Wavelength Calculator

wavelength = wave speed ÷ frequency

f

Frequency of the wave, in hertz by default — the field also takes kHz, MHz, GHz and THz. This is the number of cycles passing a fixed point each second, and it must be greater than zero

v

Speed the wave travels at, in metres per second by default — the field also takes km/h, mph, ft/s and c, the speed of light. It is set by the medium, not by the wave, so it is a property of the air, the water or the vacuum you are in

λ

Wavelength — the result, the distance from one peak to the next, in metres. Once a frequency gets high the page switches to scientific notation, because 0.0000006 m is harder to read than 5.996 × 10⁻⁷ m

Use this page whenever you know how often a wave repeats and how fast it moves, and what you actually need is a length: sizing a quarter-wave antenna, working out whether a room is big enough for a bass note to develop, finding the spacing for a microphone array or a speaker line, or checking a microwave oven's 2.45 GHz against the size of the holes in its door mesh. The one thing worth internalising is that the speed belongs to the medium and the frequency belongs to the source. A 440 Hz tone is 0.780 m in air but 3.4 m in water, because water carries sound at 1,482 m/s — the note does not change when it enters the water, the wavelength does. Three checks make the answers trustworthy. First, remember the speed of sound is roughly 343 m/s in air (331 m/s at freezing, 343 m/s at 20 °C), so a few hundred hertz is always around a metre. Second, remember that light in a vacuum is exactly 299,792,458 m/s and pick c from the unit list rather than typing it. Third, note that frequency and wavelength trade off: at a fixed speed, ten times the frequency is one tenth the wavelength, which is why the whole electromagnetic spectrum is one dial with radio at one end and gamma rays at the other.

1. Concert A in air: 440 Hz at 343 m/s

   1. Start from the formula: wavelength = wave speed ÷ frequency
   2. Put the numbers in: 343 ÷ 440
   3. Divide: 343 ÷ 440 = 0.7795454… m
   4. To three decimals, since that is what the page shows: 0.780 m
   5. Read it as a length: one full cycle of that note is 78 cm of air

   This is the default state of the page and the one case everyone can check against their own experience: a bass note is metres long and a treble note is centimetres long, and the crossover is somewhere in the middle of a piano. It also shows why the speed had to be an input rather than a constant folded into the page. The same 440 Hz in water is 1,482 ÷ 440 = 3.4 m, more than four times longer, because water carries sound faster; the note you hear is identical, and only the length changes. That is the whole distinction this page turns on: frequency from the source, speed from the medium, wavelength from the two of them together.

2. A 2.4 GHz WiFi channel, travelling at the speed of light

   1. Pick c from the wave speed unit list, so the 299,792,458 m/s is exact rather than typed
   2. Start from the formula: wavelength = wave speed ÷ frequency
   3. Put the numbers in: 299792458 ÷ 2400000000
   4. Divide: 299792458 ÷ 2400000000 = 0.1249135… m
   5. To three decimals: 0.125 m, or 12.5 cm

   The number worth carrying away is 12.5 cm, because the quarter-wave element that a 2.4 GHz antenna is actually made of is a quarter of it: about 3.1 cm, which is exactly why those antennas are the size they are and why the stub inside a phone can be printed on the board. It also shows what the GHz entry in the unit list is for. Typing the frequency into the MHz box instead means 2400, and a missing zero means 240 instead of 2400 — a wavelength ten times too long, printed with no error and no warning. This is the case where the page is most likely to be used with the wrong prefix, so it is the case the golden cases pin down.

3. Green light: 500 THz in a vacuum

   1. Pick c from the unit list again — light in a vacuum is the one case where the speed is a defined constant
   2. Divide: 299792458 ÷ 500000000000000 = 5.99584916 × 10⁻⁷ m
   3. The page prints it as 5.996 × 10⁻⁷ m, because 0.0000006 is unreadable at that size
   4. Convert to the unit people actually use for light: multiply by 10⁹ to get nanometres
   5. 5.996 × 10⁻⁷ m × 10⁹ = 599.6 nm, which is the green in the middle of the visible band

   This is the case that makes the page's number formatting a real decision rather than a style choice. A wave speed divided by a frequency this large gives 0.0000005996, and a plain decimal format with three places would show 0.001 or 0.000 — a wrong answer that looks like a right one. The page keeps ordinary decimals for anything between 10⁻³ and 10⁶ metres, so the everyday answers are unaffected, and switches to scientific notation outside that window. The nanometre conversion is left to the reader on purpose: nanometres are a display convention for light, not a unit the calculation needs, and the page would rather show one honest number in metres than guess which of nm, µm or Å you meant.

The page gives one number from two, and it assumes the wave is a simple periodic one travelling through a single uniform medium. Nothing here covers a wave crossing between media, where the frequency stays fixed and the wavelength changes — that is a different question, and the answer depends on the refractive index or the acoustic impedance of the boundary. Nothing covers dispersion either: in real media including air and glass, the speed itself varies with frequency, so the speed you enter is right for one frequency band and slightly wrong for another. Standing waves, interference and resonance are not modelled at all — the page knows nothing about the boundaries that create them, so an organ pipe's resonant length is not something it can tell you, only the wavelength of the note. Amplitude, energy and intensity are absent, because a wave's length says nothing about how much it carries: a 440 Hz whisper and a 440 Hz shout have exactly the same wavelength. Sound in air is entered as a fixed 343 m/s in the examples, which is correct at 20 °C and about 3.5 % off at freezing. Finally, the page treats the speed as independent of the direction of travel, so a moving source, a moving observer or a wind-blown medium — everything the Doppler effect lives on — is outside what it can express.

How do I calculate a wavelength from a frequency?

Divide the wave speed by the frequency: λ = v ÷ f. In air at 20 °C the speed of sound is 343 m/s, so 440 Hz gives 343 ÷ 440 = 0.780 m. The division is the whole calculation, and the only thing that catches people out is the units: a frequency in GHz has to be a frequency in Hz before you divide, so 2.4 GHz is 2,400,000,000 Hz rather than 2,400. The page does that conversion for you as soon as you pick GHz from the unit list next to the field, which is the reason the unit list goes that far up.

What is the wavelength formula?

λ = v ÷ f, where v is the speed the wave travels at and f is its frequency. It comes straight from the definition of speed: a wave covers one wavelength in the time it takes to complete one cycle, and the time for one cycle is 1 ÷ f, so the distance is v × (1 ÷ f). Rearranged, the same relation gives f = v ÷ λ for the frequency behind a known wavelength, which is how an antenna's resonant frequency is worked out from its length. The page computes the first form; the second is the same equation moved around.

Why does the same note have a different wavelength in water?

Because the frequency comes from the source and the speed comes from the medium. A 440 Hz tuning fork vibrates 440 times a second in air and 440 times a second in water — the frequency cannot change — but water carries sound at about 1,482 m/s against air's 343 m/s, so the wavelength stretches from 0.780 m to 3.4 m. Nothing about the note changes; only the distance between compressions does. It is also why the wave speed is a field on this page rather than a fixed constant: the page would otherwise be wrong by a factor of four the moment the medium changed.

What is the speed of sound I should use?

343 m/s for air at 20 °C, which is the default, and 331 m/s at 0 °C — roughly 0.6 m/s slower for every degree colder. The page's examples use 343 m/s because it is the standard reference, but the field is editable precisely because the number is a property of the medium rather than a constant: sound is 1,482 m/s in water, 5,960 m/s in steel, and faster still in a vacuum-free solid like diamond. For light in a vacuum, pick c from the unit list and the exact defined value 299,792,458 m/s is filled in for you.

Why does the answer switch to scientific notation?

Because a single decimal format cannot serve both ends of the range. Sound at 20 Hz is 17.15 m and visible light is 5.996 × 10⁻⁷ m — seven and a half orders of magnitude apart. Printed as ordinary decimals to three places, the light would come out as 0.000, an answer that is not merely imprecise but wrong-looking. The page keeps plain decimals for everything from 10⁻³ to 10⁶ metres, which covers every everyday answer, and switches outside that window. If you are working with light, the conversion people actually use is nanometres: multiply metres by 10⁹, so 5.996 × 10⁻⁷ m is 599.6 nm.

How is this different from the angular velocity calculator?

Both pages talk about speed and they mean different things by it. Here the speed is how fast the wave itself advances through a medium — a property of the air or the water, the same for every frequency — and the answer is a distance in space. On the angular velocity page the speed is how fast a point on a rotating body moves around its circle, v = ωr, which is set by the radius and is different for every point on the same object. A wave's speed does not depend on the amplitude, and a point's linear speed does not depend on the wave; the two pages share a vocabulary and nothing else.

• Speed of Sound, Frequency, and Wavelength (College Physics 2e, §17.2) — the relation v = fλ, why frequency is set by the source while speed is set by the medium, and the speed of sound in air — OpenStax
• Definitions of SI Base Units — the metre is defined by fixing the speed of light in vacuum at exactly 299,792,458 m/s, which is why the c unit option on this page is a definition rather than a measurement — National Institute of Standards and Technology
• Wavelength — the relation between wavelength, frequency and propagation speed, and the standard band boundaries of the electromagnetic spectrum — Wikipedia
• Speed of sound — measured values in air across temperature and humidity, and in water, which is why the speed is an input on this page rather than a constant — Wikipedia