Normal mode - Wikipedia

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_mechanical_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>In mechanical systems</span> </div> </a> <button aria-controls="toc-In_mechanical_systems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In mechanical systems subsection</span> </button> <ul id="toc-In_mechanical_systems-sublist" class="vector-toc-list"> <li id="toc-Coupled_oscillators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coupled_oscillators"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Coupled oscillators</span> </div> </a> <ul id="toc-Coupled_oscillators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standing_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standing_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Standing waves</span> </div> </a> <ul id="toc-Standing_waves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elastic_solids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elastic_solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Elastic solids</span> </div> </a> <ul id="toc-Elastic_solids-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>In quantum mechanics</span> </div> </a> <ul id="toc-In_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_seismology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_seismology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In seismology</span> </div> </a> <ul id="toc-In_seismology-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">5</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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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href="https://cv.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BB%C4%95_%D1%81%D1%83%D0%BB%D0%BB%D0%B0%D0%BD%D1%83%D1%81%D0%B5%D0%BC" title="Нормаллĕ сулланусем – Chuvash" lang="cv" hreflang="cv" data-title="Нормаллĕ сулланусем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlastn%C3%AD_m%C3%B3d" title="Vlastní mód – Czech" lang="cs" hreflang="cs" data-title="Vlastní mód" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eigenmode" title="Eigenmode – German" lang="de" hreflang="de" data-title="Eigenmode" 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/Normaalv%C3%B5nkumine" title="Normaalvõnkumine – Estonian" lang="et" hreflang="et" data-title="Normaalvõnkumine" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Modo_normal" title="Modo normal – Spanish" lang="es" hreflang="es" data-title="Modo normal" 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/%D9%85%D8%AF_%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="مد نرمال – Persian" lang="fa" hreflang="fa" data-title="مد نرمال" 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/Mode_normal" title="Mode normal – French" lang="fr" hreflang="fr" data-title="Mode normal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%A5%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D5%BF%D5%A1%D5%B6%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Սեփական տատանումներ – Armenian" lang="hy" hreflang="hy" data-title="Սեփական տատանումներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A4%D7%A0%D7%99_%D7%AA%D7%A0%D7%95%D7%93%D7%94_%D7%A2%D7%A6%D7%9E%D7%99%D7%99%D7%9D" title="אופני תנודה עצמיים – Hebrew" lang="he" hreflang="he" data-title="אופני תנודה עצמיים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D0%BD%D1%88%D1%96%D0%BA%D1%82%D1%96_%D0%B6%D0%B8%D1%96%D0%BB%D1%96%D0%BA" title="Меншікті жиілік – Kazakh" lang="kk" hreflang="kk" data-title="Меншікті жиілік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Normalusis_svyravimas" title="Normalusis svyravimas – Lithuanian" lang="lt" hreflang="lt" data-title="Normalusis svyravimas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%BA%E6%9C%89%E6%8C%AF%E5%8B%95" title="固有振動 – Japanese" lang="ja" hreflang="ja" data-title="固有振動" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Eigensvinging" title="Eigensvinging – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Eigensvinging" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Normal_moda" title="Normal moda – Uzbek" lang="uz" hreflang="uz" data-title="Normal moda" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Drgania_swobodne" title="Drgania swobodne – Polish" lang="pl" hreflang="pl" data-title="Drgania swobodne" 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/Modo_normal" title="Modo normal – Portuguese" lang="pt" hreflang="pt" data-title="Modo normal" 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%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5_%D0%BA%D0%BE%D0%BB%D0%B5%D0%B1%D0%B0%D0%BD%D0%B8%D1%8F" title="Нормальные колебания – Russian" lang="ru" hreflang="ru" data-title="Нормальные колебания" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Normalni_mod" title="Normalni mod – Serbian" lang="sr" hreflang="sr" data-title="Normalni mod" 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/Normalni_mod" title="Normalni mod – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Normalni mod" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D0%BA%D0%BE%D0%BB%D0%B8%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Нормальні коливання – Ukrainian" lang="uk" hreflang="uk" data-title="Нормальні коливання" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AE%80%E6%AD%A3%E6%A8%A1" title="简正模 – Chinese" lang="zh" hreflang="zh" data-title="简正模" 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 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title="Vibration mode">Vibration mode</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Pattern of oscillating motion in a system</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">December 2010</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <b>normal mode</b> of a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> is a pattern of motion in which all parts of the system move <a href="/wiki/Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidally</a> with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequencies</a> or <a href="/wiki/Resonance" title="Resonance">resonant frequencies</a>. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. </p><p>The most general motion of a linear system is a <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> to each other. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Drum_vibration_mode12.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Drum_vibration_mode12.gif/248px-Drum_vibration_mode12.gif" decoding="async" width="248" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e9/Drum_vibration_mode12.gif 1.5x" data-file-width="249" data-file-height="170" /></a><figcaption>Vibration of a single normal mode of a circular disc with a pinned boundary condition along the entire outer edge. <a href="https://commons.wikimedia.org/wiki/Category:Drum_vibration_animations" class="extiw" title="commons:Category:Drum vibration animations">See other modes</a>.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:A_cup_of_black_coffee_vibrating_in_normal_modes.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/A_cup_of_black_coffee_vibrating_in_normal_modes.jpeg/220px-A_cup_of_black_coffee_vibrating_in_normal_modes.jpeg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/A_cup_of_black_coffee_vibrating_in_normal_modes.jpeg/330px-A_cup_of_black_coffee_vibrating_in_normal_modes.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/5/54/A_cup_of_black_coffee_vibrating_in_normal_modes.jpeg 2x" data-file-width="384" data-file-height="256" /></a><figcaption> A flash photo of a cup of black coffee vibrating in normal modes</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Spherical_harmonic_in_water_drop.ogv/220px--Spherical_harmonic_in_water_drop.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="165" data-durationhint="79" data-mwtitle="Spherical_harmonic_in_water_drop.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Spherical_harmonic_in_water_drop.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/35/Spherical_harmonic_in_water_drop.ogv/Spherical_harmonic_in_water_drop.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="640" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/3/35/Spherical_harmonic_in_water_drop.ogv" type="video/ogg; codecs="theora, vorbis"" data-width="640" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/35/Spherical_harmonic_in_water_drop.ogv/Spherical_harmonic_in_water_drop.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/35/Spherical_harmonic_in_water_drop.ogv/Spherical_harmonic_in_water_drop.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="480" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/35/Spherical_harmonic_in_water_drop.ogv/Spherical_harmonic_in_water_drop.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="480" data-height="360" /></video></span><figcaption>Excitation of normal modes in a drop of water during the <a href="/wiki/Leidenfrost_effect" title="Leidenfrost effect">Leidenfrost effect</a> </figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="General_definitions">General definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=1" title="Edit section: General definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mode">Mode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=2" title="Edit section: Mode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the <a href="/wiki/Wave" title="Wave">wave theory</a> of physics and engineering, a <b>mode</b> in a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> is a <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> state of excitation, in which all the components of the system will be affected sinusoidally at a fixed frequency associated with that mode. </p><p>Because no real system can perfectly fit under the standing wave framework, the <i>mode</i> concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a <i>linear</i> fashion, in which linear <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of states can be performed. </p><p>Typical examples include: </p> <ul><li>In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, and the displacement of the rope with respect to its static state is the modal variable.</li> <li>In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, and the displacement of the air molecules is the modal variable.</li> <li>In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are the modal variable.</li> <li>In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, and thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source (a Klystron or another RF source) is the excitation and the electromagnetic field is the modal variable.</li> <li>When relating to <a href="/wiki/Music" title="Music">music</a>, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "<a href="/wiki/Overtones" class="mw-redirect" title="Overtones">overtones</a>".</li></ul> <p>The concept of normal modes also finds application in other dynamical systems, such as <a href="/wiki/Optics" title="Optics">optics</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Atmospheric_dynamics" class="mw-redirect" title="Atmospheric dynamics">atmospheric dynamics</a> and <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>. </p><p>Most dynamical systems can be excited in several modes, possibly simultaneously. Each mode is characterized by one or several frequencies,<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="wouldn't that be a superposition of two modes? (April 2020)">dubious</span></a> – <a href="/wiki/Talk:Normal_mode#Dubious" title="Talk:Normal mode">discuss</a></i>]</sup> according to the modal variable field. For example, a vibrating rope in 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in 3D space is defined by two frequencies (2D axial displacement). </p><p>For a given amplitude on the modal variable, each mode will store a specific amount of energy because of the sinusoidal excitation. </p><p>The <i>normal</i> or <i>dominant</i> mode of a system with multiple modes will be the mode storing the minimum amount of energy for a given amplitude of the modal variable, or, equivalently, for a given stored amount of energy, the dominant mode will be the mode imposing the maximum amplitude of the modal variable. </p> <div class="mw-heading mw-heading3"><h3 id="Mode_numbers">Mode numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=3" title="Edit section: Mode numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one trough) it would be vibrating in mode 2. </p><p>In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">polar coordinates</a>, we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the anti-symmetric (also called <a href="/wiki/Symmetry_in_mathematics#Skew-symmetry" title="Symmetry in mathematics">skew-symmetry</a>) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction). </p><p>In linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes. </p> <div class="mw-heading mw-heading3"><h3 id="Nodes">Nodes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=4" title="Edit section: Nodes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg/220px-Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg/330px-Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg/440px-Mode_Shape_of_a_Round_Plate_with_Node_Lines.jpg 2x" data-file-width="450" data-file-height="450" /></a><figcaption>A mode shape of a drum membrane, with nodal lines shown in pale green</figcaption></figure> <p>In a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times. </p><p>When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about halfway between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In an idealized system these lines equal zero exactly, as shown to the right. </p> <div class="mw-heading mw-heading2"><h2 id="In_mechanical_systems">In mechanical systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=5" title="Edit section: In mechanical systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the analysis of <a href="/wiki/Conservative_system" title="Conservative system">conservative systems</a> with small displacements from equilibrium, important in <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>, <a href="/wiki/Molecular_spectra" class="mw-redirect" title="Molecular spectra">molecular spectra</a>, and <a href="/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuits</a>, the system can be transformed to new coordinates called <b>normal coordinates.</b> Each normal coordinate corresponds to a single vibrational frequency of the system and the corresponding motion of the system is called the normal mode of vibration.<sup id="cite_ref-Goldstein3_1-0" class="reference"><a href="#cite_note-Goldstein3-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 332">: 332 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Coupled_oscillators">Coupled oscillators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=6" title="Edit section: Coupled oscillators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider two equal bodies (not affected by gravity), each of <a href="/wiki/Mass" title="Mass">mass</a> <span class="texhtml mvar" style="font-style:italic;">m</span>, attached to three springs, each with <a href="/wiki/Spring_constant" class="mw-redirect" title="Spring constant">spring constant</a> <span class="texhtml mvar" style="font-style:italic;">k</span>. They are attached in the following manner, forming a system that is physically symmetric: </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Coupled_Harmonic_Oscillator.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Coupled_Harmonic_Oscillator.svg/300px-Coupled_Harmonic_Oscillator.svg.png" decoding="async" width="300" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Coupled_Harmonic_Oscillator.svg/450px-Coupled_Harmonic_Oscillator.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Coupled_Harmonic_Oscillator.svg/600px-Coupled_Harmonic_Oscillator.svg.png 2x" data-file-width="597" data-file-height="156" /></a><figcaption></figcaption></figure> <p>where the edge points are fixed and cannot move. Let <span class="texhtml"><i>x</i><sub>1</sub>(<i>t</i>)</span> denote the horizontal <a href="/wiki/Displacement_(distance)" class="mw-redirect" title="Displacement (distance)">displacement</a> of the left mass, and <span class="texhtml"><i>x</i><sub>2</sub>(<i>t</i>)</span> denote the displacement of the right mass. </p><p>Denoting acceleration (the second <a href="/wiki/Derivative" title="Derivative">derivative</a> of <span class="texhtml"><i>x</i>(<i>t</i>)</span> with respect to time) as <span class="nowrap"><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 {\ddot {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\ddot {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de38fe3d6f1ac0f36ddd411bc7fc2d4450db24f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\textstyle {\ddot {x}}}"></span>,</span> the <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> are: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}m{\ddot {x}}_{1}&=-kx_{1}+k(x_{2}-x_{1})=-2kx_{1}+kx_{2}\\m{\ddot {x}}_{2}&=-kx_{2}+k(x_{1}-x_{2})=-2kx_{2}+kx_{1}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>m</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>k</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>k</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m{\ddot {x}}_{1}&=-kx_{1}+k(x_{2}-x_{1})=-2kx_{1}+kx_{2}\\m{\ddot {x}}_{2}&=-kx_{2}+k(x_{1}-x_{2})=-2kx_{2}+kx_{1}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f2987f562e70a381a9934497c6c1fce857b6eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.246ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}m{\ddot {x}}_{1}&=-kx_{1}+k(x_{2}-x_{1})=-2kx_{1}+kx_{2}\\m{\ddot {x}}_{2}&=-kx_{2}+k(x_{1}-x_{2})=-2kx_{2}+kx_{1}\end{aligned}}}"></span> </p><p>Since we expect oscillatory motion of a normal mode (where <span class="texhtml mvar" style="font-style:italic;">ω</span> is the same for both masses), we try: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x_{1}(t)&=A_{1}e^{i\omega t}\\x_{2}(t)&=A_{2}e^{i\omega t}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}(t)&=A_{1}e^{i\omega t}\\x_{2}(t)&=A_{2}e^{i\omega t}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a179717d49d29bd970ac910727abaae206e52d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.179ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}x_{1}(t)&=A_{1}e^{i\omega t}\\x_{2}(t)&=A_{2}e^{i\omega t}\end{aligned}}}"></span> </p><p>Substituting these into the equations of motion gives us: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}-\omega ^{2}mA_{1}e^{i\omega t}&=-2kA_{1}e^{i\omega t}+kA_{2}e^{i\omega t}\\-\omega ^{2}mA_{2}e^{i\omega t}&=kA_{1}e^{i\omega t}-2kA_{2}e^{i\omega t}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>k</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mi>k</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>k</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>k</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-\omega ^{2}mA_{1}e^{i\omega t}&=-2kA_{1}e^{i\omega t}+kA_{2}e^{i\omega t}\\-\omega ^{2}mA_{2}e^{i\omega t}&=kA_{1}e^{i\omega t}-2kA_{2}e^{i\omega t}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4262e99a0d66bae53a9b4f0277e3b345233fcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.322ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}-\omega ^{2}mA_{1}e^{i\omega t}&=-2kA_{1}e^{i\omega t}+kA_{2}e^{i\omega t}\\-\omega ^{2}mA_{2}e^{i\omega t}&=kA_{1}e^{i\omega t}-2kA_{2}e^{i\omega t}\end{aligned}}}"></span> </p><p>Omitting the exponential factor (because it is common to all terms) and simplifying yields: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(\omega ^{2}m-2k)A_{1}+kA_{2}&=0\\kA_{1}+(\omega ^{2}m-2k)A_{2}&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>k</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\omega ^{2}m-2k)A_{1}+kA_{2}&=0\\kA_{1}+(\omega ^{2}m-2k)A_{2}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8353d7e69f2fa7baded64c32831cd29ba816ed0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.223ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(\omega ^{2}m-2k)A_{1}+kA_{2}&=0\\kA_{1}+(\omega ^{2}m-2k)A_{2}&=0\end{aligned}}}"></span> </p><p>And in <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> representation: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\omega ^{2}m-2k&k\\k&\omega ^{2}m-2k\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\omega ^{2}m-2k&k\\k&\omega ^{2}m-2k\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18142ad4eca1263c6e240911bd4858b050c84db4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.269ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\omega ^{2}m-2k&k\\k&\omega ^{2}m-2k\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}"></span> </p><p>If the matrix on the left is invertible, the unique solution is the trivial solution <span class="texhtml">(<i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>) = (<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>) = (0, 0)</span>. The non trivial solutions are to be found for those values of <span class="texhtml mvar" style="font-style:italic;">ω</span> whereby the matrix on the left is <a href="/wiki/Singular_matrix" class="mw-redirect" title="Singular matrix">singular</a>; i.e. is not invertible. It follows that the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the matrix must be equal to 0, so: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\omega ^{2}m-2k)^{2}-k^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega ^{2}m-2k)^{2}-k^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4078f1bee64d07c57d80de91e3252ea3e8a97f9a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.985ex; height:3.176ex;" alt="{\displaystyle (\omega ^{2}m-2k)^{2}-k^{2}=0}"></span> </p><p>Solving for <span class="texhtml mvar" style="font-style:italic;">ω</span>, the two positive solutions are: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\omega _{1}&={\sqrt {\frac {k}{m}}}\\\omega _{2}&={\sqrt {\frac {3k}{m}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>k</mi> <mi>m</mi> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>3</mn> <mi>k</mi> </mrow> <mi>m</mi> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega _{1}&={\sqrt {\frac {k}{m}}}\\\omega _{2}&={\sqrt {\frac {3k}{m}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c2d7532b03a07190aa63233ec431bdcab50e15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:11.884ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\omega _{1}&={\sqrt {\frac {k}{m}}}\\\omega _{2}&={\sqrt {\frac {3k}{m}}}\end{aligned}}}"></span> </p><p>Substituting <span class="texhtml"><i>ω</i><sub>1</sub></span> into the matrix and solving for <span class="texhtml">(<i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>)</span>, yields <span class="texhtml">(1, 1)</span>. Substituting <span class="texhtml"><i>ω</i><sub>2</sub></span> results in <span class="texhtml">(1, −1)</span>. (These vectors are <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a>, and the frequencies are <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>.) </p><p>The first normal mode is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\eta }}_{1}={\begin{pmatrix}x_{1}^{1}(t)\\x_{2}^{1}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\varphi _{1})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\eta }}_{1}={\begin{pmatrix}x_{1}^{1}(t)\\x_{2}^{1}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\varphi _{1})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf74dbfe590eea6dce9b868e42028038f7a6018" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.572ex; height:6.509ex;" alt="{\displaystyle {\vec {\eta }}_{1}={\begin{pmatrix}x_{1}^{1}(t)\\x_{2}^{1}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\varphi _{1})}}"></span> </p><p>Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric. </p><p>The second normal mode is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\eta }}_{2}={\begin{pmatrix}x_{1}^{2}(t)\\x_{2}^{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\varphi _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\eta }}_{2}={\begin{pmatrix}x_{1}^{2}(t)\\x_{2}^{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\varphi _{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3df09eb8b09fd9b6bd6f7aeef1610039226d7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.38ex; height:6.509ex;" alt="{\displaystyle {\vec {\eta }}_{2}={\begin{pmatrix}x_{1}^{2}(t)\\x_{2}^{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\varphi _{2})}}"></span> </p><p>This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. This mode is called symmetric. </p><p>The general solution is a <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of the <b>normal modes</b> where <span class="texhtml"><i>c</i><sub>1</sub></span>, <span class="texhtml"><i>c</i><sub>2</sub></span>, <span class="texhtml"><i>φ</i><sub>1</sub></span>, and <span class="texhtml"><i>φ</i><sub>2</sub></span> are determined by the <a href="/wiki/Initial_condition" title="Initial condition">initial conditions</a> of the problem. </p><p>The process demonstrated here can be generalized and formulated using the formalism of <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> or <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Standing_waves">Standing waves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=7" title="Edit section: Standing waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> is a continuous form of normal mode. In a standing wave, all the space elements (i.e. <span class="texhtml">(<i>x</i>, <i>y</i>, <i>z</i>)</span> coordinates) are oscillating in the same <a href="/wiki/Frequency" title="Frequency">frequency</a> and in <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> (reaching the <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a> point together), but each has a different amplitude. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Standing-wave05.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Standing-wave05.png" decoding="async" width="588" height="308" class="mw-file-element" data-file-width="588" data-file-height="308" /></a></span> </p><p>The general form of a standing wave is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1161bbc044521e9dc6197dcfa8079a2581a02753" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.372ex; height:2.843ex;" alt="{\displaystyle \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}"></span> </p><p>where <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>, <i>z</i>)</span> represents the dependence of amplitude on location and the cosine/sine are the oscillations in time. </p><p>Physically, standing waves are formed by the <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>, <i>z</i>)</span> form of the standing wave. This space-dependence is called a <b>normal mode</b>. </p><p>Usually, for problems with continuous dependence on <span class="texhtml">(<i>x</i>, <i>y</i>, <i>z</i>)</span> there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are <a href="/wiki/Countably_many" class="mw-redirect" title="Countably many">countably many</a> normal modes (usually numbered <span class="texhtml"><i>n</i> = 1, 2, 3, ...</span>). If the problem is not bounded, there is a continuous spectrum of normal modes. </p> <div class="mw-heading mw-heading3"><h3 id="Elastic_solids">Elastic solids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=8" title="Edit section: Elastic solids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Einstein_solid" title="Einstein solid">Einstein solid</a> and <a href="/wiki/Debye_model" title="Debye model">Debye model</a></div> <p>In any solid at any temperature, the primary particles (e.g. atoms or molecules) are not stationary, but rather vibrate about mean positions. In insulators the capacity of the solid to store thermal energy is due almost entirely to these vibrations. Many physical properties of the solid (e.g. modulus of elasticity) can be predicted given knowledge of the frequencies with which the particles vibrate. The simplest assumption (by Einstein) is that all the particles oscillate about their mean positions with the same natural frequency <span class="texhtml mvar" style="font-style:italic;">ν</span>. This is equivalent to the assumption that all atoms vibrate independently with a frequency <span class="texhtml mvar" style="font-style:italic;">ν</span>. Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of <span class="texhtml mvar" style="font-style:italic;">hν</span>. The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal <a href="/wiki/Phonons" class="mw-redirect" title="Phonons">phonons</a>). </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Harmonic_partials_on_strings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/250px-Harmonic_partials_on_strings.svg.png" decoding="async" width="250" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/375px-Harmonic_partials_on_strings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/500px-Harmonic_partials_on_strings.svg.png 2x" data-file-width="620" data-file-height="590" /></a><figcaption>The <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a> and the first six <a href="/wiki/Overtone" title="Overtone">overtones</a> of a vibrating string. The mathematics of <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a> in crystalline solids consists of treating the <a href="/wiki/Harmonics" class="mw-redirect" title="Harmonics">harmonics</a> as an ideal <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> of <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> density fluctuations (or atomic displacement waves).</figcaption></figure> <p>Debye subsequently recognized that each oscillator is intimately coupled to its neighboring oscillators at all times. Thus, by replacing Einstein's identical uncoupled oscillators with the same number of coupled oscillators, Debye correlated the elastic vibrations of a one-dimensional solid with the number of mathematically special modes of vibration of a stretched string (see figure). The pure tone of lowest pitch or frequency is referred to as the fundamental and the multiples of that frequency are called its harmonic overtones. He assigned to one of the oscillators the frequency of the fundamental vibration of the whole block of solid. He assigned to the remaining oscillators the frequencies of the harmonics of that fundamental, with the highest of all these frequencies being limited by the motion of the smallest primary unit. </p><p>The normal modes of vibration of a crystal are in general superpositions of many overtones, each with an appropriate amplitude and phase. Longer wavelength (low frequency) <a href="/wiki/Phonons" class="mw-redirect" title="Phonons">phonons</a> are exactly those acoustical vibrations which are considered in the theory of sound. Both longitudinal and transverse waves can be propagated through a solid, while, in general, only longitudinal waves are supported by fluids. </p><p>In the <a href="/wiki/Longitudinal_mode" title="Longitudinal mode">longitudinal mode</a>, the displacement of particles from their positions of equilibrium coincides with the propagation direction of the wave. Mechanical longitudinal waves have been also referred to as <i><dfn>compression waves</dfn></i>. For <a href="/wiki/Transverse_mode" title="Transverse mode">transverse modes</a>, individual particles move perpendicular to the propagation of the wave. </p><p>According to quantum theory, the mean energy of a normal vibrational mode of a crystalline solid with characteristic frequency <span class="texhtml mvar" style="font-style:italic;">ν</span> is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\nu )={\frac {1}{2}}h\nu +{\frac {h\nu }{e^{h\nu /kT}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>h</mi> <mi>ν</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(\nu )={\frac {1}{2}}h\nu +{\frac {h\nu }{e^{h\nu /kT}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9877bbda6b77e34f0268b078cebed04d83e1e0d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.134ex; height:5.843ex;" alt="{\displaystyle E(\nu )={\frac {1}{2}}h\nu +{\frac {h\nu }{e^{h\nu /kT}-1}}}"></span> </p><p>The term <span class="texhtml">(1/2)<i>hν</i></span> represents the "zero-point energy", or the energy which an oscillator will have at absolute zero. <span class="texhtml"><i>E</i>(<i>ν</i>)</span> tends to the classic value <span class="texhtml mvar" style="font-style:italic;">kT</span> at high temperatures </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\nu )=kT\left[1+{\frac {1}{12}}\left({\frac {h\nu }{kT}}\right)^{2}+O\left({\frac {h\nu }{kT}}\right)^{4}+\cdots \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>O</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(\nu )=kT\left[1+{\frac {1}{12}}\left({\frac {h\nu }{kT}}\right)^{2}+O\left({\frac {h\nu }{kT}}\right)^{4}+\cdots \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8c6718db908886f694ba371f5cdb76fed8c0b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.52ex; height:7.509ex;" alt="{\displaystyle E(\nu )=kT\left[1+{\frac {1}{12}}\left({\frac {h\nu }{kT}}\right)^{2}+O\left({\frac {h\nu }{kT}}\right)^{4}+\cdots \right]}"></span> </p><p>By knowing the thermodynamic formula, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial S}{\partial E}}\right)_{N,V}={\frac {1}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>,</mo> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S}{\partial E}}\right)_{N,V}={\frac {1}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4fe33b5e6bed76eeb804eac1e1dac78158ab93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.334ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial S}{\partial E}}\right)_{N,V}={\frac {1}{T}}}"></span> </p><p>the entropy per normal mode is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S\left(\nu \right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(\nu \right){\frac {dT}{T}}\\[10pt]&={\frac {E\left(\nu \right)}{T}}-k\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> <mrow> <mo>(</mo> <mi>ν</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>ν</mi> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>T</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>ν</mi> <mo>)</mo> </mrow> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mi>k</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S\left(\nu \right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(\nu \right){\frac {dT}{T}}\\[10pt]&={\frac {E\left(\nu \right)}{T}}-k\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e24021d0faef8e3de936800d769032a13540129" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:34.718ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}S\left(\nu \right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(\nu \right){\frac {dT}{T}}\\[10pt]&={\frac {E\left(\nu \right)}{T}}-k\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)\end{aligned}}}"></span> </p><p>The free energy is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\nu )=E-TS=kT\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\nu )=E-TS=kT\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b4615e1a1769cbabf4aef7bb6dc7737391c70d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.426ex; height:6.176ex;" alt="{\displaystyle F(\nu )=E-TS=kT\log \left(1-e^{-{\frac {h\nu }{kT}}}\right)}"></span> </p><p>which, for <span class="texhtml"><i>kT</i> ≫ <i>hν</i></span>, tends to: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\nu )=kT\log \left({\frac {h\nu }{kT}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\nu )=kT\log \left({\frac {h\nu }{kT}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac494efb57564ff7e6e89922b5f10c9403c4a54" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.192ex; height:6.176ex;" alt="{\displaystyle F(\nu )=kT\log \left({\frac {h\nu }{kT}}\right)}"></span> </p><p>In order to calculate the internal energy and the specific heat, we must know the number of normal vibrational modes a frequency between the values <span class="texhtml mvar" style="font-style:italic;">ν</span> and <span class="texhtml"><i>ν</i> + <i>dν</i></span>. Allow this number to be <span class="texhtml"><i>f</i>(<i>ν</i>)<i>dν</i></span>. Since the total number of normal modes is <span class="texhtml">3<i>N</i></span>, the function <span class="texhtml"><i>f</i>(<i>ν</i>)</span> is given by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f(\nu )\,d\nu =3N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ν</mi> <mo>=</mo> <mn>3</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f(\nu )\,d\nu =3N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1a902a0782e8fd309a5c045f44409ef210177f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.06ex; height:5.676ex;" alt="{\displaystyle \int f(\nu )\,d\nu =3N}"></span> </p><p>The integration is performed over all frequencies of the crystal. Then the internal energy <span class="texhtml mvar" style="font-style:italic;">U</span> will be given by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\int f(\nu )E(\nu )\,d\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\int f(\nu )E(\nu )\,d\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bad06f52ea05c6333b4a887ab66000295390138" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.434ex; height:5.676ex;" alt="{\displaystyle U=\int f(\nu )E(\nu )\,d\nu }"></span> </p> <div class="mw-heading mw-heading2"><h2 id="In_quantum_mechanics">In quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=9" title="Edit section: In quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bound states in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> are analogous to modes. The waves in quantum systems are oscillations in probability amplitude rather than material displacement. The frequency of oscillation, <span class="texhtml mvar" style="font-style:italic;">f</span>, relates to the mode energy by <span class="texhtml"><i>E</i> = <i>hf</i></span> where <span class="texhtml mvar" style="font-style:italic;">h</span> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. Thus a system like an atom consists of a linear combination of modes of definite energy. These energies are characteristic of the particular atom. The (complex) square of the probability amplitude at a point in space gives the probability of measuring an electron at that location. The spatial distribution of this probability is characteristic of the atom.<sup id="cite_ref-FeynmanLectures_2-0" class="reference"><a href="#cite_note-FeynmanLectures-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: I49–S5">: I49–S5 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_seismology">In seismology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=10" title="Edit section: In seismology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Normal modes are generated in the Earth from long wavelength <a href="/wiki/Seismic_waves" class="mw-redirect" title="Seismic waves">seismic waves</a> from large earthquakes interfering to form standing waves. </p><p>For an elastic, isotropic, homogeneous sphere, spheroidal, toroidal and radial (or breathing) modes arise. Spheroidal modes only involve P and SV waves (like <a href="/wiki/Rayleigh_waves" class="mw-redirect" title="Rayleigh waves">Rayleigh waves</a>) and depend on overtone number <span class="texhtml mvar" style="font-style:italic;">n</span> and angular order <span class="texhtml mvar" style="font-style:italic;">l</span> but have degeneracy of azimuthal order <span class="texhtml mvar" style="font-style:italic;">m</span>. Increasing <span class="texhtml mvar" style="font-style:italic;">l</span> concentrates fundamental branch closer to surface and at large <span class="texhtml mvar" style="font-style:italic;">l</span> this tends to Rayleigh waves. Toroidal modes only involve SH waves (like <a href="/wiki/Love_waves" class="mw-redirect" title="Love waves">Love waves</a>) and do not exist in fluid outer core. Radial modes are just a subset of spheroidal modes with <span class="texhtml"><i>l</i> = 0</span>. The degeneracy does not exist on Earth as it is broken by rotation, ellipticity and 3D heterogeneous velocity and density structure. </p><p>It may be assumed that each mode can be isolated, the self-coupling approximation, or that many modes close in frequency <a href="/wiki/Resonate" class="mw-redirect" title="Resonate">resonate</a>, the cross-coupling approximation. Self-coupling will solely change the phase velocity and not the number of waves around a great circle, resulting in a stretching or shrinking of standing wave pattern. Modal cross-coupling occurs due to the rotation of the Earth, from aspherical elastic structure, or due to Earth's ellipticity and leads to a mixing of fundamental spheroidal and toroidal modes. </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=Normal_mode&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Antiresonance" title="Antiresonance">Antiresonance</a></li> <li><a href="/wiki/Critical_speed" title="Critical speed">Critical speed</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li> <li><a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">Harmonic series (music)</a></li> <li><a href="/wiki/Infrared_spectroscopy" title="Infrared spectroscopy">Infrared spectroscopy</a></li> <li><a href="/wiki/Leaky_mode" title="Leaky mode">Leaky mode</a></li> <li><a href="/wiki/Mechanical_resonance" title="Mechanical resonance">Mechanical resonance</a></li> <li><a href="/wiki/Modal_analysis" title="Modal analysis">Modal analysis</a></li> <li><a href="/wiki/Mode_(electromagnetism)" title="Mode (electromagnetism)">Mode (electromagnetism)</a></li> <li><a href="/wiki/Quasinormal_mode" title="Quasinormal mode">Quasinormal mode</a></li> <li><a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li> <li><a href="/wiki/Torsional_vibration" title="Torsional vibration">Torsional vibration</a></li> <li><a href="/wiki/Vibrations_of_a_circular_membrane" title="Vibrations of a circular membrane">Vibrations of a circular membrane</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=Normal_mode&action=edit&section=12" 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"><ol class="references"> <li id="cite_note-Goldstein3-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Goldstein3_1-0">^</a></b></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 id="CITEREFGoldsteinPooleSafko2008" class="citation book cs1">Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2008). <i>Classical mechanics</i> (3rd ed., [Nachdr.] ed.). San Francisco, Munich: Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-65702-9" title="Special:BookSources/978-0-201-65702-9"><bdi>978-0-201-65702-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+mechanics&rft.place=San+Francisco%2C+Munich&rft.edition=3rd+ed.%2C+%5BNachdr.%5D&rft.pub=Addison+Wesley&rft.date=2008&rft.isbn=978-0-201-65702-9&rft.aulast=Goldstein&rft.aufirst=Herbert&rft.au=Poole%2C+Charles+P.&rft.au=Safko%2C+John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+mode" class="Z3988"></span></span> </li> <li id="cite_note-FeynmanLectures-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FeynmanLectures_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman2011" class="citation book cs1">Feynman, Richard P. (2011). <a rel="nofollow" class="external text" href="https://www.feynmanlectures.caltech.edu/I_49.html#Ch49-S5"><i>The Feynman lectures on physics. Volume 1: Mainly mechanics, radiation, and heat</i></a> (The new millennium edition, paperback first published ed.). New York: Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-04085-8" title="Special:BookSources/978-0-465-04085-8"><bdi>978-0-465-04085-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+lectures+on+physics.+Volume+1%3A+Mainly+mechanics%2C+radiation%2C+and+heat&rft.place=New+York&rft.edition=The+new+millennium+edition%2C+paperback+first+published&rft.pub=Basic+Books&rft.date=2011&rft.isbn=978-0-465-04085-8&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft_id=https%3A%2F%2Fwww.feynmanlectures.caltech.edu%2FI_49.html%23Ch49-S5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+mode" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_mode&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlevins2001" class="citation book cs1">Blevins, Robert D. (2001). <i>Formulas for natural frequency and mode shape</i> (Reprint ed.). Malabar, Florida: Krieger Pub. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1575241845" title="Special:BookSources/978-1575241845"><bdi>978-1575241845</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formulas+for+natural+frequency+and+mode+shape&rft.place=Malabar%2C+Florida&rft.edition=Reprint&rft.pub=Krieger+Pub.&rft.date=2001&rft.isbn=978-1575241845&rft.aulast=Blevins&rft.aufirst=Robert+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+mode" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTzouBergman2008" class="citation book cs1">Tzou, H.S.; Bergman, L.A., eds. (2008). <i>Dynamics and Control of Distributed Systems</i>. Cambridge [England]: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521033749" title="Special:BookSources/978-0521033749"><bdi>978-0521033749</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+and+Control+of+Distributed+Systems.&rft.place=Cambridge+%5BEngland%5D&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0521033749&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+mode" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShearer2009" class="citation book cs1">Shearer, Peter M. (2009). <i>Introduction to seismology</i> (2nd ed.). Cambridge: Cambridge University Press. pp. 231–237. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521882101" title="Special:BookSources/9780521882101"><bdi>9780521882101</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+seismology&rft.place=Cambridge&rft.pages=231-237&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=9780521882101&rft.aulast=Shearer&rft.aufirst=Peter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+mode" class="Z3988"></span></li></ul> <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=Normal_mode&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://scholar.harvard.edu/files/david-morin/files/waves_normalmodes.pdf">Harvard lecture notes on normal modes</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Normal_mode&oldid=1245782481">https://en.wikipedia.org/w/index.php?title=Normal_mode&oldid=1245782481</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Ordinary_differential_equations" title="Category:Ordinary differential equations">Ordinary differential equations</a></li><li><a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Classical mechanics</a></li><li><a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Quantum mechanics</a></li><li><a href="/wiki/Category:Spectroscopy" title="Category:Spectroscopy">Spectroscopy</a></li><li><a href="/wiki/Category:Singular_value_decomposition" title="Category:Singular value decomposition">Singular value decomposition</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_December_2010" title="Category:Articles lacking in-text citations from December 2010">Articles lacking in-text citations from December 2010</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:All_accuracy_disputes" title="Category:All accuracy disputes">All accuracy disputes</a></li><li><a href="/wiki/Category:Articles_with_disputed_statements_from_April_2020" title="Category:Articles with disputed statements from April 2020">Articles with disputed statements from April 2020</a></li><li><a href="/wiki/Category:Articles_containing_video_clips" title="Category:Articles containing video clips">Articles containing video clips</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 15 September 2024, at 02:08<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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