Wavelength

Wavelength, written $\lambda$, is the distance between two successive points of a wave that are in the same phase, such as two neighbouring crests. For light it ranges from about 380 nm (violet) to 780 nm (deep red) across the visible band, and the full electromagnetic spectrum runs from picometres for gamma rays to kilometres for long radio waves. A helium-neon laser emits at 632.8 nm, telecom fiber links at 1310 and 1550 nm. Sound in air has wavelengths from 17 m (20 Hz) down to 1.7 cm (20 kHz).

Formula

For any periodic wave travelling at speed $v$ with frequency $f$,

$$\lambda = \frac{v}{f}.$$

For light in vacuum $v = c = 299{,}792{,}458$ m/s, so a frequency of 473.76 THz gives 632.8 nm. Three worked cases:

• Light: green light of 550 nm has $f = c/\lambda$ = 545.1 THz; one cycle lasts 1.83 fs.
• Sound: at 20 °C the speed of sound in air is about 343 m/s, so concert A at 440 Hz has $\lambda$ = 343/440 = 0.78 m, and 1 kHz has 34.3 cm.
• Radio: a 100 MHz FM signal has $\lambda$ = 3.00 m; 2.4 GHz Wi-Fi has 12.5 cm; a 1 MHz AM station has 300 m.

Wavelength, frequency, wavenumber and photon energy are interchangeable labels for the same light, and the wavelength, frequency and wavenumber entry gives the conversion shortcuts and a table of working values.

Units

The SI unit is the metre. In practice the prefix follows the region: picometres for X-rays and electron waves, nanometres for ultraviolet and visible light, micrometres for the infrared (10.6 µm for a CO₂ laser), and millimetres to metres for microwaves and radio. The ångström (1 Å = 0.1 nm) survives in crystallography and X-ray work. Spectroscopists often use wavenumber in cm⁻¹ instead, and atomic line tables conventionally list air wavelengths between 200 nm and 2 µm and vacuum wavelengths outside that range.

Wavelength in a medium

Frequency is set by the source and does not change when light enters a material; the speed drops to $c/n$, so the wavelength shrinks by the refractive index $n$:

$$\lambda_{\text{medium}} = \frac{\lambda_0}{n}.$$

The 632.8 nm HeNe line becomes 417.7 nm inside N-BK7 glass ($n$ = 1.5151) and 474.7 nm in water ($n$ = 1.333). Colour belongs to the frequency, so the light looks the same red after it leaves the glass. By convention "wavelength" in optics means the vacuum value unless the medium is stated; instruments that report air wavelengths differ from vacuum values by about 2.7 parts in 10⁴.

How wavelength is measured

In the laboratory, wavelength is measured by interference or diffraction, since both turn a length scale set by $\lambda$ into a pattern of fringes or angles.

• A wavemeter is a scanning Michelson interferometer or a static Fizeau interferometer that compares fringes of the unknown beam with those of a reference laser; moving a Michelson mirror by $\lambda/2$ (316.4 nm for 632.8 nm light) shifts the pattern by one fringe. Accuracy of parts in 10⁶ to 10⁷ is routine.
• A grating spectrometer or optical spectrum analyzer uses the diffraction grating equation, which links the diffraction angle to $\lambda$ divided by the groove period.
• The double-slit experiment gives $\lambda$ from fringe spacing, slit separation and screen distance; it is a teaching measurement accurate to a few percent.
• For the most exact work, the optical frequency is measured against a frequency comb and converted with $\lambda = c/f$.

For sound and radio, where wavelengths are centimetres to metres, standing-wave patterns can be measured directly with a ruler: adjacent nodes are $\lambda/2$ apart.

Wavelength and amplitude

Amplitude and wavelength are independent properties. Amplitude is the height of the oscillation: the peak electric field for light, the pressure excursion for sound, the displacement of a string. It sets intensity, which scales as amplitude squared. Wavelength sets the spatial period and, through $\lambda = v/f$, the frequency. A dim and a bright 532 nm laser pointer have the same wavelength and different amplitudes; for light, the energy per photon depends only on frequency ($E = hf = hc/\lambda_0$), while amplitude sets how many photons arrive per second.

Pitfalls

Mixing vacuum and in-medium wavelengths in the same calculation is the commonest error, for example using $\lambda_0$ for a thin-film thickness that should be a quarter of $\lambda_0/n$. A second is treating wavelength and frequency spacings as proportional over a wide band; the relation $\Delta f = c\,\Delta\lambda/\lambda^2$ holds only for small intervals.

Common questions

What is the formula for wavelength?

$\lambda = v/f$: wave speed divided by frequency. For light in vacuum it is $\lambda = c/f$, and in a medium $v = c/n$.

Does a longer wavelength mean more or less energy?

For light, less energy per photon, since $E = hc/\lambda$: a 1550 nm photon carries 0.80 eV, a 400 nm photon 3.10 eV.

Why do longer wavelengths bend around obstacles more?

Diffraction angles scale as $\lambda$ divided by the size of the aperture or obstacle, so metre-scale sound and radio waves spread around buildings, while light, with wavelengths near 0.5 µm, casts sharp shadows.

The wavelength calculator applies λ = v/f to light and to any other wave, and converts a wavelength to frequency, photon energy and wavenumber.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 2 and 3; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); BIPM, The International System of Units (SI brochure), 9th ed. (2019).