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.
The usage of this term can be very specific in a given discipline and describe other quantities rather than the one of its definition. In optical spectroscopy, often it describes the photon energy, assuming given speed of light and other conversion factors, and the reference distance should be assumed to be cm. For example, a particle's energy may be given as a wavenumber in cm−1, which strictly speaking is not a unit of energy. However if one assumes this corresponds to electromagnetic radiation, then it can be directly converted to any unit of energy, e.g. 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.
where λ is the wavelength of the radiation.
The historical reason for using this quantity is that it proved to be convenient in the analysis of atomic spectra. 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 wavenumber rather than frequency or energy, since spectroscopic instruments are typically calibrated in terms of wavelength, independent of the value for the speed of light or Planck's constant.
For example, the wavenumbers of the emissions lines of hydrogen atoms are given by
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 frequency via
where is the frequency, and cn is the speed of light in the medium.
In colloquial usage, the unit cm−1 is sometimes referred to as a "wavenumber", which confuses the name of a quantity with that of a unit. Furthermore, spectroscopists often express a quantity proportional to the wavenumber, such as frequency or energy, in cm−1 and leave the appropriate conversion factor as implied. Consequently, a phrase such as "the energy is 300 wavenumbers" should be interpreted or restated as "the energy corresponds to a wavenumber of 300 cm−1." (Analogous statements hold true for the unit m−1.)
- 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.