Last updated: 2026-05-18

Definition

The wave equation relates three properties of any periodic wave through a single identity:

$v = f \times \lambda$

Wave speed equals frequency times wavelength. The same relationship governs electromagnetic radiation such as radio signals and laser light, mechanical waves such as sound, and any other periodic wave. Given any two of the three quantities, the third follows by rearranging the equation, which is what this calculator computes.

The three quantities

Wave speed (v) is how fast the disturbance moves through the medium — metres per second for sound in air, kilometres per second for seismic waves, the speed of light for electromagnetic radiation.

Frequency (f) is how many complete oscillation cycles pass a fixed point every second. One cycle per second is one Hertz (Hz). A flute's top note is around 4 kHz; Wi-Fi operates at 2.4 GHz or 5 GHz; medical X-rays oscillate at roughly $10^{18}$ Hz.

Wavelength (λ) is the spatial length of one full cycle — crest to crest, or trough to trough. It is the complement of frequency: at a fixed wave speed, doubling the frequency halves the wavelength. That inverse relationship is why high-frequency waves are short and low-frequency waves are long.

The three modes of this calculator cover the three ways to rearrange the formula:

Mode	Formula	Typical use
Find wavelength	$\lambda = v / f$	Antenna sizing, optical filter design
Find frequency	$f = v / \lambda$	Reading a spectrum, identifying unknown signals
Find wave speed	$v = f \times \lambda$	Measuring propagation speed in an unknown medium

The speed of light as an exact constant

The speed of light in vacuum is exactly 299,792,458 m/s. This is not a measurement rounded off at some convenient number of decimal places — it is a definition. Since 1983, the International System of Units defines the metre as the distance light travels in exactly 1 / 299,792,458 of a second, so the speed of light in vacuum has become a defined constant with no uncertainty.

When light travels through a medium (glass, water, a diamond), it slows by a factor of the material's refractive index $n$:

$v\_{\text{medium}} = \frac{c}{n}$

Ordinary glass has $n \approx 1.5$, so light propagates at roughly 200,000 km/s inside it. Diamond has $n \approx 2.4$, slowing light to about 125,000 km/s — a large enough difference to cause the pronounced bending (refraction) that makes diamonds sparkle.

The electromagnetic spectrum

All electromagnetic waves share the same vacuum speed. What separates a radio wave from a gamma ray is only frequency — and therefore wavelength:

Band	Frequency range	Wavelength range	Common use
Radio	< 300 MHz	> 1 m	AM/FM broadcasting, NFC
Microwave	300 MHz – 300 GHz	1 m – 1 mm	Wi-Fi, radar, microwave ovens
Infrared	300 GHz – 430 THz	1 mm – 700 nm	Thermal imaging, TV remotes
Visible	430 – 790 THz	700 – 380 nm	Human vision
Ultraviolet	790 THz – 30 PHz	380 – 10 nm	Sterilisation, sunburn
X-ray	30 PHz – 30 EHz	10 nm – 0.01 nm	Medical imaging, crystallography
Gamma ray	> 30 EHz	< 0.01 nm	Cancer therapy, nuclear physics

Visible light occupies a tiny slice — roughly 380 nm (deep violet) to 700 nm (deep red). Each colour you see corresponds to a narrow wavelength band: violet sits at the short end around 400 nm, green near 550 nm, and red at the long end around 650 nm. Wavelengths shorter than 380 nm are ultraviolet (invisible but energetic enough to cause sunburn); longer than 700 nm are infrared (felt as heat).

Sound in air

Sound is a pressure wave, not an electromagnetic wave, so it travels at an entirely different speed. In dry air at 20 °C the speed of sound is approximately 343 m/s — about 880,000 times slower than light. The speed rises with temperature at roughly 0.6 m/s per °C; on a cold winter day at 0 °C it is around 331 m/s.

Because the speed is so much lower, audio frequencies produce much longer wavelengths than the equivalent radio frequencies:

Note / sound	Frequency	Wavelength in air (20 °C)
Lowest bass (pipe organ)	~16 Hz	~21 m
Standard pitch A (A4)	440 Hz	~78 cm
Highest treble (flute)	~4,000 Hz	~8.6 cm
Ultrasound (diagnostic)	~3 MHz	~0.1 mm

At A4 = 440 Hz the wavelength of 78 cm is close to the dimensions of a typical room wall or instrument body, which is why room acoustics and instrument construction are tightly coupled to the frequency range of music.

Applications

Antenna design

An antenna is most efficient when its physical length matches a specific fraction of the wavelength it is transmitting or receiving — typically one quarter or one half. An FM radio station at 100 MHz has a wavelength of about 3 m, so a quarter-wave antenna is ~75 cm. A Wi-Fi access point at 2.4 GHz works with a wavelength of 12.5 cm, small enough for the antennas to fit inside a router. The formula $\lambda = v / f$ is the starting point for antenna design.

Optical fibres

In optical communication, the wavelength of the laser signal determines which window of low attenuation is used. Modern systems operate at 1310 nm or 1550 nm (both in the infrared), where silica glass fibre is most transparent. The bit rate and channel spacing in wavelength-division multiplexing (WDM) are defined in terms of nanometre offsets from a reference wavelength.

X-ray crystallography

X-rays have wavelengths of roughly 0.05–0.25 nm — comparable to the spacing between atoms in a crystal lattice (typically 0.1–0.5 nm). This match is what makes X-ray diffraction work: the crystal acts as a diffraction grating, scattering X-rays at angles that encode the arrangement of atoms. The structure of DNA was solved this way in 1953; so were thousands of proteins since.

Medical ultrasound

Diagnostic ultrasound typically uses frequencies of 1–15 MHz. At the speed of sound in soft tissue (~1540 m/s), that gives wavelengths of roughly 0.1–1.5 mm. Spatial resolution in ultrasound imaging is approximately one wavelength, so higher frequencies (shorter wavelengths) give finer images — at the cost of less penetration depth, because higher-frequency sound is absorbed more readily.

Frequently Asked Questions (FAQ)

What is the relationship between wavelength and frequency?

Wavelength (λ) and frequency (f) are inversely proportional when the wave speed (v) is constant: λ = v / f. Doubling the frequency halves the wavelength, and vice versa. This inverse relationship holds for all wave types — light, sound, radio, microwaves — as long as the medium (and thus the wave speed) does not change.

What is the speed of light and why is it exact?

The speed of light in vacuum is exactly 299,792,458 m/s — not an approximation but an exact definition. Since 1983, the metre is defined as the distance light travels in 1/299,792,458 of a second, making c an exact integer by definition. In other media (glass, water), light slows down by a factor of the refractive index (e.g., in glass n ≈ 1.5, so v ≈ 200,000,000 m/s).

What are the wavelengths of visible light?

The human eye detects light roughly from 380 nm (deep violet) to 700 nm (deep red). In terms of frequency, that is approximately 430–790 THz. Each colour corresponds to a narrow wavelength band: violet ~380–450 nm, blue ~450–495 nm, green ~495–570 nm, yellow ~570–590 nm, orange ~590–620 nm, red ~620–700 nm. Wavelengths shorter than 380 nm are ultraviolet; longer than 700 nm are infrared.

What is the wavelength of the musical note A (440 Hz) in air?

At 20 °C, the speed of sound in air is approximately 343 m/s. For A4 (440 Hz): λ = 343 / 440 ≈ 0.780 m (78 cm). This wavelength is relevant to speaker cabinet design and room acoustics. At colder temperatures the speed of sound decreases (~0.6 m/s per °C), so the wavelength shrinks slightly.