Wavenumber

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Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length (cycles per meter). Wavenumber is the spatial analogue of 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.

1 In spectroscopy
2 In wave equations
3 In atmospheric science
4 See Also

[edit] In spectroscopy

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

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

where $λ$ is the wavelength of the radiation in vacuo. 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

$\tilde{\nu} = R\left({1\over{{n_f}^2}} - {1\over{{n_i}^2}}\right)$

where R is the Rydberg constant and $n i$ and $n f$ are the principal quantum numbers of the initial and final levels, respectively ( $n i$ is greater than $n f$ for emission).

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".

[edit] In wave equations

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

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

where λ is the wavelength in the medium, $ν$ (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.

[edit] In atmospheric science

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

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