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:

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

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:

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

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

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

The string is recalled to its natural position by a force ${\displaystyle F}$:

${\displaystyle F=\theta \,T}$

The force ${\displaystyle F}$ is also equal to the centripetal force

${\displaystyle F=m\,{\frac {v^{2}}{R}}}$

where ${\displaystyle v}$ is the speed of propagation of the wave in the string.

Let ${\displaystyle \mu }$ be the linear mass of the string. Then

${\displaystyle m=\mu \,L=\mu \,\theta \,R}$

and

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

Equating the two expressions for ${\displaystyle F}$ gives:

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

Solving for velocity v, we find

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

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

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

If the length of the string is ${\displaystyle L}$, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so ${\displaystyle L}$ is half of the wavelength of the fundamental harmonic. Hence:

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

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