User:Edblah

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Linear density, linear mass density or linear mass is a measure of mass per unit of length, and it is a characteristic of strings or other one-dimensional objects. The SI unit of linear density is the kilogram per metre (kg/m). It is defined as:

$\mu ={\frac {\partial m}{\partial x}}$

where μ is the linear density of the object, m is the mass, and x is a coordinate along the (one dimensional) object.

For the common case of a homogenous substance of length L and total mass m, this simplifies to:

$\mu ={\frac {m}{L}}$

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

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

The string is recalled to its natural position by a force $F$:

$F=\theta \,T$

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

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

and

$F=\mu \,\theta \,R\,{\frac {v^{2}}{R}}=\mu \,\theta \,v^{2}$

Equating the two expressions for $F$ gives:

$\theta \,T=\mu \,\theta \,v^{2}$

Solving for velocity v, we find

$v={\sqrt {T \over \mu }}$

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 $\lambda$ divided by the period $\tau$, or multiplied by the frequency $f$ :

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

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. Hence:

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

where $T$ is the tension, $\mu$ is the linear mass, and $L$ is the length of the vibrating part of the string.