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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 θ 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 μ 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}

[edit] 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 λ divided by the period τ, 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, μ is the linear mass, and L is the length of the vibrating part of the string.