Vibration of Wire Strings
When a stretched wire string is excited into motion by plucking or striking, a complex wave travels outward to the ends of the string. As it travels outward, this initial impulse forces the wire out of its resting position all along its length. After the impulse has passed, each part of the wire immediately begins to return toward (and overshoot) its resting position, which means vibration has been induced. Meanwhile, the initial impulse is reflected at both ends of the string and travels back toward the center. On the way, it interacts with the various vibrations it induced on the initial pass, and these interactions reduce or cancel some components of the impulse wave and reinforce others. When the reflected impulses encounter each other, their interaction again cancels some components and reinforces others.
Within a few transits of the string, all these cancellations and reinforcements sort the vibration into an orderly set of waves that vibrate over 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length of the string. These are the harmonics. As a rule, the amplitude of its vibration is less for higher harmonics than for lower, meaning that higher harmonics are softer—though the details of this differ from instrument to instrument. The exact combination of different harmonics and their amplitudes is a primary factor affecting the timbre or tone quality of a particular musical tone.
Theoretically, vibration over half the string's length will be twice as fast, and vibration over one-third of the string will be three times as fast, as the fundamental vibration over the whole string's length. In the theoretical string, however, the only force acting to return a part of the string to its rest position is the tension between its ends.
If you try bending a short piece of piano wire or guitar string slightly with your fingers, you can feel the wire's resistance to bending. In a vibrating string, that resistance adds to the effect of string tension in returning a given part of the string toward its rest position. The result is a frequency of vibration higher than the theoretical frequency. And because the wire's resistance to bending increases as its length decreases, its effect is greater in higher harmonics than in lower.
Read more about this topic: Stretched Tuning
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