String vibration

A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency in most cases 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. It should be emphasized that this articles' content is theoretical in nature, to be enhanced with actual observation.

Speed of propagation of the wave

Illustration for a vibrating string

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 $\theta$ the angle under the arc. Then $L=\theta \,R$.

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

$F=\theta \,T$

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

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

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

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

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

and

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

Equating the two expressions for $F$ gives:

$\theta \,T=\mu \,\theta \,v^{2}$

Solving for velocity v, we find

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

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

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

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:

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

where $T$ is the tension, $\mu$ 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
• 3rd bridge guitar
• Physics of music
• Musical acoustics
• Pitch
• Franz Melde

References

• Molteno, T. C. A. (2004). "An experimental investigation into the dynamics of a string". American Journal of Physics. 72 (9): 1157–1169. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
• Tufillaro, N. B. (1989). "Nonlinear and chaotic string vibrations". American Journal of Physics. 57 (5): 408.

External links

• Java simulation of waves on a string
• Physics of a harpsichord string
• A study of chaotic motion in strings