Standing Wave Node Calculator
A standing wave forms when two identical waves travel in opposite directions and interfere, creating fixed points of zero motion called nodes and fixed points of maximum motion called antinodes. On a string fixed at both ends, only certain wavelengths fit, giving a discrete set of harmonics. This calculator takes the string or air-column length, the harmonic number and the wave speed, then returns the wavelength, harmonic frequency, the number of nodes, and the spacing between adjacent nodes. It is a core tool for acoustics, musical instrument design and physics coursework.
Standing wave formulas
Wavelength: lambda = 2L / n
Frequency: f = n * v / (2L)
Number of nodes = n + 1
Node spacing = lambda / 2 = L / n
These relations apply to a string fixed at both ends or an air column open at both ends. The harmonic number n counts the antinodes; the fundamental is n = 1. Node spacing equals half a wavelength.
Standing wave context
- The two fixed ends of a string are always nodes, so the nth harmonic has n + 1 nodes total.
- Harmonic frequencies are integer multiples of the fundamental: f, 2f, 3f and so on.
- Wave speed on a string equals the square root of tension divided by linear mass density.
- The speed of sound in dry air near 20 degrees Celsius is approximately 343 metres per second.
- For a tube open at one end and closed at the other, only odd harmonics exist; this calculator covers the both-ends-fixed case.
Standing waves: frequently asked questions
What is a node in a standing wave?
A node is a point on a standing wave where the displacement is always zero because two travelling waves moving in opposite directions cancel there. For a string fixed at both ends, the two ends are always nodes. The points of maximum displacement halfway between nodes are called antinodes.
How many nodes does the nth harmonic have?
For a string fixed at both ends vibrating in its nth harmonic, there are n antinodes and n + 1 nodes (including the two fixed ends). The wavelength of the nth harmonic is 2L/n, where L is the string length, and adjacent nodes are separated by half a wavelength, L/n.
What is the formula for harmonic frequency?
The frequency of the nth harmonic on a string fixed at both ends is f = n * v / (2L), where v is the wave speed on the string and L is the length. The fundamental (n = 1) has frequency v / (2L), and higher harmonics are integer multiples of it.
What wave speed should I enter?
Wave speed depends on the medium. On a string it equals the square root of tension divided by linear mass density. For sound in air at 20 degrees Celsius it is about 343 metres per second. Enter the speed for your specific system as a user-editable input since it is not a universal constant.
How is node spacing related to wavelength?
Adjacent nodes in any standing wave are separated by exactly half a wavelength. So node spacing equals lambda divided by 2, which for the nth harmonic on a fixed string equals L/n. Antinodes also sit half a wavelength apart, offset from the nodes by a quarter wavelength.
Official sources
- NASA Glenn Research Center: Sound and waves.
- U.S. National Institute of Standards and Technology: Physical Measurement Laboratory.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.