Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length (radians per meter). Wavenumber is the spatial analogue of angular frequency. Application of a Fourier transformation on data in the time domain yields a frequency spectrum; applied on data in the spatial domain (data as a function of position) yields a spectrum as a function of wavenumber. The exact definition is dependent on the field.

In wave equations

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The circular wavenumber, k, often misleadingly abbreviated as "wavenumber", is defined as

$k\; \backslash equiv\; \backslash frac\{2\backslash pi\}\{\backslash lambda\}\; =\; \backslash frac\{2\backslash pi\backslash nu\}\{v\_p\}=\backslash frac\{\backslash omega\}\{v\_p\}=\backslash frac\{E\}\{\backslash hbar\; c\},$

where λ is the wavelength in the medium, $\backslash nu$ (Greek letter nu) is the frequency, v_p is the phase velocity of wave, ω is the angular frequency, E is the energy, ħ is the reduced Planck constant, and c is the speed of light in vacuum. The wavenumber is the scalar of the wave vector.

In spectroscopy

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In spectroscopy, the wavenumber $\backslash tilde\{\backslash nu\}$ of electromagnetic radiation is defined as

$\backslash tilde\{\backslash nu\}\; =\; 1/\backslash lambda,$

where $\backslash lambda$ as a length in the SI-system is measured in meters (m) and commonly quantified in centimetres (cm = 10⁻² m) refers to the wavelength in vacuum. The unit of this quantity is cm⁻¹, pronounced as reciprocal centimeter, or "inverse centimeter". The historical reason for using this quantity is that it is proportional to energy, but not dependent on the speed of light or Planck's constant, which were not known with sufficient accuracy (or rather not at all known).

For example, the wavenumbers of the emissions lines of hydrogen atoms are given by

$$

\tilde{\nu} = R\left({1\over - {1\over\right)

where λ is the wavelength, R is Rydberg constant, $n\_1$ and $n\_2$ are the orbit numbers, and $n\_2$ is greater than $n\_1$.

Spectroscopists often express various quantities, such as frequency and energy in cm⁻¹. In colloquial usage, the unit cm⁻¹ is sometimes referred to as a "wavenumber", which confuses the role of a unit with that of a quantity. An incorrect phrase such as "The energy is 300 wavenumbers" should be read as "The energy corresponds to a wavenumber of 300 reciprocal centimeters", or as "The energy corresponds to a wavenumber of 300 inverse centimeters".

In atmospheric science

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Wavenumber in atmospheric science is defined as length of the spatial domain divided by the wavelength, or equivalently the number of times a wave has the same phase over the spatial domain. The domain might be 2π for the non-dimensional case, or

$2\backslash pi\; R\; \backslash cos\backslash left(\backslash phi\backslash right)$

for an atmospheric wave, where R is Earth's radius and φ is latitude. Wavenumber-frequency diagrams are a common way of visualizing atmospheric waves.

Wave mechanics

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