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Standing Wave Calculator

Find standing wave harmonics on a string.

How it is calculated

For a string fixed at both ends, the tool uses fn = n × v ÷ (2 × L) and λn = 2 × L ÷ n. Here v is the wave speed along the string in m/s, L is the string length in metres and n is the harmonic number: 1 is the fundamental, 2 the first overtone and so on. The harmonic is rounded to the nearest whole number, so 2.6 is read as 3. Speed and length must be above zero and n must be at least 1. Frequency is in Hz with two decimals, wavelength in metres with four. The tool takes the wave speed as an input; it does not calculate it from tension and mass.

Worked example

A string 0.65 m long carries waves at 320 m/s. We want the third harmonic (n = 3). Twice the length is 2 × 0.65 = 1.3 m. Frequency: 3 × 320 ÷ 1.3 = 738.46 Hz. Wavelength: 1.3 ÷ 3 = 0.4333 m. The fundamental of the same string would be 320 ÷ 1.3 ≈ 246.15 Hz, so the third harmonic is exactly three times that. It fits three half-wavelengths into the string.

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Common mistakes

  • Entering the speed of sound in air, about 343 m/s, instead of the wave speed on the string. They are unrelated.
  • Using this for a pipe closed at one end. Such pipes support only odd harmonics with a different formula.
  • Treating n as the number of nodes. For a string fixed at both ends, the n-th harmonic has n − 1 nodes between the ends.
  • Entering length in centimetres, which makes frequency a hundred times too high.
  • Typing 0 for the fundamental. The fundamental is n = 1; n = 0 gives no result.

About this calculator

A standing wave forms on a string fixed at both ends when waves reflecting back and forth interfere to create stationary nodes and antinodes; only wavelengths that fit an integer number of half-wavelengths between the fixed ends survive, giving the harmonic series fₙ = n·v/(2L), where v is the wave speed on the string, L is its length, and n = 1, 2, 3… labels the fundamental and its overtones.

This relationship is the physical basis of every stringed instrument: guitar makers and piano technicians use it to relate string length, tension and mass per unit length (which set v) to pitch, and it's a standard demonstration and lab exercise in introductory physics courses on wave mechanics and acoustics.

Musicians tuning instruments, luthiers designing new ones, acoustic engineers analyzing resonance, and physics students verifying harmonic series experimentally all use this formula to predict or explain the frequencies a vibrating string can sustain.

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Comments (1)

  • Editorial team

    The speed here is the wave speed on the string, not the speed of sound in air. If you know the tension T and the mass per unit length, it is the square root of T divided by that mass per length. A string of 0.65 m with a wave speed of 200 m/s has a fundamental of 153.85 Hz. Enter harmonic 2, 3 and so on for overtones. Are you modelling a string or a pipe?

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