String vibration

A vibration in a string is a wave. Initial disturbance (such as plucking or striking) causes a vibrating string to produce a sound with constant frequency, i.e., constant pitch. The nature of this frequency selection process occurs for a stretched string with a finite length, which means that only particular frequencies can survive on this string. If the length, tension, and linear density (e.g., the thickness or material choices) of the string are correctly specified, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. For a homogeneous string, the motion is given by the wave equation.

Wave The velocity of propagation of a wave in a string ( v {\displaystyle v} ) is proportional to the square root of the force of tension of the string ( T {\displaystyle T} ) and inversely proportional to the square root of the linear density ( μ {\displaystyle \mu } ) of the string:

v = T μ . {\displaystyle v={\sqrt {T \over \mu }}.}

This relationship was discovered by Vincenzo Galilei in the late 1500s.

Derivation

Source: Let Δ x {\displaystyle \Delta x} be the length of a piece of string, m {\displaystyle m} its mass, and μ {\displaystyle \mu } its linear density. If angles α {\displaystyle \alpha } and β {\displaystyle \beta } are small, then the horizontal components of tension on either side can both be approximated by a constant T {\displaystyle T} , for which the net horizontal force is zero. Accordingly, using the small angle approximation, the horizontal tensions acting on both sides of the string segment are given by

T 1 x = T 1 cos ⁡ ( α ) ≈ T . {\displaystyle T_{1x}=T_{1}\cos(\alpha )\approx T.}

T 2 x = T 2 cos ⁡ ( β ) ≈ T . {\displaystyle T_{2x}=T_{2}\cos(\beta )\approx T.}

From Newton's second law for the vertical component, the mass (which is the product of its linear density and length) of this piece times its acceleration, a {\displaystyle a} , will be equal to the net force on the piece:

Σ F y = T 1 y − T 2 y = − T 2 sin ⁡ ( β ) + T 1 sin ⁡ ( α ) = Δ m a ≈ μ Δ x ∂ 2 y ∂ t 2 . {\displaystyle \Sigma F_{y}=T_{1y}-T_{2y}=-T_{2}\sin(\beta )+T_{1}\sin(\alpha )=\Delta ma\approx \mu \Delta x{\frac {\partial ^{2}y}{\partial t^{2}}}.}

Dividing this expression by T {\displaystyle T} and substituting the first and second equations obtains (we can choose either the first or the second equation for T {\displaystyle T} , so we conveniently choose each one with the matching angle β {\displaystyle \beta } and α {\displaystyle \alpha } )

− T 2 sin ⁡ ( β ) T 2 cos ⁡ ( β ) + T 1 sin ⁡ ( α ) T 1 cos ⁡ ( α ) = − tan ⁡ ( β ) + tan ⁡ ( α ) = μ Δ x T ∂ 2 y ∂ t 2 . {\displaystyle -{\frac {T_{2}\sin(\beta )}{T_{2}\cos(\beta )}}+{\frac {T_{1}\sin(\alpha )}{T_{1}\cos(\alpha )}}=-\tan(\beta )+\tan(\alpha )={\frac {\mu \Delta x}{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}

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