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String Vibration Calculator

Find a string's wave speed and frequency.

How it is calculated

The tool first finds the speed of a wave along the string, v = √(T ÷ μ), then the fundamental frequency of a string fixed at both ends, f = v ÷ (2 × L). Here T is the tension in newtons, μ is the linear density (mass per unit length) in kg/m and L is the vibrating length in metres. Results are in m/s and Hz. Linear density and length must be above zero and tension cannot be negative. A tension of zero gives zero speed and zero frequency. Only the lowest mode is calculated; overtones are whole-number multiples of f. Stiffness of real strings is ignored.

Worked example

A string is pulled with 100 N, has a linear density of 0.001 kg/m (1 g per metre) and a length of 0.65 m. Divide tension by density: 100 ÷ 0.001 = 100,000. The square root gives a wave speed of 316.23 m/s. The wavelength of the fundamental is twice the length, 1.3 m, so f = 316.23 ÷ 1.3 = 243.25 Hz. Quadrupling the tension would double the frequency.

Try this example →

Common mistakes

  • Entering the string's total mass in kg instead of mass per metre. Divide the mass by the length first.
  • Entering the tension in kilograms-force or pounds. The field expects newtons (1 kgf is about 9.81 N).
  • Entering length in centimetres or the full string length when only part vibrates. Use the sounding length between the fixed points, in metres.
  • Expecting frequency to rise in proportion to tension. It rises with the square root, so doubling the tension raises pitch by about 41%.
  • Taking the result as the sound's wavelength. The tool gives the wave speed along the string, not the speed of sound in air.

About this calculator

This calculator applies two results from the physics of a stretched string: the wave speed v = √(T/μ), where T is the tension and μ is the linear mass density (mass per unit length), and the fundamental frequency f = v/2L for a string of length L fixed at both ends — the lowest standing-wave mode, whose wavelength is exactly twice the string's length. Raising tension or shortening the string raises pitch, while a heavier (thicker or denser) string lowers it for the same tension.

Instrument builders and luthiers use these relationships to choose string gauges and tunings for guitars, pianos, and violins, since a given pitch can be reached by trading off tension against mass per unit length. Physics students studying standing waves and resonance, and audio engineers working with string instrument acoustics, use the same equations to predict how a string will behave before it's ever put under load.

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