Wavelength & Frequency Calculator

The Wave Equation

All waves obey the same simple equation linking their speed, frequency, and wavelength: v = f × λ. Wave speed (v, in metres per second) equals frequency (f, in hertz — cycles per second) multiplied by wavelength (λ, in metres). Rearranged, f = v/λ and λ = v/f, so knowing any two gives you the third. The relationship is intuitive: if waves travel at a fixed speed, a higher frequency must mean a shorter wavelength (more waves squeezed into the same distance per second), and vice versa. A worked example: a radio wave at 100 MHz (1 × 10⁸ Hz) travelling at the speed of light has wavelength λ = (3 × 10⁸) / (1 × 10⁸) = 3 m. That's why FM radio antennae are roughly a metre or so long — they're tuned to a fraction of the wavelength. For light in a vacuum, v equals the speed of light c, exactly 299,792,458 m/s, often rounded to 3 × 10⁸ m/s. In other media (water, glass, air) light travels slower, so for the same frequency the wavelength is shorter — which is why light bends when entering glass (refraction) and why fibre-optic communication works. For sound in air at 20°C, v ≈ 343 m/s; in water it's about 1,480 m/s. The wave equation applies to all kinds of waves: electromagnetic (light, radio, X-rays), mechanical (sound, water waves, seismic), and matter waves in quantum mechanics. This calculator handles the wave equation in all three rearrangements, so you can find the missing quantity given the other two.

The EM Spectrum

The electromagnetic spectrum covers an enormous range of wavelengths and frequencies, from radio waves with wavelengths longer than a person to gamma rays with wavelengths smaller than an atomic nucleus — all the same kind of wave, governed by the same equation, differing only in wavelength. Radio waves have the longest wavelengths (λ > 1 mm, f < 300 GHz) and are used for broadcasting, mobile phones, and radar. Microwaves (1 mm – 1 m wavelength) sit at the short end of radio and power Wi-Fi, mobile networks, microwave ovens, and satellite communication. Infrared (700 nm – 1 mm) is felt as heat and used in remote controls, night vision, and thermal imaging. Visible light occupies a tiny band from about 400 nm (violet) to 700 nm (red) — the colours of the rainbow span this narrow window. Ultraviolet (10 – 400 nm) lies just beyond visible light and causes sunburn and chemical reactions. X-rays (0.01 – 10 nm) penetrate soft tissue and are used in medical imaging and security. Gamma rays (λ < 0.01 nm) are the highest-energy form, emitted in radioactive decay and from astrophysical sources. Despite the enormous range — radio wavelengths billions of times larger than gamma — all electromagnetic waves travel at exactly c in a vacuum. Higher frequency means higher energy per photon (E = hf, where h is Planck's constant), which is why X-rays and gamma rays are ionising and dangerous, while radio waves at the same intensity are harmless. The same equation handles them all.

Sound Frequencies

Sound is a mechanical wave (a pressure wave in a medium), and the wave equation applies just as it does to light, with one big difference: sound speed depends on the medium and conditions. In air at 20°C, sound travels at about 343 m/s; in cool air it's slower, in warm air faster. In water sound moves at around 1,480 m/s, and in steel about 5,100 m/s — much faster than in gases, which is why you can hear a distant train approaching by putting your ear to the rail. Human hearing spans roughly 20 Hz to 20,000 Hz (20 kHz), though the upper limit drops with age. Middle C on a piano is 261.6 Hz, and concert A (the tuning note) is 440 Hz. A 440 Hz note in air has wavelength λ = 343 / 440 ≈ 0.78 m — about the length of an outstretched arm. Dogs hear up to about 65 kHz, which is why dog whistles work; bats and dolphins use ultrasound (above 20 kHz) for echolocation. Medical ultrasound runs at 1–20 MHz, with the short wavelengths giving high resolution for imaging soft tissue. Infrasound (below 20 Hz) is too low for human hearing but can be felt and is produced by earthquakes, large explosions, and some weather phenomena. The Doppler effect — where a moving source shifts the frequency you hear (sirens dropping in pitch as they pass) — is another consequence of how sound waves interact with motion. The wave equation applies throughout, with speed varying by medium.

Units and Significant Figures

Wave calculations require consistent SI units: frequency in hertz (Hz, cycles per second), wavelength in metres, and speed in metres per second. The most frequent error is unit mixing, especially with the very large and very small numbers involved. Frequency prefixes are common: 1 kHz = 10³ Hz, 1 MHz = 10⁶ Hz, 1 GHz = 10⁹ Hz, 1 THz = 10¹² Hz — convert these to plain hertz before substituting into v = f × λ, or you'll be wrong by factors of thousands or millions. Wavelength prefixes go the other way: 1 mm = 10⁻³ m, 1 μm = 10⁻⁶ m, 1 nm = 10⁻⁹ m — convert these to metres before computing frequency. Scientific notation makes the arithmetic far easier and reduces errors: a 500 nm wavelength is 5 × 10⁻⁷ m, and dividing the speed of light 3 × 10⁸ m/s by this gives 6 × 10¹⁴ Hz, a frequency that would be incomprehensible written out in full. Significant figures should match input precision — using c = 3 × 10⁸ m/s gives 1 significant figure, so don't quote answers to 6 figures unless you've used c = 2.998 × 10⁸ to 4 figures. For sound, the speed depends on temperature and medium, so the precision of your speed input limits the precision of the result. Sanity-check magnitudes against known reference points: visible light frequencies are around 10¹⁴-10¹⁵ Hz, audible sound 10¹-10⁴ Hz, FM radio around 10⁸ Hz — any answer wildly off these scales suggests a unit error worth rechecking. This calculator handles prefix conversion and shows results in appropriate units.