Resonance - Wikipedia

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class="vector-toc-list"> </ul> </li> <li id="toc-Linear_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Linear systems</span> </div> </a> <button aria-controls="toc-Linear_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 Linear systems subsection</span> </button> <ul id="toc-Linear_systems-sublist" class="vector-toc-list"> <li id="toc-The_driven,_damped_harmonic_oscillator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_driven,_damped_harmonic_oscillator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>The driven, damped harmonic oscillator</span> </div> </a> <ul id="toc-The_driven,_damped_harmonic_oscillator-sublist" class="vector-toc-list"> <li id="toc-The_pendulum" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_pendulum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>The pendulum</span> </div> </a> <ul id="toc-The_pendulum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-RLC_series_circuits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#RLC_series_circuits"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>RLC series circuits</span> </div> </a> <ul id="toc-RLC_series_circuits-sublist" class="vector-toc-list"> <li id="toc-Voltage_across_the_capacitor" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Voltage_across_the_capacitor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Voltage across the capacitor</span> </div> </a> <ul id="toc-Voltage_across_the_capacitor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voltage_across_the_inductor" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Voltage_across_the_inductor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Voltage across the inductor</span> </div> </a> <ul id="toc-Voltage_across_the_inductor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voltage_across_the_resistor" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Voltage_across_the_resistor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Voltage across the resistor</span> </div> </a> <ul id="toc-Voltage_across_the_resistor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Antiresonance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Antiresonance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Antiresonance</span> </div> </a> <ul id="toc-Antiresonance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationships_between_resonance_and_frequency_response_in_the_RLC_series_circuit_example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relationships_between_resonance_and_frequency_response_in_the_RLC_series_circuit_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.5</span> <span>Relationships between resonance and frequency response in the RLC series circuit example</span> </div> </a> <ul id="toc-Relationships_between_resonance_and_frequency_response_in_the_RLC_series_circuit_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizing_resonance_and_antiresonance_for_linear_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizing_resonance_and_antiresonance_for_linear_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Generalizing resonance and antiresonance for linear systems</span> </div> </a> <ul id="toc-Generalizing_resonance_and_antiresonance_for_linear_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Standing_waves" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Standing_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Standing waves</span> </div> </a> <button aria-controls="toc-Standing_waves-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 Standing waves subsection</span> </button> <ul id="toc-Standing_waves-sublist" class="vector-toc-list"> <li id="toc-Standing_waves_on_a_string" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standing_waves_on_a_string"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Standing waves on a string</span> </div> </a> <ul id="toc-Standing_waves_on_a_string-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Resonance_in_complex_networks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Resonance_in_complex_networks"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Resonance in complex networks</span> </div> </a> <ul id="toc-Resonance_in_complex_networks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-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 Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Mechanical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanical"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Mechanical</span> </div> </a> <ul id="toc-Mechanical-sublist" class="vector-toc-list"> <li id="toc-International_Space_Station" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#International_Space_Station"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>International Space Station</span> </div> </a> <ul id="toc-International_Space_Station-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Acoustic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Acoustic"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Acoustic</span> </div> </a> <ul id="toc-Acoustic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrical"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Electrical</span> </div> </a> <ul id="toc-Electrical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Optical</span> </div> </a> <ul id="toc-Optical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbital" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Orbital</span> </div> </a> <ul id="toc-Orbital-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atomic,_particle,_and_molecular" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atomic,_particle,_and_molecular"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Atomic, particle, and molecular</span> </div> </a> <ul id="toc-Atomic,_particle,_and_molecular-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Disadvantages" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Disadvantages"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Disadvantages</span> </div> </a> <ul id="toc-Disadvantages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Q_factor" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Q_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Q factor</span> </div> </a> <ul id="toc-Q_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Universal_resonance_curve" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Universal_resonance_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Universal resonance curve</span> </div> </a> <ul id="toc-Universal_resonance_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</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" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Resonance</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 64 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-64" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">64 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%86%D9%8A%D9%86_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="رنين (فيزياء) – Arabic" lang="ar" hreflang="ar" data-title="رنين (فيزياء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Rezonans" title="Rezonans – Azerbaijani" lang="az" hreflang="az" data-title="Rezonans" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%A8%E0%A6%BE%E0%A6%A6" title="অনুনাদ – Bangla" lang="bn" hreflang="bn" data-title="অনুনাদ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D1%8D%D0%B7%D0%B0%D0%BD%D0%B0%D0%BD%D1%81" title="Рэзананс – Belarusian" lang="be" hreflang="be" data-title="Рэзананс" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%D1%8D%D0%B7%D0%B0%D0%BD%D0%B0%D0%BD%D1%81" title="Рэзананс – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Рэзананс" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Bulgarian" lang="bg" hreflang="bg" data-title="Резонанс" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Rezonanca_(fizika)" title="Rezonanca (fizika) – Bosnian" lang="bs" hreflang="bs" data-title="Rezonanca (fizika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Resson%C3%A0ncia" title="Ressonància – Catalan" lang="ca" hreflang="ca" data-title="Ressonància" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Chuvash" lang="cv" hreflang="cv" data-title="Резонанс" 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/Rezonance" title="Rezonance – Czech" lang="cs" hreflang="cs" data-title="Rezonance" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyseiniant" title="Cyseiniant – Welsh" lang="cy" hreflang="cy" data-title="Cyseiniant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Resonans_(fysik)" title="Resonans (fysik) – Danish" lang="da" hreflang="da" data-title="Resonans (fysik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Resonanz" title="Resonanz – German" lang="de" hreflang="de" data-title="Resonanz" 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/Resonants" title="Resonants – Estonian" lang="et" hreflang="et" data-title="Resonants" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CF%84%CE%BF%CE%BD%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Συντονισμός – Greek" lang="el" hreflang="el" data-title="Συντονισμός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Erzya" lang="myv" hreflang="myv" data-title="Резонанс" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Resonancia" title="Resonancia – Spanish" lang="es" hreflang="es" data-title="Resonancia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Resonanco" title="Resonanco – Esperanto" lang="eo" hreflang="eo" data-title="Resonanco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erresonantzia" title="Erresonantzia – Basque" lang="eu" hreflang="eu" data-title="Erresonantzia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B4%D8%AF%DB%8C%D8%AF_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="تشدید (فیزیک) – Persian" lang="fa" hreflang="fa" data-title="تشدید (فیزیک)" 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/R%C3%A9sonance" title="Résonance – French" lang="fr" hreflang="fr" data-title="Résonance" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Athshondas_(meicnic)" title="Athshondas (meicnic) – Irish" lang="ga" hreflang="ga" data-title="Athshondas (meicnic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Resonancia" title="Resonancia – Galician" lang="gl" hreflang="gl" data-title="Resonancia" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%B5%EB%AA%85" title="공명 – Korean" lang="ko" hreflang="ko" data-title="공명" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%A5%D5%A6%D5%B8%D5%B6%D5%A1%D5%B6%D5%BD" title="Ռեզոնանս – Armenian" lang="hy" hreflang="hy" data-title="Ռեզոնանս" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%A8%E0%A4%BE%E0%A4%A6" title="अनुनाद – Hindi" lang="hi" hreflang="hi" data-title="अनुनाद" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Rezonancija" title="Rezonancija – Croatian" lang="hr" hreflang="hr" data-title="Rezonancija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Risonanza_(fisica)" title="Risonanza (fisica) – Italian" lang="it" hreflang="it" data-title="Risonanza (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%94%D7%95%D7%93%D7%94" title="תהודה – Hebrew" lang="he" hreflang="he" data-title="תהודה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%A8%E0%B3%81%E0%B2%B0%E0%B2%A3%E0%B2%A8%E0%B3%86" title="ಅನುರಣನೆ – Kannada" lang="kn" hreflang="kn" data-title="ಅನುರಣನೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%94%E1%83%96%E1%83%9D%E1%83%9C%E1%83%90%E1%83%9C%E1%83%A1%E1%83%98" title="რეზონანსი – Georgian" lang="ka" hreflang="ka" data-title="რეზონანსი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Kazakh" lang="kk" hreflang="kk" data-title="Резонанс" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Rezonans" title="Rezonans – Haitian Creole" lang="ht" hreflang="ht" data-title="Rezonans" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Kyrgyz" lang="ky" hreflang="ky" data-title="Резонанс" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Rezonansas" title="Rezonansas – Lithuanian" lang="lt" hreflang="lt" data-title="Rezonansas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rezonancia" title="Rezonancia – Hungarian" lang="hu" hreflang="hu" data-title="Rezonancia" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%86%D0%B0" title="Резонанца – Macedonian" lang="mk" hreflang="mk" data-title="Резонанца" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A8%E0%B5%81%E0%B4%A8%E0%B4%BE%E0%B4%A6%E0%B4%82" title="അനുനാദം – Malayalam" lang="ml" hreflang="ml" data-title="അനുനാദം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Resonan" title="Resonan – Malay" lang="ms" hreflang="ms" data-title="Resonan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Resonantie_(natuurkunde)" title="Resonantie (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Resonantie (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%85%B1%E9%B3%B4" title="共鳴 – Japanese" lang="ja" hreflang="ja" data-title="共鳴" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Resonans" title="Resonans – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Resonans" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Resonans" title="Resonans – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Resonans" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rezonans" title="Rezonans – Polish" lang="pl" hreflang="pl" data-title="Rezonans" 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/Resson%C3%A2ncia" title="Ressonância – Portuguese" lang="pt" hreflang="pt" data-title="Ressonância" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Rezonan%C8%9B%C4%83" title="Rezonanță – Romanian" lang="ro" hreflang="ro" data-title="Rezonanță" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Russian" lang="ru" hreflang="ru" data-title="Резонанс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rezonanca" title="Rezonanca – Albanian" lang="sq" hreflang="sq" data-title="Rezonanca" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Resonance" title="Resonance – Simple English" lang="en-simple" hreflang="en-simple" data-title="Resonance" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Resonanca" title="Resonanca – Slovenian" lang="sl" hreflang="sl" data-title="Resonanca" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Rezonancija_(fizika)" title="Rezonancija (fizika) – Serbian" lang="sr" hreflang="sr" data-title="Rezonancija (fizika)" 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/Rezonancija" title="Rezonancija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Rezonancija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Resonanssi" title="Resonanssi – Finnish" lang="fi" hreflang="fi" data-title="Resonanssi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Resonans" title="Resonans – Swedish" lang="sv" hreflang="sv" data-title="Resonans" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%B5%E0%AF%81" title="ஒத்திசைவு – Tamil" lang="ta" hreflang="ta" data-title="ஒத்திசைவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%A8%E0%B1%81%E0%B0%A8%E0%B0%BE%E0%B0%A6%E0%B0%82" title="అనునాదం – Telugu" lang="te" hreflang="te" data-title="అనునాదం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AA%E0%B8%B1%E0%B9%88%E0%B8%99%E0%B8%9E%E0%B9%89%E0%B8%AD%E0%B8%87" title="การสั่นพ้อง – Thai" lang="th" hreflang="th" data-title="การสั่นพ้อง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Tajik" lang="tg" hreflang="tg" data-title="Резонанс" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Rezonans_(fizik)" title="Rezonans (fizik) – Turkish" lang="tr" hreflang="tr" data-title="Rezonans (fizik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%81" title="Резонанс – Ukrainian" lang="uk" hreflang="uk" data-title="Резонанс" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%E1%BB%99ng_h%C6%B0%E1%BB%9Fng" title="Cộng hưởng – Vietnamese" lang="vi" hreflang="vi" data-title="Cộng hưởng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%85%B1%E6%8C%AF" title="共振 – Wu" lang="wuu" hreflang="wuu" data-title="共振" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%B1%E6%8C%AF" title="共振 – Cantonese" lang="yue" hreflang="yue" data-title="共振" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%85%B1%E6%8C%AF" title="共振 – Chinese" lang="zh" hreflang="zh" data-title="共振" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q172858#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> 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searchaux" style="display:none">Physical characteristic of oscillating systems</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">This article is about resonance in physics. For other uses, see <a href="/wiki/Resonance_(disambiguation)" class="mw-disambig" title="Resonance (disambiguation)">Resonance (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Resonate" redirects here. For other uses, see <a href="/wiki/Resonate_(disambiguation)" class="mw-disambig" title="Resonate (disambiguation)">Resonate (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Resonant" redirects here. 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Resonance%22">"Resonance"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Resonance%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Resonance%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Resonance%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Resonance%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Resonance%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2021</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>... </div> </div><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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Resonance.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Resonance.PNG/440px-Resonance.PNG" decoding="async" width="440" height="318" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Resonance.PNG/660px-Resonance.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Resonance.PNG/880px-Resonance.PNG 2x" data-file-width="3249" data-file-height="2351" /></a><figcaption>Increase of amplitude as damping decreases and frequency approaches resonant frequency of a driven <a href="/wiki/Damping_ratio" class="mw-redirect" title="Damping ratio">damped</a> <a href="/wiki/Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a>.<sup id="cite_ref-FOOTNOTEOgata2005617_1-0" class="reference"><a href="#cite_note-FOOTNOTEOgata2005617-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEGhatak20056.10_2-0" class="reference"><a href="#cite_note-FOOTNOTEGhatak20056.10-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><b>Resonance</b> is a <a href="/wiki/Phenomenon" title="Phenomenon">phenomenon</a> that occurs when an object or <a href="/wiki/System" title="System">system</a> is subjected to an external force or <a href="/wiki/Vibration" title="Vibration">vibration</a> that matches its <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a>. When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger <a href="/wiki/Amplitude" title="Amplitude">amplitude</a>. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; this frequency is known as a <b>resonant frequency</b> or <b>resonance frequency</b>. When an <a href="/wiki/Oscillation" title="Oscillation">oscillating</a> force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> (with more force) than when the same force is applied at other, non-resonant frequencies.<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005324_4-0" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005324-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a <a href="/wiki/Relative_maximum" class="mw-redirect" title="Relative maximum">relative maximum</a> within the system.<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005324_4-1" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005324-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude <a href="/wiki/Oscillation" title="Oscillation">oscillations</a> in the system due to the storage of <a href="/wiki/Vibrational_energy" class="mw-redirect" title="Vibrational energy">vibrational energy</a>. </p><p>Resonance phenomena occur with all types of vibrations or <a href="/wiki/Wave" title="Wave">waves</a>: there is <a href="/wiki/Mechanical_resonance" title="Mechanical resonance">mechanical resonance</a>, <a href="/wiki/Orbital_resonance" title="Orbital resonance">orbital resonance</a>, <a href="/wiki/Acoustic_resonance" title="Acoustic resonance">acoustic resonance</a>, <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic</a> resonance, <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a> (NMR), <a href="/wiki/Electron_paramagnetic_resonance" title="Electron paramagnetic resonance">electron spin resonance</a> (ESR) and resonance of quantum <a href="/wiki/Wave_function" title="Wave function">wave functions</a>. Resonant systems can be used to generate vibrations of a specific frequency (e.g., <a href="/wiki/Musical_instrument" title="Musical instrument">musical instruments</a>), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters). </p><p>The term <i>resonance</i> (from <a href="/wiki/Latin" title="Latin">Latin</a> <i>resonantia</i>, 'echo', from <i>resonare</i>, 'resound') originated from the field of acoustics, particularly the <a href="/wiki/Sympathetic_resonance" title="Sympathetic resonance">sympathetic resonance</a> observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Resonance occurs when a system is able to store and easily transfer energy between two or more different <a href="/w/index.php?title=Storage_mode&action=edit&redlink=1" class="new" title="Storage mode (page does not exist)">storage modes</a> (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called <a href="/wiki/Damping_ratio" class="mw-redirect" title="Damping ratio">damping</a>. When damping is small, the resonant frequency is approximately equal to the <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a> of the system, which is a frequency of unforced vibrations. Some systems have multiple and distinct resonant frequencies. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Resonance" title="Special:EditPage/Resonance">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">February 2024</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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Little_girl_on_swing.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Little_girl_on_swing.jpg/220px-Little_girl_on_swing.jpg" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Little_girl_on_swing.jpg/330px-Little_girl_on_swing.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Little_girl_on_swing.jpg/440px-Little_girl_on_swing.jpg 2x" data-file-width="2598" data-file-height="1995" /></a><figcaption>Pushing a person in a <a href="/wiki/Swing_(seat)" title="Swing (seat)">swing</a> is a common example of resonance. The loaded swing, a <a href="/wiki/Pendulum" title="Pendulum">pendulum</a>, has a <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a> of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate.</figcaption></figure> <p>A familiar example is a playground <a href="/wiki/Swing_(seat)" title="Swing (seat)">swing</a>, which acts as a <a href="/wiki/Pendulum" title="Pendulum">pendulum</a>. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs.<sup id="cite_ref-Hüwel_5-0" class="reference"><a href="#cite_note-Hüwel-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.2–24">: p.2–24 </span></sup> This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations. </p><p>Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all <a href="/wiki/Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> waves and vibrations are generated. For example, when hard objects like <a href="/wiki/Metal" title="Metal">metal</a>, <a href="/wiki/Glass" title="Glass">glass</a>, or <a href="/wiki/Wood" title="Wood">wood</a> are struck, there are brief resonant vibrations in the object.<sup id="cite_ref-Hüwel_5-1" class="reference"><a href="#cite_note-Hüwel-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p.2–24">: p.2–24 </span></sup> Light and other short wavelength <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> is produced by resonance on an <a href="/wiki/Atomic_spacing" title="Atomic spacing">atomic scale</a>, such as <a href="/wiki/Electron" title="Electron">electrons</a> in atoms. Other examples of resonance include: </p> <ul><li>Timekeeping mechanisms of modern clocks and watches, e.g., the <a href="/wiki/Balance_wheel" title="Balance wheel">balance wheel</a> in a mechanical <a href="/wiki/Watch" title="Watch">watch</a> and the <a href="/wiki/Quartz_crystal" class="mw-redirect" title="Quartz crystal">quartz crystal</a> in a <a href="/wiki/Quartz_watch" class="mw-redirect" title="Quartz watch">quartz watch</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a> of the <a href="/wiki/Bay_of_Fundy" title="Bay of Fundy">Bay of Fundy</a></li> <li><a href="/wiki/Acoustic_resonance" title="Acoustic resonance">Acoustic resonances</a> of <a href="/wiki/Musical_instruments" class="mw-redirect" title="Musical instruments">musical instruments</a> and the human <a href="/wiki/Vocal_tract" title="Vocal tract">vocal tract</a></li> <li>Shattering of a crystal wineglass when exposed to a musical tone of the right pitch (its resonant frequency)</li> <li><a href="/wiki/Friction_idiophone" title="Friction idiophone">Friction idiophones</a>, such as making a glass object (glass, bottle, vase) <a href="/wiki/Vibration" title="Vibration">vibrate</a> by rubbing around its rim with a fingertip</li> <li><a href="/wiki/Electrical_resonance" title="Electrical resonance">Electrical resonance</a> of <a href="/wiki/Tuned_circuit" class="mw-redirect" title="Tuned circuit">tuned circuits</a> in <a href="/wiki/Radio" title="Radio">radios</a> and <a href="/wiki/TV" class="mw-redirect" title="TV">TVs</a> that allow radio frequencies to be selectively received</li> <li>Creation of <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent</a> light by <a href="/w/index.php?title=Optical_resonance&action=edit&redlink=1" class="new" title="Optical resonance (page does not exist)">optical resonance</a> in a <a href="/wiki/Laser" title="Laser">laser</a> <a href="/wiki/Optical_cavity" title="Optical cavity">cavity</a></li> <li><a href="/wiki/Orbital_resonance" title="Orbital resonance">Orbital resonance</a> as exemplified by some <a href="/wiki/Natural_satellite" title="Natural satellite">moons</a> of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>'s <a href="/wiki/Giant_planet" title="Giant planet">giant planets</a> and resonant groups such as the <a href="/wiki/Plutino" title="Plutino">plutinos</a></li> <li>Material resonances in atomic scale are the basis of several <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopic</a> techniques that are used in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a> <ul><li><a href="/wiki/Electron_spin_resonance" class="mw-redirect" title="Electron spin resonance">Electron spin resonance</a></li> <li><a href="/wiki/M%C3%B6ssbauer_effect" title="Mössbauer effect">Mössbauer effect</a></li> <li><a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">Nuclear magnetic resonance</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Linear_systems">Linear systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=3" title="Edit section: Linear systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Resonance manifests itself in many <a href="/wiki/Linear_system" title="Linear system">linear</a> and <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinear systems</a> as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large. </p><p>Since many linear and nonlinear systems that oscillate are modeled as <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillators</a> near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An <a href="/wiki/RLC_circuit" title="RLC circuit">RLC circuit</a> is used to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, these connections for higher-order linear systems with multiple inputs and outputs are generalized. </p> <div class="mw-heading mw-heading3"><h3 id="The_driven,_damped_harmonic_oscillator"><span id="The_driven.2C_damped_harmonic_oscillator"></span>The driven, damped harmonic oscillator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=4" title="Edit section: The driven, damped harmonic oscillator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Harmonic_oscillator#Driven_harmonic_oscillators" title="Harmonic oscillator">Harmonic oscillator § Driven harmonic oscillators</a></div> <p>Consider a damped mass on a spring driven by a sinusoidal, externally applied force. <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> takes the form </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde479ede693fe7d16c7c2d77102aa8b482e3ba6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.871ex; height:6.009ex;" alt="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}},}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>where <i>m</i> is the mass, <i>x</i> is the displacement of the mass from the equilibrium point, <i>F</i><sub>0</sub> is the driving amplitude, <i>ω</i> is the driving angular frequency, <i>k</i> is the spring constant, and <i>c</i> is the viscous damping coefficient. This can be rewritten in the form </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F_{0}}{m}}\sin(\omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>m</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F_{0}}{m}}\sin(\omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803db72528e97e044ed1c9b88dbc960cb0219fb2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.707ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F_{0}}{m}}\sin(\omega t),}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \omega _{0}={\sqrt {k/m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{0}={\sqrt {k/m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79077021f977e6d7a9dc72fe5481306088e3596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.336ex; height:3.343ex;" alt="{\textstyle \omega _{0}={\sqrt {k/m}}}"></span> is called the <i>undamped <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the oscillator</i> or the <i>natural frequency</i>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta ={\frac {c}{2{\sqrt {mk}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> <mi>k</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {c}{2{\sqrt {mk}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5cba92dfb2c3942fb2b9446f3966d0f90f44db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.38ex; height:5.676ex;" alt="{\displaystyle \zeta ={\frac {c}{2{\sqrt {mk}}}}}"></span> is called the <i>damping ratio</i>.</li></ul> <p>Many sources also refer to <i>ω</i><sub>0</sub> as the <i>resonant frequency</i>. However, as shown below, when analyzing oscillations of the displacement <i>x</i>(<i>t</i>), the resonant frequency is close to but not the same as <i>ω</i><sub>0</sub>. In general the resonant frequency is close to but not necessarily the same as the natural frequency.<sup id="cite_ref-FOOTNOTEHardt2004_6-0" class="reference"><a href="#cite_note-FOOTNOTEHardt2004-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The RLC circuit example in the next section gives examples of different resonant frequencies for the same system. </p><p>The general solution of Equation (<b><a href="#math_2">2</a></b>) is the sum of a <a href="/wiki/Transient_(oscillation)" class="mw-redirect" title="Transient (oscillation)">transient</a> solution that depends on initial conditions and a <a href="/wiki/Steady_state" title="Steady state">steady state</a> solution that is independent of initial conditions and depends only on the driving amplitude <i>F</i><sub>0</sub>, driving frequency <i>ω</i>, undamped angular frequency <i>ω</i><sub>0</sub>, and the damping ratio <i>ζ</i>. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution. </p><p>It is possible to write the steady-state solution for <i>x</i>(<i>t</i>) as a function proportional to the driving force with an induced <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> change <i>φ</i>, </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 x(t)={\frac {F_{0}}{m{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}}\sin(\omega t+\varphi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ζ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)={\frac {F_{0}}{m{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}}\sin(\omega t+\varphi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d58889f1fa01d9300ed4daa0e04b8c7400e483" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:47.234ex; height:8.176ex;" alt="{\displaystyle x(t)={\frac {F_{0}}{m{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}}\sin(\omega t+\varphi ),}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ζ</mi> </mrow> <mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>n</mi> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a273779e28823798b25b49d52fe5a216142f5b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.657ex; height:7.509ex;" alt="{\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi .}"></span> </p><p>The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mplwp_resonance_zeta_envelope.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Mplwp_resonance_zeta_envelope.svg/300px-Mplwp_resonance_zeta_envelope.svg.png" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Mplwp_resonance_zeta_envelope.svg/450px-Mplwp_resonance_zeta_envelope.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Mplwp_resonance_zeta_envelope.svg/600px-Mplwp_resonance_zeta_envelope.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>Steady-state variation of amplitude with relative frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega /\omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega /\omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c152d9a23c80a377f843ca1190c8982bdfa90ff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.108ex; height:2.843ex;" alt="{\displaystyle \omega /\omega _{0}}"></span> and damping <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span> of a driven <a href="/wiki/Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a></figcaption></figure> <p>Resonance occurs when, at certain driving frequencies, the steady-state amplitude of <i>x</i>(<i>t</i>) is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies. Looking at the amplitude of <i>x</i>(<i>t</i>) as a function of the driving frequency <i>ω</i>, the amplitude is maximal at the driving frequency <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 _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d296309e40828eeedfc8ba4555f88528292d3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.326ex; height:4.843ex;" alt="{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}"></span> </p><p><i>ω</i><sub><i>r</i></sub> is the <b>resonant frequency</b> for this system. Again, the resonant frequency does not equal the undamped angular frequency <i>ω</i><sub>0</sub> of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including <i>ω</i><sub>0</sub>, but the maximum response is at the resonant frequency. </p><p>Also, <i>ω</i><sub><i>r</i></sub> is only real and non-zero if <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 \zeta <1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ζ</mi> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \zeta <1/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9d44d6ebbabd1ea56a1ac12c5272f4e7329334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.617ex; height:3.176ex;" alt="{\textstyle \zeta <1/{\sqrt {2}}}"></span>, so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large. </p> <div class="mw-heading mw-heading4"><h4 id="The_pendulum">The pendulum</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=5" title="Edit section: The pendulum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains <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 _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea780fa44c6c32c55ca1601e7543ca25024ff932" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.326ex; height:4.843ex;" alt="{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},}"></span> but the definitions of <i>ω</i><sub>0</sub> and <i>ζ</i> change based on the physics of the system. For a pendulum of length <i>ℓ</i> and small displacement angle <i>θ</i>, Equation (<b><a href="#math_1">1</a></b>) becomes <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 m\ell {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-mg\theta -c\ell {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mi>g</mi> <mi>θ</mi> <mo>−</mo> <mi>c</mi> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\ell {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-mg\theta -c\ell {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30fd0b56d0e14137736f6ce968693b3e6189a44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.391ex; height:6.009ex;" alt="{\displaystyle m\ell {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-mg\theta -c\ell {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}"></span> </p><p>and therefore </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}={\sqrt {\frac {g}{\ell }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>g</mi> <mi>ℓ</mi> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}={\sqrt {\frac {g}{\ell }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0a7cebd5b4d72a267e63d25f5c4108093ff856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.521ex; height:6.343ex;" alt="{\displaystyle \omega _{0}={\sqrt {\frac {g}{\ell }}},}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta ={\frac {c}{2m}}{\sqrt {\frac {\ell }{g}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ℓ</mi> <mi>g</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {c}{2m}}{\sqrt {\frac {\ell }{g}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b30af02cefc0151ed3c5496d893f90c24b4ffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.155ex; height:7.509ex;" alt="{\displaystyle \zeta ={\frac {c}{2m}}{\sqrt {\frac {\ell }{g}}}.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="RLC_series_circuits">RLC series circuits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=6" title="Edit section: RLC series circuits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Summarize_section plainlinks metadata ambox ambox-style" 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/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.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 section <b>may be <a href="/wiki/Help:Section#Section_size_policies" title="Help:Section">too long</a> and excessively detailed.</b><span class="hide-when-compact"> Please consider summarizing the material.</span> <span class="date-container"><i>(<span class="date">January 2021</span>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/RLC_Circuit#Series_circuit" class="mw-redirect" title="RLC Circuit">RLC Circuit § Series circuit</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rajz_RLC_soros.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Rajz_RLC_soros.svg/220px-Rajz_RLC_soros.svg.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Rajz_RLC_soros.svg/330px-Rajz_RLC_soros.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Rajz_RLC_soros.svg/440px-Rajz_RLC_soros.svg.png 2x" data-file-width="425" data-file-height="283" /></a><figcaption>An RLC series circuit</figcaption></figure> <p>Consider a <a href="/wiki/Electrical_network" title="Electrical network">circuit</a> consisting of a <a href="/wiki/Resistor" title="Resistor">resistor</a> with resistance <i>R</i>, an <a href="/wiki/Inductor" title="Inductor">inductor</a> with inductance <i>L</i>, and a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> with capacitance <i>C</i> connected in series with current <i>i</i>(<i>t</i>) and driven by a <a href="/wiki/Voltage" title="Voltage">voltage</a> source with voltage <i>v</i><sub><i>in</i></sub>(<i>t</i>). The voltage drop around the circuit is </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" 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="{\displaystyle L{\frac {di(t)}{dt}}+Ri(t)+V(0)+{\frac {1}{C}}\int _{0}^{t}i(\tau )d\tau =v_{\text{in}}(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>R</mi> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>C</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>i</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L{\frac {di(t)}{dt}}+Ri(t)+V(0)+{\frac {1}{C}}\int _{0}^{t}i(\tau )d\tau =v_{\text{in}}(t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69787553b7d28bd56b63cf82b00a86c29194e3d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.605ex; height:6.176ex;" alt="{\displaystyle L{\frac {di(t)}{dt}}+Ri(t)+V(0)+{\frac {1}{C}}\int _{0}^{t}i(\tau )d\tau =v_{\text{in}}(t).}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span>)</b></td></tr></tbody></table> <p>Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze the frequency response of this circuit. Taking the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> of Equation (<b><a href="#math_4">4</a></b>), <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 sLI(s)+RI(s)+{\frac {1}{sC}}I(s)=V_{\text{in}}(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>L</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>R</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sLI(s)+RI(s)+{\frac {1}{sC}}I(s)=V_{\text{in}}(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6e421569473e8566988dc761bd15f9f50013dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.629ex; height:5.343ex;" alt="{\displaystyle sLI(s)+RI(s)+{\frac {1}{sC}}I(s)=V_{\text{in}}(s),}"></span> where <i>I</i>(<i>s</i>) and <i>V</i><sub><i>in</i></sub>(<i>s</i>) are the Laplace transform of the current and input voltage, respectively, and <i>s</i> is a <a href="/wiki/Complex_number" title="Complex number">complex</a> frequency parameter in the Laplace domain. Rearranging terms, <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 I(s)={\frac {s}{s^{2}L+Rs+{\frac {1}{C}}}}V_{\text{in}}(s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> <mo>+</mo> <mi>R</mi> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>C</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(s)={\frac {s}{s^{2}L+Rs+{\frac {1}{C}}}}V_{\text{in}}(s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/983a388021c23c0298b6cb451eadb8d09a783296" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:28.859ex; height:6.343ex;" alt="{\displaystyle I(s)={\frac {s}{s^{2}L+Rs+{\frac {1}{C}}}}V_{\text{in}}(s).}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Voltage_across_the_capacitor">Voltage across the capacitor</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=7" title="Edit section: Voltage across the capacitor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An RLC circuit in series presents several options for where to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage 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 V_{\text{out}}(s)={\frac {1}{sC}}I(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)={\frac {1}{sC}}I(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06d32c5297f459e0fa116015957aa64595f9bcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.726ex; height:5.343ex;" alt="{\displaystyle V_{\text{out}}(s)={\frac {1}{sC}}I(s)}"></span> or <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 V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mi>C</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>L</mi> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a24c16b7172aaa16a5efdbc42858f1ab87cb7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.765ex; height:6.843ex;" alt="{\displaystyle V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).}"></span> </p><p>Define for this circuit a natural frequency and a damping ratio, <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 _{0}={\frac {1}{\sqrt {LC}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60485db680546e424106c33f9b23ac93737663f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.367ex; height:6.176ex;" alt="{\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},}"></span> <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 \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>C</mi> <mi>L</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10046992fd4e30d39eac42c77a157a85f6294dfe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.367ex; height:6.176ex;" alt="{\displaystyle \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.}"></span> </p><p>The ratio of the output voltage to the input voltage becomes <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 H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3fdf4be41628918122b5ce40e70af394c85769" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.869ex; height:7.176ex;" alt="{\displaystyle H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}}"></span> </p><p><i>H</i>(<i>s</i>) is the <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> between the input voltage and the output voltage. This transfer function has two <a href="/wiki/Zeros_and_poles" title="Zeros and poles">poles</a>–roots of the polynomial in the transfer function's denominator–at </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>±</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44faa2cc8e68ebaa6545b27089e452056bb3451b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.234ex; height:4.843ex;" alt="{\displaystyle s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span>)</b></td></tr></tbody></table> <p>and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for <span class="texhtml"><i>ζ</i> ≤ 1</span>, the magnitude of these poles is the natural frequency <i>ω</i><sub>0</sub> and that for <span class="texhtml"><i>ζ</i> < 1/<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span></span>, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis. </p><p>Evaluating <i>H</i>(<i>s</i>) along the imaginary axis <span class="texhtml"><i>s</i> = <i>iω</i></span>, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of Equation (<b><a href="#math_4">4</a></b>) instead of the Laplace transform. The transfer function, which is also complex, can be written as a gain and phase, <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 H(i\omega )=G(\omega )e^{i\Phi (\omega )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(i\omega )=G(\omega )e^{i\Phi (\omega )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7537bd3c3c5334d9da51eb970a8a84795c2b9c78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.32ex; height:3.343ex;" alt="{\displaystyle H(i\omega )=G(\omega )e^{i\Phi (\omega )}.}"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_Series_Circuit_Bode_Magnitude_Plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/RLC_Series_Circuit_Bode_Magnitude_Plot.svg/300px-RLC_Series_Circuit_Bode_Magnitude_Plot.svg.png" decoding="async" width="300" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/RLC_Series_Circuit_Bode_Magnitude_Plot.svg/450px-RLC_Series_Circuit_Bode_Magnitude_Plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/RLC_Series_Circuit_Bode_Magnitude_Plot.svg/600px-RLC_Series_Circuit_Bode_Magnitude_Plot.svg.png 2x" data-file-width="1971" data-file-height="937" /></a><figcaption>Bode magnitude plot for the voltage across the elements of an RLC series circuit. Natural frequency <span class="texhtml"><i>ω</i><sub>0</sub> = 1 rad/s</span>, damping ratio <span class="texhtml"><i>ζ</i> = 0.4</span>. The capacitor voltage peaks below the circuit's natural frequency, the inductor voltage peaks above the natural frequency, and the resistor voltage peaks at the natural frequency with a peak gain of one. The gain for the voltage across the capacitor and inductor combined in series shows antiresonance, with gain going to zero at the natural frequency.</figcaption></figure> <p>A sinusoidal input voltage at frequency <i>ω</i> results in an output voltage at the same frequency that has been scaled by <i>G</i>(<i>ω</i>) and has a phase shift <i>Φ</i>(<i>ω</i>). The gain and phase can be plotted versus frequency on a <a href="/wiki/Bode_plot" title="Bode plot">Bode plot</a>. For the RLC circuit's capacitor voltage, the gain of the transfer function <i>H</i>(<i>iω</i>) is </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 G(\omega )={\frac {\omega _{0}^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ζ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {\omega _{0}^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f411097ba31fe2631e89a8f6428ec0c20d3c7d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:34.598ex; height:8.843ex;" alt="{\displaystyle G(\omega )={\frac {\omega _{0}^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span>)</b></td></tr></tbody></table> <p>Note the similarity between the gain here and the amplitude in Equation (<b><a href="#math_3">3</a></b>). Once again, the gain is maximized at the <b>resonant frequency</b> <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 _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d296309e40828eeedfc8ba4555f88528292d3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.326ex; height:4.843ex;" alt="{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}"></span> </p><p>Here, the resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies. </p> <div class="mw-heading mw-heading4"><h4 id="Voltage_across_the_inductor">Voltage across the inductor</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=8" title="Edit section: Voltage across the inductor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The resonant frequency need not always take the form given in the examples above. For the RLC circuit, suppose instead that the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain the voltage across the inductor 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 V_{\text{out}}(s)=sLI(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>L</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)=sLI(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37c01c367d6b03d6845c7f64e50ee3811df6c006" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.353ex; height:2.843ex;" alt="{\displaystyle V_{\text{out}}(s)=sLI(s),}"></span> <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 V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>L</mi> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9e3f66e4829a6e8e404e0ed2d7761c95723978" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.507ex; height:7.343ex;" alt="{\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s),}"></span> <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 V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}V_{\text{in}}(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}V_{\text{in}}(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754f6118b990e92a83af091c345b8fe3439ce6fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.476ex; height:6.843ex;" alt="{\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}V_{\text{in}}(s),}"></span> </p><p>using the same definitions for <i>ω</i><sub>0</sub> and <i>ζ</i> as in the previous example. The transfer function between <i>V</i><sub>in</sub>(<i>s</i>) and this new <i>V</i><sub>out</sub>(<i>s</i>) across the inductor 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 H(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c4310e9b45f15240e14971a649ceece400ae64" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.718ex; height:6.843ex;" alt="{\displaystyle H(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}"></span> </p><p>This transfer function has the same poles as the transfer function in the previous example, but it also has two zeroes in the numerator at <span class="nowrap"><i>s</i> = 0</span>. Evaluating <i>H</i>(<i>s</i>) along the imaginary axis, its gain becomes <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 G(\omega )={\frac {\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ζ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5daa610d97193f149a89cb869f7afa5bd6a19e5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:34.598ex; height:8.509ex;" alt="{\displaystyle G(\omega )={\frac {\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"></span> </p><p>Compared to the gain in Equation (<b><a href="#math_6">6</a></b>) using the capacitor voltage as the output, this gain has a factor of <i>ω</i><sup>2</sup> in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency 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 \omega _{r}={\frac {\omega _{0}}{\sqrt {1-2\zeta ^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msqrt> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{r}={\frac {\omega _{0}}{\sqrt {1-2\zeta ^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddfa2e9aa1c04cd28a16e844dcea121cef266282" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.663ex; height:6.009ex;" alt="{\displaystyle \omega _{r}={\frac {\omega _{0}}{\sqrt {1-2\zeta ^{2}}}},}"></span> </p><p>So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now <i>larger</i> than the natural frequency, though it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different choices of output is not contradictory. As shown in Equation (<b><a href="#math_4">4</a></b>), the voltage drop across the circuit is divided among the three circuit elements, and each element has different dynamics. The capacitor's voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, whereas the inductor's voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency where it tends to oscillate, the different dynamics of each circuit element make each element resonate at a slightly different frequency. </p> <div class="mw-heading mw-heading4"><h4 id="Voltage_across_the_resistor">Voltage across the resistor</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=9" title="Edit section: Voltage across the resistor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor 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 V_{\text{out}}(s)=RI(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)=RI(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bfec05c73737b22fc92efed6986e4cc9746b39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.444ex; height:2.843ex;" alt="{\displaystyle V_{\text{out}}(s)=RI(s),}"></span> <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 V_{\text{out}}(s)={\frac {Rs}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V_{\text{in}}(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>s</mi> </mrow> <mrow> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>L</mi> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)={\frac {Rs}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V_{\text{in}}(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e624dcd8f556d9070728aa0022d02aada0cec6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:36.252ex; height:7.843ex;" alt="{\displaystyle V_{\text{out}}(s)={\frac {Rs}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V_{\text{in}}(s),}"></span> </p><p>and using the same natural frequency and damping ratio as in the capacitor example the transfer function 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 H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfee81d46359aa67986d2aa93259ff940a36514" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.718ex; height:6.509ex;" alt="{\displaystyle H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}"></span> </p><p>This transfer function also has the same poles as the previous RLC circuit examples, but it only has one zero in the numerator at <i>s</i> = 0. For this transfer function, its gain 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 G(\omega )={\frac {2\zeta \omega _{0}\omega }{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ω</mi> </mrow> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ζ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {2\zeta \omega _{0}\omega }{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b42dbcb38c086eeb0db323d718227951efd5ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:34.598ex; height:8.176ex;" alt="{\displaystyle G(\omega )={\frac {2\zeta \omega _{0}\omega }{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"></span> </p><p>The resonant frequency that maximizes this gain 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 \omega _{r}=\omega _{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{r}=\omega _{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a262b6ae6eb5acd599fbd1ba147b72e87baeb076" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.665ex; height:2.009ex;" alt="{\displaystyle \omega _{r}=\omega _{0},}"></span> and the gain is one at this frequency, so the voltage across the resistor resonates <i>at</i> the circuit's natural frequency and at this frequency the amplitude of the voltage across the resistor equals the input voltage's amplitude. </p> <div class="mw-heading mw-heading4"><h4 id="Antiresonance">Antiresonance</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=10" title="Edit section: Antiresonance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Antiresonance" title="Antiresonance">Antiresonance</a></div> <p>Some systems exhibit antiresonance that can be analyzed in the same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately <i>small</i> rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined. </p><p>Suppose that the output voltage of interest in the RLC circuit is the voltage across the inductor <i>and</i> the capacitor combined in series. Equation (<b><a href="#math_4">4</a></b>) showed that the sum of the voltages across the three circuit elements sums to the input voltage, so measuring the output voltage as the sum of the inductor and capacitor voltages combined is the same as <i>v</i><sub><i>in</i></sub> minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of the voltage drop across the resistor <i>equals</i> the amplitude of <i>v</i><sub><i>in</i></sub>, and therefore the voltage across the inductor and capacitor combined has zero amplitude. We can show this with the transfer function. </p><p>The sum of the inductor and capacitor voltages 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 V_{\text{out}}(s)=(sL+{\frac {1}{sC}})I(s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>L</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)=(sL+{\frac {1}{sC}})I(s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0029e768b9f7595f827c03793245f264d0f20037" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.696ex; height:5.343ex;" alt="{\displaystyle V_{\text{out}}(s)=(sL+{\frac {1}{sC}})I(s),}"></span> <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 V_{\text{out}}(s)={\frac {s^{2}+{\frac {1}{LC}}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>L</mi> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{out}}(s)={\frac {s^{2}+{\frac {1}{LC}}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79eeb734c29893ce0ad88413de0ca15d0e64184" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.507ex; height:8.176ex;" alt="{\displaystyle V_{\text{out}}(s)={\frac {s^{2}+{\frac {1}{LC}}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s).}"></span> </p><p>Using the same natural frequency and damping ratios as the previous examples, the transfer function 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 H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15e3d551dffe47852cc299d2bb9898577421c82" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.718ex; height:7.176ex;" alt="{\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}"></span> </p><p>This transfer has the same poles as the previous examples but has zeroes at </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 s=\pm i\omega _{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>±</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\pm i\omega _{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca74893ee6811798a04a2471181a2f06e9f8c882" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.946ex; height:2.509ex;" alt="{\displaystyle s=\pm i\omega _{0}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span>)</b></td></tr></tbody></table> <p>Evaluating the transfer function along the imaginary axis, its gain 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 G(\omega )={\frac {\omega _{0}^{2}-\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ζ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {\omega _{0}^{2}-\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea87dad9b6adb84e3b03241144dc4b45af30e34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:34.598ex; height:8.843ex;" alt="{\displaystyle G(\omega )={\frac {\omega _{0}^{2}-\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}"></span> </p><p>Rather than look for resonance, i.e., peaks of the gain, notice that the gain goes to zero at <i>ω</i> = <i>ω</i><sub>0</sub>, which complements our analysis of the resistor's voltage. This is called <b>antiresonance</b>, which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to the zeroes of the transfer function, which were shown in Equation (<b><a href="#math_7">7</a></b>) and were on the imaginary axis. </p> <div class="mw-heading mw-heading4"><h4 id="Relationships_between_resonance_and_frequency_response_in_the_RLC_series_circuit_example">Relationships between resonance and frequency response in the RLC series circuit example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=11" title="Edit section: Relationships between resonance and frequency response in the RLC series circuit example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These RLC circuit examples illustrate how resonance is related to the frequency response of the system. Specifically, these examples illustrate: </p> <ul><li>How resonant frequencies can be found by looking for peaks in the gain of the transfer function between the input and output of the system, for example in a Bode magnitude plot</li> <li>How the resonant frequency for a single system can be different for different choices of system output</li> <li>The connection between the system's natural frequency, the system's damping ratio, and the system's resonant frequency</li> <li>The connection between the system's natural frequency and the magnitude of the transfer function's poles, pointed out in Equation (<b><a href="#math_5">5</a></b>), and therefore a connection between the poles and the resonant frequency</li> <li>A connection between the transfer function's zeroes and the shape of the gain as a function of frequency, and therefore a connection between the zeroes and the resonant frequency that maximizes gain</li> <li>A connection between the transfer function's zeroes and antiresonance</li></ul> <p>The next section extends these concepts to resonance in a general linear system. </p> <div class="mw-heading mw-heading3"><h3 id="Generalizing_resonance_and_antiresonance_for_linear_systems">Generalizing resonance and antiresonance for linear systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=12" title="Edit section: Generalizing resonance and antiresonance for linear systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Next consider an arbitrary linear system with multiple inputs and outputs. For example, in <a href="/wiki/State-space_representation" title="State-space representation">state-space representation</a> a third order <a href="/wiki/Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant system</a> with three inputs and two outputs might be written as <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}{\dot {x}}_{1}\\{\dot {x}}_{2}\\{\dot {x}}_{3}\end{bmatrix}}=A{\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{bmatrix}}+B{\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{bmatrix}},}"> <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> <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> </mtr> <mtr> <mtd> <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> </mtr> <mtr> <mtd> <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>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <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> </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> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\{\dot {x}}_{3}\end{bmatrix}}=A{\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{bmatrix}}+B{\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6005ca08749ed29cb559938e2a6a4c0efe8cf634" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:34.099ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\{\dot {x}}_{3}\end{bmatrix}}=A{\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{bmatrix}}+B{\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{bmatrix}},}"></span> <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}y_{1}(t)\\y_{2}(t)\end{bmatrix}}=C{\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{bmatrix}}+D{\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{bmatrix}},}"> <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> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <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> </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> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}y_{1}(t)\\y_{2}(t)\end{bmatrix}}=C{\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{bmatrix}}+D{\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d200f1a74d29a115af881c8f0670ff71b578a2b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:36.095ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}y_{1}(t)\\y_{2}(t)\end{bmatrix}}=C{\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{bmatrix}}+D{\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{bmatrix}},}"></span> where <i>u</i><sub><i>i</i></sub>(<i>t</i>) are the inputs, <i>x</i><sub><i>i</i></sub>(t) are the state variables, <i>y</i><sub><i>i</i></sub>(<i>t</i>) are the outputs, and <i>A</i>, <i>B</i>, <i>C</i>, and <i>D</i> are matrices describing the dynamics between the variables. </p><p>This system has a <a href="/wiki/Transfer_function_matrix" title="Transfer function matrix">transfer function matrix</a> whose elements are the transfer functions between the various inputs and outputs. For example, <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}Y_{1}(s)\\Y_{2}(s)\end{bmatrix}}={\begin{bmatrix}H_{11}(s)&H_{12}(s)&H_{13}(s)\\H_{21}(s)&H_{22}(s)&H_{23}(s)\end{bmatrix}}{\begin{bmatrix}U_{1}(s)\\U_{2}(s)\\U_{3}(s)\end{bmatrix}}.}"> <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> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}Y_{1}(s)\\Y_{2}(s)\end{bmatrix}}={\begin{bmatrix}H_{11}(s)&H_{12}(s)&H_{13}(s)\\H_{21}(s)&H_{22}(s)&H_{23}(s)\end{bmatrix}}{\begin{bmatrix}U_{1}(s)\\U_{2}(s)\\U_{3}(s)\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e727b00d321b7fb9222cbcca786cf899639c585" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:49.623ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}Y_{1}(s)\\Y_{2}(s)\end{bmatrix}}={\begin{bmatrix}H_{11}(s)&H_{12}(s)&H_{13}(s)\\H_{21}(s)&H_{22}(s)&H_{23}(s)\end{bmatrix}}{\begin{bmatrix}U_{1}(s)\\U_{2}(s)\\U_{3}(s)\end{bmatrix}}.}"></span> </p><p>Each <i>H</i><sub><i>ij</i></sub>(<i>s</i>) is a scalar transfer function linking one of the inputs to one of the outputs. The RLC circuit examples above had one input voltage and showed four possible output voltages–across the capacitor, across the inductor, across the resistor, and across the capacitor and inductor combined in series–each with its own transfer function. If the RLC circuit were set up to measure all four of these output voltages, that system would have a 4×1 transfer function matrix linking the single input to each of the four outputs. </p><p>Evaluated along the imaginary axis, each <i>H</i><sub><i>ij</i></sub>(<i>iω</i>) can be written as a gain and phase shift, <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 H_{ij}(i\omega )=G_{ij}(\omega )e^{i\Phi _{ij}(\omega )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{ij}(i\omega )=G_{ij}(\omega )e^{i\Phi _{ij}(\omega )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c8478b8316edd9b3e90b0f61c0c04b42d90393" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.317ex; height:3.509ex;" alt="{\displaystyle H_{ij}(i\omega )=G_{ij}(\omega )e^{i\Phi _{ij}(\omega )}.}"></span> </p><p>Peaks in the gain at certain frequencies correspond to resonances between that transfer function's input and output, assuming the system is <a href="/wiki/Exponential_stability" title="Exponential stability">stable</a>. </p><p>Each transfer function <i>H</i><sub><i>ij</i></sub>(<i>s</i>) can also be written as a fraction whose numerator and denominator are polynomials of <i>s</i>. <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 H_{ij}(s)={\frac {N_{ij}(s)}{D_{ij}(s)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{ij}(s)={\frac {N_{ij}(s)}{D_{ij}(s)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bfa3d990b40dd512a28f8a4014bb43b41c3bf9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.191ex; height:6.509ex;" alt="{\displaystyle H_{ij}(s)={\frac {N_{ij}(s)}{D_{ij}(s)}}.}"></span> </p><p>The complex roots of the numerator are called zeroes, and the complex roots of the denominator are called poles. For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonate and at which frequencies. In particular, any stable or <a href="/wiki/Marginal_stability" title="Marginal stability">marginally stable</a>, complex conjugate pair of poles with imaginary components can be written in terms of a natural frequency and a damping ratio as <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 s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>±</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a53ff490eceeac48817b42f23d3fa3a3fa0b6e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.881ex; height:4.843ex;" alt="{\displaystyle s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}},}"></span> as in Equation (<b><a href="#math_5">5</a></b>). The natural frequency <i>ω</i><sub>0</sub> of that pole is the magnitude of the position of the pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general,<sup id="cite_ref-FOOTNOTEHardt2004_6-1" class="reference"><a href="#cite_note-FOOTNOTEHardt2004-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Complex conjugate pairs of <i>poles</i> near the imaginary axis correspond to a peak or resonance in the frequency response in the vicinity of the pole's natural frequency. If the pair of poles is <i>on</i> the imaginary axis, the gain is infinite at that frequency.</li> <li>Complex conjugate pairs of <i>zeroes</i> near the imaginary axis correspond to a notch or antiresonance in the frequency response in the vicinity of the zero's frequency, i.e., the frequency equal to the magnitude of the zero. If the pair of zeroes is <i>on</i> the imaginary axis, the gain is zero at that frequency.</li></ul> <p>In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation (<b><a href="#math_5">5</a></b>). The second generalization relating zeroes to antiresonance is observed in Equation (<b><a href="#math_7">7</a></b>). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition ζ < 1/<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Standing_waves">Standing waves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=13" title="Edit section: Standing waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Standing_wave" title="Standing wave">Standing wave</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Animaci%C3%B3n1.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Animaci%C3%B3n1.gif/220px-Animaci%C3%B3n1.gif" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Animaci%C3%B3n1.gif/330px-Animaci%C3%B3n1.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/80/Animaci%C3%B3n1.gif 2x" data-file-width="440" data-file-height="262" /></a><figcaption>A mass on a spring has one <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a>, as it has a single <a href="/wiki/Degrees_of_freedom_(engineering)" class="mw-redirect" title="Degrees of freedom (engineering)">degree of freedom</a></figcaption></figure> <p>A physical system can have as many natural frequencies as it has <a href="/wiki/Degrees_of_freedom_(engineering)" class="mw-redirect" title="Degrees of freedom (engineering)">degrees of freedom</a> and can resonate near each of those natural frequencies. A mass on a spring, which has one degree of freedom, has one natural frequency. A <a href="/wiki/Double_pendulum" title="Double pendulum">double pendulum</a>, which has two degrees of freedom, can have two natural frequencies. As the number of coupled harmonic oscillators increases, the time it takes to transfer energy from one to the next becomes significant. Systems with very large numbers of degrees of freedom can be thought of as <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuous</a> rather than as having discrete oscillators.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2021)">citation needed</span></a></i>]</sup> </p><p>Energy transfers from one oscillator to the next in the form of waves. For example, the string of a guitar or the surface of water in a bowl can be modeled as a continuum of small coupled oscillators and waves can travel along them. In many cases these systems have the potential to resonate at certain frequencies, forming <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a> with large-amplitude oscillations at fixed positions. Resonance in the form of standing waves underlies many familiar phenomena, such as the sound produced by musical instruments, electromagnetic cavities used in lasers and microwave ovens, and energy levels of atoms.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Standing_waves_on_a_string">Standing waves on a string</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=14" title="Edit section: Standing waves on a string"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Standing_wave_2.gif" class="mw-file-description"><img alt="animation of a standing wave" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Standing_wave_2.gif/220px-Standing_wave_2.gif" decoding="async" width="220" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Standing_wave_2.gif/330px-Standing_wave_2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Standing_wave_2.gif/440px-Standing_wave_2.gif 2x" data-file-width="750" data-file-height="250" /></a><figcaption>A <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> (in black), created when two waves moving from left and right meet and superimpose</figcaption></figure> <p>When a string of fixed length is driven at a particular frequency, a wave propagates along the string at the same frequency. The waves <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">reflect</a> off the ends of the string, and eventually a <a href="/wiki/Steady_state" title="Steady state">steady state</a> is reached with waves traveling in both directions. The waveform is the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of the waves.<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005432_7-0" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005432-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>At certain frequencies, the steady state waveform does not appear to travel along the string. At fixed positions called <a href="/wiki/Node_(physics)" title="Node (physics)">nodes</a>, the string is never <a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">displaced</a>. Between the nodes the string oscillates and exactly halfway between the nodes–at positions called anti-nodes–the oscillations have their largest amplitude.<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005431–432_8-0" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005431–432-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESerwayFaughn1992472_9-0" class="reference"><a href="#cite_note-FOOTNOTESerwayFaughn1992472-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Standing_waves_on_a_string.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Standing_waves_on_a_string.gif/170px-Standing_waves_on_a_string.gif" decoding="async" width="170" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Standing_waves_on_a_string.gif/255px-Standing_waves_on_a_string.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Standing_waves_on_a_string.gif/340px-Standing_waves_on_a_string.gif 2x" data-file-width="360" data-file-height="339" /></a><figcaption>Standing waves in a string – the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a> mode and the first 5 <a href="/wiki/Harmonic" title="Harmonic">harmonics</a>.</figcaption></figure> <p>For a string of length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> with fixed ends, the displacement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/728a21d017a0d932521303cf109fc328ce876212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.168ex; height:2.843ex;" alt="{\displaystyle y(x,t)}"></span> of the string perpendicular to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-axis at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005432_7-1" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005432-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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 y(x,t)=2y_{\text{max}}\sin(kx)\cos(2\pi ft),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x,t)=2y_{\text{max}}\sin(kx)\cos(2\pi ft),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b166faa21ef6e36d666f5bd01b50c7e35ea70d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.019ex; height:2.843ex;" alt="{\displaystyle y(x,t)=2y_{\text{max}}\sin(kx)\cos(2\pi ft),}"></span> </p><p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{\text{max}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{\text{max}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f990a65e495ff47ae797bc7fb8516068b932cb2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.43ex; height:2.009ex;" alt="{\displaystyle y_{\text{max}}}"></span> is the <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> of the left- and right-traveling waves interfering to form the standing wave,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the <a href="/wiki/Wave_number" class="mw-redirect" title="Wave number">wave number</a>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the <a href="/wiki/Frequency" title="Frequency">frequency</a>.</li></ul> <p>The frequencies that resonate and form standing waves relate to the length of the string as<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005434_11-0" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005434-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESerwayFaughn1992472_9-1" class="reference"><a href="#cite_note-FOOTNOTESerwayFaughn1992472-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <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={\frac {nv}{2L}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>v</mi> </mrow> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {nv}{2L}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a4133b000b230e75de398326ecbb6bc31f6f77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.605ex; height:4.676ex;" alt="{\displaystyle f={\frac {nv}{2L}},}"></span> <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 n=1,2,3,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,3,\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b207567215497887a3250644b8876c23dc3ebf10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;" alt="{\displaystyle n=1,2,3,\dots }"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is the speed of the wave and the integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> denotes different modes or <a href="/wiki/Harmonic" title="Harmonic">harmonics</a>. The standing wave with <span class="texhtml"><i>n</i> = 1</span> oscillates at the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a> and has a wavelength that is twice the length of the string. The possible modes of oscillation form a <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a>.<sup id="cite_ref-FOOTNOTEHallidayResnickWalker2005434_11-1" class="reference"><a href="#cite_note-FOOTNOTEHallidayResnickWalker2005434-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Resonance_in_complex_networks">Resonance in complex networks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=15" title="Edit section: Resonance in complex networks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A generalization to complex networks of coupled harmonic oscillators shows that such systems have a finite number of natural resonant frequencies, related to the topological structure of the network itself. In particular, such frequencies result related to the eigenvalues of the network's Laplacian matrix. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f7b46d371ea1fd658b16a62e2b07022d9f9295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\bf {A}}}"></span> be the <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> describing the topological structure of the network and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {L}}={\bf {K}}-{\bf {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {L}}={\bf {K}}-{\bf {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ac587e0e67717fe41b99f27bb3d93c9d5343b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.661ex; height:2.343ex;" alt="{\displaystyle {\bf {L}}={\bf {K}}-{\bf {A}}}"></span> the corresponding <a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian matrix</a>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {K}}={\rm {diag}}\,\{k_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">{</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {K}}={\rm {diag}}\,\{k_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60ecfe0e6d723b7fb30e2b1173f59d3eff7ddfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.179ex; height:2.843ex;" alt="{\displaystyle {\bf {K}}={\rm {diag}}\,\{k_{i}\}}"></span> is the diagonal matrix of the degrees of the network's nodes. Then, for a network of classical and identical harmonic oscillators, when a sinusoidal driving force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=F_{0}\sin \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=F_{0}\sin \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5ef8f710eaf062ada6b07597f5067dcae534cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.49ex; height:2.843ex;" alt="{\displaystyle f(t)=F_{0}\sin \omega t}"></span> is applied to a specific node, the global resonant frequencies of the network are given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{i}={\sqrt {1+\mu _{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{i}={\sqrt {1+\mu _{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a5f3bb2d00ff0deca1b1d04353491a8acfd425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.872ex; height:3.509ex;" alt="{\displaystyle \omega _{i}={\sqrt {1+\mu _{i}}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0a0293841cce9eef98b55e53a92b82ae59ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}"></span> are the <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> of the Laplacian <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9065157c3abcaed3d13ac4df954b199449f3f5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;" alt="{\displaystyle {\bf {L}}}"></span>.<sup id="cite_ref-Bartesaghi2023_12-0" class="reference"><a href="#cite_note-Bartesaghi2023-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=16" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mechanical">Mechanical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=17" title="Edit section: Mechanical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Mechanical_resonance" title="Mechanical resonance">Mechanical resonance</a> and <a href="/wiki/String_resonance" class="mw-redirect" title="String resonance">String resonance</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Resonating_mass_experiment.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Resonating_mass_experiment.jpg/170px-Resonating_mass_experiment.jpg" decoding="async" width="170" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Resonating_mass_experiment.jpg/255px-Resonating_mass_experiment.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Resonating_mass_experiment.jpg/340px-Resonating_mass_experiment.jpg 2x" data-file-width="659" data-file-height="850" /></a><figcaption>School resonating mass experiment</figcaption></figure> <p><a href="/wiki/Mechanical_resonance" title="Mechanical resonance">Mechanical resonance</a> is the tendency of a <a href="/wiki/Mechanics" title="Mechanics">mechanical system</a> to absorb more energy when the <a href="/wiki/Frequency" title="Frequency">frequency</a> of its oscillations matches the system's natural frequency of <a href="/wiki/Vibration" title="Vibration">vibration</a> than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, <a href="/wiki/Engineers" class="mw-redirect" title="Engineers">engineers</a> must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as <a href="/wiki/Mechanical_resonance#Resonance_disaster" title="Mechanical resonance">resonance disaster</a>. </p><p>Avoiding resonance disasters is a major concern in every building, tower, and <a href="/wiki/Bridge" title="Bridge">bridge</a> <a href="/wiki/Construction" title="Construction">construction</a> project. As a countermeasure, <a href="/wiki/Shock_mount" title="Shock mount">shock mounts</a> can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The <a href="/wiki/Taipei_101" title="Taipei 101">Taipei 101</a> building relies on a 660-tonne pendulum (730-short-ton)—a <a href="/wiki/Tuned_mass_damper" title="Tuned mass damper">tuned mass damper</a>—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that does not typically occur. Buildings in <a href="/wiki/Seismic" class="mw-redirect" title="Seismic">seismic</a> zones are often constructed to take into account the oscillating frequencies of expected <a href="/wiki/Ground_motion" title="Ground motion">ground motion</a>. In addition, <a href="/wiki/Engineer" title="Engineer">engineers</a> designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts. </p><p><a href="/wiki/Clock" title="Clock">Clocks</a> keep time by mechanical resonance in a <a href="/wiki/Balance_wheel" title="Balance wheel">balance wheel</a>, pendulum, or <a href="/wiki/Quartz_clock" title="Quartz clock">quartz crystal</a>. </p><p>The cadence of runners has been hypothesized to be energetically favorable due to resonance between the elastic energy stored in the lower limb and the mass of the runner.<sup id="cite_ref-FOOTNOTESnyderFarley2011_13-0" class="reference"><a href="#cite_note-FOOTNOTESnyderFarley2011-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="International_Space_Station">International Space Station</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=18" title="Edit section: International Space Station"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Rocket_engine" title="Rocket engine">rocket engines</a> for the <a href="/wiki/International_Space_Station" title="International Space Station">International Space Station</a> (ISS) are controlled by an <a href="/wiki/Autopilot" title="Autopilot">autopilot</a>. Ordinarily, uploaded parameters for controlling the engine control system for the Zvezda module make the rocket engines boost the International Space Station to a higher orbit. The rocket engines are <a href="/wiki/Hinge" title="Hinge">hinge</a>-mounted, and ordinarily the crew does not notice the operation. On January 14, 2009, however, the uploaded parameters made the autopilot swing the rocket engines in larger and larger oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video, and lasted for 142 seconds.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Acoustic">Acoustic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=19" title="Edit section: Acoustic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Acoustic_resonance" title="Acoustic resonance">Acoustic resonance</a></div> <p><a href="/wiki/Acoustic_resonance" title="Acoustic resonance">Acoustic resonance</a> is a branch of mechanical resonance that is concerned with the mechanical vibrations across the frequency range of human hearing, in other words <a href="/wiki/Sound" title="Sound">sound</a>. For humans, hearing is normally limited to frequencies between about 20 <a href="/wiki/Hertz" title="Hertz">Hz</a> and 20,000 Hz (20 <a href="/wiki/KHz" class="mw-redirect" title="KHz">kHz</a>),<sup id="cite_ref-FOOTNOTEOlson1967248–249_15-0" class="reference"><a href="#cite_note-FOOTNOTEOlson1967248–249-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Many objects and materials act as resonators with resonant frequencies within this range, and when struck vibrate mechanically, pushing on the surrounding air to create sound waves. This is the source of many percussive sounds we hear. </p><p>Acoustic resonance is an important consideration for instrument builders, as most acoustic <a href="/wiki/Musical_instrument" title="Musical instrument">instruments</a> use <a href="/wiki/Resonator" title="Resonator">resonators</a>, such as the <a href="/wiki/String_resonance" class="mw-redirect" title="String resonance">strings</a> and body of a <a href="/wiki/Violin" title="Violin">violin</a>, the length of tube in a <a href="/wiki/Flute" title="Flute">flute</a>, and the shape of, and tension on, a drum membrane. </p><p>Like mechanical resonance, acoustic resonance can result in catastrophic failure of the object at resonance. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass, although this is difficult in practice.<sup id="cite_ref-UCLA_16-0" class="reference"><a href="#cite_note-UCLA-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electrical">Electrical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=20" title="Edit section: Electrical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Electrical_resonance" title="Electrical resonance">Electrical resonance</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tuned_circuit_animation_3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Tuned_circuit_animation_3.gif/240px-Tuned_circuit_animation_3.gif" decoding="async" width="240" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Tuned_circuit_animation_3.gif/360px-Tuned_circuit_animation_3.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1d/Tuned_circuit_animation_3.gif 2x" data-file-width="408" data-file-height="335" /></a><figcaption>Animation illustrating electrical resonance in a <a href="/wiki/Tuned_circuit" class="mw-redirect" title="Tuned circuit">tuned circuit</a>, consisting of a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> (C) and an <a href="/wiki/Inductor" title="Inductor">inductor</a> (L) connected together. Charge flows back and forth between the capacitor plates through the inductor. Energy oscillates back and forth between the capacitor's <a href="/wiki/Electric_field" title="Electric field">electric field</a> (<span class="texhtml mvar" style="font-style:italic;">E</span>) and the inductor's <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> (<span class="texhtml mvar" style="font-style:italic;">B</span>). </figcaption></figure> <p><a href="/wiki/Electrical_resonance" title="Electrical resonance">Electrical resonance</a> occurs in an electric circuit at a particular <i>resonant frequency</i> when the <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> of the circuit is at a minimum in a series circuit or at maximum in a parallel circuit (usually when the transfer function peaks in absolute value). Resonance in circuits are used for both transmitting and receiving wireless communications such as television, cell phones and radio. </p> <div class="mw-heading mw-heading3"><h3 id="Optical">Optical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=21" title="Edit section: Optical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Optical_cavity" title="Optical cavity">Optical cavity</a></div> <p>An <a href="/wiki/Optical_cavity" title="Optical cavity">optical cavity</a>, also called an <i>optical resonator</i>, is an arrangement of <a href="/wiki/Mirror" title="Mirror">mirrors</a> that forms a standing wave <a href="/wiki/Cavity_resonator" class="mw-redirect" title="Cavity resonator">cavity resonator</a> for <a href="/wiki/Light_wave" class="mw-redirect" title="Light wave">light waves</a>. Optical cavities are a major component of <a href="/wiki/Laser" title="Laser">lasers</a>, surrounding the <a href="/wiki/Gain_medium" class="mw-redirect" title="Gain medium">gain medium</a> and providing <a href="/wiki/Feedback" title="Feedback">feedback</a> of the laser light. They are also used in <a href="/wiki/Optical_parametric_oscillator" title="Optical parametric oscillator">optical parametric oscillators</a> and some <a href="/wiki/Interferometer" class="mw-redirect" title="Interferometer">interferometers</a>. Light confined in the cavity reflects multiple times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called "modes". <a href="/wiki/Longitudinal_mode" title="Longitudinal mode">Longitudinal modes</a> differ only in frequency while <a href="/wiki/Transverse_mode" title="Transverse mode">transverse modes</a> differ for different frequencies and have different intensity patterns across the cross-section of the beam. <a href="/wiki/Optical_ring_resonators" title="Optical ring resonators">Ring resonators</a> and <a href="/wiki/Whispering_gallery" title="Whispering gallery">whispering galleries</a> are examples of optical resonators that do not form standing waves. </p><p>Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them; flat mirrors are not often used because of the difficulty of aligning them precisely. The geometry (resonator type) must be chosen so the beam remains stable, i.e., the beam size does not continue to grow with each reflection. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity. </p><p>Optical cavities are designed to have a very large <a href="/wiki/Q_factor" title="Q factor"><i>Q</i> factor</a>.<sup id="cite_ref-q_factor_17-0" class="reference"><a href="#cite_note-q_factor-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> A beam reflects a large number of times with little <a href="/wiki/Attenuation" title="Attenuation">attenuation</a>—therefore the frequency <a href="/wiki/Line_width" class="mw-redirect" title="Line width">line width</a> of the beam is small compared to the frequency of the laser. </p><p>Additional optical resonances are <a href="/wiki/Guided-mode_resonance" title="Guided-mode resonance">guided-mode resonances</a> and <a href="/wiki/Surface_plasmon_resonance" title="Surface plasmon resonance">surface plasmon resonance</a>, which result in anomalous reflection and high evanescent fields at resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are usually excited by a subwavelength grating. </p> <div class="mw-heading mw-heading3"><h3 id="Orbital">Orbital</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=22" title="Edit section: Orbital"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orbital_resonance" title="Orbital resonance">Orbital resonance</a></div> <p>In <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, an <a href="/wiki/Orbital_resonance" title="Orbital resonance">orbital resonance</a> occurs when two <a href="/wiki/Orbit" title="Orbit">orbiting</a> bodies exert a regular, periodic gravitational influence on each other, usually due to their <a href="/wiki/Orbital_period" title="Orbital period">orbital periods</a> being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an <i>unstable</i> interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a>'s moons <a href="/wiki/Ganymede_(moon)" title="Ganymede (moon)">Ganymede</a>, <a href="/wiki/Europa_(moon)" title="Europa (moon)">Europa</a>, and <a href="/wiki/Io_(moon)" title="Io (moon)">Io</a>, and the 2:3 resonance between <a href="/wiki/Pluto" title="Pluto">Pluto</a> and <a href="/wiki/Neptune" title="Neptune">Neptune</a>. Unstable resonances with <a href="/wiki/Saturn" title="Saturn">Saturn</a>'s inner moons give rise to gaps in the <a href="/wiki/Rings_of_Saturn" title="Rings of Saturn">rings of Saturn</a>. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to <a href="/wiki/Clearing_the_neighbourhood" title="Clearing the neighbourhood">clear the neighborhood</a> around their orbits by ejecting nearly everything else around them; this effect is used in the current <a href="/wiki/Definition_of_planet" title="Definition of planet">definition of a planet</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Atomic,_particle,_and_molecular"><span id="Atomic.2C_particle.2C_and_molecular"></span>Atomic, particle, and molecular</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=23" title="Edit section: Atomic, particle, and molecular"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">Nuclear magnetic resonance</a> and <a href="/wiki/Resonance_(particle_physics)" title="Resonance (particle physics)">Resonance (particle physics)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HWB-NMR_-_900MHz_-_21.2_Tesla.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/HWB-NMR_-_900MHz_-_21.2_Tesla.jpg/170px-HWB-NMR_-_900MHz_-_21.2_Tesla.jpg" decoding="async" width="170" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/f9/HWB-NMR_-_900MHz_-_21.2_Tesla.jpg 1.5x" data-file-width="250" data-file-height="309" /></a><figcaption><a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">NMR</a> Magnet at HWB-NMR, Birmingham, UK. In its strong 21.2-<a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a> field, the proton resonance is at 900 MHz.</figcaption></figure> <p><a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">Nuclear magnetic resonance</a> (NMR) is the name given to a physical resonance phenomenon involving the observation of specific <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> <a href="/wiki/Magnetism" title="Magnetism">magnetic</a> properties of an <a href="/wiki/Atom" title="Atom">atomic</a> <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nucleus</a> in the presence of an applied, external magnetic field. Many scientific techniques exploit NMR phenomena to study <a href="/wiki/Molecular_physics" title="Molecular physics">molecular physics</a>, <a href="/wiki/Crystallography" title="Crystallography">crystals</a>, and non-crystalline materials through <a href="/wiki/NMR_spectroscopy" class="mw-redirect" title="NMR spectroscopy">NMR spectroscopy</a>. NMR is also routinely used in advanced medical imaging techniques, such as in <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> (MRI). </p><p>All nuclei containing odd numbers of <a href="/wiki/Nucleon" title="Nucleon">nucleons</a> have an intrinsic <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moment</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. A key feature of NMR is that the resonant frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonant frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency. </p><p><a href="/wiki/Electron_paramagnetic_resonance" title="Electron paramagnetic resonance">Electron paramagnetic resonance</a>, otherwise known as <i>electron spin resonance</i> (ESR), is a spectroscopic technique similar to NMR, but uses unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be <a href="/wiki/Paramagnetic" class="mw-redirect" title="Paramagnetic">paramagnetic</a>. </p><p>The <a href="/wiki/M%C3%B6ssbauer_effect" title="Mössbauer effect">Mössbauer effect</a> is the resonant and <a href="/wiki/Recoil" title="Recoil">recoil</a>-free emission and absorption of <a href="/wiki/Gamma_ray" title="Gamma ray">gamma ray</a> photons by atoms bound in a solid form. </p><p><a href="/wiki/Resonance_(particle_physics)" title="Resonance (particle physics)">Resonance in particle physics</a> appears in similar circumstances to <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a> at the level of quantum mechanics and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. Resonances can also be thought of as unstable particles, with the formula in the <a href="#Universal_resonance_curve">Universal resonance curve</a> section of this article applying if <i>Γ</i> is the particle's <a href="/wiki/Particle_decay#Decay_rate" title="Particle decay">decay rate</a> and <i>Ω</i> is the particle's mass <i>M</i>. In that case, the formula comes from the particle's <a href="/wiki/Propagator_(Quantum_Theory)" class="mw-redirect" title="Propagator (Quantum Theory)">propagator</a>, with its mass replaced by the complex number <i>M</i> + <i>iΓ</i>. The formula is further related to the particle's decay rate by the <a href="/wiki/Optical_theorem" title="Optical theorem">optical theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Disadvantages">Disadvantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=24" title="Edit section: Disadvantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A column of soldiers marching in regular step on a narrow and structurally flexible bridge can set it into dangerously large <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> <a href="/wiki/Oscillation" title="Oscillation">oscillations</a>. On April 12, 1831, the <a href="/wiki/Broughton_Suspension_Bridge" title="Broughton Suspension Bridge">Broughton Suspension Bridge</a> near <a href="/wiki/Salford,_England" class="mw-redirect" title="Salford, England">Salford, England</a> collapsed while a group of British soldiers were marching across.<sup id="cite_ref-Bishop_18-0" class="reference"><a href="#cite_note-Bishop-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Since then, the British Army has had a standing order for soldiers to break stride when marching across bridges, to avoid resonance from their regular marching pattern affecting the bridge.<sup id="cite_ref-MEN_19-0" class="reference"><a href="#cite_note-MEN-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Vibrations of a motor or engine can induce resonant vibration in its supporting structures if their <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a> is close to that of the vibrations of the engine. A common example is the rattling sound of a bus body when the engine is left idling. </p><p>Structural resonance of a suspension bridge induced by winds can lead to its catastrophic collapse. Several early suspension bridges in <a href="/wiki/Europe" title="Europe">Europe</a> and <a href="/wiki/United_States" title="United States">United States</a> were destroyed by structural resonance induced by modest winds. The collapse of the <a href="/wiki/Tacoma_Narrows_Bridge_(1940)" title="Tacoma Narrows Bridge (1940)">Tacoma Narrows Bridge</a> on 7 November 1940 is characterized in physics as a classic example of resonance.<sup id="cite_ref-Forbes_21-0" class="reference"><a href="#cite_note-Forbes-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> It has been argued by <a href="/wiki/Robert_H._Scanlan" title="Robert H. Scanlan">Robert H. Scanlan</a> and others that the destruction was instead caused by <a href="/wiki/Aeroelasticity#Flutter" title="Aeroelasticity">aeroelastic flutter</a>, a complicated interaction between the bridge and the winds passing through it—an example of a <a href="/wiki/Self_oscillation" class="mw-redirect" title="Self oscillation">self oscillation</a>, or a kind of "self-sustaining vibration" as referred to in the nonlinear theory of vibrations.<sup id="cite_ref-FOOTNOTEBillahScanlan1991_22-0" class="reference"><a href="#cite_note-FOOTNOTEBillahScanlan1991-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Q_factor">Q factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=25" title="Edit section: Q factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Q_factor" title="Q factor">Q factor</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:High_and_low_Q_factor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/High_and_low_Q_factor.svg/220px-High_and_low_Q_factor.svg.png" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/High_and_low_Q_factor.svg/330px-High_and_low_Q_factor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/High_and_low_Q_factor.svg/440px-High_and_low_Q_factor.svg.png 2x" data-file-width="606" data-file-height="410" /></a><figcaption>High and low Q factor</figcaption></figure> <p>The <i>Q</i> factor or <i>quality factor</i> is a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> parameter that describes how under-damped an <a href="/wiki/Oscillation" title="Oscillation">oscillator</a> or resonator is, and characterizes the <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a> of a resonator relative to its center frequency.<sup id="cite_ref-FOOTNOTEHarlow20042.216_23-0" class="reference"><a href="#cite_note-FOOTNOTEHarlow20042.216-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTETooley200677–78_24-0" class="reference"><a href="#cite_note-FOOTNOTETooley200677–78-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> A high value for <i>Q</i> indicates a lower rate of energy loss relative to the stored energy, i.e., the system is lightly damped. The parameter is defined by the equation: <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 Q=2\pi {\text{ }}{\frac {\text{ maximum energy stored}}{\text{total energy lost per cycle at resonance}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext> maximum energy stored</mtext> <mtext>total energy lost per cycle at resonance</mtext> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=2\pi {\text{ }}{\frac {\text{ maximum energy stored}}{\text{total energy lost per cycle at resonance}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdedbe66a9d00fcc4b10e8124e18f69a7952ba7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.512ex; height:5.843ex;" alt="{\displaystyle Q=2\pi {\text{ }}{\frac {\text{ maximum energy stored}}{\text{total energy lost per cycle at resonance}}}}"></span>.<sup id="cite_ref-MIT_25-0" class="reference"><a href="#cite_note-MIT-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>The higher the Q factor, the greater the amplitude at the resonant frequency, and the smaller the <i>bandwidth</i>, or range of frequencies around resonance occurs. In electrical resonance, a high-<i>Q</i> circuit in a radio receiver is more difficult to tune, but has greater <a href="/wiki/Selectivity_(radio)" title="Selectivity (radio)">selectivity</a>, and so would be better at filtering out signals from other stations. High Q oscillators are more stable.<sup id="cite_ref-MIT_25-1" class="reference"><a href="#cite_note-MIT-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Examples that normally have a low Q factor include <a href="/wiki/Door_closer" title="Door closer">door closers</a> (Q=0.5). Systems with high Q factors include <a href="/wiki/Tuning_fork" title="Tuning fork">tuning forks</a> (Q=1000), <a href="/wiki/Atomic_clock" title="Atomic clock">atomic clocks</a> and lasers (Q≈10<sup>11</sup>).<sup id="cite_ref-NIST_26-0" class="reference"><a href="#cite_note-NIST-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Universal_resonance_curve">Universal resonance curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=26" title="Edit section: Universal resonance curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Universal_Resonance_Curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Universal_Resonance_Curve.svg/290px-Universal_Resonance_Curve.svg.png" decoding="async" width="290" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Universal_Resonance_Curve.svg/435px-Universal_Resonance_Curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Universal_Resonance_Curve.svg/580px-Universal_Resonance_Curve.svg.png 2x" data-file-width="512" data-file-height="362" /></a><figcaption>"Universal Resonance Curve", a symmetric approximation to the normalized response of a resonant circuit; <a href="/wiki/Abscissa" class="mw-redirect" title="Abscissa">abscissa</a> values are deviation from center frequency, in units of center frequency divided by 2Q; <a href="/wiki/Ordinate" class="mw-redirect" title="Ordinate">ordinate</a> is relative amplitude, and phase in cycles; dashed curves compare the range of responses of real two-pole circuits for a <i>Q</i> value of 5; for higher <i>Q</i> values, there is less deviation from the universal curve. Crosses mark the edges of the 3 dB bandwidth (gain 0.707, phase shift 45° or 0.125 cycle).</figcaption></figure> <p>The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system, and is usually not exactly symmetric about the resonant frequency, as illustrated for the <a href="/wiki/Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a> above. </p><p>For a lightly damped linear oscillator with a resonance frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}"></span>, the <i>intensity</i> of oscillations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> when the system is driven with a driving frequency <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span></i> is typically approximated by the following formula that is symmetric about the resonance frequency:<sup id="cite_ref-FOOTNOTESiegman1986105–108_27-0" class="reference"><a href="#cite_note-FOOTNOTESiegman1986105–108-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><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 I(\omega )\equiv |\chi |^{2}\propto {\frac {1}{(\omega -\omega _{0})^{2}+\left({\frac {\Gamma }{2}}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>χ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\omega )\equiv |\chi |^{2}\propto {\frac {1}{(\omega -\omega _{0})^{2}+\left({\frac {\Gamma }{2}}\right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2f9d1508bbb61876143cefd0679cb81b3474b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:34.093ex; height:8.343ex;" alt="{\displaystyle I(\omega )\equiv |\chi |^{2}\propto {\frac {1}{(\omega -\omega _{0})^{2}+\left({\frac {\Gamma }{2}}\right)^{2}}}.}"></span> </p><p>Where the susceptibility <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eaafe487da26a1b95abc0db0fe578216cc31395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:2.843ex;" alt="{\displaystyle \chi (\omega )}"></span> links the amplitude of the oscillator to the driving force in frequency space:<sup id="cite_ref-FOOTNOTEAspelmeyerKippenbergMarquardt2014_28-0" class="reference"><a href="#cite_note-FOOTNOTEAspelmeyerKippenbergMarquardt2014-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><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 x(\omega )=\chi (\omega )F(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\omega )=\chi (\omega )F(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eee56dc887f9663dd064f90a1d84b1c7ed6aca2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.389ex; height:2.843ex;" alt="{\displaystyle x(\omega )=\chi (\omega )F(\omega )}"></span> </p><p>The intensity is defined as the square of the amplitude of the oscillations. This is a <a href="/wiki/Lorentzian_function" class="mw-redirect" title="Lorentzian function">Lorentzian function</a>, or <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>, and this response is found in many physical situations involving resonant systems. <span class="texhtml">Γ</span> is a parameter dependent on the damping of the oscillator, and is known as the <i>linewidth</i> of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonant frequency. The linewidth is <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">inversely proportional</a> to the <i>Q</i> factor, which is a measure of the sharpness of the resonance. </p><p>In <a href="/wiki/Radio_engineering" class="mw-redirect" title="Radio engineering">radio engineering</a> and <a href="/wiki/Electronics_engineering" class="mw-redirect" title="Electronics engineering">electronics engineering</a>, this approximate symmetric response is known as the <i>universal resonance curve</i>, a concept introduced by <a href="/wiki/Frederick_E._Terman" class="mw-redirect" title="Frederick E. Terman">Frederick E. Terman</a> in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and <i>Q</i> values.<sup id="cite_ref-FOOTNOTETerman1932_29-0" class="reference"><a href="#cite_note-FOOTNOTETerman1932-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESiebert1986113_30-0" class="reference"><a href="#cite_note-FOOTNOTESiebert1986113-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Resonance&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 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Vol. part 2 (7th ed.). 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Retrieved <span class="nowrap">18 April</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=2.14+Analysis+and+Design+of+Feedback+Control+Systems&rft.atitle=Understanding+Poles+and+Zeros&rft.date=2004&rft.aulast=Hardt&rft.aufirst=David&rft_id=https%3A%2F%2Fweb.mit.edu%2F2.14%2Fwww%2FHandouts%2FPoleZero.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarlow2004" class="citation book cs1">Harlow, James H., ed. (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DANXjaoaucYC"><i>Electric Power Transformer Engineering</i></a>. London: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-1704-0" title="Special:BookSources/978-0-8493-1704-0"><bdi>978-0-8493-1704-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electric+Power+Transformer+Engineering&rft.place=London&rft.pub=CRC+Press&rft.date=2004&rft.isbn=978-0-8493-1704-0&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDANXjaoaucYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOgata2005" class="citation book cs1">Ogata, Katsuhiko (2005). <i>System Dynamics</i> (4th ed.). Harlow: Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-292-02608-4" title="Special:BookSources/978-1-292-02608-4"><bdi>978-1-292-02608-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=System+Dynamics&rft.place=Harlow&rft.edition=4th&rft.pub=Pearson&rft.date=2005&rft.isbn=978-1-292-02608-4&rft.aulast=Ogata&rft.aufirst=Katsuhiko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOlson1967" class="citation book cs1"><a href="/wiki/Harry_F._Olson" title="Harry F. Olson">Olson, Harry F.</a> (1967). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/musicphysicsengi0000olso"><i>Music, Physics and Engineering</i></a></span>. Vol. 2. New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-21769-7" title="Special:BookSources/978-0-486-21769-7"><bdi>978-0-486-21769-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Music%2C+Physics+and+Engineering&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1967&rft.isbn=978-0-486-21769-7&rft.aulast=Olson&rft.aufirst=Harry+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmusicphysicsengi0000olso&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerwayFaughn1992" class="citation book cs1">Serway, Raymond A.; Faughn, Jerry S. (1992). <i>College Physics</i> (3rd ed.). Saunders College Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-076377-0" title="Special:BookSources/0-03-076377-0"><bdi>0-03-076377-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Physics&rft.edition=3rd&rft.pub=Saunders+College+Publishing&rft.date=1992&rft.isbn=0-03-076377-0&rft.aulast=Serway&rft.aufirst=Raymond+A.&rft.au=Faughn%2C+Jerry+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiebert1986" class="citation book cs1">Siebert, William McC. (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zBTUiIrb2WIC"><i>Circuits, Signals, and Systems</i></a>. London; New York: MIT Press' McGraw Hill Book Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-19229-3" title="Special:BookSources/978-0-262-19229-3"><bdi>978-0-262-19229-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Circuits%2C+Signals%2C+and+Systems&rft.place=London%3B+New+York&rft.pub=MIT+Press%27+McGraw+Hill+Book+Company&rft.date=1986&rft.isbn=978-0-262-19229-3&rft.aulast=Siebert&rft.aufirst=William+McC.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzBTUiIrb2WIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiegman1986" class="citation book cs1"><a href="/wiki/Anthony_E._Siegman" title="Anthony E. Siegman">Siegman, A. E.</a> (1986). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/lasers0000sieg/page/n5/mode/2up"><i>Lasers</i></a></span>. University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-935702-11-8" title="Special:BookSources/978-0-935702-11-8"><bdi>978-0-935702-11-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lasers&rft.pub=University+Science+Books&rft.date=1986&rft.isbn=978-0-935702-11-8&rft.aulast=Siegman&rft.aufirst=A.+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flasers0000sieg%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSnyderFarley2011" class="citation journal cs1">Snyder, Kristine L.; Farley, Claire T. (2011). <a rel="nofollow" class="external text" href="https://jeb.biologists.org/content/214/12/2089">"Energetically optimal stride frequency in running: the effects of incline and decline"</a>. <i><a href="/wiki/The_Journal_of_Experimental_Biology" title="The Journal of Experimental Biology">The Journal of Experimental Biology</a></i>. <b>214</b> (12): 2089–2095. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1242%2Fjeb.053157">10.1242/jeb.053157</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21613526">21613526</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Experimental+Biology&rft.atitle=Energetically+optimal+stride+frequency+in+running%3A+the+effects+of+incline+and+decline&rft.volume=214&rft.issue=12&rft.pages=2089-2095&rft.date=2011&rft_id=info%3Adoi%2F10.1242%2Fjeb.053157&rft_id=info%3Apmid%2F21613526&rft.aulast=Snyder&rft.aufirst=Kristine+L.&rft.au=Farley%2C+Claire+T.&rft_id=https%3A%2F%2Fjeb.biologists.org%2Fcontent%2F214%2F12%2F2089&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTerman1932" class="citation book cs1"><a href="/wiki/Frederick_Terman" title="Frederick Terman">Terman, Frederick Emmons</a> (1932). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/radioengineering00term/page/n7/mode/2up"><i>Radio Engineering</i></a></span> (1st ed.). New York: McGraw-Hill Book Company. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1036819790">1036819790</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Radio+Engineering&rft.place=New+York&rft.edition=1st&rft.pub=McGraw-Hill+Book+Company&rft.date=1932&rft_id=info%3Aoclcnum%2F1036819790&rft.aulast=Terman&rft.aufirst=Frederick+Emmons&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fradioengineering00term%2Fpage%2Fn7%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTooley2006" class="citation book cs1">Tooley, Michael H. (2006). <i>Electronic Circuits: Fundamentals and Applications</i>. Oxford: Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-6923-8" title="Special:BookSources/978-0-7506-6923-8"><bdi>978-0-7506-6923-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Circuits%3A+Fundamentals+and+Applications&rft.place=Oxford&rft.pub=Taylor+%26+Francis&rft.date=2006&rft.isbn=978-0-7506-6923-8&rft.aulast=Tooley&rft.aufirst=Michael+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AResonance" 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=Resonance&action=edit&section=30" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Resonance" class="extiw" title="commons:Category:Resonance">Resonance</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_23.html">The Feynman Lectures on Physics Vol. I Ch. 23: Resonance</a></li> <li><a rel="nofollow" class="external text" href="http://www.lightandmatter.com/html_books/lm/ch18/ch18.html">Resonance</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170103120102/http://www.lightandmatter.com/html_books/lm/ch18/ch18.html">Archived</a> 2017-01-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> - a chapter from an online textbook</li> <li><a href="/wiki/Brian_Greene" title="Brian Greene">Greene, Brian</a>, "<i><a rel="nofollow" class="external text" href="https://www.pbs.org/wgbh/nova/elegant/resonance.html">Resonance in strings</a></i>". <a href="/wiki/The_Elegant_Universe" title="The Elegant Universe">The Elegant Universe</a>, <a href="/wiki/Nova_(American_TV_program)" title="Nova (American TV program)">NOVA</a> (<a href="/wiki/Public_Broadcasting_Service" class="mw-redirect" title="Public Broadcasting Service">PBS</a>)</li> <li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/sound/rescon.html#c1">Hyperphysics section on resonance concepts</a></li> <li><a rel="nofollow" class="external text" href="http://users.ece.gatech.edu/~mleach/misc/resonance.html">Resonance versus resonant</a> (usage of terms)</li> <li><a rel="nofollow" class="external text" href="http://www.johnsankey.ca/bottom.html">Wood and Air Resonance in a Harpsichord</a></li> <li><a rel="nofollow" class="external text" href="http://www.acoustics.salford.ac.uk/acoustics_info/glass">Breaking glass with sound</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081202121519/http://www.acoustics.salford.ac.uk/acoustics_info/glass">Archived</a> 2008-12-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, including high-speed footage of glass breaking</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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