Wavelength & Frequency Calculator
The wave equation v = fλ relates speed, frequency, and wavelength. Enter any two values to calculate the third for any type of wave.
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What do I need to know about 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.
What's the key thing to understand about 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.
What's the key thing to understand about 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.
What's the key thing to understand about 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.
The Electromagnetic Spectrum: Full Reference Table
Every band of the electromagnetic spectrum obeys the same wave equation, v = f × λ, with v fixed at the speed of light in a vacuum — the bands differ only in wavelength and frequency, which is why a single reference table can summarise the whole spectrum.
| Band | Wavelength range | Frequency range | Common uses |
|---|---|---|---|
| Radio | > 1 mm | < 300 GHz | Broadcasting, mobile networks, radar |
| Microwave | 1 mm – 1 m | 300 MHz – 300 GHz | Wi-Fi, satellite links, microwave ovens |
| Infrared | 700 nm – 1 mm | 300 GHz – 430 THz | Heat radiation, remote controls, night vision |
| Visible light | 400 – 700 nm | 430 – 750 THz | Human vision |
| Ultraviolet | 10 – 400 nm | 750 THz – 30 PHz | Sunburn, sterilisation, fluorescent lighting |
| X-ray | 0.01 – 10 nm | 30 PHz – 30 EHz | Medical imaging, airport security scanning |
| Gamma ray | < 0.01 nm | > 30 EHz | Radioactive decay, astrophysical sources |
The boundaries between bands are conventions rather than hard physical cut-offs — there's no sudden change in behaviour exactly at 400 nm, for instance, it's simply the point where human eyes stop being able to detect the light as visible violet.
Worked Example: Wavelength to Frequency to Photon Energy
A single worked example ties the wave equation to photon energy, showing how wavelength, frequency, and energy are all linked for electromagnetic radiation. Take green light with a wavelength of 500 nm (500 × 10⁻⁹ m). Step 1 — find frequency using f = c ÷ λ: f = (2.998 × 10⁸) ÷ (500 × 10⁻⁹) ≈ 6.00 × 10¹⁴ Hz (600 terahertz). Step 2 — find photon energy using E = h × f, where h is Planck's constant, 6.626 × 10⁻³⁴ J·s: E = 6.626 × 10⁻³⁴ × 6.00 × 10¹⁴ ≈ 3.97 × 10⁻¹⁹ joules. That's an extremely small number of joules, as expected for a single photon, so physicists usually convert to electronvolts (1 eV = 1.602 × 10⁻¹⁹ J) for convenience: 3.97 × 10⁻¹⁹ ÷ 1.602 × 10⁻¹⁹ ≈ 2.48 eV — a figure that matches published values for green light almost exactly. This example also shows why shorter-wavelength light is more energetic: since E = h × (c ÷ λ), energy is inversely proportional to wavelength, so violet light (shorter wavelength than green) carries more energy per photon, and ultraviolet, X-ray, and gamma radiation (progressively shorter still) are correspondingly more energetic and more biologically hazardous.
Using the Exact vs Rounded Speed of Light
The speed of light in a vacuum is exactly 299,792,458 m/s by international definition (the metre itself is now defined in terms of this constant), but most everyday calculations round it to 3 × 10⁸ m/s for convenience. That rounding introduces a small but sometimes meaningful error: 299,792,458 is about 0.07% less than 300,000,000, so a frequency or wavelength calculated with the rounded constant will be off by roughly that same 0.07% — usually negligible for everyday radio and optics work, but significant for precision applications like GPS timing, laser interferometry, or astronomical distance measurement, where even a fraction of a percent compounds into metres or microseconds of error. As a rule of thumb for this calculator: use 3 × 10⁸ m/s when you only need 2-3 significant figures of accuracy (most practical radio, Wi-Fi, and optics questions), and use the full 2.998 × 10⁸ m/s (or the exact value) whenever your inputs are already given to 4 or more significant figures, so the precision of the constant doesn't become the weakest link in the calculation.