String vibration

Wave in a string

Speed of propagation of the wave

File:VibratingString.gif

Let $L$ be the length of the string, $m$ its mass and $T$ the tension.

When the string is touched it bends as an arc of circle. Let $R$ the the radius and $\theta$ the angle under the arc. Then $L=\theta \,R$.

The string is recalled to its natural position by a force $F$ which is equal to $\theta \,T$.

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 $m=\mu \,L=\mu \,\theta \,R$.

If we equal the two expressions of $F$ we have: $\theta \,T=\mu \,\theta \,R\,{\frac {v^{2}}{R}}$

So $v^{2}={\frac {T}{\mu }}$

Frequency of the wave

Once we know the speed of propagation, it is almost immediate to find the frequency of the sound produced by the string. In fact we know that the speed of propagation of a wave is equal to the wavelength $\lambda$ divided by the period $T$, or multiplied by the frequency $f$ :

$v={\frac {\lambda }{T}}=\lambda f$

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.

So we have: $f={\frac {v}{2L}}={\frac {\sqrt {\frac {T}{\mu }}}{2L}}$