String vibration

If the length or tension of the string is correctly adjusted, the sound produced is a musical tone.

For an homogenous string, the motion is given by the wave equation.

) is proportional to the square root of the force of tension of the string (

) and inversely proportional to the square root of the linear density (

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

are small, then the horizontal components of tension on either side can both be approximated by a constant

Accordingly, using the small angle approximation, the horizontal tensions acting on both sides of the string segment are given by From Newton's second law for the vertical component, the mass (which is the product of its linear density and length) of this piece times its acceleration,

, will be equal to the net force on the piece: Dividing this expression by

) According to the small-angle approximation, the tangents of the angles at the ends of the string piece are equal to the slopes at the ends, with an additional minus sign due to the definition of

approaches zero, the left hand side is the definition of the second derivative of

is not a good approximation for the length of the string piece, the horizontal component of tension is not necessarily constant.

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

, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so

is the linear density (that is, the mass per unit length), and

, then we easily get an expression for the frequency of the nth harmonic: And for a string under a tension T with linear density

, then 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.

(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.

Otherwise, one can use bending 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 97.999 Hz.

A slight adjustment can alter it to 100 Hz, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia, 50 Hz.

In most countries of the Americas—where the AC frequency is 60 Hz—altering A# on the fifth string, first fret from 116.54 Hz to 120 Hz produces a similar effect.