Vibrating string

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.

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Wave

The speed of propagation of a wave in a string ($v$) is proportional to the square root of the tension of the string ($T$) and inversely proportional to the square root of the linear mass ($mu$) of the string:

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

imple derivation

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

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

:$F\; =\; heta,T.$

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

:$m\; =\; mu,L\; =\; mu,\; heta,R$

and

:$F\; =\; mu,\; heta,R,frac\{v^2\}\{R\}\; =\; mu,\; heta,v^2.$

Equating the two expressions for $F$ gives::$heta,T\; =\; mu,\; heta,v^2.$

Solving for velocity "v", we find

:$v\; =\; sqrt\{T\; over\; mu\}.$

Complicated but more general derivation

Let $Delta\; x$ be the length of a piece of string, $m$ its mass, and $mu$ its linear mass. If the horizontal component of tension in the string is a constant, $T$, then the tension acting on each side of the string piece are related by

:

This implies that there is no horizontal component of the net force on the string piece. From Newton's second law, the mass of this piece times its acceleration, $a$, will be equal to the net force on the piece:

:

Dividing this by the prior expressions (that both equal $T$) provides

:

The slopes at the ends of the string piece are by definition equal to the tangents of the angles at the ends. Using this fact and rearranging provides

:$frac\{1\}\{Delta\; x\}left(left.frac\{partial\; y\}\{partial\; x\}\; ight|^\{x+Delta\; x\}-left.frac\{partial\; y\}\{partial\; x\}\; ight|^x\; ight)=frac\{mu\}\{T\}frac\{partial^2\; y\}\{partial\; t^2\}$

In the limit that $Delta\; x$ approaches zero, the left hand side is the definition of the second derivative of $y$:

:$frac\{partial^2\; y\}\{partial\; x^2\}=frac\{mu\}\{T\}frac\{partial^2\; y\}\{partial\; t^2\}.$

This is a standard differential equation for $y(x,t)$, and the coefficient of the acceleration always plays the role of $v^\{-2\}$; thus

:$v=sqrt\{Tovermu\},$

where $v$ is the speed of propagation of the wave in the string. (See the article on the wave equation for more about this). However, this derivation is only valid for vibrations of small amplitude; for those of large amplitude, $Delta\; x$ is not a good approximation for the length of the string piece, and the horizontal component of tension is not necessarily constant.

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 $au$, or multiplied by the frequency $f$ :

:$v\; =\; frac\{lambda\}\{\; au\}\; =\; lambda\; f.$

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 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

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 or, perhaps more easily, by adjusting the machine heads, 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 adjustment, 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 (music)
* Vibrations of a circular drum
* Melde's experiment

References

*cite journal | last=Molteno | first=T. C. A. | coauthors=N. B. Tufillaro | title=An experimental investigation into the dynamics of a string | journal=American Journal of Physics | month=September | year=2004 | volume=72 | issue=9 | pages=1157–1169 | doi=10.1119/1.1764557
*cite journal | last=Tufillaro | first=N. B. | title=Nonlinear and chaotic string vibrations | journal=American Journal of Physics | year=1989 | volume=57 | issue=5 | pages=408 | doi=10.1119/1.16011

External links

* [http://www.falstad.com/loadedstring/ Java simulation of waves on a string]
* [http://sankey.ws/string.html Physics of a harpsichord string]
* [http://www.drchaos.net/drchaos/string_web_page/index.html A study of chaotic motion in strings]
* [http://www.acoustics.salford.ac.uk/feschools/waves/string.htm#fullGuitar A friendly explanation of standing waves and fundamental frequency]
* " [http://demonstrations.wolfram.com/TheVibratingString/ The Vibrating String] " by Alain Goriely and Mark Robertson-Tessi, The Wolfram Demonstrations Project.