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.
The velocity of propagation of a wave in a string (
v
T
\mu
v=\sqrt{T\over\mu}.
This relationship was discovered by Vincenzo Galilei in the late 1500s.
Source:[1]
Let
\Deltax
m
\mu
\alpha
\beta
T
T1x=T1\cos(\alpha) ≈ T.
T2x=T2\cos(\beta) ≈ 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,
a
\SigmaFy=T1y-T2y=-T2\sin(\beta)+T1\sin(\alpha)=\Deltama ≈ \mu\Deltax
\partial2y | |
\partialt2 |
.
Dividing this expression by
T
T
\beta
\alpha
- | T2\sin(\beta) | + |
T2\cos(\beta) |
T1\sin(\alpha) | =-\tan(\beta)+\tan(\alpha)= | |
T1\cos(\alpha) |
\mu\Deltax | |
T |
\partial2y | |
\partialt2 |
.
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
\alpha
\beta
1 | \left(\left. | |
\Deltax |
\partialy | |
\partialx |
\right|x+\Delta-\left.
\partialy | |
\partialx |
| ||||
\right| |
\partial2y | |
\partialt2 |
.
In the limit that
\Deltax
y
\partial2y | = | |
\partialx2 |
\mu | |
T |
\partial2y | |
\partialt2 |
.
This is the wave equation for
y(x,t)
1 | |
v2 |
v=\sqrt{T\over\mu},
Where
v
\Deltax
T
λ
\tau
f
v=
λ | |
\tau |
=λf.
If the length of the string is
L
L
f=
v | |
2L |
={1\over2L}\sqrt{T\over\mu}
where
T
\mu
L
Moreover, if we take the nth harmonic as having a wavelength given by
λn=2L/n
fn=
nv | |
2L |
And for a string under a tension T with linear density
\mu
fn=
n | \sqrt{ | |
2L |
T | |
\mu |
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.(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 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.
See also: Guitar tunings, 12 equal temperament and A440 (pitch standard).
A Wikipedia user's Jackson Professional Soloist XL electric guitar has a nut-to-bridge distance (corresponding to
L
- | 1 | 0.00899 | 13.1 | 7.726 (steel alloy) | - | 2 | 0.0110 | 11.0 | " | - | 3 | 0.0160 | 14.7 | " | - | 4 | 0.0241 | 15.8 | 6.533 (nickel-wound steel alloy) | - | 5 | 0.0322 | 15.8 | " | - | 6 | 0.0416 | 14.8 | " |
f
To answer this, we can start with the formula in the preceding section, with
n=1
f=
1 | \sqrt{ | |
2L |
T | |
\mu |
\mu
\rho
\mu=\pir2\rho=\pid2\rho/4
r
d
f=
1 | \sqrt{ | |
2L |
T | |
\pid2\rho/4 |
T
T=ma
m
T
g0=980.665
fHz=
1 | \sqrt{ | |
Lin x 2.54 cm/in x din x 2.54 cm/in |
| ||||||||
|
f
f1=
1 | \sqrt{ | |
25.625 in x 2.54 cm/in x 0.00899 in x 2.54 cm/in |
| ||||
|
- | 1 | 330 | E4 (= 440 ÷ 25/12 ≈ 329.628 Hz) | - | 2 | 247 | B3 (= 440 ÷ 210/12 ≈ 246.942 Hz) | - | 3 | 196 | G3 (= 440 ÷ 214/12 ≈ 195.998 Hz) | - | 4 | 147 | D3 (= 440 ÷ 219/12 ≈ 146.832 Hz) | - | 5 | 110 | A2 (= 440 ÷ 224/12 = 110 Hz) | - | 6 | 82.4 | E2 (= 440 ÷ 229/12 ≈ 82.407 Hz) |