Diagram illustrating the relationship between the wavenumber and the other properties of harmonic waves.
In the physical sciences, the wavenumber (also wave number or repetency) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.
Wavenumber can be used to specify quantities other than spatial frequency. In optical spectroscopy, it is often used as a unit of temporal frequency assuming a certain speed of light.
Wavenumber, as used in spectroscopy and most chemistry fields, is defined as the number of wavelengths per unit distance, typically centimeters (cm−1):
where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber". It equals the spatial frequency. A wavenumber in inverse cm can be converted to a frequency in GHz by multiplying by 29.9792458 (the speed of light in centimeters per nanosecond). An electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space.
In theoretical physics, a wave number defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often 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.
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 ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and vp 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 a vacuum, in which the wave propagates at the speed of light, k is given by:
The historical reason for using this spectroscopic wavenumber rather than frequency is that it is a convenient unit when studying atomic spectra by counting fringes per cm with an interferometer : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:
which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from diffraction gratings and the distance between fringes in interferometers, when those instruments are operated in air or vacuum. Such 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 spectroscopic wavenumber rather than frequency or energy.
It can also be converted into wavelength of light:
where n is the refractive index of the medium. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.
Conventionally, inverse centimeter (cm−1) units are used for , so often that such spatial frequencies are stated by some authors "in wavenumbers", incorrectly transferring the name of the quantity to the CGS unit cm−1 itself.
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