Avibration in astring is awave. Initial disturbance (such as plucking or striking) causes avibrating string to produce asound with constantfrequency, i.e., constantpitch. The nature of thisfrequency selection process occurs for a stretched string with a finite length, which means that only particular frequencies can survive on this string. If the length, tension, andlinear density (e.g., the thickness or material choices) of the string are correctly specified, the sound produced is amusical tone. Vibrating strings are the basis ofstring instruments such asguitars,cellos, andpianos. For a homogeneous string, the motion is given by thewave equation.

The velocity of propagation of a wave in a string (${\displaystyle v}$) is proportional to thesquare root of the force of tension of the string (${\displaystyle T}$) and inversely proportional to the square root of the linear density (${\displaystyle \mu }$) of the string:

${\displaystyle v=\sqrt{\frac{T}{\mu }}.}$

This relationship was discovered byVincenzo Galilei in the late 1500s.^{[citation needed]}

Source:^[1]

Let${\displaystyle \mathrm{\Delta }x}$ be thelength of a piece of string,${\displaystyle m}$ itsmass, and${\displaystyle \mu }$ itslinear density. If angles${\displaystyle \alpha }$ and${\displaystyle \beta }$ are small, then the horizontal components oftension on either side can both be approximated by a constant${\displaystyle T}$, for which the net horizontal force is zero. Accordingly, using thesmall angle approximation, the horizontal tensions acting on both sides of the string segment are given by

${\displaystyle T_{1x}=T_1\cos (\alpha )\approx T.}$

${\displaystyle T_{2x}=T_2\cos (\beta )\approx T.}$

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,${\displaystyle a}$, will be equal to the net force on the piece:

${\displaystyle \mathrm{\Sigma }{F}_y=T_{1y}-T_{2y}=-T_2\sin (\beta )+T_1\sin (\alpha )=\mathrm{\Delta }ma\approx \mu \mathrm{\Delta }x\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}.}$

Dividing this expression by${\displaystyle T}$ and substituting the first and second equations obtains (we can choose either the first or the second equation for${\displaystyle T}$, so we conveniently choose each one with the matching angle${\displaystyle \beta }$ and${\displaystyle \alpha }$)

${\displaystyle -\frac{T_2\sin (\beta )}{T_2\cos (\beta )}+\frac{T_1\sin (\alpha )}{T_1\cos (\alpha )}=-\tan (\beta )+\tan (\alpha )=\frac{\mu \mathrm{\Delta }x}{T}\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}.}$

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${\displaystyle \alpha }$ and${\displaystyle \beta }$. Using this fact and rearranging provides

${\displaystyle \frac{1}{\mathrm{\Delta }x}\left({\frac{\mathrm{\partial }y}{\mathrm{\partial }x}|}^{x+\mathrm{\Delta }x}-{\frac{\mathrm{\partial }y}{\mathrm{\partial }x}|}^x\right)=\frac{\mu }{T}\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}.}$

In the limit that${\displaystyle \mathrm{\Delta }x}$ approaches zero, the left hand side is the definition of the second derivative of${\displaystyle y}$,

${\displaystyle \frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{x}^2}=\frac{\mu }{T}\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}.}$

this equation is known as thewave equation, and the coefficient of the second time derivative term is equal to${\displaystyle \frac{1}{v^2}}$; thus

${\displaystyle v=\sqrt{\frac{T}{\mu }},}$

Where${\displaystyle v}$ is thespeed of propagation of the wave in the string. However, this derivation is only valid for small amplitude vibrations; for those of large amplitude,${\displaystyle \mathrm{\Delta }x}$ is not a good approximation for the length of the string piece, the horizontal component of tension is not necessarily constant. The horizontal tensions are not well approximated by${\displaystyle T}$.

Once the speed of propagation is known, thefrequency of thesound produced by the string can be calculated. Thespeed of propagation of a wave is equal to thewavelength${\displaystyle \lambda }$ divided by theperiod${\displaystyle \tau }$, or multiplied by thefrequency${\displaystyle f}$:

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

If the length of the string is${\displaystyle L}$, thefundamental harmonic is the one produced by the vibration whosenodes are the two ends of the string, so${\displaystyle L}$ is half of the wavelength of the fundamental harmonic. Hence one obtainsMersenne's laws:

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

where${\displaystyle T}$ is thetension (in Newtons),${\displaystyle \mu }$ is thelinear density (that is, themass per unit length), and${\displaystyle L}$ is thelength 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

Moreover, if we take the nth harmonic as having a wavelength given by${\displaystyle {\lambda }_n=2L/n}$, then we easily get an expression for the frequency of the nth harmonic:

${\displaystyle f_n=\frac{nv}{2L}}$

And for a string under a tension T with linear density${\displaystyle \mu }$, then

${\displaystyle f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu }}}$

One can see thewaveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of aCRT screen such as one of atelevision or acomputer (not of an analog oscilloscope).This effect is called thestroboscopic effect, and the rate at which the string seems to vibrate is the difference between the frequency of the string and therefresh rate of the screen. The same can happen with afluorescent lamp, at a rate that is the difference between the frequency of the string and the frequency of thealternating 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 topersistence of vision.

A similar but more controllable effect can be obtained using astroboscope. This device allows matching the frequency of thexenon flash lamp to the frequency of vibration of the string. In a dark room, this clearly shows the waveform. Otherwise, one can usebending 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.

• Fretted instruments
• Musical acoustics
• Vibrations of a circular drum
• Melde's experiment
• 3rd bridge (harmonic resonance based on equal string divisions)
• String resonance
• Reflection phase change
• Hilbert space

• Molteno, T. C. A.; N. B. Tufillaro (September 2004). "An experimental investigation into the dynamics of a string".American Journal of Physics.72 (9):1157–1169.Bibcode:2004AmJPh..72.1157M.doi:10.1119/1.1764557.
• Tufillaro, N. B. (1989). "Nonlinear and chaotic string vibrations".American Journal of Physics.57 (5): 408.Bibcode:1989AmJPh..57..408T.doi:10.1119/1.16011.

Specific

• "The Vibrating String" byAlain Goriely and Mark Robertson-Tessi,The Wolfram Demonstrations Project.

• Mechanical vibrations
• Sound
• String instrument construction

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from November 2017