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Harmonics and Resonance Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 12, 2011

Standing Waves

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

Problem

How far apart is a standing-wave loop from an adjacent node in terms of degrees of phase?

Solution

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.

Irregular Waves

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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