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class="vector-toc-list"> </ul> </li> <li id="toc-In_spectroscopy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_spectroscopy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>In spectroscopy</span> </div> </a> <ul id="toc-In_spectroscopy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" 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href="https://be.wikipedia.org/wiki/%D0%A5%D0%B2%D0%B0%D0%BB%D0%B5%D0%B2%D1%8B_%D0%BB%D1%96%D0%BA" title="Хвалевы лік – Belarusian" lang="be" hreflang="be" data-title="Хвалевы лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%8A%D0%BB%D0%BD%D0%BE%D0%B2%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Вълново число – Bulgarian" lang="bg" hreflang="bg" data-title="Вълново число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Talasni_broj" title="Talasni broj – Bosnian" lang="bs" hreflang="bs" data-title="Talasni broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_d%27ona" title="Nombre d'ona – Catalan" lang="ca" hreflang="ca" data-title="Nombre d'ona" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlnov%C3%A9_%C4%8D%C3%ADslo" title="Vlnové číslo – Czech" lang="cs" hreflang="cs" data-title="Vlnové číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/B%C3%B8lgetal" title="Bølgetal – Danish" lang="da" hreflang="da" data-title="Bølgetal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Wellenzahl" title="Wellenzahl – German" lang="de" hreflang="de" data-title="Wellenzahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lainearv" title="Lainearv – Estonian" lang="et" hreflang="et" data-title="Lainearv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%85%CE%BC%CE%B1%CF%84%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Κυματαριθμός – Greek" lang="el" hreflang="el" data-title="Κυματαριθμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_onda" title="Número de onda – Spanish" lang="es" hreflang="es" data-title="Número de onda" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D9%88%D8%AC" title="عدد موج – Persian" lang="fa" hreflang="fa" data-title="عدد موج" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_d%27onde" title="Nombre d'onde – French" lang="fr" hreflang="fr" data-title="Nombre d'onde" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_de_onda" title="Número de onda – Galician" lang="gl" hreflang="gl" data-title="Número de onda" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8C%8C%EC%88%98" title="파수 – Korean" lang="ko" hreflang="ko" data-title="파수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%AC%D5%AB%D6%84%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A9%D5%AB%D5%BE" title="Ալիքային թիվ – Armenian" lang="hy" hreflang="hy" data-title="Ալիքային թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Valni_broj" title="Valni broj – Croatian" lang="hr" hreflang="hr" data-title="Valni broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_gelombang" title="Bilangan gelombang – Indonesian" lang="id" hreflang="id" data-title="Bilangan gelombang" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_d%27onda" title="Numero d'onda – Italian" lang="it" hreflang="it" data-title="Numero d'onda" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%92%D7%9C" title="מספר גל – Hebrew" lang="he" hreflang="he" data-title="מספר גל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%90%E1%83%9A%E1%83%A6%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ტალღური რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="ტალღური რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BB%D2%9B%D1%8B%D0%BD%D0%B4%D1%8B%D2%9B_%D1%81%D0%B0%D0%BD" title="Толқындық сан – Kazakh" lang="kk" hreflang="kk" data-title="Толқындық сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hull%C3%A1msz%C3%A1m" title="Hullámszám – Hungarian" lang="hu" hreflang="hu" data-title="Hullámszám" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Golfgetal" title="Golfgetal – Dutch" lang="nl" hreflang="nl" data-title="Golfgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B3%A2%E6%95%B0" title="波数 – Japanese" lang="ja" hreflang="ja" data-title="波数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/B%C3%B8lgetall" title="Bølgetall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bølgetall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/B%C3%B8lgjetal" title="Bølgjetal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Bølgjetal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba_falowa" title="Liczba falowa – Polish" lang="pl" hreflang="pl" data-title="Liczba falowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_de_onda" title="Número de onda – Portuguese" lang="pt" hreflang="pt" data-title="Número de onda" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%BE%D0%BB%D0%BD%D0%BE%D0%B2%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Волновое число – Russian" lang="ru" hreflang="ru" data-title="Волновое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vlnov%C3%A9_%C4%8D%C3%ADslo" title="Vlnové číslo – Slovak" lang="sk" hreflang="sk" data-title="Vlnové číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Valovno_%C5%A1tevilo" title="Valovno število – Slovenian" lang="sl" hreflang="sl" data-title="Valovno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Talasni_broj" title="Talasni broj – Serbian" lang="sr" hreflang="sr" data-title="Talasni broj" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Valni_broj" title="Valni broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Valni broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aaltoluku" title="Aaltoluku – Finnish" lang="fi" hreflang="fi" data-title="Aaltoluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/V%C3%A5gtal" title="Vågtal – Swedish" lang="sv" hreflang="sv" data-title="Vågtal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B2%E0%AF%88_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="அலை எண் – Tamil" lang="ta" hreflang="ta" data-title="அலை எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A5%D0%B2%D0%B8%D0%BB%D1%8C%D0%BE%D0%B2%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Хвильове число – Ukrainian" lang="uk" hreflang="uk" data-title="Хвильове число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_s%C3%B3ng" title="Số sóng – Vietnamese" lang="vi" hreflang="vi" data-title="Số sóng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B3%A2%E6%95%B8" title="波數 – Chinese" lang="zh" hreflang="zh" data-title="波數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q192510#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Commutative_diagram_of_harmonic_wave_properties.svg/300px-Commutative_diagram_of_harmonic_wave_properties.svg.png" decoding="async" width="300" height="376" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Commutative_diagram_of_harmonic_wave_properties.svg/450px-Commutative_diagram_of_harmonic_wave_properties.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Commutative_diagram_of_harmonic_wave_properties.svg/600px-Commutative_diagram_of_harmonic_wave_properties.svg.png 2x" data-file-width="689" data-file-height="864" /></a><figcaption>Diagram illustrating the relationship between the wavenumber and the other properties of harmonic waves.</figcaption></figure> <p>In the <a href="/wiki/Physical_science" class="mw-redirect" title="Physical science">physical sciences</a>, the <b>wavenumber</b> (or <b>wave number</b>), also known as <b>repetency</b>,<sup id="cite_ref-iso80000-3_1-0" class="reference"><a href="#cite_note-iso80000-3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is the <i><a href="/wiki/Spatial_frequency" title="Spatial frequency">spatial frequency</a></i> of a <a href="/wiki/Wave" title="Wave">wave</a>, measured in <a href="/wiki/Cycle_(angular_unit)" class="mw-redirect" title="Cycle (angular unit)">cycles</a> per unit distance (<b>ordinary wavenumber</b>) or <a href="/wiki/Radian" title="Radian">radians</a> per unit distance (<b>angular wavenumber</b>).<sup id="cite_ref-Rodrigues_Sardinha_Pita_2021_p._73_2-0" class="reference"><a href="#cite_note-Rodrigues_Sardinha_Pita_2021_p._73-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Solimini_2016_p._679_3-0" class="reference"><a href="#cite_note-Solimini_2016_p._679-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Robinson_Treitel_2008_p._9_4-0" class="reference"><a href="#cite_note-Robinson_Treitel_2008_p._9-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It is analogous to temporal <a href="/wiki/Frequency" title="Frequency">frequency</a>, which is defined as the number of wave cycles per unit time (<i>ordinary frequency</i>) or radians per unit time (<i>angular frequency</i>). </p><p>In <a href="/wiki/Multidimensional_systems" class="mw-redirect" title="Multidimensional systems">multidimensional systems</a>, the wavenumber is the magnitude of the <i><a href="/wiki/Wave_vector" title="Wave vector">wave vector</a></i>. The space of wave vectors is called <i><a href="/wiki/Reciprocal_space" class="mw-redirect" title="Reciprocal space">reciprocal space</a></i>. Wave numbers and wave vectors play an essential role in <a href="/wiki/Optics" title="Optics">optics</a> and the physics of wave <a href="/wiki/Scattering" title="Scattering">scattering</a>, such as <a href="/wiki/X-ray_diffraction" title="X-ray diffraction">X-ray diffraction</a>, <a href="/wiki/Neutron_diffraction" title="Neutron diffraction">neutron diffraction</a>, <a href="/wiki/Electron_diffraction" title="Electron diffraction">electron diffraction</a>, and <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particle</a> physics. For <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> waves, the wavenumber multiplied by the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a> is the <i><a href="/wiki/Momentum_operator" title="Momentum operator">canonical momentum</a></i>. </p><p>Wavenumber can be used to specify quantities other than spatial frequency. For example, in <a href="/wiki/Optical_spectroscopy" class="mw-redirect" title="Optical spectroscopy">optical spectroscopy</a>, it is often used as a unit of temporal frequency assuming a certain <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wavenumber, as used in <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a> and most chemistry fields, is defined as the number of <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a> per unit distance, typically centimeters (cm<sup>−1</sup>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2bec8816dd6a9197a6f041a75aab959459bf42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.543ex; height:5.343ex;" alt="{\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}"></span></dd></dl> <p>where <i>λ</i> is the wavelength. It is sometimes called the "spectroscopic wavenumber".<sup id="cite_ref-iso80000-3_1-1" class="reference"><a href="#cite_note-iso80000-3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It equals the <a href="/wiki/Spatial_frequency" title="Spatial frequency">spatial frequency</a>. </p><p>For example, a wavenumber in inverse centimeters can be converted to a frequency expressed in the unit gigahertz by multiplying by <span class="nowrap"><span data-sort-value="7008299792458000000♠"></span>29.979<span style="margin-left:.25em;">2458</span> cm/ns</span> (the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, in centimeters per nanosecond);<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> conversely, an electromagnetic wave at 29.9792458 GHz has a wavelength of 1 cm in free space. </p><p>In theoretical physics, a wave number, defined as the number of radians per unit distance, sometimes called "angular wavenumber", is more often used:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\;=\;{\frac {2\pi }{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\;=\;{\frac {2\pi }{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8858be973eace693ed39a7e4f1e4631c2a7c2172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.931ex; height:5.343ex;" alt="{\displaystyle k\;=\;{\frac {2\pi }{\lambda }}}"></span></dd></dl> <p>When wavenumber is represented by the symbol <span class="texhtml"><a href="/wiki/Nu_(letter)" title="Nu (letter)"><var>ν</var></a></span>, a <a href="/wiki/Frequency" title="Frequency">frequency</a> is still being represented, albeit indirectly. As described in the spectroscopy section, this is done through the relationship <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\nu _{\text{s}}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>c</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>≡</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\nu _{\text{s}}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99fbee0cfff025bf0cc46b940df9ba0797b8a4ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.874ex; height:3.676ex;" alt="{\textstyle {\frac {\nu _{\text{s}}}{c}}\;=\;{\frac {1}{\lambda }}\;\equiv \;{\tilde {\nu }},}"></span> where <span class="texhtml"><var>ν</var></span><sub>s</sub> is a frequency expressed in the unit <a href="/wiki/Hertz" title="Hertz">hertz</a>. This is done for convenience as frequencies tend to be very large.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Wavenumber has <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensions</a> of <a href="/wiki/Reciprocal_length" title="Reciprocal length">reciprocal length</a>, so its <a href="/wiki/International_System_of_Units" title="International System of Units">SI unit</a> is the reciprocal of meters (m<sup>−1</sup>). In <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a> it is usual to give wavenumbers in <a href="/wiki/Cgs_unit" class="mw-redirect" title="Cgs unit">cgs unit</a> (i.e., reciprocal centimeters; cm<sup>−1</sup>); in this context, the wavenumber was formerly called the <i>kayser</i>, after <a href="/wiki/Heinrich_Kayser" title="Heinrich Kayser">Heinrich Kayser</a> (some older scientific papers used this unit, abbreviated as <i>K</i>, where 1<span class="nowrap"> </span>K = 1<span class="nowrap"> </span>cm<sup>−1</sup>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The angular wavenumber may be expressed in the unit <a href="/wiki/Radian" title="Radian">radian</a> per meter (rad⋅m<sup>−1</sup>), or as above, since the <a href="/wiki/Radian" title="Radian">radian</a> is <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a>. </p><p>For <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> in vacuum, wavenumber is directly proportional to frequency and to <a href="/wiki/Photon" title="Photon">photon</a> energy. Because of this, wavenumbers are used as a convenient <a href="/wiki/Unit_of_energy" class="mw-redirect" title="Unit of energy">unit of energy</a> in spectroscopy. </p> <div class="mw-heading mw-heading3"><h3 id="Complex">Complex</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=2" title="Edit section: Complex"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A complex-valued wavenumber can be defined for a medium with complex-valued relative <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287bce0849b56661bbaebca9942dfe3a6cadc21c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.057ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{r}}"></span>, relative <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e2dc5a760017c379bd66f0043d213db98bf77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.375ex; height:2.176ex;" alt="{\displaystyle \mu _{r}}"></span> and <a href="/wiki/Refraction_index" class="mw-redirect" title="Refraction index">refraction index</a> <i>n</i> as:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=k_{0}{\sqrt {\varepsilon _{r}\mu _{r}}}=k_{0}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=k_{0}{\sqrt {\varepsilon _{r}\mu _{r}}}=k_{0}n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8a017256a357edf783aeb7c6f2c5fd72329e96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.702ex; height:3.009ex;" alt="{\displaystyle k=k_{0}{\sqrt {\varepsilon _{r}\mu _{r}}}=k_{0}n}"></span></dd></dl> <p>where <i>k</i><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 <a href="/wiki/Evanescent_field" title="Evanescent field">evanescent fields</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Plane_waves_in_linear_media">Plane waves in linear media</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=3" title="Edit section: Plane waves in linear media"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The propagation factor of a sinusoidal plane wave propagating in the positive x direction in a linear material is given by<sup id="cite_ref-Harrington_1961_10-0" class="reference"><a href="#cite_note-Harrington_1961-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 51">: 51 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=e^{-jkx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>j</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=e^{-jkx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97d24dd9ee3325cc21f1b1bb96dc93e85e7088c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.912ex; height:2.676ex;" alt="{\displaystyle P=e^{-jkx}}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>j</mi> <msup> <mi>k</mi> <mo>″</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <msup> <mi>μ</mi> <mo>″</mo> </msup> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <msup> <mi>μ</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>σ</mi> <mo>+</mo> <mi>ω</mi> <msup> <mi>ε</mi> <mo>″</mo> </msup> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <msup> <mi>ε</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26870fcd660116c237c80bffb28ebd09323ac1d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:50.785ex; height:4.676ex;" alt="{\displaystyle k=k'-jk''={\sqrt {-\left(\omega \mu ''+j\omega \mu '\right)\left(\sigma +\omega \varepsilon ''+j\omega \varepsilon '\right)}}\;}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k'=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k'=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d662504990cb8c46708301cdb1b54ca317d06d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.349ex; height:2.509ex;" alt="{\displaystyle k'=}"></span> <a href="/wiki/Phase_constant" class="mw-redirect" title="Phase constant">phase constant</a> in the units of <a href="/wiki/Radian" title="Radian">radians</a>/meter</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k''=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mo>″</mo> </msup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k''=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7dfd9af434ae3d00681b629f1e5ee4fccf8cf7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.802ex; height:2.509ex;" alt="{\displaystyle k''=}"></span> <a href="/wiki/Attenuation_constant" class="mw-redirect" title="Attenuation constant">attenuation constant</a> in the units of <a href="/wiki/Neper" title="Neper">nepers</a>/meter</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7830275522ed715a3400458c3af80a67e4246fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.899ex; height:1.676ex;" alt="{\displaystyle \omega =}"></span> angular frequency</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f52c38d57d310f38e2bbc90a27860263a3bb7309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.783ex; height:1.676ex;" alt="{\displaystyle x=}"></span> distance traveled in the <i>x</i> direction</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26bca8bd6da6ec8db793ef3e7158cc6c5a94881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.783ex; height:1.676ex;" alt="{\displaystyle \sigma =}"></span> <a href="/wiki/Electrical_resistivity_and_conductivity" title="Electrical resistivity and conductivity">conductivity</a> in <a href="/wiki/Siemens_(unit)" title="Siemens (unit)">Siemens</a>/meter</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <msup> <mi>ε</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>j</mi> <msup> <mi>ε</mi> <mo>″</mo> </msup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95235c7fb971d5d56a158695bee897481066794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.423ex; height:2.843ex;" alt="{\displaystyle \varepsilon =\varepsilon '-j\varepsilon ''=}"></span> <a href="/wiki/Permittivity#Complex_permittivity" title="Permittivity">complex permittivity</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu '-j\mu ''=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msup> <mi>μ</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>j</mi> <msup> <mi>μ</mi> <mo>″</mo> </msup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu '-j\mu ''=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754e0a3ffb86dacd2b348d6a6ed79d229d5a2436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.377ex; height:3.009ex;" alt="{\displaystyle \mu =\mu '-j\mu ''=}"></span> <a href="/wiki/Permeability_(electromagnetism)#Complex_permeability" title="Permeability (electromagnetism)">complex permeability</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j={\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec2c043c1bedcc4aba77e29a310a0adcb96a929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:8.99ex; height:3.009ex;" alt="{\displaystyle j={\sqrt {-1}}}"></span></li></ul> <p>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. </p><p><a href="/wiki/Wavelength" title="Wavelength">Wavelength</a>, <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>, and <a href="/wiki/Skin_effect" title="Skin effect">skin depth</a> have simple relationships to the components of the wavenumber: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {2\pi }{k'}}\qquad v_{p}={\frac {\omega }{k'}}\qquad \delta ={\frac {1}{k''}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <msup> <mi>k</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <msup> <mi>k</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mspace width="2em" /> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mo>″</mo> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {2\pi }{k'}}\qquad v_{p}={\frac {\omega }{k'}}\qquad \delta ={\frac {1}{k''}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dffb1bbe0422339b0d5977bc50494ca87c03bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.424ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {2\pi }{k'}}\qquad v_{p}={\frac {\omega }{k'}}\qquad \delta ={\frac {1}{k''}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="In_wave_equations">In wave equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=4" title="Edit section: In wave equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 <a href="/wiki/Wavepacket" class="mw-redirect" title="Wavepacket">wavepacket</a> for discussion of the case when these quantities are not constant. </p><p>In general, the angular wavenumber <i>k</i> (i.e. the <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> of the <a href="/wiki/Wave_vector" title="Wave vector">wave vector</a>) is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>ν</mi> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c788109ff6f42bb7dbb9b9c25df4da34793a5f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.51ex; height:5.843ex;" alt="{\displaystyle k={\frac {2\pi }{\lambda }}={\frac {2\pi \nu }{v_{\mathrm {p} }}}={\frac {\omega }{v_{\mathrm {p} }}}}"></span></dd></dl> <p>where <i>ν</i> is the frequency of the wave, <i>λ</i> is the wavelength, <i>ω</i> = 2<i>πν</i> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the wave, and <i>v</i><sub>p</sub> is the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a>. </p><p>For the special case of an <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a> in a vacuum, in which the wave propagates at the speed of light, <i>k</i> is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {E}{\hbar c}}={\frac {\omega }{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {E}{\hbar c}}={\frac {\omega }{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2890e1e97447d6946789354979a4172ca16691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.84ex; height:5.343ex;" alt="{\displaystyle k={\frac {E}{\hbar c}}={\frac {\omega }{c}}}"></span></dd></dl> <p>where <i>E</i> is the <a href="/wiki/Energy" title="Energy">energy</a> of the wave, <i>ħ</i> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, and <i>c</i> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in a vacuum. </p><p>For the special case of a <a href="/wiki/Matter_wave" title="Matter wave">matter wave</a>, 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): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\equiv {\frac {2\pi }{\lambda }}={\frac {p}{\hbar }}={\frac {\sqrt {2mE}}{\hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mi>m</mi> <mi>E</mi> </msqrt> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\equiv {\frac {2\pi }{\lambda }}={\frac {p}{\hbar }}={\frac {\sqrt {2mE}}{\hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0a38f022fbe3a7175c5a8d081b9ad74e94dd89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.73ex; height:6.009ex;" alt="{\displaystyle k\equiv {\frac {2\pi }{\lambda }}={\frac {p}{\hbar }}={\frac {\sqrt {2mE}}{\hbar }}}"></span></dd></dl> <p>Here <i>p</i> is the <a href="/wiki/Momentum" title="Momentum">momentum</a> of the particle, <i>m</i> is the <a href="/wiki/Mass" title="Mass">mass</a> of the particle, <i>E</i> is the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> of the particle, and <i>ħ</i> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>. </p><p>Wavenumber is also used to define the <a href="/wiki/Group_velocity" title="Group velocity">group velocity</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_spectroscopy">In spectroscopy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=5" title="Edit section: In spectroscopy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a>, "wavenumber" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6b66a3de6d14b026bff87c519315d260be4c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.316ex; height:2.176ex;" alt="{\displaystyle {\tilde {\nu }}}"></span> (in <a href="/wiki/Reciprocal_meter" class="mw-redirect" title="Reciprocal meter">reciprocal centimeters</a>, cm<sup>−1</sup>) refers to a temporal frequency (in hertz) which has been divided by the <a href="/wiki/Speed_of_light_in_vacuum" class="mw-redirect" title="Speed of light in vacuum">speed of light in vacuum</a> (usually in centimeters per second, cm⋅s<sup>−1</sup>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d967f15817ddec4b1ef8143aaf22c96b392489c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.566ex; height:4.676ex;" alt="{\displaystyle {\tilde {\nu }}={\frac {\nu }{c}}={\frac {\omega }{2\pi c}}.}"></span></dd></dl> <p>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 <a href="/wiki/Interferometer" class="mw-redirect" title="Interferometer">interferometer</a> : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\rm {vac}}={\frac {1}{\tilde {\nu }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\rm {vac}}={\frac {1}{\tilde {\nu }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb0ff7d6c030180926eff0cc5ad2fe3e7350fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.905ex; height:5.176ex;" alt="{\displaystyle \lambda _{\rm {vac}}={\frac {1}{\tilde {\nu }}},}"></span></dd></dl> <p>which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction gratings</a> and the distance between fringes in <a href="/wiki/Interferometer" class="mw-redirect" title="Interferometer">interferometers</a>, when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of <a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Johannes Rydberg</a> in the 1880s. The <a href="/wiki/Rydberg%E2%80%93Ritz_combination_principle" title="Rydberg–Ritz combination principle">Rydberg–Ritz combination principle</a> of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum theory</a> 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. </p><p>For example, the spectroscopic wavenumbers of the <a href="/wiki/Hydrogen_spectral_series" title="Hydrogen spectral series">emission spectrum of atomic hydrogen</a> are given by the <a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg formula</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089707c8342c060a9268b4707083e4be7fd48bcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.965ex; height:6.176ex;" alt="{\displaystyle {\tilde {\nu }}=R\left({\frac {1}{{n_{\text{f}}}^{2}}}-{\frac {1}{{n_{\text{i}}}^{2}}}\right),}"></span></dd></dl> <p>where <i>R</i> is the <a href="/wiki/Rydberg_constant" title="Rydberg constant">Rydberg constant</a>, and <i>n</i><sub>i</sub> and <i>n</i><sub>f</sub> are the <a href="/wiki/Principal_quantum_number" title="Principal quantum number">principal quantum numbers</a> of the initial and final levels respectively (<i>n</i><sub>i</sub> is greater than <i>n</i><sub>f</sub> for emission). </p><p>A spectroscopic wavenumber can be converted into <a href="/wiki/Photon_energy" title="Photon energy">energy per photon</a> <i>E</i> by <a href="/wiki/Planck%27s_relation" class="mw-redirect" title="Planck's relation">Planck's relation</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=hc{\tilde {\nu }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>h</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=hc{\tilde {\nu }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b260eda3e07d72eb26ead5cd259a353ac3414a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.183ex; height:2.176ex;" alt="{\displaystyle E=hc{\tilde {\nu }}.}"></span></dd></dl> <p>It can also be converted into wavelength of light: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {1}{n{\tilde {\nu }}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {1}{n{\tilde {\nu }}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e473dba5a33f88b9ee8d9eccaade0387d1fc102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.648ex; height:5.176ex;" alt="{\displaystyle \lambda ={\frac {1}{n{\tilde {\nu }}}},}"></span></dd></dl> <p>where <i>n</i> is the <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a> of the <a href="/wiki/Optical_medium" title="Optical medium">medium</a>. Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant. </p><p>Often spatial frequencies are stated by some authors "in wavenumbers",<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> incorrectly transferring the name of the quantity to the CGS unit cm<sup>−1</sup> itself.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Angular_wavelength" class="mw-redirect" title="Angular wavelength">Angular wavelength</a></li> <li><a href="/wiki/Spatial_frequency" title="Spatial frequency">Spatial frequency</a></li> <li><a href="/wiki/Refractive_index" title="Refractive index">Refractive index</a></li> <li><a href="/wiki/Zonal_wavenumber" title="Zonal wavenumber">Zonal wavenumber</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-iso80000-3-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-iso80000-3_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-iso80000-3_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.iso.org/standard/64974.html"><i>ISO 80000-3:2019 Quantities and units – Part 3: Space and time</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=ISO+80000-3%3A2019+Quantities+and+units+%E2%80%93+Part+3%3A+Space+and+time&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F64974.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-Rodrigues_Sardinha_Pita_2021_p._73-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rodrigues_Sardinha_Pita_2021_p._73_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRodriguesSardinhaPita2021" class="citation book cs1">Rodrigues, A.; Sardinha, R.A.; Pita, G. (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jVYlEAAAQBAJ&pg=PA73"><i>Fundamental Principles of Environmental Physics</i></a>. Springer International Publishing. p. 73. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-69025-0" title="Special:BookSources/978-3-030-69025-0"><bdi>978-3-030-69025-0</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-12-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Principles+of+Environmental+Physics&rft.pages=73&rft.pub=Springer+International+Publishing&rft.date=2021&rft.isbn=978-3-030-69025-0&rft.aulast=Rodrigues&rft.aufirst=A.&rft.au=Sardinha%2C+R.A.&rft.au=Pita%2C+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjVYlEAAAQBAJ%26pg%3DPA73&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-Solimini_2016_p._679-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Solimini_2016_p._679_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolimini2016" class="citation book cs1">Solimini, D. (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dSMGDAAAQBAJ&pg=PA679"><i>Understanding Earth Observation: The Electromagnetic Foundation of Remote Sensing</i></a>. Remote Sensing and Digital Image Processing. Springer International Publishing. p. 679. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-25633-7" title="Special:BookSources/978-3-319-25633-7"><bdi>978-3-319-25633-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-12-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Earth+Observation%3A+The+Electromagnetic+Foundation+of+Remote+Sensing&rft.series=Remote+Sensing+and+Digital+Image+Processing&rft.pages=679&rft.pub=Springer+International+Publishing&rft.date=2016&rft.isbn=978-3-319-25633-7&rft.aulast=Solimini&rft.aufirst=D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdSMGDAAAQBAJ%26pg%3DPA679&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-Robinson_Treitel_2008_p._9-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Robinson_Treitel_2008_p._9_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinsonTreitel2008" class="citation book cs1">Robinson, E.A.; Treitel, S. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IH-Pu3PlJgAC&pg=PA9"><i>Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing</i></a>. Geophysical references. Society of Exploration Geophysicists. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56080-148-1" title="Special:BookSources/978-1-56080-148-1"><bdi>978-1-56080-148-1</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-12-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Imaging+and+Deconvolution%3A+The+ABCs+of+Seismic+Exploration+and+Processing&rft.series=Geophysical+references&rft.pages=9&rft.pub=Society+of+Exploration+Geophysicists&rft.date=2008&rft.isbn=978-1-56080-148-1&rft.aulast=Robinson&rft.aufirst=E.A.&rft.au=Treitel%2C+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIH-Pu3PlJgAC%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.nist.gov/PhysRefData/wavenum/html/tables.html">"NIST: Wavenumber Calibration Tables - Description"</a>. <i>physics.nist.gov</i><span class="reference-accessdate">. Retrieved <span class="nowrap">19 March</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=physics.nist.gov&rft.atitle=NIST%3A+Wavenumber+Calibration+Tables+-+Description&rft_id=https%3A%2F%2Fphysics.nist.gov%2FPhysRefData%2Fwavenum%2Fhtml%2Ftables.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFW." class="citation web cs1">W., Weisstein, Eric. <a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/Wavenumber.html">"Wavenumber -- from Eric Weisstein's World of Physics"</a>. <i>scienceworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">19 March</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=scienceworld.wolfram.com&rft.atitle=Wavenumber+--+from+Eric+Weisstein%27s+World+of+Physics&rft.aulast=W.&rft.aufirst=Weisstein%2C+Eric&rft_id=http%3A%2F%2Fscienceworld.wolfram.com%2Fphysics%2FWavenumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/637882/wave-number">"Wave number"</a>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">19 April</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclop%C3%A6dia+Britannica&rft.atitle=Wave+number&rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F637882%2Fwave-number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurthyLakshman1981" class="citation journal cs1">Murthy, V. L. R.; Lakshman, S. V. J. (1981). "Electronic absorption spectrum of cobalt antipyrine complex". <i>Solid State Communications</i>. <b>38</b> (7): 651–652. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981SSCom..38..651M">1981SSCom..38..651M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0038-1098%2881%2990960-1">10.1016/0038-1098(81)90960-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Solid+State+Communications&rft.atitle=Electronic+absorption+spectrum+of+cobalt+antipyrine+complex&rft.volume=38&rft.issue=7&rft.pages=651-652&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0038-1098%2881%2990960-1&rft_id=info%3Abibcode%2F1981SSCom..38..651M&rft.aulast=Murthy&rft.aufirst=V.+L.+R.&rft.au=Lakshman%2C+S.+V.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="http://www.ece.rutgers.edu/~orfanidi/ewa/ch02.pdf">[1]</a>, eq.(2.13.3)</span> </li> <li id="cite_note-Harrington_1961-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Harrington_1961_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarrington1961" class="citation cs2">Harrington, Roger F. (1961), <i>Time-Harmonic Electromagnetic Fields</i> (1st ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-026745-6" title="Special:BookSources/0-07-026745-6"><bdi>0-07-026745-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time-Harmonic+Electromagnetic+Fields&rft.edition=1st&rft.pub=McGraw-Hill&rft.date=1961&rft.isbn=0-07-026745-6&rft.aulast=Harrington&rft.aufirst=Roger+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">See for example, <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFiechtner2001" class="citation journal cs1">Fiechtner, G. (2001). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1259655">"Absorption and the dimensionless overlap integral for two-photon excitation"</a>. <i><a href="/w/index.php?title=Journal_of_Quantitative_Spectroscopy_and_Radiative_Transfer&action=edit&redlink=1" class="new" title="Journal of Quantitative Spectroscopy and Radiative Transfer (page does not exist)">Journal of Quantitative Spectroscopy and Radiative Transfer</a></i>. <b>68</b> (5): 543–557. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JQSRT..68..543F">2001JQSRT..68..543F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0022-4073%2800%2900044-3">10.1016/S0022-4073(00)00044-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Quantitative+Spectroscopy+and+Radiative+Transfer&rft.atitle=Absorption+and+the+dimensionless+overlap+integral+for+two-photon+excitation&rft.volume=68&rft.issue=5&rft.pages=543-557&rft.date=2001&rft_id=info%3Adoi%2F10.1016%2FS0022-4073%2800%2900044-3&rft_id=info%3Abibcode%2F2001JQSRT..68..543F&rft.aulast=Fiechtner&rft.aufirst=G.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1259655&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></li> <li><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><span class="citation patent" id="CITEREFRay,_James_C.Asari,_Logan_R.1991"><a rel="nofollow" class="external text" href="https://worldwide.espacenet.com/textdoc?DB=EPODOC&IDX=US5046846">US 5046846</a>, Ray, James C. & Asari, Logan R., "Method and apparatus for spectroscopic comparison of compositions", published 1991-09-10</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Apatent&rft.number=5046846&rft.cc=US&rft.title=Method+and+apparatus+for+spectroscopic+comparison+of+compositions&rft.inventor=Ray%2C+James+C.&rft.pubdate=1991-09-10"><span style="display: none;"> </span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1">"Boson Peaks and Glass Formation". <i><a href="/wiki/Science_(journal)" title="Science (journal)">Science</a></i>. <b>308</b> (5726): 1221. 2005. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.308.5726.1221a">10.1126/science.308.5726.1221a</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220096687">220096687</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Boson+Peaks+and+Glass+Formation&rft.volume=308&rft.issue=5726&rft.pages=1221&rft.date=2005&rft_id=info%3Adoi%2F10.1126%2Fscience.308.5726.1221a&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220096687%23id-name%3DS2CID&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></li></ul> </span></li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHollas2004" class="citation book cs1">Hollas, J. Michael (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lVyXQZkcKKkC&pg=PR22"><i>Modern spectroscopy</i></a>. John Wiley & Sons. p. xxii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470844151" title="Special:BookSources/978-0470844151"><bdi>978-0470844151</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+spectroscopy&rft.pages=xxii&rft.pub=John+Wiley+%26+Sons&rft.date=2004&rft.isbn=978-0470844151&rft.aulast=Hollas&rft.aufirst=J.+Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlVyXQZkcKKkC%26pg%3DPR22&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWavenumber" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wavenumber&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span 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