Wavenumber

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Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m⁻¹). Wavenumber is the spatial analogue of frequency. Application of a Fourier transformation on data as a function of time yields a frequency spectrum; application on data as a function of position yields a wavenumber spectrum. The exact definition depends 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 wavenumber has dimensions of inverse length and SI units of reciprocal meters (m⁻¹). The cgs unit of this quantity is cm⁻¹, pronounced as reciprocal centimeter or inverse centimeter and historically synonymous with kayser. 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

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

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).

In colloquial usage, the unit cm⁻¹ is sometimes referred to as a "wavenumber", which confuses the role of a dimension with that of the name of a quantity. 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, an incorrect phrase such as "The energy is 300 wavenumbers" should be interpreted or restated as "The energy corresponds to a wavenumber of 300 reciprocal centimeters", or as "The energy corresponds to a wavenumber of 300 inverse centimeters", or as "The energy corresponds to a wavenumber of 300 per centimeter". The analogous statements hold true for the unit m⁻¹.

[edit] In wave equations

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

$k \equiv \frac{2\pi}{\lambda}$

for a wave of wavelength $λ$ .

For the special case of an electromagnetic wave,

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

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

For the special case of a matter wave, for example an electron wave, in the non-relativistic approximation:

$k \equiv \frac{2\pi}{\lambda} = \frac{p}{\hbar}= \frac{\sqrt{2 m E }}{\hbar}.$

Here $p$ is the momentum of the particle, $m$ is the mass of the particle, $E$ is the kinetic energy of the particle, and $\hbar$ is the reduced Planck's constant.

[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

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