About
Wave equation: c = λ × f. For light, c = 3×10⁸ m/s. Visible light: 400-700 nm wavelength, 430-770 THz frequency. Radio waves: 1 mm to 100 km. Photon energy E = hf where h = 6.626×10⁻³⁴ J·s.
Formula
Frequently asked questions
What is the formula linking wavelength and frequency?
The wave equation is c = lambda x f, where c is the wave speed, lambda is the wavelength, and f is the frequency. Rearranged, wavelength equals speed divided by frequency (lambda = c / f), and frequency equals speed divided by wavelength (f = c / lambda). For light in a vacuum c is about 3 x 10^8 metres per second, so wavelength and frequency are inversely related: as one rises, the other falls.
What speed should I use for sound versus light?
Use the wave's speed in its medium. Light and other electromagnetic waves travel at about 3 x 10^8 m/s in a vacuum. Sound in dry air at 20 degrees C travels at about 343 m/s, and sound in water at roughly 1,480 m/s. The wave equation is the same; only the speed changes, which is why a 1,000 Hz sound wave and a 1,000 Hz radio wave have wildly different wavelengths.
How do I convert frequency to wavelength?
Divide the wave speed by the frequency. For a 100 MHz FM radio signal, wavelength = (3 x 10^8 m/s) / (100 x 10^6 Hz) = 3 metres. Keep units consistent: speed in metres per second and frequency in hertz give wavelength in metres. To go the other way, divide speed by wavelength to get frequency.
What is the wavelength range of visible light?
Visible light spans roughly 400 to 700 nanometres, from violet at the short-wavelength end to red at the long end. In frequency terms that is about 430 to 770 terahertz. Shorter wavelengths (higher frequency) carry more energy per photon, which is why ultraviolet light beyond 400 nm can damage skin while longer infrared wavelengths are felt as heat.
How is photon energy related to frequency?
Photon energy is E = h x f, where h is Planck's constant, about 6.626 x 10^-34 joule-seconds. Because frequency and wavelength are linked by c = lambda x f, you can also write E = h x c / lambda. Higher frequency (shorter wavelength) means more energetic photons, which is the reason gamma rays and X-rays are ionising while radio waves are not.
About wavelength and frequency
Every wave, whether it is light, sound, a radio signal, or a ripple on water, carries two linked measurements: wavelength and frequency. Wavelength is the physical distance between two matching points on the wave, such as crest to crest, usually measured in metres or nanometres. Frequency is how many of those cycles pass a fixed point each second, measured in hertz. The two are tied together by the speed of the wave, and this calculator converts between them in either direction.
Because the product of wavelength and frequency always equals the wave's speed, the two are inversely proportional in any given medium. Double the frequency and the wavelength halves. That single relationship explains why a deep bass note has a long wavelength while a high treble note is short, and why blue light oscillates faster than red. The same equation runs from radio engineering to spectroscopy to musical acoustics.
How the conversion works
c = lambda x f (the wave equation) Wavelength: lambda = c / f Frequency: f = c / lambda Photon energy: E = h x f = h x c / lambda c (light, vacuum) = 3 x 10^8 m/s c (sound, dry air 20C) = 343 m/s h (Planck constant) = 6.626 x 10^-34 J-s
- c is the wave speed, which depends on the medium. Pick the wave type to set it.
- lambda is wavelength, the distance per cycle.
- f is frequency, the cycles per second in hertz.
- E is the energy of a single photon, relevant for electromagnetic waves.
Worked example
What is the wavelength of a 100 MHz FM radio station, an electromagnetic wave travelling at the speed of light?
- Pick the equation: we want wavelength, so lambda = c / f.
- Insert the speed of light: c = 3 x 10^8 m/s.
- Insert the frequency: f = 100 MHz = 100 x 10^6 Hz = 1 x 10^8 Hz.
- Divide: lambda = (3 x 10^8) / (1 x 10^8) = 3 metres.
Wavelength and frequency across the spectrum
| Wave | Typical wavelength | Typical frequency |
|---|---|---|
| Radio (FM) | about 3 m | about 100 MHz |
| Microwave (oven) | about 12 cm | about 2.45 GHz |
| Infrared | 700 nm to 1 mm | 300 GHz to 430 THz |
| Visible light | 400 to 700 nm | 430 to 770 THz |
| Ultraviolet | 10 to 400 nm | 750 THz to 30 PHz |
| Audible sound (air) | 1.7 cm to 17 m | 20 Hz to 20 kHz |
Common pitfalls
- Using the speed of light for sound. Sound travels at 343 m/s in air, not 3 x 10^8 m/s. Match the speed to the wave.
- Mixing unit prefixes. Convert MHz, GHz, nm, and cm to base units (hertz and metres) before dividing.
- Forgetting the medium changes c. Light slows in glass or water, which shifts the wavelength even though frequency stays fixed.
- Confusing frequency with angular frequency. The wave equation uses ordinary frequency f in hertz, not omega in radians per second.
- Assuming wavelength alone sets photon energy. Energy depends on frequency; only for a fixed speed does wavelength map cleanly to energy.