# Wavenumber (with radians)

## Description

In the physical sciences, the wavenumber (also wave number) is the spatial frequency of a wave, either in cycles per unit distance or radians per unit distance. It can be envisaged as the number of waves that exist over a specified distance (analogous to frequency being the number of cycles or radians per unit time).

In multidimensional systems, the wavenumber is the magnitude of the wave vector. The space of wave vectors is called reciprocal space. Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, and elementary particle physics.

For quantum mechanical waves, wavenumber multiplied by Planck’s constant is the canonical momentum.

Wavenumber can be used to specify quantities other than spatial frequency. In optical spectroscopy, it is often used as a unit of temporal frequency assuming a certain speed of light. In this context, it is the number of cycles—not radians—per unit length, and the reference distance should be assumed to be cm. In the same domain, wavenumber can also be used as a unit of energy; 1 cm−1 of energy is the amount of energy in a single photon with a wavelength of 1 cm, the conversion being done using Planck’s relation. For example, 1 cm−1 implies 1.23984×10−4 eV and 8065.54 cm−1 implies 1 eV.

The first definition of the formula, with the number of wavelengths per unit distance (equivalently, the number of cycles per wavelength), is shown here.

Related formulas## Variables

k | wavenumber, aslo termed the angular wavenumber or circular wavenumber (1/m) |

π | pi |

λ | wavelength, also termed the spectroscopic wavenumber (m) |