Wave Speed Calculator

Background.

Take the turntable out of a microwave oven, lay a flat bar of chocolate inside and run the oven for a few seconds. The chocolate softens in spots rather than evenly, and the spots sit about 6.1 cm apart, because the oven fills with a standing wave whose hot spots are half a wavelength apart. The rating plate says 2,450 MHz. Multiply that frequency by the full wavelength, 12.2 cm, and you get 2.989 × 10⁸ m/s, within about 0.3% of the speed of light. A kitchen appliance and a ruler have just measured c.

That is all v = fλ says. In one period a wave's pattern moves forward by exactly one wavelength, so the speed of its crests is frequency times wavelength. Frequency is cycles per second, wavelength is metres per cycle, and the cycles cancel to leave metres per second. The same relation holds for a guitar string, a sound in a concert hall, a radio signal and swell rolling in from a distant storm, because it follows from what the two words mean rather than from anything about the material.

The material decides something else: which pairs of frequency and wavelength can exist in it. Air at 20 °C carries sound at about 343 m/s, so a 440 Hz tuning note has a wavelength near 0.78 m. Send the same note through water, where sound runs at roughly 1,480 m/s, and the wavelength stretches to about 3.36 m while the pitch stays the same. Feed in a pair you measured, and the speed that comes back tells you what kind of medium could have carried it, or warns you that one of the two measurements is wrong.

What you get is the phase speed, the rate at which a single crest of a single-frequency wave moves. When different frequencies travel at different speeds, as with deep-water waves or light in glass, a pulse moves at a different rate called the group velocity, and frequency times wavelength will not give you that.

What is wave speed calculator?

Wave speed in this sense is phase speed: the rate at which a point of fixed phase, such as a crest, a trough or a zero crossing, moves through space. A wave of frequency f repeats every period T = 1/f, and during each period the pattern advances one wavelength λ, so the crest covers λ/T = fλ metres each second. In a non-dispersive medium, such as air for audible sound or a uniform stretched string, every frequency shares the same phase speed, and that speed is a property of the medium. In a dispersive medium the phase speed changes with frequency, and a pulse, together with the energy it carries, travels at the group velocity dω/dk instead, which can be slower or faster than fλ.

How to use this calculator.

1. 01Enter the frequency in hertz. For a musical tone it is the pitch (concert A is 440 Hz); for a radio or microwave source it is on the label or the datasheet; for water waves it is 1 divided by the seconds between crests passing a fixed post.
2. 02Enter the wavelength in metres, measured crest to crest. On a standing wave, such as a vibrating string, a resonance tube or the oven hot spots, neighbouring nodes are half a wavelength apart, so double the gap before you enter it.
3. 03Read the wave speed in m/s and compare it with what the medium should carry: about 343 m/s for air at 20 °C, roughly 1,480 m/s for fresh water, 299,792,458 m/s for light in vacuum.
4. 04If the speed comes out a whole-number factor away from the value you expected, half or double or a third, recount the mode: a string sounding its second or third harmonic has nodes closer together than its fundamental.
5. 05For waves on water, time ten crests passing a fixed post and divide by ten to get the period, then enter 1 ÷ period as the frequency; judge the crest spacing against something of known length, such as a moored boat.

The formula.

v = f × λ, with v in metres per second, f in hertz and λ in metres. Follow one crest: it sits at some position now and one wavelength further on after one period T = 1/f, so its speed is λ/T. In angular terms, with angular frequency ω = 2πf and wavenumber k = 2π/λ, the same statement reads v = ω/k. Nothing else goes into the product, so the answer only means something if the frequency and wavelength belong to the same wave in the same medium.

The physics of the medium enters through a second equation that fixes v. For a stretched string, v is the square root of tension divided by mass per metre. For sound in a gas, v = √(γRT/M), with T the absolute temperature and M the molar mass, which is why air carries sound at about 331 m/s at 0 °C and 343 m/s at 20 °C, roughly 0.6 m/s more per degree. In a liquid, v = √(K/ρ), bulk modulus over density; in a solid the stiffness term is an elastic modulus that depends on the kind of wave and the shape of the piece, but the speed is still stiffness over density under a square root. For light, v = c/n, where n is the refractive index: about 2.25 × 10⁸ m/s in water and 2.0 × 10⁸ m/s in ordinary glass. Water waves in shallow water, where the depth d is less than about a twentieth of the wavelength, travel at √(gd); in deep water the phase speed depends on the wavelength itself, √(gλ/2π), which is exactly what makes a wave dispersive.

Motion is the other thing the product does not sort out for you. When a siren drives towards you, the wavelength ahead of it is squeezed and you hear a higher pitch, but the higher frequency times the squeezed wavelength is still the speed of sound in still air. Wind is different: it carries the whole pattern, so a frequency and wavelength measured from the ground multiply to the speed of sound plus the wind component along the path.

Rearranged, the identity gives λ = v/f when you know the medium and the frequency, and f = v/λ when you know the medium and the wavelength. On paper the speed is proportional to both inputs; in a real non-dispersive medium the speed stays fixed, so doubling the frequency halves the wavelength instead.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 80000-3:2019, Quantities and units — Space and time (BIBLIOGRAPHIC/PAYWALL).
2. [2]OpenStax, University Physics Volume 1, section 16.1, Traveling Waves.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]NIST Special Publication 330, The International System of Units.