Standing Wave Harmonics Calculator
A standing wave forms when a wave reflects back on itself inside a bounded medium and the two travelling waves reinforce at fixed points. Only certain wavelengths survive this process, because the ends of a string fixed at both ends, or the open ends of a pipe, impose strict boundary conditions. The frequencies that fit are called harmonics, and they fall at neat whole-number multiples of the lowest resonant frequency, known as the fundamental. This calculator finds the frequency of any harmonic you choose. You enter the harmonic number, the speed of the wave in the medium, and the length of the string or pipe, and it returns the resonant frequency of that harmonic in hertz. The relationship is exact and depends only on those three quantities, so the first harmonic sets the pitch and every higher harmonic stacks evenly above it. Musicians use this to understand why a string sounds a particular note, while engineers use it to predict resonance in pipes, ducts and structural members. Every figure here is computed deterministically from the standard physics formula shown below, with a worked example that reconciles exactly to the calculator defaults so you can follow each step and check the result for yourself.
The harmonic frequency equals the harmonic number times the wave speed divided by twice the length: f_n = n x v / (2 x L). For the first harmonic with a wave speed of 340 m/s and a length of 0.5 m, the resonant frequency is 340.00 Hz.
Standing wave harmonics formula
f_n = n x v / (2 x L)
n = harmonic number (1, 2, 3, ...)
v = wave speed in the medium (m/s)
L = length of the string or pipe (m)
Valid for a string fixed at both ends or a pipe open at both ends
The fundamental frequency is the case n equals 1, giving v divided by 2L. Each higher harmonic is simply that fundamental multiplied by the harmonic number, so the resonant frequencies climb in equal steps. The wavelength of harmonic n is 2L divided by n.
Worked example
Take the first harmonic, with a wave speed of 340 meters per second in air and a length of 0.5 meters.
- f_n = n x v / (2 x L) = 1 x 340 / (2 x 0.5)
- Denominator = 2 x 0.5 = 1
- f_n = 340 / 1 = 340.00 Hz
The harmonic frequency is 340.00 Hz. These are the calculator's default inputs, so the result above matches the widget exactly.
First few harmonics
| Harmonic n | Multiple of fundamental | Frequency (Hz) |
|---|---|---|
| 1 (fundamental) | 1x | 340.00 |
| 2 | 2x | 680.00 |
| 3 | 3x | 1,020.00 |
| 4 | 4x | 1,360.00 |
Frequencies shown for v = 340 m/s and L = 0.5 m. Each harmonic is a whole-number multiple of the 340 Hz fundamental.
Standing wave harmonics calculator: frequently asked questions
What is a standing wave harmonic?
A harmonic is one of the discrete resonant frequencies a string or air column can sustain. When a wave reflects back on itself and the two travelling waves reinforce, only certain wavelengths fit the boundary conditions. Each whole-number multiple of the fundamental frequency is a harmonic, numbered n equals 1, 2, 3 and so on.
What is the fundamental frequency?
The fundamental is the lowest resonant frequency, the harmonic where n equals 1. For a string fixed at both ends or an open pipe of length L, it equals the wave speed divided by twice the length, v divided by 2L. Every higher harmonic is a whole-number multiple of this fundamental.
Does this formula work for a pipe closed at one end?
No. The formula f_n equals n times v divided by 2L applies to a string fixed at both ends and to a pipe open at both ends, which support all integer harmonics. A pipe closed at one end supports only odd harmonics and uses f_n equals n times v divided by 4L, so use a dedicated tool for that case.
What wave speed should I use?
Use the speed of the wave in the medium. For sound in air at about 20 degrees Celsius the speed is roughly 340 meters per second, which is the default here. For a vibrating string the wave speed depends on tension and mass per unit length, so measure or compute it for your specific string before entering it.
How are the higher harmonics spaced?
They are evenly spaced. Because each harmonic is n times the fundamental, the frequencies climb in equal steps the size of the fundamental. With a 340 hertz fundamental the harmonics fall at 340, 680, 1,020 and 1,360 hertz, each one fundamental apart from the next.
Official sources
- Acoustics, waves and SI units: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.