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class="vector-toc-numb">2</span> <span>Sinusoid form</span> </div> </a> <ul id="toc-Sinusoid_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_function_of_both_position_and_time" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#As_a_function_of_both_position_and_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>As a function of both position and time</span> </div> </a> <button aria-controls="toc-As_a_function_of_both_position_and_time-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 As a function of both position and time subsection</span> </button> <ul id="toc-As_a_function_of_both_position_and_time-sublist" class="vector-toc-list"> <li id="toc-Standing_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standing_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Standing waves</span> </div> </a> <ul id="toc-Standing_waves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple_spatial_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_spatial_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Multiple spatial dimensions</span> </div> </a> <ul id="toc-Multiple_spatial_dimensions-sublist" class="vector-toc-list"> <li id="toc-Sinusoidal_plane_wave" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sinusoidal_plane_wave"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Sinusoidal plane wave</span> </div> </a> <ul id="toc-Sinusoidal_plane_wave-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Fourier_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fourier_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Fourier analysis</span> </div> </a> <ul id="toc-Fourier_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differentiation_and_integration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Differentiation_and_integration"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Differentiation and integration</span> </div> </a> <button aria-controls="toc-Differentiation_and_integration-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 Differentiation and integration subsection</span> </button> <ul id="toc-Differentiation_and_integration-sublist" class="vector-toc-list"> <li id="toc-Differentiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differentiation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Differentiation</span> </div> </a> <ul id="toc-Differentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integration"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Integration</span> </div> </a> <ul id="toc-Integration-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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">Sine wave</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 52 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-52" 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">52 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/%D9%85%D9%88%D8%AC%D8%A9_%D8%AC%D9%8A%D8%A8%D9%8A%D8%A9" title="موجة جيبية – Arabic" lang="ar" hreflang="ar" data-title="موجة جيبية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%A4%E0%A6%B0%E0%A6%99%E0%A7%8D%E0%A6%97" title="সাইন তরঙ্গ – Bangla" lang="bn" hreflang="bn" data-title="সাইন তরঙ্গ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Sinusoida" title="Sinusoida – Bosnian" lang="bs" hreflang="bs" data-title="Sinusoida" 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/Sinusoide" title="Sinusoide – Catalan" lang="ca" hreflang="ca" data-title="Sinusoide" 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%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0" title="Синусоида – Chuvash" lang="cv" hreflang="cv" data-title="Синусоида" 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/Sinusoida" title="Sinusoida – Czech" lang="cs" hreflang="cs" data-title="Sinusoida" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Sinusb%C3%B8lge" title="Sinusbølge – Danish" lang="da" hreflang="da" data-title="Sinusbølge" 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/Sinusoid" title="Sinusoid – German" lang="de" hreflang="de" data-title="Sinusoid" 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/Sinusoid" title="Sinusoid – Estonian" lang="et" hreflang="et" data-title="Sinusoid" 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%97%CE%BC%CE%B9%CF%84%CE%BF%CE%BD%CE%BF%CE%B5%CE%B9%CE%B4%CE%AE%CF%82_%CE%BA%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B7" title="Ημιτονοειδής καμπύλη – Greek" lang="el" hreflang="el" data-title="Ημιτονοειδής καμπύλη" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sinusoide" title="Sinusoide – Spanish" lang="es" hreflang="es" data-title="Sinusoide" 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/Sinusoido" title="Sinusoido – Esperanto" lang="eo" hreflang="eo" data-title="Sinusoido" 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/Sinusoide" title="Sinusoide – Basque" lang="eu" hreflang="eu" data-title="Sinusoide" 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/%D9%85%D9%88%D8%AC_%D8%B3%DB%8C%D9%86%D9%88%D8%B3%DB%8C" title="موج سینوسی – Persian" lang="fa" hreflang="fa" data-title="موج سینوسی" 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/Signal_sinuso%C3%AFdal" title="Signal sinusoïdal – French" lang="fr" hreflang="fr" data-title="Signal sinusoïdal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%EC%9D%B8%ED%8C%8C" title="사인파 – Korean" lang="ko" hreflang="ko" data-title="사인파" 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%8D%D5%AB%D5%B6%D5%B8%D6%82%D5%BD%D5%B8%D5%AB%D5%A4" title="Սինուսոիդ – Armenian" lang="hy" hreflang="hy" data-title="Սինուսոիդ" 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%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97" title="ज्या तरंग – Hindi" lang="hi" hreflang="hi" data-title="ज्या तरंग" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Sinusoido" title="Sinusoido – Ido" lang="io" hreflang="io" data-title="Sinusoido" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gelombang_sinus" title="Gelombang sinus – Indonesian" lang="id" hreflang="id" data-title="Gelombang sinus" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Onda_sinusoidale" title="Onda sinusoidale – Italian" lang="it" hreflang="it" data-title="Onda sinusoidale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%9C%E1%83%A3%E1%83%A1%E1%83%9D%E1%83%98%E1%83%93%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A2%E1%83%90%E1%83%9A%E1%83%A6%E1%83%90" title="სინუსოიდალური ტალღა – Georgian" lang="ka" hreflang="ka" data-title="სინუსოიდალური ტალღა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ond_sinisoyidal" title="Ond sinisoyidal – Haitian Creole" lang="ht" hreflang="ht" data-title="Ond sinisoyidal" 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-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0" title="Синусоида – Lezghian" lang="lez" hreflang="lez" data-title="Синусоида" data-language-autonym="Лезги" data-language-local-name="Lezghian" class="interlanguage-link-target"><span>Лезги</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sinusoid%C4%97" title="Sinusoidė – Lithuanian" lang="lt" hreflang="lt" data-title="Sinusoidė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%B5%D0%BD_%D0%B1%D1%80%D0%B0%D0%BD" title="Синусен бран – Macedonian" lang="mk" hreflang="mk" data-title="Синусен бран" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Gelombang_sinus" title="Gelombang sinus – Malay" lang="ms" hreflang="ms" data-title="Gelombang sinus" 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-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0%D1%81%D1%8C" title="Синусоидась – Moksha" lang="mdf" hreflang="mdf" data-title="Синусоидась" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sinuso%C3%AFde" title="Sinusoïde – Dutch" lang="nl" hreflang="nl" data-title="Sinusoïde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E5%BC%A6%E6%B3%A2" title="正弦波 – Japanese" lang="ja" hreflang="ja" data-title="正弦波" 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/Sinuskurve" title="Sinuskurve – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sinuskurve" 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/Sinuskurve" title="Sinuskurve – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Sinuskurve" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%BE%E0%A8%88%E0%A8%A8_%E0%A8%B5%E0%A9%87%E0%A8%B5" title="ਸਾਈਨ ਵੇਵ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਾਈਨ ਵੇਵ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B3%D8%A7%D8%A6%DB%8C%D9%86_%D9%88%DB%8C%D9%88" title="سائین ویو – Western Punjabi" lang="pnb" hreflang="pnb" data-title="سائین ویو" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Fala_sinusoidalna" title="Fala sinusoidalna – Polish" lang="pl" hreflang="pl" data-title="Fala sinusoidalna" 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/Senoide" title="Senoide – Portuguese" lang="pt" hreflang="pt" data-title="Senoide" 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/Sinusoid%C4%83" title="Sinusoidă – Romanian" lang="ro" hreflang="ro" data-title="Sinusoidă" 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%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0" title="Синусоида – Russian" lang="ru" hreflang="ru" data-title="Синусоида" 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/Vala_sinusoidale" title="Vala sinusoidale – Albanian" lang="sq" hreflang="sq" data-title="Vala sinusoidale" 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/Sine_wave" title="Sine wave – Simple English" lang="en-simple" hreflang="en-simple" data-title="Sine wave" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0" title="Синусоида – Serbian" lang="sr" hreflang="sr" data-title="Синусоида" 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/Sinusoida" title="Sinusoida – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Sinusoida" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Gelombang_sinus" title="Gelombang sinus – Sundanese" lang="su" hreflang="su" data-title="Gelombang sinus" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Siniaalto" title="Siniaalto – Finnish" lang="fi" hreflang="fi" data-title="Siniaalto" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv 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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">"Sinusoid" redirects here. Not to be confused with <a href="/wiki/Sinusoid_(blood_vessel)" class="mw-redirect" title="Sinusoid (blood vessel)">Sinusoid (blood vessel)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-One_source plainlinks metadata ambox ambox-content ambox-one_source" 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 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 article <b>relies largely or entirely on a <a href="/wiki/Wikipedia:Articles_with_a_single_source" title="Wikipedia:Articles with a single source">single source</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Sine_wave##" title="Talk:Sine wave">talk page</a>. Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">introducing citations to additional sources</a>.<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=%22Sine+wave%22">"Sine wave"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Sine+wave%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Sine+wave%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=%22Sine+wave%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Sine+wave%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Sine+wave%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2024</span>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:One_positive_frequency_component,_cosine_and_sine,_from_rotating_vector_(fast_animation).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/One_positive_frequency_component%2C_cosine_and_sine%2C_from_rotating_vector_%28fast_animation%29.gif/282px-One_positive_frequency_component%2C_cosine_and_sine%2C_from_rotating_vector_%28fast_animation%29.gif" decoding="async" width="282" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/One_positive_frequency_component%2C_cosine_and_sine%2C_from_rotating_vector_%28fast_animation%29.gif/423px-One_positive_frequency_component%2C_cosine_and_sine%2C_from_rotating_vector_%28fast_animation%29.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/One_positive_frequency_component%2C_cosine_and_sine%2C_from_rotating_vector_%28fast_animation%29.gif/564px-One_positive_frequency_component%2C_cosine_and_sine%2C_from_rotating_vector_%28fast_animation%29.gif 2x" data-file-width="1210" data-file-height="500" /></a><figcaption>Tracing the y component of a <a href="/wiki/Circle" title="Circle">circle</a> while going around the circle results in a sine wave (red). Tracing the x component results in a <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> wave (blue). Both waves are sinusoids of the same frequency but different phases.</figcaption></figure> <p>A <b>sine wave</b>, <b>sinusoidal wave</b>, or <b>sinusoid</b> (symbol: <b>∿</b>) is a <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> <a href="/wiki/Wave" title="Wave">wave</a> whose <a href="/wiki/Waveform" title="Waveform">waveform</a> (shape) is the <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric</a> <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine function</a>. In <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, as a linear <a href="/wiki/Motion" title="Motion">motion</a> over time, this is <i><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">simple harmonic motion</a></i>; as <a href="/wiki/Rotation" title="Rotation">rotation</a>, it corresponds to <i><a href="/wiki/Uniform_circular_motion" class="mw-redirect" title="Uniform circular motion">uniform circular motion</a></i>. Sine waves occur often in <a href="/wiki/Physics" title="Physics">physics</a>, including <a href="/wiki/Wind_wave" title="Wind wave">wind waves</a>, <a href="/wiki/Sound" title="Sound">sound</a> waves, and <a href="/wiki/Light" title="Light">light</a> waves, such as <a href="/wiki/Monochromatic_radiation" title="Monochromatic radiation">monochromatic radiation</a>. In <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. </p><p>When any two sine waves of the same <a href="/wiki/Frequency" title="Frequency">frequency</a> (but arbitrary <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>) are <a href="/wiki/Linear_combination" title="Linear combination">linearly combined</a>, the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the <i>sine</i> and <i>cosine</i> <a href="/wiki/Vector_component" class="mw-redirect" title="Vector component">components</a>, respectively. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Audio_example">Audio example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=1" title="Edit section: Audio example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1096940132">.mw-parser-output .listen .side-box-text{line-height:1.1em}.mw-parser-output .listen-plain{border:none;background:transparent}.mw-parser-output .listen-embedded{width:100%;margin:0;border-width:1px 0 0 0;background:transparent}.mw-parser-output .listen-header{padding:2px}.mw-parser-output .listen-embedded .listen-header{padding:2px 0}.mw-parser-output .listen-file-header{padding:4px 0}.mw-parser-output .listen .description{padding-top:2px}.mw-parser-output .listen .mw-tmh-player{max-width:100%}@media(max-width:719px){.mw-parser-output .listen{clear:both}}@media(min-width:720px){.mw-parser-output .listen:not(.listen-noimage){width:320px}.mw-parser-output .listen-left{overflow:visible;float:left}.mw-parser-output .listen-center{float:none;margin-left:auto;margin-right:auto}}</style><div class="side-box side-box-right listen noprint"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-center" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/50px-Gnome-mime-sound-openclipart.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/75px-Gnome-mime-sound-openclipart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/100px-Gnome-mime-sound-openclipart.svg.png 2x" data-file-width="160" data-file-height="160" /></span><figcaption></figcaption></figure></div> <div class="side-box-text plainlist"><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:220_Hz_sine_wave.ogg" title="File:220 Hz sine wave.ogg">Sine wave</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_0" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="5" data-mwtitle="220_Hz_sine_wave.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/d/d1/220_Hz_sine_wave.ogg" type="audio/ogg; codecs="vorbis"" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d1/220_Hz_sine_wave.ogg/220_Hz_sine_wave.ogg.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description">Five seconds of a 220 Hz sine wave. This is the <a href="/wiki/Sound#Waves" title="Sound">sound wave</a> described by a sine function with <i>f</i> = 220 oscillations per second.</div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing this file? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <p>A sine wave represents a single <a href="/wiki/Frequency" title="Frequency">frequency</a> with no <a href="/wiki/Harmonic" title="Harmonic">harmonics</a> and is considered an <a href="/wiki/Acoustics" title="Acoustics">acoustically</a> <a href="/wiki/Pure_tone" title="Pure tone">pure tone</a>. Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a> causes variation in the <a href="/wiki/Timbre" title="Timbre">timbre</a>, which is the reason why the same <a href="/wiki/Pitch_(music)" title="Pitch (music)">musical pitch</a> played on different instruments sounds different. </p> <div class="mw-heading mw-heading2"><h2 id="Sinusoid_form">Sinusoid form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=2" title="Edit section: Sinusoid form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sine waves of arbitrary phase and amplitude are called <i>sinusoids</i> and have the general form:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<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(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <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> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f7f3ec4cdc5ad694b390c5adfcdc0a858f1457" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.211ex; height:2.843ex;" alt="{\displaystyle y(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )}"></span> where: </p> <ul><li><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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i>, <i><a href="/wiki/Amplitude" title="Amplitude">amplitude</a></i>, the peak deviation of the function from zero.</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 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>, the <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a>, usually representing <a href="/wiki/Time" title="Time">time</a> in <a href="/wiki/Seconds" class="mw-redirect" title="Seconds">seconds</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 \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><a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a></i>, the rate of change of the function argument in units of <a href="/wiki/Radians_per_second" class="mw-redirect" title="Radians per second">radians per second</a>.</li> <li><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 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></i>, <i><a href="/wiki/Ordinary_frequency" class="mw-redirect" title="Ordinary frequency">ordinary frequency</a></i>, the <i><a href="/wiki/Real_number" title="Real number">number</a></i> of oscillations (<a href="/wiki/Turn_(angle)" title="Turn (angle)">cycles</a>) that occur each second of time.</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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>, <i><a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a></i>, specifies (in <a href="/wiki/Radian" title="Radian">radians</a>) where in its cycle the oscillation is at <i>t</i> = 0. <ul><li>When <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is non-zero, the entire waveform appears to be shifted backwards in time by the amount <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 {\tfrac {\varphi }{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>φ</mi> <mi>ω</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\varphi }{\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c84c658594464cc4246e11afcfba081cdac5ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.911ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\varphi }{\omega }}}"></span> seconds. A negative value represents a delay, and a positive value represents an advance.</li> <li>Adding or subtracting <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span> (one cycle) to the phase results in an equivalent wave.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="As_a_function_of_both_position_and_time">As a function of both position and time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=3" title="Edit section: As a function of both position and time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Animated-mass-spring.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Animated-mass-spring.gif/123px-Animated-mass-spring.gif" decoding="async" width="123" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/25/Animated-mass-spring.gif 1.5x" data-file-width="160" data-file-height="320" /></a><figcaption>The displacement of an undamped <a href="/wiki/Spring_mass_system" class="mw-redirect" title="Spring mass system">spring-mass system</a> oscillating around the equilibrium over time is a sine wave.</figcaption></figure> <p>Sinusoids that exist in both position and time also have: </p> <ul><li>a spatial variable <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> that represents the <i>position</i> on the dimension on which the wave propagates.</li> <li>a <a href="/wiki/Wave_number" class="mw-redirect" title="Wave number">wave number</a> (or angular wave number) <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>, which represents the proportionality between the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> and the linear speed (<a href="/wiki/Phase_velocity" title="Phase velocity">speed of propagation</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>: <ul><li>wavenumber is related to the angular frequency 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="{\textstyle k{=}{\frac {\omega }{v}}{=}{\frac {2\pi f}{v}}{=}{\frac {2\pi }{\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>v</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>f</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k{=}{\frac {\omega }{v}}{=}{\frac {2\pi f}{v}}{=}{\frac {2\pi }{\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb85eecafc3508df95d29528d5b7c620541d918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.598ex; height:4.176ex;" alt="{\textstyle k{=}{\frac {\omega }{v}}{=}{\frac {2\pi f}{v}}{=}{\frac {2\pi }{\lambda }}}"></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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> (<a href="/wiki/Lambda" title="Lambda">lambda</a>) is the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>.</li></ul></li></ul> <p>Depending on their direction of travel, they can take the form: </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(x,t)=A\sin(kx-\omega t+\varphi )}"> <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> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x,t)=A\sin(kx-\omega t+\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab3502270349ff4b9c819170eb370c7d35a440a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.089ex; height:2.843ex;" alt="{\displaystyle y(x,t)=A\sin(kx-\omega t+\varphi )}"></span>, if the wave is moving to the right, or</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 y(x,t)=A\sin(kx+\omega t+\varphi )}"> <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> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x,t)=A\sin(kx+\omega t+\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b125c02254130171f1ae41b4ea0abe04843b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.089ex; height:2.843ex;" alt="{\displaystyle y(x,t)=A\sin(kx+\omega t+\varphi )}"></span>, if the wave is moving to the left.</li></ul> <p>Since sine waves propagate without changing form in <i>distributed linear systems</i>,<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="You can help -- (August 2019)">definition needed</span></a></i>]</sup> they are often used to analyze <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Standing_waves">Standing waves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=4" 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> <p>When two waves with the same <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> and <a href="/wiki/Frequency" title="Frequency">frequency</a> traveling in opposite directions <a href="/wiki/Superposition_principle" title="Superposition principle">superpose</a> each other, then a <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> pattern is created. </p><p>On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's <a href="/wiki/Resonant" class="mw-redirect" title="Resonant">resonant</a> frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a>) and integer divisions of that (corresponding to higher harmonics). </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_spatial_dimensions">Multiple spatial dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=5" title="Edit section: Multiple spatial dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The earlier equation gives 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> of the wave at a position <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> 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> along a single line. This could, for example, be considered the value of a wave along a wire. </p><p>In two or three spatial dimensions, the same equation describes a travelling <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> if position <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> and wavenumber <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> are interpreted as vectors, and their product as a <a href="/wiki/Dot_product" title="Dot product">dot product</a>. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. </p> <div class="mw-heading mw-heading4"><h4 id="Sinusoidal_plane_wave">Sinusoidal plane wave</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=6" title="Edit section: Sinusoidal plane wave"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Sinusoidal_plane_wave" title="Sinusoidal plane wave">Sinusoidal plane wave</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Sinusoidal_plane_wave&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> In <a href="/wiki/Physics" title="Physics">physics</a>, a <a href="/wiki/Sinusoidal_plane_wave" title="Sinusoidal plane wave">sinusoidal plane wave</a> is a special case of <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a>: a <a href="/wiki/Field_(physics)" title="Field (physics)">field</a> whose value varies as a <a href="/wiki/Sinusoidal_function" class="mw-redirect" title="Sinusoidal function">sinusoidal function</a> of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constant <a href="/wiki/Frequency" title="Frequency">frequency</a> (as in <a href="/wiki/Monochromatic_radiation" title="Monochromatic radiation">monochromatic radiation</a>).</div></div> <div class="mw-heading mw-heading2"><h2 id="Fourier_analysis">Fourier analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=7" title="Edit section: Fourier analysis"><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/Fourier_series" title="Fourier series">Fourier series</a>, <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, and <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></div> <p>French mathematician <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including <a href="/wiki/Square_wave" title="Square wave">square waves</a>. These <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> are frequently used in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> and the statistical analysis of <a href="/wiki/Time_series" title="Time series">time series</a>. The <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> then extended Fourier series to handle general functions, and birthed the field of <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Differentiation_and_integration">Differentiation and integration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=8" title="Edit section: Differentiation and integration"><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">See also: <a href="/wiki/Phasor#Differentiation_and_integration" title="Phasor">Phasor § Differentiation and integration</a></div> <div class="mw-heading mw-heading3"><h3 id="Differentiation">Differentiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=9" title="Edit section: Differentiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Derivative" title="Derivative">Differentiating</a> any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: </p><p><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 {\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>A</mi> <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 stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> <mi>ω</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> <mi>ω</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1d774c1775716120ab0ded159150b63b8c4a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.617ex; margin-bottom: -0.221ex; width:42.016ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}}"></span> </p><p>A <a href="/wiki/Differentiator" title="Differentiator">differentiator</a> has a <a href="/wiki/Zeros_and_poles" title="Zeros and poles">zero</a> at the origin of the <a href="/wiki/Complex_frequency" class="mw-redirect" title="Complex frequency">complex frequency</a> plane. The <a href="/wiki/Gain_(electronics)" title="Gain (electronics)">gain</a> of its <a href="/wiki/Frequency_response" title="Frequency response">frequency response</a> increases at a rate of +20 <a href="/wiki/Decibel" title="Decibel">dB</a> per <a href="/wiki/Decade_(log_scale)" title="Decade (log scale)">decade</a> of frequency (for <a href="/wiki/Root-power" class="mw-redirect" title="Root-power">root-power</a> quantities), the same positive slope as a 1<sup>st</sup> order <a href="/wiki/High-pass_filter" title="High-pass filter">high-pass filter</a>'s <a href="/wiki/Stopband" title="Stopband">stopband</a>, although a differentiator doesn't have a <a href="/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequency</a> or a flat <a href="/wiki/Passband" title="Passband">passband</a>. A n<sup>th</sup>-order high-pass filter approximately applies the n<sup>th</sup> time derivative of <a href="/wiki/Signals" class="mw-redirect" title="Signals">signals</a> whose frequency band is significantly lower than the filter's cutoff frequency. </p> <div class="mw-heading mw-heading3"><h3 id="Integration">Integration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=10" title="Edit section: Integration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Integral" title="Integral">Integrating</a> any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: </p><p><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 {\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>ω</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>ω</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>ω</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f27fb900539e12f857c9c575510c3f84c6b4d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:47.238ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}}"></span> </p><p>The <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> will be zero if the <a href="/wiki/Bounds_of_integration" class="mw-redirect" title="Bounds of integration">bounds of integration</a> is an integer multiple of the sinusoid's period. </p><p>An <a href="/wiki/Integrator" title="Integrator">integrator</a> has a <a href="/wiki/Zeros_and_poles" title="Zeros and poles">pole</a> at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1<sup>st</sup> order <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>'s stopband, although an integrator doesn't have a cutoff frequency or a flat passband. A n<sup>th</sup>-order low-pass filter approximately performs the n<sup>th</sup> time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency. </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=Sine_wave&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">Crest (physics)</a></li> <li><a href="/wiki/Complex_exponential" class="mw-redirect" title="Complex exponential">Complex exponential</a></li> <li><a href="/wiki/Damped_sine_wave" class="mw-redirect" title="Damped sine wave">Damped sine wave</a></li> <li><a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic series (mathematics)</a></li> <li><a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">Harmonic series (music)</a></li> <li><a href="/wiki/Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a></li> <li><a href="/wiki/Instantaneous_phase" class="mw-redirect" title="Instantaneous phase">Instantaneous phase</a></li> <li><a href="/wiki/In-phase_and_quadrature_components" title="In-phase and quadrature components">In-phase and quadrature components</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Oscilloscope" title="Oscilloscope">Oscilloscope</a></li> <li><a href="/wiki/Phasor" title="Phasor">Phasor</a></li> <li><a href="/wiki/Pure_tone" title="Pure tone">Pure tone</a></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Sinusoidal_model" title="Sinusoidal model">Sinusoidal model</a></li> <li><a href="/wiki/Wave_(physics)" class="mw-redirect" title="Wave (physics)">Wave (physics)</a></li> <li><a href="/wiki/Wave_equation" title="Wave equation">Wave equation</a></li> <li><a href="/wiki/Tilde#Electronics" title="Tilde">∿</a> the sine wave symbol (U+223F)</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSmith" class="citation web cs1">Smith, Julius Orion. <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/st/Sinusoids.html">"Sinusoids"</a>. <i>ccrma.stanford.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ccrma.stanford.edu&rft.atitle=Sinusoids&rft.aulast=Smith&rft.aufirst=Julius+Orion&rft_id=https%3A%2F%2Fccrma.stanford.edu%2F~jos%2Fst%2FSinusoids.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASine+wave" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sine_wave&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathematicalmysteries.org/sine-wave/">"Sine Wave"</a>. <i>Mathematical Mysteries</i>. 2021-11-17<span class="reference-accessdate">. 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