Editing String vibration (section)

== 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]] <math>\lambda</math> divided by the [[Wave period|period]] <math>\tau</math>, or multiplied by the [[frequency]] <span style="white-space:nowrap;"><math>f</math>:</span>

:<math>v = \frac{\lambda}{\tau} = \lambda f.</math>

If the length of the string is <math>L</math>, the [[Fundamental frequency|fundamental harmonic]] is the one produced by the vibration whose [[node (physics)|node]]s are the two ends of the string, so <math>L</math> is half of the wavelength of the fundamental harmonic.  Hence one obtains [[Mersenne's laws]]:

:<math>f = \frac{v}{2L} = { 1 \over 2L } \sqrt{T \over \mu}</math>

where <math>T</math> is the [[Tension (mechanics)|tension]] (in Newtons), <math>\mu</math> is the [[linear density]] (that is, the [[mass]] per unit length), and <math>L</math> 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

Moreover, if we take the nth harmonic as having a wavelength given by <math>\lambda_n = 2L/n</math>, then we easily get an expression for the frequency of the nth harmonic:

:<math>f_n = \frac{nv}{2L}</math>

And for a string under a tension T with linear density <math>\mu</math>, then

:<math>f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}</math>