What is the formula for standing waves on a string?

For a string fixed at both ends (or an open pipe): λₙ = 2L/n and fₙ = nv/(2L), where n = 1, 2, 3, … is the harmonic number, L is the length, and v is the wave speed.

How do closed pipes differ from open pipes?

A closed pipe (closed at one end) only supports odd harmonics: λₙ = 4L/n and fₙ = nv/(4L) for n = 1, 3, 5, … Its fundamental wavelength is twice that of an open pipe of the same length, so its fundamental pitch is one octave lower.

What wave speed should I use for sound in air?

At 20 °C (68 °F) the speed of sound in air is approximately 343 m/s. It increases by about 0.6 m/s per degree Celsius. At 0 °C use 331 m/s; at 37 °C (body temperature) use 353 m/s.

What is the fundamental frequency?

The fundamental (first harmonic, n = 1) is the lowest resonant frequency for that medium. All higher harmonics (overtones) are integer multiples of the fundamental for strings and open pipes, and odd-integer multiples for closed pipes.

How is wave speed on a string calculated?

For a stretched string, v = √(T/μ), where T is tension in newtons and μ (mu) is linear mass density in kg/m. Use that result as the wave speed input in this calculator.

Why do musical instruments produce harmonic overtones?

Instruments create standing waves within fixed-length cavities or strings. The multiple resonant modes (harmonics) vibrate simultaneously; their relative amplitudes shape the timbre (tone color) that makes a violin sound different from a flute.

Standing Wave Calculator

Enter wave speed and length to instantly get harmonic wavelengths and frequencies for strings, open pipes, and closed pipes.

Wave Parameters

Medium Type

String / Open Pipe (both ends) All harmonics (n = 1, 2, 3 …)

Closed Pipe (one end closed) Odd harmonics only (n = 1, 3, 5 …)

Wave Speed (m/s)

Presets:

Length (m)

Show Harmonics Up To (n)

Harmonics Table

Enter values and click Calculate

Harmonic (n)	Wavelength	Frequency

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Summary

Standing waves form when a wave reflects back on itself and the two waves interfere constructively at certain resonant frequencies — the harmonics. This calculator covers three common boundary-condition cases: a string (or open pipe) fixed at both ends, an open pipe open at both ends, and a closed pipe open at one end. Enter the wave speed and the length of the medium to see all harmonics up to the 10th.

How it works

Select the medium type: string/open pipe (both ends fixed or both open) or closed pipe (one end closed, one open).
Enter the wave speed in m/s — use ~343 m/s for sound in air at 20 °C, or the known string wave speed.
Enter the length of the string or pipe in meters.
Click Calculate to generate harmonic number, wavelength, and frequency for each mode.
Use the mode table to identify the fundamental frequency (n = 1) and overtones.
Copy any individual result via the copy button next to that row.

Use cases

Determine the resonant frequencies of a guitar or violin string.
Design organ pipe lengths to hit target musical pitches.
Calculate harmonics in a closed-end brass instrument.
Verify physics homework on wave resonance problems.
Explore how shortening a string raises pitch in stringed instruments.
Analyze acoustic resonance in duct and HVAC systems.
Understand why only odd harmonics exist in a closed-end pipe.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu