Fourier, part 2.5

Fourier Approximations and Music

Part 2: Musical notes

2.5 Harmonics and complex vibrations

As we have noted, a string can be made to vibrate so as to produce a pure tone -- if the string starts from rest in the shape of a sine graph. But suppose the string is either plucked or bowed, such as those on a guitar or violin. In both of these cases the string will neither start out at rest nor have the initial profile of a sine graph. It is not surprising that the resulting pressure function will not resemble a sine graph, and the sound produced will not be a pure tone.

In fact, we argue that, when a string is plucked or bowed, the string vibrates in a number of pure tone modes simultaneously, each with a characteristic amplitude. The resulting complex vibration is a combination of these pure tone vibrations. If you have a guitar handy, there is a simple experiment you can perform to demonstrate this. Pluck a guitar string, then gently touch the string at its midpoint (without out pushing it to the fingerboard). The tone will change, becoming quieter and with higher pitch. Those who are musically knowledgeable will recognize the tone produced as being one octave higher than the original. If we pluck the string, then touch it slightly off its midpoint, the sound is completely eliminated. When the string is plucked, then touched 1/3 or 1/4 of the way from either end, we again hear a high pitched ringing tone.

When you touched the string slightly off its midpoint, you prevented the string from vibrating and damped out all sound. When you touched the string at its midpoint, you did not damp all of the vibration -- you heard the high pitched tone. The sound produced was quieter because you damped some of the vibration. The plucked string must have a manner of vibrating that leaves its midpoint and the points 1/3 and 1/4 of the way from the ends still. In fact, you will recall that a string vibrating in the 2nd harmonic mode leaves the midpoint of the string unmoved.

What points on a vibrating string remain still when the string vibrates in its third harmonic mode?

When you touched the string at its midpoint, you prevented the string from vibrating in any mode except those that leave the midpoint unmoved. You stopped the fundamental and all the odd harmonic vibrations, but the even harmonic vibrations were not affected. The sound you heard was primarily that of the sound associated with the second harmonic. Pictured below are two graphs recorded using a Texas Instrument CBL system. The first graph represents the sound of a plucked guitar string. The second graph was made from the sound produced when that same guitar string was lightly touched at its midpoint after it was plucked.

CBL record of a plucked A string on an acoustic guitar

Same string plucked, then touched at midpoint

Using the graphs from the guitar experiment, estimate the period of the initial pressure function.
Which harmonics are eliminated when the string is plucked, then lightly touched 1/3 of the way from its end?

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