Standing wave calculator
Choose a string, an open pipe or a pipe closed at one end. For a string, enter its tension and mass per metre; for a pipe, the speed of sound. Add the length and the harmonic number to get the frequency, the wavelength, the fundamental and, for a string, the number of nodes and antinodes.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Pick a string, an open pipe or a closed pipe.
- For a string, enter the tension in newtons and the linear density in g/m or kg/m. For a pipe, enter the speed of sound.
- Enter the vibrating length and the harmonic number.
- String wave speed: v = √(T ÷ μ).
- String or open pipe: fₙ = n v ÷ (2L), λₙ = 2L ÷ n, any n = 1, 2, 3 …
- Closed pipe: fₙ = n v ÷ (4L), λₙ = 4L ÷ n, n = 1, 3, 5 …
- All harmonics are whole multiples of the fundamental f₁.
Worked example
A string with 100 N of tension and 5 g/m, 0.65 m long: v = √(100 ÷ 0.005) = 141.42 m/s and f₁ = 141.42 ÷ (2 × 0.65) = 108.79 Hz, with λ₁ = 1.3 m.
An open pipe 0.5 m long at 343 m/s in its second harmonic: f₂ = 2 × 343 ÷ 1 = 686 Hz, fundamental 343 Hz.
A closed pipe 0.25 m long at 340 m/s, third harmonic: f₃ = 3 × 340 ÷ 1 = 1,020 Hz, fundamental 340 Hz.
Tips
- Tighter is higher. Frequency grows with the square root of tension: four times the tension doubles the pitch.
- Heavier is lower. Bass strings are thicker or wound to raise μ.
- End correction. A real pipe sounds as if it were a little longer than it is, by roughly 0.6 times its radius at each open end; add that to L for better accuracy.
Frequently asked questions
How do you find the frequencies of a vibrating string?
The wave speed is v = √(T ÷ μ), from the tension T and the mass per unit length μ. A string fixed at both ends fits whole half-wavelengths, so λₙ = 2L ÷ n and fₙ = n × v ÷ (2L). The first harmonic, n = 1, is the fundamental.
What is the difference between an open and a closed pipe?
A pipe open at both ends behaves like a string, fₙ = n v ÷ (2L) for every n. A pipe closed at one end has a node at the closed end and an antinode at the open end, so it fits quarter-wavelengths, fₙ = n v ÷ (4L), with odd n only. A closed pipe sounds an octave lower than an open pipe of the same length.
What are nodes and antinodes?
Nodes are the points that do not move; antinodes are where the motion is largest. On a string in its n-th harmonic there are n + 1 nodes, including the two fixed ends, and n antinodes.
What speed of sound should I use?
343 m/s is air at 20 °C. The speed rises by about 0.6 m/s per degree, so a warm room makes wind instruments play slightly sharp. The speed of sound calculator gives it for any temperature.