Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. Wavenumber is the spatial analogue of frequency. Application of a Fourier transformation on data in the time domain yields a spectrum as a function of frequency; 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.

Physics

Here, wavenumber, k, is most frequently defined as

<math>k \equiv \frac{2\pi}{\lambda} = \frac{2\pi\nu}{v_p}=\frac{\omega}{v_p},<math>

where λ is the wavelength in the medium, ν is the frequency, v_p is the phase velocity of wave, and ω is the angular frequency. The wavenumber is closely related to the concept of the wave vector.

Spectroscopy

In spectroscopy, the wavenumber <math>\tilde{\nu}<math> is defined as

<math>\tilde{\nu} = 1/\lambda,<math>

where <math>\lambda<math> is measured in cm and refers to the wavelength in vacuum. The unit of this quantity is cm^-1, pronounced as reciprocal centimeter. The historical reason for using this quantity is that this quantity is proportional to energy, but not dependent on the speed of light or Planck's constant, which were not known with sufficient accuracy (or not even known at all).

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

<math>

\tilde{\nu} = R\left({1\over{{n_1}^2}} - {1\over{{n_2}^2}}\right) <math>

where λ is the wavelength, R is Rydberg constant, <math>n_1<math> and <math>n_2<math> are the orbit numbers, and <math>n_2<math> is greater than <math>n_1<math>.

Spectroscopists often express various quantities, such as frequency and energy in cm^-1. In colloquial usage, the unit cm^-1 is sometimes pronounced as "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".

Atmospheric science

Wavenumber in atmospheric science is defined as wavelength divided by the length of the spatial domain, 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

<math>2\pi R \cos\left(\phi\right)<math>

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.

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