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, cello, or piano.

__TOC__

Speed of propagation of the wave

---

Let $L$ be the length of the string, $m$ its mass and $T$ the tension.

When the string is deflected it bends as an approximate arc of circle. Let $R$ be the radius and $\backslash theta$ the angle under the arc. Then $L\; =\; \backslash theta\backslash ,R$.

The string is recalled to its natural position by a force $F$:

$F\; =\; \backslash theta\backslash ,T$

The force $F$ is also equal to the centripetal force

$F\; =\; m\backslash ,\backslash frac\{v^2\}\{R\}$

where $v$ is the speed of propagation of the wave in the string.

Let $\backslash mu$ be the linear mass of the string. Then

$m\; =\; \backslash mu\backslash ,L\; =\; \backslash mu\backslash ,\backslash theta\backslash ,R$

and

$F\; =\; \backslash mu\backslash ,\backslash theta\backslash ,R\backslash ,\backslash frac\{v^2\}\{R\}\; =\; \backslash mu\backslash ,\backslash theta\backslash ,v^2$

Equating the two expressions for $F$ gives:

$\backslash theta\backslash ,T\; =\; \backslash mu\backslash ,\backslash theta\backslash ,v^2$

Solving for velocity v, we find

$v\; =\; \backslash sqrt\{T\; \backslash over\; \backslash mu\}$

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 $\backslash lambda$ divided by the period $\backslash tau$, or multiplied by the frequency $f$ :

$v\; =\; \backslash frac\{\backslash lambda\}\{\backslash tau\}\; =\; \backslash lambda\; f$

If the length of the string is $L$, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so $L$ is half of the wavelength of the fundamental harmonic. Hence:

$f\; =\; \backslash frac\{v\}\{2L\}\; =\; \{\; 1\; \backslash over\; 2L\; \}\; \backslash sqrt\{T\; \backslash over\; \backslash mu\}$

where $T$ is the tension, $\backslash mu$ is the linear mass, and $L$ 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 lighter the string, the higher the note

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 oscilloscope). This effect is called temporal aliasing, 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 which 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, this effect does not occur and the string will appear to be still, but thicker and lighter, due to persistence of vision.

A similar but more controllable effect can be obtained using a stroboscope. This device allows the frequency of the xenon flash lamp to be exactly matched to the frequency of vibration of the string; in a darkened room, this clearly shows the waveform. Otherwise, one can use bending to obtain the same frequency, or a multiple of, the AC frequency to achieve the same effect. For example, in the case of a guitar, the bass string pressed to the third fret gives a G at 97.999 Hz; with a slight bend, a frequency of 100 Hz can be obtained, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia. In most countries of the Americas, where the AC frequency is 60 Hz, one can start from A# at 116.54 Hz, on the fifth string at the first fret, to obtain a frequency of 120 Hz.

See also

---

• String instruments
• Fretted instruments
• Physics of music
• Musical acoustics
• Pitch
• Franz Melde

External links

---

• Java simulation of waves on a string
• Physics of a harpsichord string

Classical mechanics | Sound | Introductory physics | String instrument construction

Corda (musica) | מיתר (כלי מיתר) | Húr | Snaar