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Wikipedia / String vibration

A vibration in a string is a wave. Initial disturbance (such as plucking or striking) causes a vibrating string to produce a sound with constant frequency, i.e., constant pitch. The nature of this frequency selection process occurs for a stretched string with a finite length, which means that only particular frequencies can survive on this string. If the length, tension, and linear density (e.g., the thickness or material choices) of the string are correctly specified, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. For a homogeneous string, the motion is given by the wave equation.

Wave

) is proportional to the square root) of the string:

This relationship was discovered by Vincenzo Galilei in the late 1500s.

Derivation

Source:

be the length its mass its linear density are small, then the horizontal components of tension, for which the net horizontal force is zero. Accordingly, using the small angle approximation, the horizontal tensions acting on both sides of the string segment are given by

, will be equal to the net force on the piece:

)

. Using this fact and rearranging provides

,

this equation is known as the wave equation; thus

is the speed.

Frequency of the wave

Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated. The speed of propagation of a wave is equal to the wavelength divided by the period, or multiplied by the frequency :

, the fundamental harmonic is the one produced by the vibration whose nodes is half of the wavelength of the fundamental harmonic. Hence one obtains Mersenne's laws:

is the tension is the linear density (that is, the mass is the length of the vibrating part of the string. Therefore:

  • the shorter the string, the higher the frequency of the fundamental

  • the higher the tension, the higher the frequency of the fundamental

  • the lighter the string, the higher the frequency of the fundamental

, then we easily get an expression for the frequency of the nth harmonic:

, then

Observing string vibrations

One can see the waveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of a CRT screen such as one of a television or a computer (not of an analog oscilloscope). This effect is called the stroboscopic effect, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the refresh rate of the screen. The same can happen with a fluorescent lamp, at a rate that is the difference between the frequency of the string and the frequency of the alternating current. (If the refresh rate of the screen equals the frequency of the string or an integer multiple thereof, the string will appear still but deformed.) In daylight and other non-oscillating light sources, this effect does not occur and the string appears still but thicker, and lighter or blurred, due to persistence of vision.

A similar but more controllable effect can be obtained using a stroboscope. This device allows matching the frequency of the xenon flash lamp to the frequency of vibration of the string. In a dark room, this clearly shows the waveform. Otherwise, one can use bendinging or, perhaps more easily, by adjusting the machine heads, to obtain the same, or a multiple, of the AC frequency to achieve the same effect. For example, in the case of a guitar, the 6th (lowest pitched) string pressed to the third fret gives a G at 97.999Hz. A slight adjustment can alter it to 100Hz, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia, 50Hz. In most countries of the Americas—where the AC frequency is 60Hz—altering A# on the fifth string, first fret from 116.54Hz to 120Hz produces a similar effect.

See also

References

  1. Specific

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