String vibration

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 tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos.
Wave.
The velocity of propagation of a wave in a string (formula_1) is proportional to the square root of the force of tension of the string (formula_2) and inversely proportional to the square root of the linear density (formula_3) of the string:
formula_4
This relationship was discovered by Vincenzo Galilei in the late 1500s. 
Derivation.
Source:
Let formula_5 be the length of a piece of string, formula_6 its mass, and formula_3 its linear density. If angles formula_8 and formula_9 are small, then the horizontal components of tension on either side can both be approximated by a constant formula_2, for which the net horizontal force is zero. Accordingly, using the small angle approximation, the horizontal tensions acting on both sides of the string segment are given by
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, formula_13, will be equal to the net force on the piece:
Dividing this expression by formula_2 and substituting the first and second equations obtains (we can choose either the first or the second equation for formula_2, so we conveniently choose each one with the matching angle formula_9 and formula_8)
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 formula_8 and formula_9. Using this fact and rearranging provides
In the limit that formula_5 approaches zero, the left hand side is the definition of the second derivative of formula_24:
This is the wave equation for formula_26, and the coefficient of the second time derivative term is equal to formula_27; thus
Where formula_1 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 small amplitude vibrations; for those of large amplitude, formula_5 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 formula_2.
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 formula_32 divided by the period formula_33, or multiplied by the frequency formula_34:
If the length of the string is formula_36, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so formula_36 is half of the wavelength of the fundamental harmonic. Hence one obtains Mersenne's laws:
where formula_2 is the tension (in Newtons), formula_3 is the linear density (that is, the mass per unit length), and formula_36 is the length of the vibrating part of the string. Therefore:
Moreover, if we take the nth harmonic as having a wavelength given by formula_42, then we easily get an expression for the frequency of the nth harmonic:
And for a string under a tension T with linear density formula_3, then
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.
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.
Real-world example.
A Wikipedia user's Jackson Professional Soloist XL electric guitar has a nut-to-bridge distance (corresponding to formula_36 above) of 25 in. and D'Addario XL Nickel-wound Super-light-gauge EXL-120 electric guitar strings with the following manufacturer specs:
Given the above specs, what would the computed vibrational frequencies (formula_34) of the above strings' fundamental harmonics be if the strings be strung at the tensions recommended by the manufacturer?
To answer this, we can start with the formula in the preceding section, with formula_48:
The linear density formula_3 can be expressed in terms of the spatial (mass/volume) density formula_51 via the relation formula_52, where formula_53 is the radius of the string and formula_54 is the diameter (aka thickness) in the table above:
For purposes of computation, we can substitute for the tension formula_2 above, via Newton's second law (Force = mass × acceleration), the expression formula_57, where formula_6 is the mass that, at the Earth's surface, would have the equivalent weight corresponding to the tension values formula_2 in the table above, as related through the standard acceleration due to gravity at the Earth's surface, formula_60 cm/s2. (This substitution is convenient here since the string tensions provided by the manufacturer above are in pounds of force, which can be most conveniently converted to equivalent masses in kilograms via the familiar conversion factor 1 lb. = 453.59237 g.) The above formula then explicitly becomes:
Using this formula to compute formula_34 for string no. 1 above yields:
Repeating this computation for all six strings results in the following frequencies. Shown next to each frequency is the musical note (in scientific pitch notation) in standard guitar tuning whose frequency is closest, confirming that stringing the above strings at the manufacturer-recommended tensions does indeed result in the standard pitches of a guitar: