The note A at 440 Hz
plain
Any wave: v = 343 m/s, f = 440 Hzλ = v/f = 343 / 440 = 0.780 m, about 78 cm. The lowest note people hear, 20 Hz, is 343 / 20 = 17.15 m long; the highest, 20 kHz, only 1.7 cm.
Wavelength Calculator
Wavelength, frequency and wave speed, and the energy of a photon of light.
Last updated
Want to solve these without the calculator?
The Coddy physics course teaches the method: you solve each problem in steps on an interactive board, and it tells you which step went wrong.
The wave speed formula, and what it says
Every wave, sound, water or light, obeys the same formula: v = fλ. The frequency f is how many waves pass a point each second, in hertz; the wavelength λ is the distance from one crest to the next, in metres; and v is how fast the wave travels, in metres per second. Sound moves through air at about 343 m/s at 20 °C, so a 440 Hz note has a wavelength of 343 / 440 = 0.78 m.
At a fixed wave speed, frequency and wavelength trade off: double the frequency and the wavelength halves. Light in a vacuum always travels at c = 299,792,458 m/s, so c = fλ turns any wavelength into a frequency. Light also comes in photons, and each one carries an energy E = hf, where h = 6.62607015 × 10⁻³⁴ J·s is Planck's constant. A higher frequency, which is a shorter wavelength, means more energy per photon.
The calculator has two tabs. Any wave solves v = fλ: give two of speed, frequency and wavelength and it finds the third. Light needs only one value: give a wavelength, a frequency or a photon energy and it finds the other two from c = fλ and E = hf. Nanometres, micrometres, kilohertz to terahertz and electronvolts are converted to SI first, as their own step, and each answer is also given in the other units.
What to notice
- v = fλ gives metres per second only when f is in hertz and λ in metres. 500 nm is 500 × 10⁻⁹ m, and 2.4 GHz is 2.4 × 10⁹ Hz.
- When a wave passes into a new material, its frequency stays the same; the speed and the wavelength change. Sound going from air into water speeds up and its wavelength grows about four times.
- For light in a vacuum the speed is always c, so one value is enough: the wavelength fixes the frequency, and the frequency fixes the photon energy.
- Photon energy goes up with frequency and down with wavelength. Violet light carries more energy per photon than red, and an X-ray photon far more than either.
- A handy shortcut for light: the photon energy in electronvolts is about 1240 divided by the wavelength in nanometres. 620 nm gives 2.0 eV.
How to calculate wavelength and frequency
Find the wave speed
343 m/s for sound in air at 20 °C, about 1500 m/s in water, 299,792,458 m/s for light in a vacuum. A problem usually gives it.
Put the frequency in hertz and the wavelength in metres
kHz multiply by 1000, MHz by 10⁶, GHz by 10⁹. nm multiply by 10⁻⁹, μm by 10⁻⁶. 100 MHz is 100,000,000 Hz.
Divide for the one you want
λ = v/f for the wavelength, f = v/λ for the frequency, v = fλ for the speed. 343 / 440 = 0.78 m.
For light, multiply the frequency by h for the energy
E = hf, in joules. Divide by 1.602176634 × 10⁻¹⁹ for electronvolts: 5.996 × 10¹⁴ Hz gives 3.973 × 10⁻¹⁹ J, which is 2.48 eV.
The electromagnetic spectrum
Every band travels at c in a vacuum, so wavelength and frequency go together through c = fλ. The edges between bands are conventions, not sharp lines; these are the usual ones.
| Band | Wavelength | Frequency | Where you meet it |
|---|---|---|---|
| Radio | longer than 1 m | below 300 MHz | FM radio at 100 MHz, 3 m |
| Microwaves | 1 mm to 1 m | 300 MHz to 300 GHz | Wi-Fi at 2.4 GHz, 12.5 cm |
| Infrared | 750 nm to 1 mm | 300 GHz to 400 THz | Heat from warm objects, TV remotes |
| Visible light | 380 to 750 nm | 400 to 790 THz | Red at the long end, violet at the short end |
| Ultraviolet | 10 to 380 nm | 790 THz to 30 PHz | Sunburn, black lights |
| X-rays | 0.01 to 10 nm | 30 PHz to 30 EHz | Medical and airport scans |
| Gamma rays | shorter than 0.01 nm | above 30 EHz | Radioactive decay |
Worked examples
A 3 m wave in water
plain
Any wave: v = 1500 m/s, λ = 3 mf = v/λ = 1500 / 3 = 500 Hz. In air the same 500 Hz sound would have a wavelength of 343 / 500 = 0.686 m: the frequency stays, the wavelength changes with the speed.
Green light at 500 nm
plain
Light: λ = 500 nm500 nm is 5 × 10⁻⁷ m. f = c/λ = 299,792,458 / (5 × 10⁻⁷) = 5.996 × 10¹⁴ Hz, about 600 THz. Then E = hf = 6.62607015 × 10⁻³⁴ · 5.996 × 10¹⁴ = 3.973 × 10⁻¹⁹ J, which is 2.48 eV.
An FM station at 100 MHz
plain
Light: f = 100 MHzRadio waves are light too, so c = fλ applies. 100 MHz is 10⁸ Hz, and λ = c/f = 299,792,458 / 10⁸ = 2.998 m. Each photon carries only E = hf = 6.626 × 10⁻²⁶ J, about six million times less than a photon of green light.
A photon of 2 eV
plain
Light: E = 2 eVConvert first: 2 eV = 2 · 1.602176634 × 10⁻¹⁹ = 3.204 × 10⁻¹⁹ J. Then f = E/h = 4.836 × 10¹⁴ Hz, and λ = c/f = 6.199 × 10⁻⁷ m, which is 619.9 nm: the edge between orange and red.
Common mistakes
- Leaving nanometres as they are. 500 nm is 5 × 10⁻⁷ m; putting 500 into c = fλ gives a frequency a billion times too small.
- Mixing up MHz and Hz. 100 MHz is 100,000,000 Hz, not 100 Hz.
- Using the speed of light for sound, or 343 m/s for light. Each kind of wave has its own speed, and it depends on what the wave travels through.
- Thinking the frequency changes when a wave enters water or glass. The frequency is set by the source; the speed and wavelength change.
- Confusing frequency with period. The period T is the time for one wave, T = 1/f; a 50 Hz wave has a period of 0.02 s.
- Mixing joules and electronvolts in E = hf. With h in J·s the energy comes out in joules; divide by 1.602176634 × 10⁻¹⁹ for eV.
Wavelength and frequency FAQ
What is the wave speed formula?
v = fλ: the wave speed equals the frequency times the wavelength. With f in hertz and λ in metres, v comes out in metres per second. A wave of 2 Hz and 0.5 m travels at 1 m/s.
How do you calculate wavelength from frequency?
Divide the wave speed by the frequency: λ = v/f. For light use c = 299,792,458 m/s, so λ = c/f. Wi-Fi at 2.4 GHz has a wavelength of 299,792,458 / (2.4 × 10⁹) = 0.125 m, about 12.5 cm.
How do you calculate frequency from wavelength?
Divide the speed by the wavelength: f = v/λ, or f = c/λ for light. Red light at 650 nm has a frequency of 299,792,458 / (650 × 10⁻⁹) = 4.61 × 10¹⁴ Hz, about 461 THz.
How do you calculate the energy of a photon?
E = hf, where h = 6.62607015 × 10⁻³⁴ J·s is Planck's constant. From a wavelength, combine it with c = fλ: E = hc/λ. A 500 nm photon carries 3.97 × 10⁻¹⁹ J, or 2.48 eV.
What wavelength is each colour of light?
Roughly: violet 380 to 450 nm, blue 450 to 495 nm, green 495 to 570 nm, yellow 570 to 590 nm, orange 590 to 620 nm, red 620 to 750 nm. The colours blend into each other, so the edges are approximate.
What is the speed of sound?
About 343 m/s in dry air at 20 °C, and 331 m/s at 0 °C; it rises with temperature. In water it is about 1500 m/s, more than four times faster, and in solids faster still.
Why is c exactly 299,792,458 m/s?
Since 1983 the metre has been defined from the speed of light: it is the distance light travels in a vacuum in 1/299,792,458 of a second. So c is exact by definition, and so is Planck's constant since 2019.
