Frequency of a string under tension (nth harmonic)

Description

A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical note. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos.

Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated.

- 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

For a string under a tension T with density μ, the frequency formula is shown here.

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 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.

Related formulas

Variables

fnfrequency of a string under tension (hz)
nnth harmonic (dimensionless)
Llength of the string (m)
Ttension (N)
μlinear density of the string (kg/m)