Wavelength and Frequency Calculator (v = f λ)
Convert frequency to wavelength and back with v = f λ. Enter any two of wavelength, frequency and wave speed, with the exact speed of light as default.
Wavelength and Frequency Calculator
Wave speed, frequency and wavelength are tied together by v = f x λ. Enter two of them and this page works out the third.
Result
Wavelength is wave speed divided by frequency: λ = v ÷ f.
The wavelength in other units
Documentation
Wavelength and Frequency Calculator
The wavelength of a wave is the distance from one crest to the next. Its frequency is how many cycles pass a fixed point each second. The two are tied together by how fast the wave travels, so this calculator takes any two of wavelength, frequency and wave speed and works out the third.
What the calculator does
Choose which of the three quantities to calculate. The other two become the input fields. The page converts both entries into base units, applies the wave relation once, and converts the answer back into the unit chosen for it. The same answer is also listed in every other unit of its kind, and a copy button puts all three quantities on the clipboard.
| Quantity | Units offered |
|---|---|
| Wavelength | nanometres, micrometres, millimetres, centimetres, metres, kilometres |
| Frequency | hertz, kilohertz, megahertz, gigahertz, terahertz |
| Wave speed | metres per second, kilometres per hour, miles per hour, feet per second, knots |
Every factor comes from the site's shared unit registry, so the metre used here is the metre used on every other page. Results are shown to ten significant figures. A value below 0.0001, or 1,000,000,000 and above, is shown in scientific notation.
The wavelength and frequency formula
A wave travels exactly one wavelength in one cycle. In one second it completes cycles, so it covers wavelengths:
where is the wave speed in metres per second, is the frequency in hertz (cycles per second) and is the wavelength in metres.
The same relation turned around gives the other two answers:
Each arrangement is a single multiplication or a single division, so nothing here builds up the drift that a long chain of sums does.
For light travelling through empty space, the wave speed is the speed of light:
where is the speed of light in vacuum. The wave speed field starts at that value and can be changed to any other. Sound in dry air near 20 °C travels about 343 m/s, and light inside ordinary glass moves at roughly two thirds of its vacuum speed.
Which constants are exact
The numbers behind these conversions are definitions agreed by standards bodies, not measurements that a better experiment could revise.
| Constant | Value | Exact? | Source |
|---|---|---|---|
| Speed of light in vacuum | 299,792,458 m/s | exact by definition | SI Brochure, 9th edition (2019); the metre is defined from the second and this number |
| Hertz | 1 Hz is one cycle per second | exact by definition | SI Brochure, 9th edition |
| Prefixes kilo, mega, giga, tera | 10³, 10⁶, 10⁹, 10¹² | exact by definition | SI Brochure, 9th edition |
| Nanometre | 0.000000001 m | exact by definition | SI Brochure, 9th edition |
| Foot | 0.3048 m | exact by definition | International Yard and Pound Agreement (1959) |
| Mile per hour | 0.44704 m/s | exact by definition | International Yard and Pound Agreement (1959) |
| Knot | 1,852 m in 3,600 s | exact by definition | International Hydrographic Conference, Monaco (1929) |
| Kilometre per hour | 1,000 m in 3,600 s | exact by definition | SI Brochure, 9th edition |
The speed of sound is not a constant of this kind. It changes with the temperature and the gas, so the page treats it as a number the reader types rather than one it defines.
Worked example: an FM radio wavelength
A station broadcasts at 100 MHz. The wave speed is the speed of light, because radio waves in air travel very close to their vacuum speed.
100 MHz is 100,000,000 Hz. Dividing 299,792,458 by 100,000,000 gives 2.99792458 m. That division only moves the decimal point, so the answer is exact, with nothing rounded away. The page shows 2.99792458 m, and the same wavelength as 299.792458 cm, 2,997.92458 mm and 2.99792458E9 nm.
Running it backwards checks the arrangement. Multiplying 2.99792458 m by 100 MHz returns 299,792,458 m/s, the speed that went in.
More examples
| Wave | Given | Answer |
|---|---|---|
| Wi-Fi | 2.4 GHz at the speed of light | 0.1249135242 m, or 12.49135242 cm |
| Radar | 1 GHz at the speed of light | 0.299792458 m, or 29.9792458 cm |
| Red light | 700 nm at the speed of light | 428.27494 THz |
| Concert pitch in air | 440 Hz at 343 m/s | 0.7795454545 m, close to 78 cm |
| A wave of unknown speed | 0.5 m and 440 Hz | 220 m/s |
Radio wavelengths shrink as the frequency climbs, which is why a 100 MHz signal needs an aerial about a metre long while a 2.4 GHz one fits inside a phone.
How the arithmetic is arranged
Computers store these numbers in binary, and most decimal fractions have no exact binary form. Ten significant figures is wide enough to show that dust, so the page is arranged to keep it out of the answer where it can.
Only the calculated quantity is converted. The two values that were typed are shown exactly as typed. Sending one out to a base unit and back multiplies and divides by the same factor, which can shift its last digit: 7.7 km/h returns from metres per second as 7.699999999999999.
Some inputs cannot land on an exact number by any arrangement. A wavelength of 700 nm is held as 0.0000007000000000000001 m rather than 0.0000007 m, so the frequency comes out as 428,274,939,999,999.94 Hz instead of the exact 428,274,940,000,000 Hz. At ten significant figures both print 428.27494 THz, which is the right answer. Entering 0.1 GHz plus 0.2 GHz gives 300,000,000.00000006 Hz, and the page reports the value it was given rather than tidying it into the round number nobody typed.
Ten figures is also the narrowest width that prints the speed of light without rounding one of its nine digits away.
What the calculator refuses
An input box passes through whatever is typed or pasted, and a shared link can carry any value, so the checks live in the calculation code rather than in the form.
| Situation | Why there is no answer |
|---|---|
| Only one value given | Two of the three quantities are needed to fix the third. |
| A value of zero or less | A wave with no length, no frequency or no speed is not a slow wave, it is not a wave, and dividing by zero gives no answer. |
| A value too small to survive its conversion | A wavelength of 1e-320 nm becomes zero metres, and the division after it would report an infinite frequency for a number the reader entered as positive. |
| An answer outside the number range | Both entries can be in range while the result is not, such as a metre-per-second speed divided by a wavelength near the smallest number a computer can hold. |
| A unit the page does not have | A hand-edited or stale link can name a unit belonging to a different quantity. |
What the page does not model
It applies one speed to the whole wave. Real materials are dispersive: the speed of light in glass, and the speed of sound in some materials, changes slightly with frequency, which is how a prism splits white light into colours. A single figure cannot show that, so the page reports the wavelength that goes with the one speed it was given.
The relation also does not distinguish phase velocity from group velocity, which differ inside a dispersive medium. For a wave in vacuum, or for sound in air over ordinary conditions, the two are the same and the answer stands.
Frequently asked questions
How do you convert frequency to wavelength? Divide the wave speed by the frequency. For a radio wave, the speed is the speed of light, 299,792,458 m/s. At 100 MHz the wavelength is 2.99792458 m.
How do you convert wavelength to frequency? Divide the wave speed by the wavelength. Light of 700 nm gives a frequency of 428.27494 THz.
What is the wavelength of a 2.4 GHz Wi-Fi signal? About 12.5 cm. Dividing 299,792,458 m/s by 2,400,000,000 Hz gives 0.1249135242 m, which the page shows as 12.49135242 cm.
Is the speed of light exact? Yes. Since 1983 the metre has been defined from the second and from the figure 299,792,458 m/s, and the 2019 SI kept that definition. The number is fixed by agreement, so no future measurement can change it.
Does this work for sound as well as light? Yes. The relation holds for any travelling wave. Replace the wave speed with the speed of sound in the material, near 343 m/s for dry air at 20 °C, and the page works out the sound wavelength. That speed rises with temperature, so it is a value to enter rather than a constant.
Why does 100 MHz give exactly 2.99792458 m? Because dividing by 100 MHz only shifts the decimal point of an exact defined value. The speed of light is 299,792,458 m/s exactly, so the wavelength is 2.99792458 m exactly, with no rounding at any step.