope move up and down a lot, some move up and down a little, and others stand completely still, only rotating slightly as the rest of the rope wags. The points where the rope moves up and down the furthest are called loops; the points where the rope doesn’t move are known as nodes . There are always two loops and two nodes in a complete standing-wave cycle. They are all equally spaced from one another.
Standing Waves Practice Problem
How far apart is a standing-wave loop from an adjacent node in terms of degrees of phase?
As you should remember, there are 360° of phase in a complete cycle. From the preceding description, there are two loops and two nodes in a complete cycle, all equally spaced from each other; this means that they are all one-quarter cycle, or 90°, apart. Any given loop is 90° from the node on either side; any given node is 90° from the loop on either side.
Not all waves are sine waves. Some nonsinusoidal waves are simple but are seen rarely in nature. Some of these waves have abrupt transitions; unlike the smooth sinusoid, they jump or jerk back and forth. If you’ve used a laboratory oscilloscope, you’re familiar with waves like this. The simplest nonsinusoids are the square wave , the ramp wave , the sawtooth wave , and the triangular wave . These can be generated with an electronic music synthesizer, and they have a certain mathematical perfection, but you’ll never see them on the sea. Irregular waves come in myriad shapes, like fingerprints or snowflakes. The sea is filled with these. In the world of waves, simplicity is scarce, and chaos is common.
Most musical instruments produce irregular waves, like the chop on the surface of a lake. These are complex combinations of sine waves. Any waveform can be broken down into sinusoid components, although the mathematics that define this can become complicated. Cycles superimpose themselves on longer cycles, which in turn superimpose themselves on still longer cycles, ad infinitum . Even square, ramp, sawtooth, and triangular waves, with their straight edges and sharp corners, are composites of smooth sinusoids that exist in precise proportions. Waves of this sort are easier on the ear than sine waves. They are also easier to generate. Try setting a music synthesizer or signal generator to produce square, ramp, sawtooth, triangular, and irregular waves, and listen to the differences in the way they sound. They all have the same pitch, but the timbre , or tone, of the sound is different.
Practice problems of these concepts can be found at: Wave Phenomena Practice Test
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