Vibrating string

A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note.
Vibrating strings are the basis of any string instrument like guitar or piano.

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Speed of propagation of the wave

Let be the length of the string, its mass and the tension.

When the string is touched it bends as an arc of circle. Let be the radius and the angle under the arc. Then .

The string is recalled to its natural position by a force which is equal to .

The force is also equal to the centripetal force , where is the speed of propagation of the wave in the string.

Let be the linear mass of the string. Then .

If we equate the two expressions of we have:

Frequency of the wave

Once we know the speed of propagation, it is almost immediate to find the frequency of the sound produced by the string. In fact we know that the speed of propagation of a wave is equal to the wavelength divided by the period , or multiplied by the frequency :

If the length of the string is , the fundamental harmonic is the one produced by the vibration whose nodess are the two ends of the string, so is half of the wavelength of the fundamental harmonic.

Hence:

where is the tension, is the linear mass, and is the length of the vibrating part of the string. Therefore:

the shorter the string, the higher the note

the higher the tension, the higher the note

the heavier the string, the lower the note

External Links

Java simulation of waves on a string

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