Vibrating String

A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note. Vibrating strings are the basis of any string instrument like guitar or piano.

Speed of propagation of the wave

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 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 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 equate 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. Hence: f = \frac{v}{2L} = \frac{\sqrt{\frac{T}{\mu}}}{2L} where T is the tension, \mu is the linear mass, and L is the length of the vibrating part of the string. Therefore:
  • the shorter the string, the higher the note
  • the higher the tension, the higher the note
  • the heavier the string, the lower the note

See also

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