Editing Wavenumber

{{Short description|Spatial frequency of a wave}}
[[File:Commutative diagram of harmonic wave properties.svg|thumb|Diagram illustrating the relationship between the wavenumber and the other properties of harmonic waves.]]

In the [[physical science]]s, the '''wavenumber''' (or '''wave number'''), also known as '''repetency''',<ref name="iso80000-3">{{Cite ISO standard |csnumber=64974|title=ISO 80000-3:2019 Quantities and units – Part 3: Space and time}}</ref> is the [[spatial frequency]] of a [[wave]]. '''Ordinary wavenumber''' is defined as the number of [[wave cycle]]s divided by length; it is a [[physical quantity]] with [[physical dimension|dimension]] of [[reciprocal length]], expressed in [[SI]] [[unit of measurement|units]] of [[Cycle (angular unit)|cycles]] per metre or [[reciprocal metre]] (m<sup>−1</sup>). '''Angular wavenumber''', defined as the [[wave phase]] divided by time, is a quantity with dimension of [[angle]] per length and SI units of [[radian]]s per metre.<ref name="Rodrigues Sardinha Pita 2021 p. 73">{{cite book | last1=Rodrigues | first1=A. | last2=Sardinha | first2=R.A. | last3=Pita | first3=G. | title=Fundamental Principles of Environmental Physics | publisher=Springer International Publishing | year=2021 | isbn=978-3-030-69025-0 | url=https://books.google.com/books?id=jVYlEAAAQBAJ&pg=PA73 | access-date=2022-12-04 | page=73}}</ref><ref name="Solimini 2016 p. 679">{{cite book | last=Solimini | first=D. | title=Understanding Earth Observation: The Electromagnetic Foundation of Remote Sensing | publisher=Springer International Publishing | series=Remote Sensing and Digital Image Processing | year=2016 | isbn=978-3-319-25633-7 | url=https://books.google.com/books?id=dSMGDAAAQBAJ&pg=PA679 | access-date=2022-12-04 | page=679}}</ref><ref name="Robinson Treitel 2008 p. 9">{{cite book | last1=Robinson | first1=E.A. | last2=Treitel | first2=S. | title=Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing | publisher=Society of Exploration Geophysicists | series=Geophysical references | year=2008 | isbn=978-1-56080-148-1 | url=https://books.google.com/books?id=IH-Pu3PlJgAC&pg=PA9 | access-date=2022-12-04 | page=9}}</ref> They are analogous to temporal [[frequency]], respectively the ''[[ordinary frequency]]'', defined as the number of wave cycles divided by time (in cycles per second or [[reciprocal second]]s), and the ''[[angular frequency]]'', defined as the phase angle divided by time (in radians per second).

In [[multidimensional systems]], the wavenumber is the magnitude of the ''[[wave vector]]''. The space of wave vectors is called ''[[reciprocal space]]''. Wave numbers and wave vectors play an essential role in [[optics]] and the physics of wave [[scattering]], such as [[X-ray diffraction]], [[neutron diffraction]], [[electron diffraction]], and [[elementary particle]] physics. For [[quantum mechanical]] waves, the wavenumber multiplied by the [[reduced Planck constant]] is the ''[[momentum operator|canonical momentum]]''.

Wavenumber can be used to specify quantities other than spatial frequency. For example, in [[optical spectroscopy]], it is often used as a unit of temporal frequency assuming a certain [[speed of light]].

== Definition ==
Wavenumber, as used in [[spectroscopy]] and most chemistry fields,<ref>{{Cite book |url=https://goldbook.iupac.org/ |title=The IUPAC Compendium of Chemical Terminology: The Gold Book |date=2019 |publisher=International Union of Pure and Applied Chemistry (IUPAC) |editor-last=Gold |editor-first=Victor |edition=4 |location=Research Triangle Park, NC |language=en |doi=10.1351/goldbook.w06664}}</ref> is defined as the number of [[wavelength]]s per unit distance: 
: <math>\tilde{\nu} \;=\; \frac{1}{\lambda},</math>
where ''λ'' is the wavelength. It is sometimes called the "spectroscopic wavenumber".<ref name="iso80000-3" /> It equals the [[spatial frequency]].<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |date=2017 |publisher=Pearson Education, Inc |isbn=978-0-13-397722-6 |edition=5 |location=Boston |pages=16 |chapter=2.2 Harmonic Waves}}</ref>

In theoretical physics, an angular wave number, defined as the number of radians per unit distance is more often used:<ref>{{cite web|url=http://scienceworld.wolfram.com/physics/Wavenumber.html|title=Wavenumber -- from Eric Weisstein's World of Physics|first=Eric W.|last=Weisstein|website=scienceworld.wolfram.com|access-date=19 March 2018 | archive-url=https://web.archive.org/web/20190627132558/https://scienceworld.wolfram.com/physics/Wavenumber.html |archive-date=27 June 2019}}</ref>
: <math>k \;=\; \frac{2\pi}{\lambda} = 2\pi\tilde{\nu}</math>.

===Units===

The [[International System of Units|SI unit]] of spectroscopic  wavenumber is the reciprocal m, written m<sup>&minus;1</sup>.
However, it is more common, especially in [[spectroscopy]], to give wavenumbers in [[cgs units]] i.e., reciprocal centimeters or cm<sup>&minus;1</sup>, with

:<math>1~\mathrm{cm}^{-1} = 100~\mathrm{m}^{-1}</math>.

Occasionally in older references, the unit ''kayser''  (after [[Heinrich Kayser]]) is used;<ref>{{cite book | title=Scientific Unit Conversion - A Practical Guide to Metrication | year=1997 | author=François Cardarelli |page=209}}</ref> it is abbreviated as ''K'' or ''Ky'', where 1{{nbsp}}K = 1{{nbsp}}cm<sup>−1</sup>.<ref>{{cite journal| last1= Murthy| first1 = V. L. R.| last2 = Lakshman| first2= S. V. J. | date = 1981| title=Electronic absorption spectrum of cobalt antipyrine complex| journal=Solid State Communications| volume=38| issue = 7| pages=651–652| bibcode =1981SSCom..38..651M | doi  =10.1016/0038-1098(81)90960-1}}</ref> 

Angular wavenumber may be expressed in the unit [[radian]] per meter (rad&sdot;m<sup>−1</sup>), or as above, since the [[radian]] is [[dimensionless]].

====Unit conversions====
The frequency of light with wavenumber <math>\tilde{\nu}</math> is
: <math>f = \frac{c}{\lambda} = c \tilde{\nu}</math>,
where <math>c</math> is the [[speed of light]].
The conversion from spectroscopic wavenumber to frequency is therefore<ref>{{cite web |url= http://www.britannica.com/EBchecked/topic/637882/wave-number
|title= Wave number|author=<!--Staff writer(s); no by-line.--> |website= [[Encyclopædia Britannica]]|access-date= 19 April 2015}}</ref>

: <math>1~\mathrm{cm}^{-1}  :=29.979245~\mathrm{GHz}. </math>

Wavenumber can also be used as [[unit of energy]], since a photon of frequency <math>f</math> has energy <math>hf</math>, where <math>h</math> is [[Planck's constant]].
The energy of a photon with wavenumber <math> \tilde{\nu} </math> is
: <math>E = hf = hc \tilde{\nu}</math>.
The conversion from spectroscopic wavenumber to energy is therefore
: <math>1~\mathrm{cm}^{-1} 
:= 1.986446 \times 10^{-23}~\mathrm{J}
= 1.239842 \times 10^{-4}~\mathrm{eV}
</math>
where energy is expressed either in [[Joule|J]] or 
[[Electronvolt|eV]].

=== Complex ===
A complex-valued wavenumber can be defined for a medium with complex-valued relative [[permittivity]] <math>\varepsilon_r</math>, relative [[Permeability (electromagnetism)|permeability]] <math>\mu_r</math> and [[refraction index]] ''n'' as:<ref>[http://www.ece.rutgers.edu/~orfanidi/ewa/ch02.pdf], eq.(2.13.3)</ref>
: <math>k = k_0 \sqrt{\varepsilon_r\mu_r} = k_0 n</math>
where ''k''<sub>0</sub> is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decaying [[evanescent field]]s.

=== Plane waves in linear media ===
The propagation factor of a [[sinusoidal plane wave]] propagating in the positive x direction in a linear material is given by<ref name="Harrington_1961">{{Citation |last=Harrington |first= Roger F. |year= 1961 |title= Time-Harmonic Electromagnetic Fields |edition= 1st |publisher= McGraw-Hill |isbn=0-07-026745-6  }}</ref>{{rp|p=51}}
: <math>  P = e^{-jkx}  </math>
where
* <math>k = k' - jk'' = \sqrt{-\left(\omega \mu '' + j \omega \mu' \right) \left(\sigma + \omega \varepsilon '' + j \omega \varepsilon  ' \right) }\;</math>
* <math>k' =</math> [[phase constant]] in the units of [[radian]]s/meter
* <math>k'' =</math> [[attenuation constant]] in the units of [[neper]]s/meter
* <math>\omega =</math> angular frequency
* <math>x =</math> distance traveled in the ''x'' direction
* <math>\sigma =</math> [[Electrical resistivity and conductivity|conductivity]] in [[Siemens (unit)|Siemens]]/meter
* <math>\varepsilon = \varepsilon' - j\varepsilon'' =</math> [[Permittivity#Complex permittivity|complex permittivity]]
* <math>\mu = \mu' - j\mu'' =</math> [[Permeability (electromagnetism)#Complex permeability|complex permeability]]
* <math>j=\sqrt{-1}</math>

The [[sign convention]] is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x-direction.

[[Wavelength]], [[phase velocity]], and [[skin effect|skin depth]] have simple relationships to the components of the wavenumber:
: <math> \lambda = \frac {2 \pi} {k'} \qquad  v_p = \frac {\omega} {k'}   \qquad  \delta = \frac 1 {k''}  </math>

== In wave equations ==
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.

In general, the angular wavenumber ''k'' (i.e. the [[magnitude (mathematics)|magnitude]] of the [[wave vector]]) is given by
: <math>k = \frac{2\pi}{\lambda} = \frac{2\pi\nu}{v_\mathrm{p}}=\frac{\omega}{v_\mathrm{p}}</math>
where ''ν'' is the frequency of the wave, ''λ'' is the wavelength, ''ω'' = 2''πν'' is the [[angular frequency]] of the wave, and ''v''<sub>p</sub> 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:
: <math>k = \frac{E}{\hbar c} = \frac{\omega}{c}</math>
where ''E'' is the [[energy]] of the wave, ''ħ'' is the [[reduced Planck constant]], and ''c'' is the [[speed of light]] in a vacuum.

For the special case of a [[matter wave]], for example an electron wave, in the non-relativistic approximation (in the case of a [[free particle]], that is, the particle has no potential energy):
: <math>k \equiv \frac{2\pi}{\lambda} = \frac{p}{\hbar}= \frac{\sqrt{2 m E }}{\hbar} </math>
Here ''p'' is the [[momentum]] of the particle, ''m'' is the [[mass]] of the particle, ''E'' is the [[kinetic energy]] of the particle, and ''ħ'' is the [[reduced Planck constant]].

Wavenumber is also used to define the [[group velocity]].

== In spectroscopy ==
In [[spectroscopy]], "wavenumber" <math>\tilde{\nu}</math> (in [[Reciprocal meter|reciprocal centimeters]], cm<sup>−1</sup>) refers to a temporal frequency (in hertz) which has been divided by the [[speed of light in vacuum]] (usually in centimeters per second, cm⋅s<sup>−1</sup>):
: <math> \tilde{\nu} = \frac{\nu}{c} = \frac{\omega}{2\pi c}. </math>
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:
: <math>\lambda_{\rm vac} = \frac{1}{\tilde \nu},</math>
which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from [[diffraction grating]]s and the distance between fringes in [[interferometer]]s, 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 mechanics|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.

For example, the spectroscopic wavenumbers of the [[Hydrogen spectral series|emission spectrum of atomic hydrogen]] are given by the [[Rydberg formula]]:
: <math> \tilde{\nu} = R\left(\frac{1}{{n_\text{f}}^2} - \frac{1}{{n_\text{i}}^2}\right), </math>
where ''R'' is the [[Rydberg constant]], and ''n''<sub>i</sub> and ''n''<sub>f</sub> are the [[principal quantum number]]s of the initial and final levels respectively (''n''<sub>i</sub> is greater than ''n''<sub>f</sub> for emission).

A spectroscopic wavenumber can be converted into [[photon energy|energy per photon]] ''E'' by [[Planck's relation]]:
: <math>E = hc\tilde{\nu}.</math>
It can also be converted into wavelength of light:
: <math>\lambda = \frac{1}{n \tilde \nu},</math>
where ''n'' is the [[refractive index]] of the [[optical medium|medium]]. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.

Often spatial frequencies are stated by some authors "in wavenumbers",<ref>See for example,
* {{cite journal |last1=Fiechtner |first1=G.
 |year=2001
 |title=Absorption and the dimensionless overlap integral for two-photon excitation
 |journal=[[Journal of Quantitative Spectroscopy and Radiative Transfer]]
 |volume=68 |issue=5 |pages=543–557
 |doi=10.1016/S0022-4073(00)00044-3
 |bibcode = 2001JQSRT..68..543F |url=https://zenodo.org/record/1259655
}}
* {{cite patent
 |invent1=Ray, James C.
 |invent2=Asari, Logan R.
 |pubdate=1991-09-10
 |title=Method and apparatus for spectroscopic comparison of compositions
 |country=US 
 |number=5046846
}}
* {{cite journal
 |year=2005
 |title=Boson Peaks and Glass Formation
 |journal=[[Science (journal)|Science]]
 |volume=308 |issue=5726 |pages=1221
 |doi=10.1126/science.308.5726.1221a
 |s2cid=220096687
}}</ref> incorrectly transferring the name of the quantity to the CGS unit cm<sup>−1</sup> itself.<ref>
{{cite book
 |last=Hollas
 |first=J. Michael
 |date=2004
 |title=Modern spectroscopy
 |url=https://books.google.com/books?id=lVyXQZkcKKkC&pg=PR22
 |publisher=John Wiley & Sons
 |page=xxii
 |isbn=978-0470844151
}}</ref>

== See also ==
* [[Angular wavelength]]
* [[Spatial frequency]]
* [[Refractive index]]
* [[Zonal wavenumber]]

== References ==
{{reflist}}

== External links ==
* {{Commons category-inline|Wavenumber}}

{{Authority control}}

[[Category:Wave mechanics]]
[[Category:Scalar physical quantities]]
[[Category:Units of frequency]]
[[Category:Quotients]]