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. Multiplied by Planck's constant, it is the momentum of a wave, and therefore is employed in all wave mechanics, including quantum mechanics, electrodynamics, etc. The space of wave vectors is called reciprocal space or momentum space and spans 3 dimensions orthogonal to real space, spanning the 6-dimensional phase space, which also describes classical mechanics.
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.
It can be defined as either:
- , the number of wavelengths per unit distance (equivalently, the number of cycles per wavelength), where λ is the wavelength, sometimes termed the spectroscopic wavenumber, or
- ,the number of radians per unit distance, sometimes termed the angular wavenumber or circular wavenumber, but more often simply wavenumber.
There are four total symbols for wavenumber. Under the first definition either ν, , or σ may be used; for the second, k should be used.
When wavenumber is represented by the symbol ν, a frequency is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship , where νs is a frequency in hertz. This is done for convenience as frequencies tend to be very large. 
It has dimensions of reciprocal length, so its SI unit is the reciprocal of meters (m−1). In spectroscopy it is usual to give wavenumbers in cgs unit, i.e., reciprocal centimeters (cm−1); in this context the wavenumber was formerly called the kayser, after Heinrich Kayser. The angular wavenumber may be expressed in radians per meter (rad·m−1), or as above, since the radian is dimensionless.
where k0 is the free-space wavenumber, as above.
In wave equations
Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants. See wavepacket for discussion of the case when these quantities are not constant.
where is the frequency of the wave, is the wavelength, is the angular frequency of the wave, and is the phase velocity of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a dispersion relation.
For the special case of an electromagnetic wave in vacuum, where vp = c, k is given by
For the special case of a matter wave, for example an electron wave, in the non-relativistic approximation (in the case of a free particle, that is, the particle has no potential energy):
Wavenumber is also used to define the group velocity.
The historical reason for using this spectroscopic wavenumber rather than frequency is that it proved to be convenient in the measurement of atomic spectra: the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum,
which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from diffraction gratings, or the distance between fringes in interferometers, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of Johannes Rydberg in the 1880s. The Rydberg–Ritz combination principle of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in quantum theory as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy.
where R is the Rydberg constant and ni and nf are the principal quantum numbers of the initial and final levels, respectively (ni is greater than nf for emission).
It can also be converted into wavelength of light via
- NIST Reference on Constants, Units and Uncertainty (CODATA 2010), specifically 100/m and 1 eV. Retrieved April 25, 2013.
- "Wave number". Encyclopedia Britannica. Retrieved 19 April 2015.
- , eq.(2.13.3)
- See for example,
- Fiechtner, G. (2001). "Absorption and the dimensionless overlap integral for two-photon excitation". Journal of Quantitative Spectroscopy and Radiative Transfer 68 (5): 543. Bibcode:2001JQSRT..68..543F. doi:10.1016/S0022-4073(00)00044-3.
- US 5046846, Ray, James C. & Asari, Logan R., "Method and apparatus for spectroscopic comparison of compositions", published 1991-09-10
- "Boson Peaks and Glass Formation". Science 308 (5726): 1221. 2005. doi:10.1126/science.308.5726.1221a.