Wavelength Calculator

The one relationship behind every wavelength answer

Wavelength is the distance from one crest to the next. Frequency counts how many crests pass a point each second. Speed says how far a crest travels in that second. Those three describe the same wave, so any two of them fix the third.

In symbols, wavelength equals speed divided by frequency. The units show why the division works: metres per second divided by cycles per second leaves metres per cycle. A tone of 440 Hz in air at 20 C travels at 343.42 m/s, so 343.42 ÷ 440 = 0.7805 m. Ask the question the other way and a 25 cm wave in the same air has a frequency of 343.42 ÷ 0.25 = 1373.7 Hz.

Period is the fourth quantity that gets mixed into this conversation. One cycle takes 1 ÷ frequency seconds, so a 440 Hz tone repeats every 0.00227 s, or 2.27 ms. Wavelength is the cycle measured in space and period is the cycle measured in time. The period calculator handles the time side of the same wave.

Nothing in the formula is specific to sound or to light. The only thing that changes between them is the speed in the denominator, which is why the next section spends its time on that number.

Which speed belongs in the denominator

Light in a vacuum moves at 299,792,458 m/s. That value is not a measurement average: the metre is defined from it, so it is exact by definition, and NIST lists it among the CODATA fundamental constants with no uncertainty. Using 300,000,000 instead is fine for a mental estimate and gives a 0.07 percent error.

Light in air is slower by a factor of about 1.0003, which puts it near 299,705,000 m/s. That difference is invisible in everyday work. For radio it changes a 12.49 cm wavelength in the fifth significant figure, which matters to a phase-array designer and to nobody trimming a wire antenna.

Sound is a different story because its speed depends on the material and, in a gas, on temperature. A linear fit that is accurate to about 0.3 percent between 0 C and 30 C is 331.3 + 0.606 × T, with T in degrees Celsius. That gives 331.3 m/s at freezing, 343.4 m/s at room temperature and 349.5 m/s at 30 C. Pressure barely enters the formula: at audible frequencies the speed changes by less than 0.1 percent between sea level and a mountain pass, because density and stiffness rise together. Humidity does matter a little, and wet air carries sound about 0.3 percent faster than dry air at the same reading.

Liquids and solids are faster still, and their speeds are handbook values rather than definitions. Sound crosses fresh water at roughly 1480 m/s and steel at roughly 5000 m/s, which is why a train is heard through the rails before it is heard through the air.

The widget on this page uses these numbers, and the last option lets you type a measured speed when the material is not on the list.

Running the formula backwards to get frequency

Frequency equals speed divided by wavelength. Nothing new happens, but this direction is the one that shows how compact the numbers get at radio frequencies.

An FM station at 100 MHz has a wavelength of 299,792,458 ÷ 100,000,000 = 2.998 m. Move to the 2.4 GHz Wi-Fi band and the answer falls to 12.49 cm, close enough to a hand span that a quarter-wave antenna is about 3 cm of wire. The 5 GHz band gives 6.00 cm, and the 60 GHz short range band gives 5.0 mm, which is why those links need a clear line of sight and lose signal when a hand covers the module.

The classic mistake here is a unit slip, and it usually costs a factor of a thousand. 2.4 GHz is 2,400,000,000 Hz, not 2.4 million. 100 MHz is 100,000,000 Hz. Enter 2.4 with the unit box set to Hz and the answer comes back as 1.25 hundred million metres, which is a strong hint that something is wrong before you put it in a report.

Radio wavelengths are quoted in metres, microwave and Wi-Fi in centimetres or millimetres, and optics in nanometres. The page prints all three so a number can be lifted straight into a datasheet without a conversion step.

Photon energy from the same wavelength

For light, wavelength also fixes the energy carried by each photon, and the bridge is the Planck constant. A photon's energy is the Planck constant times its frequency, so substituting the wave formula gives energy equals the Planck constant times the speed of light, divided by wavelength. The Planck constant is 6.62607015e-34 joule seconds, exact since the 2019 revision of the SI, and the product of the two constants is 1.9864459e-25 joule metres.

Optics people skip the constants and use a shortcut: energy in electron volts is 1239.84 divided by wavelength in nanometres. A 500 nm green photon carries 1239.84 ÷ 500 = 2.48 eV. A 400 nm violet photon carries 3.10 eV, a 940 nm remote control pulse carries 1.32 eV, and a 1550 nm fibre signal carries 0.800 eV. The Joules-per-photon column in the widget gives 3.97e-19 J for the green photon, which is the same number written without the electron volt conversion.

This is where the everyday intuition about short wavelengths comes from. Ultraviolet photons carry enough energy per photon to break chemical bonds, which is why they fade paint and damage skin, while infrared photons carry too little to do either and mostly warm whatever absorbs them. The NASA Goddard spectrum chart lists the same wavelength, frequency and energy limits for each band.

Sound has no equivalent. A sound wave is a pressure oscillation in a material, not a stream of particles, so the energy in it depends on amplitude rather than on wavelength. The widget says so instead of printing a meaningless number.

Cutting antennas and pipes to length

Antenna builders use wavelength as a ruler. A half-wave dipole is half a wavelength long, so at 100 MHz in free space it is 0.5 × 2.998 = 1.499 m. Cut exactly that and it usually resonates a little low, because the wire has thickness and the ends interact with their surroundings. The usual correction is to shorten to about 95 percent, which gives 1.424 m. A quarter-wave vertical for the same frequency is 0.25 × 2.998 = 0.7495 m in free space, trimmed to about 0.712 m.

Coaxial cable adds a second correction, the velocity factor. A signal in RG-58 with a velocity factor of 0.66 travels at 66 percent of the free-space speed, so a 2.4 GHz wave inside it has a wavelength of 12.49 × 0.66 = 8.24 cm. Phase-matched cable runs and baluns are cut against that figure, not the free-space one.

Pipes follow the same geometry in reverse. An open pipe open at both ends resonates at its fundamental when its length is half a wavelength, so a 440 Hz organ pipe of that type is 0.390 m long. Stopped at one end the fundamental sits at a quarter wavelength, 0.195 m. That is why a stopped pipe sounds an octave lower than an open pipe of the same length.

The useful habit is to keep the unit and the precision honest. At 100 MHz, one centimetre of error is 0.33 percent of the wavelength, and on a tuned antenna that is enough to see on a standing wave meter.

Wavelengths across the whole spectrum

The formula is unchanged from a 17 m sound wave to a gamma ray, and the range it covers is hard to picture until the numbers sit side by side. The lowest note most people hear, 20 Hz, has a wavelength of 17.2 m in room temperature air. The highest, 20 kHz, is 1.72 cm. Visible light spans 400 to 700 nm, which is more than 24,000 times shorter than the 1.72 cm wave at the top of hearing.

Read the energy column from top to bottom and the reason for the ordering becomes clear: a factor of a billion in wavelength is a factor of a billion in photon energy. A radio photon at 1 m carries 1.24 microelectronvolts, roughly the thermal energy of a single molecule at room temperature, while a gamma photon at one picometre carries 1.24 megaelectronvolts, enough to ionise atoms and damage DNA.

Where the simple formula stops working

Wavelength equals speed divided by frequency assumes the speed is a constant for the wave you are following. Three situations break that assumption and are worth knowing before a number gets used in a design.

The first is dispersion. In glass or water the refractive index depends on wavelength, so different colours travel at different speeds. The frequency stays fixed when light enters the material, and only the wavelength changes: 500 nm light in water with an index of 1.333 becomes 500 ÷ 1.333 = 375 nm while still oscillating at 599.6 THz. That shift is why a lens focuses blue light at a slightly different point than red, and why prism spectrometers work at all.

The second is a waveguide. Inside a metal pipe or a fibre, the guide wavelength is longer than the free-space value, and near the cutoff frequency it stretches without limit. A 2.4 GHz design that was cut for 12.49 cm in air shows a different electrical length once it is inside a guide.

The third is a moving source or a moving listener. The Doppler effect shifts the received frequency, so the wavelength measured by the listener is not the wavelength emitted. An ambulance siren at 440 Hz does not stay at 440 Hz as it passes, and the shift is proportional to the ratio of speeds, which is why police radar can read a car's speed from a frequency difference.

None of this invalidates the calculation. It tells you which speed belongs in it, and that is the part the formula cannot decide for you.

Unit and rounding traps worth a second look

Most wrong wavelength answers are unit errors rather than arithmetic errors, and they cluster in four places. Hertz against kilohertz, megahertz and gigahertz is the first: the prefixes are factors of 1000, so 2.4 GHz is 2.4 billion hertz. Centimetres against metres is the second, and it bites when the frequency is given in MHz and the answer is expected in cm. Nanometres against micrometres is the third, a factor of 1000 that turns a green photon into an ultraviolet one. Mixing wavelength with period is the fourth, since both can be described as the length of one cycle.

Rounding has a rule that saves trouble. Keep more digits than you need until the last step, then round the answer to three or four significant figures. Rounding 343.42 to 343 before dividing changes 0.7805 m to 0.7795 m, which is fine on a blackboard and wrong on a calibration sheet.

One sanity check catches most errors in a second. A 440 Hz tone should come out near 0.78 m in air, since the speed of sound is close to 340 m/s and 340 ÷ 440 is a little under 0.8. If your answer is 0.78 cm or 78 m, a prefix or a decimal point has moved.