Editing Wavenumber (section)

== 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>