Group velocity - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Group velocity - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"cab88f1f-5ddb-42c6-a1ed-ef14dc739880","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Group_velocity","wgTitle":"Group velocity","wgCurRevisionId":1246175866,"wgRevisionId":1246175866,"wgArticleId":12778,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["All articles with dead external links","Articles with dead external links from June 2024","Articles with permanently dead external links","Articles with short description","Short description is different from Wikidata","Articles with excerpts","Webarchive template wayback links","Radio frequency propagation","Optical quantities","Wave mechanics","Physical quantities","Mathematical physics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext", "wgRelevantPageName":"Group_velocity","wgRelevantArticleId":12778,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Group_speed","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgInternalRedirectTargetUrl":"/wiki/Group_velocity","wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId": "Q217361","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site", "mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/d/d8/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="600"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif/800px-Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="400"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif/640px-Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="320"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Group velocity - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Group_velocity"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Group_velocity&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Group_velocity"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Group_velocity rootpage-Group_velocity skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Group+velocity" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Group+velocity" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Group+velocity" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Group+velocity" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_and_interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_and_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition and interpretation</span> </div> </a> <button aria-controls="toc-Definition_and_interpretation-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 Definition and interpretation subsection</span> </button> <ul id="toc-Definition_and_interpretation-sublist" class="vector-toc-list"> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_expressions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Other expressions</span> </div> </a> <ul id="toc-Other_expressions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dispersion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dispersion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Dispersion</span> </div> </a> <ul id="toc-Dispersion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_phase_velocity,_refractive_index_and_transmission_speed" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_phase_velocity,_refractive_index_and_transmission_speed"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relation to phase velocity, refractive index and transmission speed</span> </div> </a> <ul id="toc-Relation_to_phase_velocity,_refractive_index_and_transmission_speed-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_three_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In three dimensions</span> </div> </a> <ul id="toc-In_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_lossy_or_gainful_media" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_lossy_or_gainful_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In lossy or gainful media</span> </div> </a> <button aria-controls="toc-In_lossy_or_gainful_media-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In lossy or gainful media subsection</span> </button> <ul id="toc-In_lossy_or_gainful_media-sublist" class="vector-toc-list"> <li id="toc-Superluminal_group_velocities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Superluminal_group_velocities"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Superluminal group velocities</span> </div> </a> <ul id="toc-Superluminal_group_velocities-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">7</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">8</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </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">9</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">Group velocity</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 29 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-29" 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">29 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%B3%D8%B1%D8%B9%D8%A9_%D8%A7%D9%84%D8%B2%D9%85%D8%B1%D8%A9" title="سرعة الزمرة – Arabic" lang="ar" hreflang="ar" data-title="سرعة الزمرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0%D0%B2%D0%B0%D1%8F_%D1%81%D0%BA%D0%BE%D1%80%D0%B0%D1%81%D1%86%D1%8C" title="Групавая скорасць – Belarusian" lang="be" hreflang="be" data-title="Групавая скорасць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Velocitat_de_grup" title="Velocitat de grup – Catalan" lang="ca" hreflang="ca" data-title="Velocitat de grup" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Grupov%C3%A1_rychlost" title="Grupová rychlost – Czech" lang="cs" hreflang="cs" data-title="Grupová rychlost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gruppengeschwindigkeit" title="Gruppengeschwindigkeit – German" lang="de" hreflang="de" data-title="Gruppengeschwindigkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Velocidad_de_grupo" title="Velocidad de grupo – Spanish" lang="es" hreflang="es" data-title="Velocidad de grupo" 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/Grupa_rapido" title="Grupa rapido – Esperanto" lang="eo" hreflang="eo" data-title="Grupa rapido" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D8%B1%D8%B9%D8%AA_%DA%AF%D8%B1%D9%88%D9%87" title="سرعت گروه – Persian" lang="fa" hreflang="fa" data-title="سرعت گروه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0%EC%86%8D%EB%8F%84" title="군속도 – Korean" lang="ko" hreflang="ko" data-title="군속도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BD%D5%B4%D5%A2%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A1%D6%80%D5%A1%D5%A3%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Խմբային արագություն – Armenian" lang="hy" hreflang="hy" data-title="Խմբային արագություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Velocit%C3%A0_di_gruppo" title="Velocità di gruppo – Italian" lang="it" hreflang="it" data-title="Velocità di gruppo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%94%D7%99%D7%A8%D7%95%D7%AA_%D7%97%D7%91%D7%95%D7%A8%D7%94" title="מהירות חבורה – Hebrew" lang="he" hreflang="he" data-title="מהירות חבורה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AF%E1%83%92%E1%83%A3%E1%83%A4%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A1%E1%83%98%E1%83%A9%E1%83%A5%E1%83%90%E1%83%A0%E1%83%94" title="ჯგუფური სიჩქარე – Georgian" lang="ka" hreflang="ka" data-title="ჯგუფური სიჩქარე" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D1%82%D1%8B%D2%9B_%D0%B6%D1%8B%D0%BB%D0%B4%D0%B0%D0%BC%D0%B4%D1%8B%D2%9B" title="Топтық жылдамдық – Kazakh" lang="kk" hreflang="kk" data-title="Топтық жылдамдық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Grupinis_greitis" title="Grupinis greitis – Lithuanian" lang="lt" hreflang="lt" data-title="Grupinis greitis" 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%93%D1%80%D1%83%D0%BF%D0%BD%D0%B0_%D0%B1%D1%80%D0%B7%D0%B8%D0%BD%D0%B0" title="Групна брзина – Macedonian" lang="mk" hreflang="mk" data-title="Групна брзина" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%BE%A4%E9%80%9F%E5%BA%A6" title="群速度 – Japanese" lang="ja" hreflang="ja" data-title="群速度" 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/Gruppefart" title="Gruppefart – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gruppefart" 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/Gruppefart" title="Gruppefart – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Gruppefart" 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/Pr%C4%99dko%C5%9B%C4%87_grupowa" title="Prędkość grupowa – Polish" lang="pl" hreflang="pl" data-title="Prędkość grupowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%BE%D0%B2%D0%B0%D1%8F_%D1%81%D0%BA%D0%BE%D1%80%D0%BE%D1%81%D1%82%D1%8C" title="Групповая скорость – Russian" lang="ru" hreflang="ru" data-title="Групповая скорость" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Group_velocity" title="Group velocity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Group velocity" 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/Skupinska_hitrost" title="Skupinska hitrost – Slovenian" lang="sl" hreflang="sl" data-title="Skupinska hitrost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ryhm%C3%A4nopeus" title="Ryhmänopeus – Finnish" lang="fi" hreflang="fi" data-title="Ryhmänopeus" 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/Grupphastighet" title="Grupphastighet – Swedish" lang="sv" hreflang="sv" data-title="Grupphastighet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Grup_h%C4%B1z%C4%B1" title="Grup hızı – Turkish" lang="tr" hreflang="tr" data-title="Grup hızı" 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%93%D1%80%D1%83%D0%BF%D0%BE%D0%B2%D0%B0_%D1%88%D0%B2%D0%B8%D0%B4%D0%BA%D1%96%D1%81%D1%82%D1%8C" title="Групова швидкість – Ukrainian" lang="uk" hreflang="uk" data-title="Групова швидкість" 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/V%E1%BA%ADn_t%E1%BB%91c_nh%C3%B3m" title="Vận tốc nhóm – Vietnamese" lang="vi" hreflang="vi" data-title="Vận tốc nhóm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BE%A4%E9%80%9F%E5%BA%A6" title="群速度 – Chinese" lang="zh" hreflang="zh" data-title="群速度" 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/Q217361#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Group_velocity" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Group_velocity" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Group_velocity"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Group_velocity&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Group_velocity&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Group_velocity"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Group_velocity&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Group_velocity&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Group_velocity" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Group_velocity" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Group_velocity&oldid=1246175866" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Group_velocity&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Group_velocity&id=1246175866&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGroup_velocity"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGroup_velocity"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Group_velocity&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Group_velocity&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Group_velocity" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q217361" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Group_speed&redirect=no" class="mw-redirect" title="Group speed">Group speed</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Velocity at which the overall shape of a wave's amplitudes propagates</div> <figure typeof="mw:File/Frame"><a href="/wiki/File:Wave_group.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/Wave_group.gif" decoding="async" width="465" height="36" class="mw-file-element" data-file-width="465" data-file-height="36" /></a><figcaption><a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">Frequency dispersion</a> in groups of <a href="/wiki/Gravity_wave" title="Gravity wave">gravity waves</a> on the surface of deep water. The <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><span class="legend-color mw-no-invert" style="background-color:red; color:black;"> </span> red square moves with the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>, and the <span style="border-radius: 25px; background-color:#77ac30; border:1px solid;">     </span> green circles propagate with the group velocity. In this deep-water case, <i>the phase velocity is twice the group velocity</i>. The red square overtakes two green circles when moving from the left to the right of the figure.<div class="paragraphbreak" style="margin-top:0.5em"></div> New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.<div class="paragraphbreak" style="margin-top:0.5em"></div> For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_packet_propagation_(phase_faster_than_group,_nondispersive).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif/220px-Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif/330px-Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif/440px-Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif 2x" data-file-width="816" data-file-height="408" /></a><figcaption>Propagation of a wave packet demonstrating a phase velocity greater than the group velocity.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_opposite-group-phase-velocity.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Wave_opposite-group-phase-velocity.gif/220px-Wave_opposite-group-phase-velocity.gif" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c7/Wave_opposite-group-phase-velocity.gif 1.5x" data-file-width="250" data-file-height="154" /></a><figcaption>This shows a wave with the group velocity and phase velocity going in different directions.<sup id="cite_ref-nemirovsky2012negative_1-0" class="reference"><a href="#cite_note-nemirovsky2012negative-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The group velocity is positive (i.e., the <a href="/wiki/Envelope_(waves)" title="Envelope (waves)">envelope</a> of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward).</figcaption></figure> <p>The <b>group velocity</b> of a <a href="/wiki/Wave" title="Wave">wave</a> is the <a href="/wiki/Velocity" title="Velocity">velocity</a> with which the overall envelope shape of the wave's <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a>—known as the <i>modulation</i> or <i><a href="/wiki/Envelope_(waves)" title="Envelope (waves)">envelope</a></i> of the wave—propagates through space. </p><p>For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a <a href="/wiki/Capillary_wave" title="Capillary wave">capillary wave</a>. The expanding ring of waves is the <b>wave group</b> or <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a>, within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The idea of a group velocity distinct from a wave's <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> was first proposed by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">W.R. Hamilton</a> in 1839, and the first full treatment was by <a href="/wiki/John_Strutt,_3rd_Baron_Rayleigh" class="mw-redirect" title="John Strutt, 3rd Baron Rayleigh">Rayleigh</a> in his "Theory of Sound" in 1877.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definition_and_interpretation">Definition and interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=2" title="Edit section: Definition and interpretation"><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:Wave_packet.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wave_packet.svg/220px-Wave_packet.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wave_packet.svg/330px-Wave_packet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wave_packet.svg/440px-Wave_packet.svg.png 2x" data-file-width="5478" data-file-height="3964" /></a><figcaption><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:dodgerblue solid;"> </span> A <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:red dashed;"> </span> The <i>envelope</i> of the wave packet. The envelope moves at the group velocity.</div></figcaption></figure> <p>The group velocity <span class="texhtml"><i>v</i><sub>g</sub></span> is defined by the equation:<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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\rm {g}}\ \equiv \ {\frac {\partial \omega }{\partial k}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>≡</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {g}}\ \equiv \ {\frac {\partial \omega }{\partial k}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5cfaa899791133a4808c1799ed6a0f7976a34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.429ex; height:5.509ex;" alt="{\displaystyle v_{\rm {g}}\ \equiv \ {\frac {\partial \omega }{\partial k}}\,}"></span></dd></dl> <p>where <span class="texhtml"><i>ω</i></span> is the wave's <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> (usually expressed in <a href="/wiki/Radians_per_second" class="mw-redirect" title="Radians per second">radians per second</a>), and <span class="texhtml"><i>k</i></span> is the <a href="/wiki/Angular_wavenumber" class="mw-redirect" title="Angular wavenumber">angular wavenumber</a> (usually expressed in radians per meter). The <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> is: <span class="texhtml"><i>v</i><sub>p</sub> = <i>ω</i>/<i>k</i></span>. </p><p>The <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"><i>ω</i>(<i>k</i>)</span>, which gives <span class="texhtml"><i>ω</i></span> as a function of <span class="texhtml"><i>k</i></span>, is known as the <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a>. </p> <ul><li>If <span class="texhtml"><i>ω</i></span> is <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">directly proportional</a> to <span class="texhtml"><i>k</i></span>, then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity.</li> <li>If <i>ω</i> is a linear function of <i>k</i>, but not directly proportional <span class="texhtml">(<i>ω</i> = <i>ak</i> + <i>b</i>)</span>, then the group velocity and phase velocity are different. The envelope of a <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a> (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity.</li> <li>If <span class="texhtml"><i>ω</i></span> is not a linear function of <span class="texhtml"><i>k</i></span>, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies (and hence different values of <span class="texhtml"><i>k</i></span>), the group velocity <span class="texhtml"><i>∂ω/∂k</i></span> will be different for different values of <span class="texhtml"><i>k</i></span>. Therefore, the envelope does not move at a single velocity, but its wavenumber components (<span class="texhtml"><i>k</i></span>) move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and <span class="texhtml"><i>ω</i>(<i>k</i>)</span> is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. See further discussion <a href="#Higher-order_terms_in_dispersion">below</a>. For example, for <a href="/wiki/Gravity_wave#Deep_water" title="Gravity wave">deep water</a> <a href="/wiki/Gravity_waves" class="mw-redirect" title="Gravity waves">gravity waves</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="{\textstyle \omega ={\sqrt {gk}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>g</mi> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega ={\sqrt {gk}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aff3ffc2733a55be0159aeaf838d0f26b6a48b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.195ex; height:3.343ex;" alt="{\textstyle \omega ={\sqrt {gk}}}"></span>, and hence <span class="texhtml"><i>v</i><sub>g</sub> = <i>v</i><sub>p</sub> /2</span>.<div class="paragraphbreak" style="margin-top:0.5em"></div> This underlies the <i><a href="/wiki/Kelvin_wake_pattern" title="Kelvin wake pattern">Kelvin wake pattern</a></i> for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, on each side the wake forms an angle of 19.47° = arcsin(1/3) with the line of travel.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <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">Further information: <a href="/wiki/Airy_wave_theory#Table_of_wave_quantities" title="Airy wave theory">Airy wave theory § Table of wave quantities</a></div> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=3" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One derivation of the formula for group velocity is as follows.<sup id="cite_ref-Griffiths_8-0" class="reference"><a href="#cite_note-Griffiths-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Consider a <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a> as a function of position <span class="texhtml"><i>x</i></span> and time <span class="texhtml"><i>t</i>: <i>α</i>(<i>x</i>,<i>t</i>)</span>. </p><p>Let <span class="texhtml"><i>A</i>(<i>k</i>)</span> be its Fourier transform at time <span class="nowrap"><span class="texhtml"><i>t</i> = 0</span></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x,0)=\int _{-\infty }^{\infty }dk\,A(k)e^{ikx}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mi>A</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x,0)=\int _{-\infty }^{\infty }dk\,A(k)e^{ikx}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b66789652a97ffcdc75dcafbb337a48a0f830ba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.66ex; height:6.009ex;" alt="{\displaystyle \alpha (x,0)=\int _{-\infty }^{\infty }dk\,A(k)e^{ikx}.}"></span></dd></dl> <p>By the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>, the wavepacket at any time <span class="texhtml"><i>t</i></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x,t)=\int _{-\infty }^{\infty }dk\,A(k)e^{i(kx-\omega t)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mi>A</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x,t)=\int _{-\infty }^{\infty }dk\,A(k)e^{i(kx-\omega t)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51d2afe66fbfb18e05c182311617ae7ea9611ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.511ex; height:6.009ex;" alt="{\displaystyle \alpha (x,t)=\int _{-\infty }^{\infty }dk\,A(k)e^{i(kx-\omega t)},}"></span></dd></dl> <p>where <span class="texhtml"><i>ω</i></span> is implicitly a function of <span class="texhtml"><i>k</i></span>. </p><p>Assume that the wave packet <span class="texhtml"><i>α</i></span> is almost <a href="/wiki/Monochromatic" class="mw-redirect" title="Monochromatic">monochromatic</a>, so that <span class="texhtml"><i>A</i>(<i>k</i>)</span> is sharply peaked around a central <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> <span class="texhtml"><i>k</i><sub>0</sub></span>. </p><p>Then, <a href="/wiki/Linearization" title="Linearization">linearization</a> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (k)\approx \omega _{0}+\left(k-k_{0}\right)\omega '_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (k)\approx \omega _{0}+\left(k-k_{0}\right)\omega '_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e91ddcb8c8afc6a48a5d2af02d6dfdaa2ea3397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.919ex; height:3.009ex;" alt="{\displaystyle \omega (k)\approx \omega _{0}+\left(k-k_{0}\right)\omega '_{0}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}=\omega (k_{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> <mo>=</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}=\omega (k_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c6680b1b53758d07425fd5ce21661be988d687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.119ex; height:2.843ex;" alt="{\displaystyle \omega _{0}=\omega (k_{0})}"></span> 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 \omega '_{0}=\left.{\frac {\partial \omega (k)}{\partial k}}\right|_{k=k_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega '_{0}=\left.{\frac {\partial \omega (k)}{\partial k}}\right|_{k=k_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5950cd6aaca61087187b96a3f69eda1859a2441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.921ex; height:6.843ex;" alt="{\displaystyle \omega '_{0}=\left.{\frac {\partial \omega (k)}{\partial k}}\right|_{k=k_{0}}}"></span></dd></dl> <p>(see next section for discussion of this step). Then, after some algebra, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x,t)=e^{i\left(k_{0}x-\omega _{0}t\right)}\int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mi>A</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x,t)=e^{i\left(k_{0}x-\omega _{0}t\right)}\int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59da4e9d40e716d2ac2d643938f970c1a57ef9e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.719ex; height:6.009ex;" alt="{\displaystyle \alpha (x,t)=e^{i\left(k_{0}x-\omega _{0}t\right)}\int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}.}"></span></dd></dl> <p>There are two factors in this expression. The first factor, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\left(k_{0}x-\omega _{0}t\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\left(k_{0}x-\omega _{0}t\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cef174c4739aacb3a9fe27b10f131a7f3900ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.517ex; height:2.843ex;" alt="{\displaystyle e^{i\left(k_{0}x-\omega _{0}t\right)}}"></span>, describes a perfect monochromatic wave with wavevector <span class="texhtml"><i>k</i><sub>0</sub></span>, with peaks and troughs moving at the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</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 _{0}/k_{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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}/k_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca195b23a402e89003bdba25e52d2b92c9be3af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.928ex; height:2.843ex;" alt="{\displaystyle \omega _{0}/k_{0}}"></span> within the envelope of the wavepacket. </p><p>The other factor, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mi>A</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bdada3f5e546d09d60922700feec2fc4c29406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.569ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }dk\,A(k)e^{i(k-k_{0})\left(x-\omega '_{0}t\right)}}"></span>,</dd></dl> <p>gives the envelope of the wavepacket. This envelope function depends on position and time <i>only</i> through the combination <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-\omega '_{0}t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-\omega '_{0}t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed73e759bc11fc47d54690f105b9bc67f9ccd6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.319ex; height:3.009ex;" alt="{\displaystyle (x-\omega '_{0}t)}"></span>. </p><p>Therefore, the envelope of the wavepacket travels at velocity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega '_{0}=\left.{\frac {d\omega }{dk}}\right|_{k=k_{0}}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ω</mi> </mrow> <mrow> <mi>d</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega '_{0}=\left.{\frac {d\omega }{dk}}\right|_{k=k_{0}}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77040167425496133d8273bfff56e094d63773c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.026ex; height:6.176ex;" alt="{\displaystyle \omega '_{0}=\left.{\frac {d\omega }{dk}}\right|_{k=k_{0}}~,}"></span></dd></dl> <p>which explains the group velocity formula. </p> <div class="mw-heading mw-heading3"><h3 id="Other_expressions">Other expressions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=4" title="Edit section: Other expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For light, the refractive index <span class="texhtml"><i>n</i></span>, vacuum wavelength <span class="texhtml"><i>λ<sub>0</sub></i></span>, and wavelength in the medium <span class="texhtml"><i>λ</i></span>, are related by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}={\frac {2\pi c}{\omega }},\;\;\lambda ={\frac {2\pi }{k}}={\frac {2\pi v_{\rm {p}}}{\omega }},\;\;n={\frac {c}{v_{\rm {p}}}}={\frac {\lambda _{0}}{\lambda }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>c</mi> </mrow> <mi>ω</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mi>ω</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>λ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{0}={\frac {2\pi c}{\omega }},\;\;\lambda ={\frac {2\pi }{k}}={\frac {2\pi v_{\rm {p}}}{\omega }},\;\;n={\frac {c}{v_{\rm {p}}}}={\frac {\lambda _{0}}{\lambda }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7803fbe4a8760e2a910694c9347c9191a7f83ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.575ex; height:6.176ex;" alt="{\displaystyle \lambda _{0}={\frac {2\pi c}{\omega }},\;\;\lambda ={\frac {2\pi }{k}}={\frac {2\pi v_{\rm {p}}}{\omega }},\;\;n={\frac {c}{v_{\rm {p}}}}={\frac {\lambda _{0}}{\lambda }},}"></span></dd></dl> <p>with <span class="texhtml"><i>v</i><sub>p</sub> = <i>ω</i>/<i>k</i></span> the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>. </p><p>The group velocity, therefore, can be calculated by any of the following formulas, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v_{\rm {g}}&={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}={\frac {c}{n-\lambda _{0}{\frac {\partial n}{\partial \lambda _{0}}}}}\\&=v_{\rm {p}}\left(1+{\frac {\lambda }{n}}{\frac {\partial n}{\partial \lambda }}\right)=v_{\rm {p}}-\lambda {\frac {\partial v_{\rm {p}}}{\partial \lambda }}=v_{\rm {p}}+k{\frac {\partial v_{\rm {p}}}{\partial k}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>n</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v_{\rm {g}}&={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}={\frac {c}{n-\lambda _{0}{\frac {\partial n}{\partial \lambda _{0}}}}}\\&=v_{\rm {p}}\left(1+{\frac {\lambda }{n}}{\frac {\partial n}{\partial \lambda }}\right)=v_{\rm {p}}-\lambda {\frac {\partial v_{\rm {p}}}{\partial \lambda }}=v_{\rm {p}}+k{\frac {\partial v_{\rm {p}}}{\partial k}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb59dd13395d71cc1c83ae7e0852ceda0081601d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:50.391ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}v_{\rm {g}}&={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}={\frac {c}{n-\lambda _{0}{\frac {\partial n}{\partial \lambda _{0}}}}}\\&=v_{\rm {p}}\left(1+{\frac {\lambda }{n}}{\frac {\partial n}{\partial \lambda }}\right)=v_{\rm {p}}-\lambda {\frac {\partial v_{\rm {p}}}{\partial \lambda }}=v_{\rm {p}}+k{\frac {\partial v_{\rm {p}}}{\partial k}}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Dispersion">Dispersion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=5" title="Edit section: Dispersion"><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/Group_velocity_dispersion" class="mw-redirect" title="Group velocity dispersion">Group velocity dispersion</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_disp.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/95/Wave_disp.gif" decoding="async" width="388" height="74" class="mw-file-element" data-file-width="388" data-file-height="74" /></a><figcaption>Distortion of wave groups by higher-order dispersion effects, for <a href="/wiki/Surface_gravity_wave" class="mw-redirect" title="Surface gravity wave">surface gravity waves</a> on deep water (with <span class="texhtml"><i>v</i><sub>g</sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>v</i><sub>p</sub></span>).<div class="paragraphbreak" style="margin-top:0.5em"></div> This shows the superposition of three wave components—with respectively 22, 25 and 29 wavelengths fitting in a <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> horizontal domain of 2 km length. The wave <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a> of the components are respectively 1, 2 and 1 meter.</figcaption></figure> <p>Part of the previous derivation is the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series approximation</a> that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (k)\approx \omega _{0}+(k-k_{0})\omega '_{0}(k_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (k)\approx \omega _{0}+(k-k_{0})\omega '_{0}(k_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d070d2c365a660286ec72a51aeaa58c1d662702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.606ex; height:3.009ex;" alt="{\displaystyle \omega (k)\approx \omega _{0}+(k-k_{0})\omega '_{0}(k_{0})}"></span></dd></dl> <p>If the wavepacket has a relatively large frequency spread, or if the dispersion <span class="texhtml"><i>ω(k)</i></span> has sharp variations (such as due to a <a href="/wiki/Resonance" title="Resonance">resonance</a>), or if the packet travels over very long distances, this assumption is not valid, and higher-order terms in the Taylor expansion become important. </p><p>As a result, the envelope of the wave packet not only moves, but also <i>distorts,</i> in a manner that can be described by the material's <a href="/wiki/Group_velocity_dispersion" class="mw-redirect" title="Group velocity dispersion">group velocity dispersion</a>. Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back. Eventually, the wave packet gets stretched out. This is an important effect in the propagation of signals through <a href="/wiki/Optical_fiber" title="Optical fiber">optical fibers</a> and in the design of high-power, short-pulse lasers. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_phase_velocity,_refractive_index_and_transmission_speed"><span id="Relation_to_phase_velocity.2C_refractive_index_and_transmission_speed"></span>Relation to phase velocity, refractive index and transmission speed</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=6" title="Edit section: Relation to phase velocity, refractive index and transmission speed"><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/Phase_velocity#Group_velocity" title="Phase velocity">Phase velocity § Group velocity</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=Phase_velocity&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wavepacket1.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3e/Wavepacket1.gif" decoding="async" width="200" height="150" class="mw-file-element" data-file-width="200" data-file-height="150" /></a><figcaption>A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line).</figcaption></figure> <p>The group velocity of a collection of waves is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7c0a2350b8fdd8eff7819595b008f6764b58ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.494ex; height:5.509ex;" alt="{\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}.}"></span></dd></dl> <p>When multiple sinusoidal waves are propagating together, the resultant superposition of the waves can result in an "envelope" wave as well as a "carrier" wave that lies inside the envelope. This commonly appears in wireless communication when <a href="/wiki/Modulation" title="Modulation">modulation</a> (a change in amplitude and/or phase) is employed to send data. To gain some intuition for this definition, we consider a superposition of (cosine) waves <span class="texhtml mvar" style="font-style:italic;">f(x, t)</span> with their respective angular frequencies and wavevectors. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x,t)&=\cos(k_{1}x-\omega _{1}t)+\cos(k_{2}x-\omega _{2}t)\\&=2\cos \left({\frac {(k_{2}-k_{1})x-(\omega _{2}-\omega _{1})t}{2}}\right)\cos \left({\frac {(k_{2}+k_{1})x-(\omega _{2}+\omega _{1})t}{2}}\right)\\&=2f_{1}(x,t)f_{2}(x,t).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x,t)&=\cos(k_{1}x-\omega _{1}t)+\cos(k_{2}x-\omega _{2}t)\\&=2\cos \left({\frac {(k_{2}-k_{1})x-(\omega _{2}-\omega _{1})t}{2}}\right)\cos \left({\frac {(k_{2}+k_{1})x-(\omega _{2}+\omega _{1})t}{2}}\right)\\&=2f_{1}(x,t)f_{2}(x,t).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cddfe3ee3615adeb00ee0ecea9077edbc9957e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:74.495ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}f(x,t)&=\cos(k_{1}x-\omega _{1}t)+\cos(k_{2}x-\omega _{2}t)\\&=2\cos \left({\frac {(k_{2}-k_{1})x-(\omega _{2}-\omega _{1})t}{2}}\right)\cos \left({\frac {(k_{2}+k_{1})x-(\omega _{2}+\omega _{1})t}{2}}\right)\\&=2f_{1}(x,t)f_{2}(x,t).\end{aligned}}}"></span></dd></dl> <p>So, we have a product of two waves: an envelope wave formed by <span class="texhtml"> <i>f</i><sub>1</sub> </span> and a carrier wave formed by <span class="texhtml"> <i>f</i><sub>2</sub> </span>. We call the velocity of the envelope wave the group velocity. We see that the phase velocity of <span class="texhtml"> <i>f</i><sub>1</sub> </span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\omega _{2}-\omega _{1}}{k_{2}-k_{1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\omega _{2}-\omega _{1}}{k_{2}-k_{1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84351831ae79583733ada7b1bd130061226d434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.323ex; height:5.509ex;" alt="{\displaystyle {\frac {\omega _{2}-\omega _{1}}{k_{2}-k_{1}}}.}"></span></dd></dl> In the continuous differential case, this becomes the definition of the group velocity.</div></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><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/Phase_velocity#Refractive_index" title="Phase velocity">Phase velocity § Refractive index</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=Phase_velocity&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>In the context of electromagnetics and optics, the frequency is some function <span class="texhtml"><i>ω</i>(<i>k</i>)</span> of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light <i>c</i> and the phase velocity <i>v</i><sub><i>p</i></sub> is known as the <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a>, <span class="texhtml"><i>n</i> = <i>c</i> / <i>v</i><sub><i>p</i></sub> = <i>ck</i> / <i>ω</i></span>. </p><p>In this way, we can obtain another form for group velocity for electromagnetics. Writing <span class="texhtml"> <i>n</i> = <i>n</i>(ω)</span>, a quick way to derive this form is to observe </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {1}{c}}\omega n(\omega )\implies dk={\frac {1}{c}}\left(n(\omega )+\omega {\frac {\partial }{\partial \omega }}n(\omega )\right)d\omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mi>ω</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>d</mi> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {1}{c}}\omega n(\omega )\implies dk={\frac {1}{c}}\left(n(\omega )+\omega {\frac {\partial }{\partial \omega }}n(\omega )\right)d\omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860d78e29d2e4b66a19292e2a7081633c3c47209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.004ex; height:6.176ex;" alt="{\displaystyle k={\frac {1}{c}}\omega n(\omega )\implies dk={\frac {1}{c}}\left(n(\omega )+\omega {\frac {\partial }{\partial \omega }}n(\omega )\right)d\omega .}"></span></dd></dl> <p>We can then rearrange the above to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{g}={\frac {\partial w}{\partial k}}={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{g}={\frac {\partial w}{\partial k}}={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d449c479df74fded1774509d3700e07e353f370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.119ex; height:7.176ex;" alt="{\displaystyle v_{g}={\frac {\partial w}{\partial k}}={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}.}"></span></dd></dl> From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of 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="{\textstyle \partial n/\partial \omega =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \partial n/\partial \omega =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d40b7fa0b8a50d98a4a88c950c1a86696e63722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.9ex; height:2.843ex;" alt="{\textstyle \partial n/\partial \omega =0}"></span>. When this occurs, the medium is called non-dispersive, as opposed to <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersive</a>, where various properties of the medium depend on the frequency <span class="texhtml mvar" style="font-style:italic;">ω</span>. The relation <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 (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb1537a1f4d4185fe800d4682abd78bdab60402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.466ex; height:2.843ex;" alt="{\displaystyle \omega (k)}"></span> is known as the <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a> of the medium.</div></div> <div class="mw-heading mw-heading2"><h2 id="In_three_dimensions">In three dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=7" title="Edit section: In three dimensions"><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/Plane_wave" title="Plane wave">Plane wave</a></div> <p>For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way:<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> <ul><li>One dimension: <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_{\rm {p}}=\omega /k,\quad v_{\rm {g}}={\frac {\partial \omega }{\partial k}},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mo>,</mo> <mspace width="1em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {p}}=\omega /k,\quad v_{\rm {g}}={\frac {\partial \omega }{\partial k}},\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282186d9efc602f2bd71d4b6514fe1daec16da50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.463ex; height:5.509ex;" alt="{\displaystyle v_{\rm {p}}=\omega /k,\quad v_{\rm {g}}={\frac {\partial \omega }{\partial k}},\,}"></span></li> <li>Three dimensions: <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_{\rm {p}})_{i}={\frac {\omega }{{k}_{i}}},\quad \mathbf {v} _{\rm {g}}={\vec {\nabla }}_{\mathbf {k} }\,\omega \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>ω</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{\rm {p}})_{i}={\frac {\omega }{{k}_{i}}},\quad \mathbf {v} _{\rm {g}}={\vec {\nabla }}_{\mathbf {k} }\,\omega \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6672915c28bf453958a03c8e95f0688fcbf7cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.134ex; height:5.176ex;" alt="{\displaystyle (v_{\rm {p}})_{i}={\frac {\omega }{{k}_{i}}},\quad \mathbf {v} _{\rm {g}}={\vec {\nabla }}_{\mathbf {k} }\,\omega \,}"></span></li></ul> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}_{\mathbf {k} }\,\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}_{\mathbf {k} }\,\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc8acb1059761668b37c49d2c197280ae8681bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.999ex; height:3.176ex;" alt="{\displaystyle {\vec {\nabla }}_{\mathbf {k} }\,\omega }"></span> means the <a href="/wiki/Gradient" title="Gradient">gradient</a> of the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> <span class="texhtml mvar" style="font-style:italic;">ω</span> as a function of the wave vector <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 \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }"></span>, 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 {\hat {\mathbf {k} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {k} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c643fb60ea71542145705fe801c7ab8c769507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {k} }}}"></span> is the <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> in direction <b>k</b>. </p><p>If the waves are propagating through an <a href="/wiki/Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a> (i.e., not rotationally symmetric) medium, for example a <a href="/wiki/Crystal" title="Crystal">crystal</a>, then the phase velocity vector and group velocity vector may point in different directions. </p> <div class="mw-heading mw-heading2"><h2 id="In_lossy_or_gainful_media">In lossy or gainful media</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=8" title="Edit section: In lossy or gainful media"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The group velocity is often thought of as the velocity at which <a href="/wiki/Energy" title="Energy">energy</a> or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the <a href="/wiki/Signal_velocity" title="Signal velocity">signal velocity</a> of the <a href="/wiki/Wave" title="Wave">waveform</a>. However, if the wave is travelling through an absorptive or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity. </p><p>In his text "Wave Propagation in Periodic Structures",<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/L%C3%A9on_Brillouin" title="Léon Brillouin">Brillouin</a> argued that in a lossy medium the group velocity ceases to have a clear physical meaning. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Another example is mechanical waves in the <a href="/wiki/Solar_photosphere" class="mw-redirect" title="Solar photosphere">solar photosphere</a>: The waves are damped (by radiative heat flow from the peaks to the troughs), and related to that, the energy velocity is often substantially lower than the waves' group velocity.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Despite this ambiguity, a common way to extend the concept of group velocity to complex media is to consider spatially damped plane wave solutions inside the medium, which are characterized by a <i>complex-valued</i> wavevector. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i.e., </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\rm {g}}=\left({\frac {\partial \left(\operatorname {Re} k\right)}{\partial \omega }}\right)^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>Re</mi> <mo>⁡</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {g}}=\left({\frac {\partial \left(\operatorname {Re} k\right)}{\partial \omega }}\right)^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9f4b0f539748e7f08ba44db3958e6a513540ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.373ex; height:6.676ex;" alt="{\displaystyle v_{\rm {g}}=\left({\frac {\partial \left(\operatorname {Re} k\right)}{\partial \omega }}\right)^{-1}.}"></span></dd></dl> <p>Or, equivalently, in terms of the real part of complex <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a>, <span class="texhtml"><u><i>n</i></u> = <i>n</i> + <i>iκ</i></span>, one has<sup id="cite_ref-Boyd1170885_14-0" class="reference"><a href="#cite_note-Boyd1170885-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c}{v_{\rm {g}}}}=n+\omega {\frac {\partial n}{\partial \omega }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c}{v_{\rm {g}}}}=n+\omega {\frac {\partial n}{\partial \omega }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6074f5e202b84321bb8f9b90e78571b68e585c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.044ex; height:6.009ex;" alt="{\displaystyle {\frac {c}{v_{\rm {g}}}}=n+\omega {\frac {\partial n}{\partial \omega }}.}"></span></dd></dl> <p>It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real <span class="texhtml mvar" style="font-style:italic;">k</span>, complex <span class="texhtml mvar" style="font-style:italic;">ω</span>), or, allow group velocity to be a complex-valued quantity.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Different considerations yield distinct velocities, yet all definitions agree for the case of a lossless, gainless medium. </p><p>The above generalization of group velocity for complex media can behave strangely, and the example of <a href="/wiki/Anomalous_dispersion" class="mw-redirect" title="Anomalous dispersion">anomalous dispersion</a> serves as a good illustration. At the edges of a region of anomalous dispersion, <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_{\rm {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f46c0d4015dee2ca67dae03efab6b245539141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.182ex; height:2.343ex;" alt="{\displaystyle v_{\rm {g}}}"></span> becomes infinite (surpassing even the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum), 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 v_{\rm {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f46c0d4015dee2ca67dae03efab6b245539141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.182ex; height:2.343ex;" alt="{\displaystyle v_{\rm {g}}}"></span> may easily become negative (its sign opposes Re<span class="texhtml mvar" style="font-style:italic;">k</span>) inside the band of anomalous dispersion.<sup id="cite_ref-DEWSL06_18-0" class="reference"><a href="#cite_note-DEWSL06-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BLSB06_19-0" class="reference"><a href="#cite_note-BLSB06-19"><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> <div class="mw-heading mw-heading3"><h3 id="Superluminal_group_velocities">Superluminal group velocities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=9" title="Edit section: Superluminal group velocities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the 1980s, various experiments have verified that it is possible for the group velocity (as defined above) of <a href="/wiki/Laser" title="Laser">laser</a> light pulses sent through lossy materials, or gainful materials, to significantly exceed the <a href="/wiki/Speed_of_light_in_vacuum" class="mw-redirect" title="Speed of light in vacuum">speed of light in vacuum</a> <span class="texhtml mvar" style="font-style:italic;">c</span>. The peaks of wavepackets were also seen to move faster than <span class="texhtml mvar" style="font-style:italic;">c</span>. </p><p>In all these cases, however, there is no possibility that signals could be carried <a href="/wiki/Faster_than_light" class="mw-redirect" title="Faster than light">faster than the speed of light in vacuum</a>, since the high value of <span class="texhtml mvar" style="font-style:italic;">v</span><sub><span class="texhtml mvar" style="font-style:italic;">g</span></sub> does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal. Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium. In a wide band analysis it is seen that the apparently paradoxical speed of propagation of the signal envelope is actually the result of local interference of a wider band of frequencies over many cycles, all of which propagate perfectly causally and at phase velocity. The result is akin to the fact that shadows can travel faster than light, even if the light causing them always propagates at light speed; since the phenomenon being measured is only loosely connected with causality, it does not necessarily respect the rules of causal propagation, even if it under normal circumstances does so and leads to a common intuition.<sup id="cite_ref-Boyd1170885_14-1" class="reference"><a href="#cite_note-Boyd1170885-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-DEWSL06_18-1" class="reference"><a href="#cite_note-DEWSL06-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BLSB06_19-1" class="reference"><a href="#cite_note-BLSB06-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-GSBKB06_21-0" class="reference"><a href="#cite_note-GSBKB06-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SLBBJ05_22-0" class="reference"><a href="#cite_note-SLBBJ05-22"><span class="cite-bracket">[</span>22<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=Group_velocity&action=edit&section=10" 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" style="column-width: 22em;"> <ul><li><a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">Wave propagation</a></li> <li><a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">Dispersion (water waves)</a></li> <li><a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">Dispersion (optics)</a></li> <li><a href="/wiki/Wave_propagation_speed" class="mw-redirect" title="Wave propagation speed">Wave propagation speed</a></li> <li><a href="/wiki/Group_delay" class="mw-redirect" title="Group delay">Group delay</a></li> <li><a href="/wiki/Group_velocity_dispersion" class="mw-redirect" title="Group velocity dispersion">Group velocity dispersion</a></li> <li><a href="/wiki/Group_delay_dispersion" class="mw-redirect" title="Group delay dispersion">Group delay dispersion</a></li> <li><a href="/wiki/Phase_delay" class="mw-redirect" title="Phase delay">Phase delay</a></li> <li><a href="/wiki/Phase_velocity" title="Phase velocity">Phase velocity</a></li> <li><a href="/wiki/Signal_velocity" title="Signal velocity">Signal velocity</a></li> <li><a href="/wiki/Slow_light" title="Slow light">Slow light</a></li> <li><a href="/wiki/Front_velocity" title="Front velocity">Front velocity</a></li> <li><a href="/wiki/Matter_wave#Group_velocity" title="Matter wave">Matter wave#Group velocity</a></li> <li><a href="/wiki/Soliton" title="Soliton">Soliton</a></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=Group_velocity&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=12" title="Edit section: Notes"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-nemirovsky2012negative-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-nemirovsky2012negative_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFNemirovskyRechtsman,_Mikael_CSegev,_Mordechai2012" class="citation journal cs1">Nemirovsky, Jonathan; Rechtsman, Mikael C; Segev, Mordechai (9 April 2012). <a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOE.20.008907">"Negative radiation pressure and negative effective refractive index via dielectric birefringence"</a>. <i>Optics Express</i>. <b>20</b> (8): 8907–8914. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012OExpr..20.8907N">2012OExpr..20.8907N</a>. <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/10.1364%2FOE.20.008907">10.1364/OE.20.008907</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/22513601">22513601</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Optics+Express&rft.atitle=Negative+radiation+pressure+and+negative+effective+refractive+index+via+dielectric+birefringence&rft.volume=20&rft.issue=8&rft.pages=8907-8914&rft.date=2012-04-09&rft_id=info%3Apmid%2F22513601&rft_id=info%3Adoi%2F10.1364%2FOE.20.008907&rft_id=info%3Abibcode%2F2012OExpr..20.8907N&rft.aulast=Nemirovsky&rft.aufirst=Jonathan&rft.au=Rechtsman%2C+Mikael+C&rft.au=Segev%2C+Mordechai&rft_id=https%3A%2F%2Fdoi.org%2F10.1364%252FOE.20.008907&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrillouin1960" class="citation cs2">Brillouin, Léon (1960), <i>Wave Propagation and Group Velocity</i>, New York: Academic Press Inc., <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/537250">537250</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wave+Propagation+and+Group+Velocity&rft.place=New+York&rft.pub=Academic+Press+Inc.&rft.date=1960&rft_id=info%3Aoclcnum%2F537250&rft.aulast=Brillouin&rft.aufirst=L%C3%A9on&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrillouin2003" class="citation cs2"><a href="/wiki/L%C3%A9on_Brillouin" title="Léon Brillouin">Brillouin, Léon</a> (2003) [1946], <i>Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices</i>, Dover, p. 75, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49556-9" title="Special:BookSources/978-0-486-49556-9"><bdi>978-0-486-49556-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wave+Propagation+in+Periodic+Structures%3A+Electric+Filters+and+Crystal+Lattices&rft.pages=75&rft.pub=Dover&rft.date=2003&rft.isbn=978-0-486-49556-9&rft.aulast=Brillouin&rft.aufirst=L%C3%A9on&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLighthill2001" class="citation cs2"><a href="/wiki/James_Lighthill" title="James Lighthill">Lighthill, James</a> (2001) [1978], <i>Waves in fluids</i>, Cambridge University Press, p. 242, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-01045-0" title="Special:BookSources/978-0-521-01045-0"><bdi>978-0-521-01045-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Waves+in+fluids&rft.pages=242&rft.pub=Cambridge+University+Press&rft.date=2001&rft.isbn=978-0-521-01045-0&rft.aulast=Lighthill&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFLighthill1965">Lighthill (1965)</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFHayes1973">Hayes (1973)</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">G.B. Whitham (1974). <i>Linear and Nonlinear Waves</i> (John Wiley & Sons Inc., 1974) pp 409–410 <a rel="nofollow" class="external text" href="https://archive.org/details/LinearAndNonlinearWaves">Online scan</a></span> </li> <li id="cite_note-Griffiths-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Griffiths_8-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths,_David_J.1995" class="citation book cs1">Griffiths, David J. (1995). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoqu00grif_200"><i>Introduction to Quantum Mechanics</i></a></span>. <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoqu00grif_200/page/n61">48</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780131244054" title="Special:BookSources/9780131244054"><bdi>9780131244054</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Mechanics&rft.pages=48&rft.pub=Prentice+Hall&rft.date=1995&rft.isbn=9780131244054&rft.au=Griffiths%2C+David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoqu00grif_200&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_K._Ferry2001" class="citation book cs1">David K. Ferry (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=imvYBULWPMQC&pg=PA18"><i>Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers</i></a> (2nd ed.). CRC Press. pp. 18–19. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001qmid.book.....F">2001qmid.book.....F</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7503-0725-3" title="Special:BookSources/978-0-7503-0725-3"><bdi>978-0-7503-0725-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+An+Introduction+for+Device+Physicists+and+Electrical+Engineers&rft.pages=18-19&rft.edition=2nd&rft.pub=CRC+Press&rft.date=2001&rft_id=info%3Abibcode%2F2001qmid.book.....F&rft.isbn=978-0-7503-0725-3&rft.au=David+K.+Ferry&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DimvYBULWPMQC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=cC0Kye7nHEEC&pg=PA239">Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation, by Geoffrey K. Vallis, p239</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrillouin,_L.1946" class="citation book cs1">Brillouin, L. (1946). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.166889"><i>Wave Propagation in Periodic Structures</i></a>. New York: McGraw Hill. p. 75.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wave+Propagation+in+Periodic+Structures&rft.place=New+York&rft.pages=75&rft.pub=McGraw+Hill&rft.date=1946&rft.au=Brillouin%2C+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.166889&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoudon,_R.1973" class="citation book cs1">Loudon, R. (1973). <i>The Quantum Theory of Light</i>. Oxford.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Light&rft.pub=Oxford&rft.date=1973&rft.au=Loudon%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWorrall,_G.2012" class="citation journal cs1">Worrall, G. (2012). "On the Effect of Radiative Relaxation on the Flux of Mechanical-Wave Energy in the Solar Atmosphere". <i>Solar Physics</i>. <b>279</b> (1): 43–52. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012SoPh..279...43W">2012SoPh..279...43W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11207-012-9982-z">10.1007/s11207-012-9982-z</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119595058">119595058</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Solar+Physics&rft.atitle=On+the+Effect+of+Radiative+Relaxation+on+the+Flux+of+Mechanical-Wave+Energy+in+the+Solar+Atmosphere&rft.volume=279&rft.issue=1&rft.pages=43-52&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119595058%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11207-012-9982-z&rft_id=info%3Abibcode%2F2012SoPh..279...43W&rft.au=Worrall%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-Boyd1170885-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Boyd1170885_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Boyd1170885_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoydGauthier2009" class="citation journal cs1">Boyd, R. W.; Gauthier, D. J. (2009). <a rel="nofollow" class="external text" href="http://www.phy.duke.edu/~qelectron/pubs/Science326_1074_2009.pdf">"Controlling the velocity of light pulses"</a> <span class="cs1-format">(PDF)</span>. <i>Science</i>. <b>326</b> (5956): 1074–7. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009Sci...326.1074B">2009Sci...326.1074B</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.630.2223">10.1.1.630.2223</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1170885">10.1126/science.1170885</a>. <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/19965419">19965419</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2370109">2370109</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Controlling+the+velocity+of+light+pulses&rft.volume=326&rft.issue=5956&rft.pages=1074-7&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2370109%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2009Sci...326.1074B&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.630.2223%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F19965419&rft_id=info%3Adoi%2F10.1126%2Fscience.1170885&rft.aulast=Boyd&rft.aufirst=R.+W.&rft.au=Gauthier%2C+D.+J.&rft_id=http%3A%2F%2Fwww.phy.duke.edu%2F~qelectron%2Fpubs%2FScience326_1074_2009.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorin2009" class="citation web cs1">Morin, David (2009). <a rel="nofollow" class="external text" href="https://scholar.harvard.edu/files/david-morin/files/waves_dispersion.pdf">"Dispersion"</a> <span class="cs1-format">(PDF)</span>. <i>people.fas.harvard.edu</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120521232240/http://www.people.fas.harvard.edu/~djmorin/waves/dispersion.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2012-05-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-07-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=people.fas.harvard.edu&rft.atitle=Dispersion&rft.date=2009&rft.aulast=Morin&rft.aufirst=David&rft_id=https%3A%2F%2Fscholar.harvard.edu%2Ffiles%2Fdavid-morin%2Ffiles%2Fwaves_dispersion.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMuschiettiDum1993" class="citation journal cs1">Muschietti, L.; Dum, C. T. (1993). "Real group velocity in a medium with dissipation". <i>Physics of Fluids B: Plasma Physics</i>. <b>5</b> (5): 1383. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhFlB...5.1383M">1993PhFlB...5.1383M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.860877">10.1063/1.860877</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids+B%3A+Plasma+Physics&rft.atitle=Real+group+velocity+in+a+medium+with+dissipation&rft.volume=5&rft.issue=5&rft.pages=1383&rft.date=1993&rft_id=info%3Adoi%2F10.1063%2F1.860877&rft_id=info%3Abibcode%2F1993PhFlB...5.1383M&rft.aulast=Muschietti&rft.aufirst=L.&rft.au=Dum%2C+C.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerasikStastna2010" class="citation journal cs1">Gerasik, Vladimir; Stastna, Marek (2010). "Complex group velocity and energy transport in absorbing media". <i>Physical Review E</i>. <b>81</b> (5): 056602. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010PhRvE..81e6602G">2010PhRvE..81e6602G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.81.056602">10.1103/PhysRevE.81.056602</a>. <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/20866345">20866345</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+E&rft.atitle=Complex+group+velocity+and+energy+transport+in+absorbing+media&rft.volume=81&rft.issue=5&rft.pages=056602&rft.date=2010&rft_id=info%3Apmid%2F20866345&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.81.056602&rft_id=info%3Abibcode%2F2010PhRvE..81e6602G&rft.aulast=Gerasik&rft.aufirst=Vladimir&rft.au=Stastna%2C+Marek&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-DEWSL06-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-DEWSL06_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DEWSL06_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDollingEnkrichWegenerSoukoulis2006" class="citation cs2">Dolling, Gunnar; Enkrich, Christian; Wegener, Martin; Soukoulis, Costas M.; Linden, Stefan (2006), <a rel="nofollow" class="external text" href="https://publikationen.bibliothek.kit.edu/1000010972/901157">"Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial"</a>, <i>Science</i>, <b>312</b> (5775): 892–894, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006Sci...312..892D">2006Sci...312..892D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1126021">10.1126/science.1126021</a>, <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/16690860">16690860</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29012046">29012046</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Simultaneous+Negative+Phase+and+Group+Velocity+of+Light+in+a+Metamaterial&rft.volume=312&rft.issue=5775&rft.pages=892-894&rft.date=2006&rft_id=info%3Adoi%2F10.1126%2Fscience.1126021&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A29012046%23id-name%3DS2CID&rft_id=info%3Apmid%2F16690860&rft_id=info%3Abibcode%2F2006Sci...312..892D&rft.aulast=Dolling&rft.aufirst=Gunnar&rft.au=Enkrich%2C+Christian&rft.au=Wegener%2C+Martin&rft.au=Soukoulis%2C+Costas+M.&rft.au=Linden%2C+Stefan&rft_id=https%3A%2F%2Fpublikationen.bibliothek.kit.edu%2F1000010972%2F901157&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-BLSB06-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-BLSB06_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BLSB06_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBigelowLepeshkinShinBoyd2006" class="citation cs2">Bigelow, Matthew S.; Lepeshkin, Nick N.; Shin, Heedeuk; Boyd, Robert W. (2006), "Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities", <i>Journal of Physics: Condensed Matter</i>, <b>18</b> (11): 3117–3126, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JPCM...18.3117B">2006JPCM...18.3117B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0953-8984%2F18%2F11%2F017">10.1088/0953-8984/18/11/017</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:38556364">38556364</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics%3A+Condensed+Matter&rft.atitle=Propagation+of+a+smooth+and+discontinuous+pulses+through+materials+with+very+large+or+very+small+group+velocities&rft.volume=18&rft.issue=11&rft.pages=3117-3126&rft.date=2006&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A38556364%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0953-8984%2F18%2F11%2F017&rft_id=info%3Abibcode%2F2006JPCM...18.3117B&rft.aulast=Bigelow&rft.aufirst=Matthew+S.&rft.au=Lepeshkin%2C+Nick+N.&rft.au=Shin%2C+Heedeuk&rft.au=Boyd%2C+Robert+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWithayachumnankulFischerFergusonDavis2010" class="citation cs2">Withayachumnankul, W.; Fischer, B. M.; Ferguson, B.; Davis, B. R.; Abbott, D. (2010), "A Systemized View of Superluminal Wave Propagation", <i>Proceedings of the IEEE</i>, <b>98</b> (10): 1775–1786, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJPROC.2010.2052910">10.1109/JPROC.2010.2052910</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15100571">15100571</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IEEE&rft.atitle=A+Systemized+View+of+Superluminal+Wave+Propagation&rft.volume=98&rft.issue=10&rft.pages=1775-1786&rft.date=2010&rft_id=info%3Adoi%2F10.1109%2FJPROC.2010.2052910&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15100571%23id-name%3DS2CID&rft.aulast=Withayachumnankul&rft.aufirst=W.&rft.au=Fischer%2C+B.+M.&rft.au=Ferguson%2C+B.&rft.au=Davis%2C+B.+R.&rft.au=Abbott%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-GSBKB06-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-GSBKB06_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehringSchweinsbergBarsiKostinski2006" class="citation cs2">Gehring, George M.; Schweinsberg, Aaron; Barsi, Christopher; Kostinski, Natalie; Boyd, Robert W. (2006), "Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity", <i>Science</i>, <b>312</b> (5775): 895–897, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006Sci...312..895G">2006Sci...312..895G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1124524">10.1126/science.1124524</a>, <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/16690861">16690861</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:28800603">28800603</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Observation+of+a+Backward+Pulse+Propagation+Through+a+Medium+with+a+Negative+Group+Velocity&rft.volume=312&rft.issue=5775&rft.pages=895-897&rft.date=2006&rft_id=info%3Adoi%2F10.1126%2Fscience.1124524&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28800603%23id-name%3DS2CID&rft_id=info%3Apmid%2F16690861&rft_id=info%3Abibcode%2F2006Sci...312..895G&rft.aulast=Gehring&rft.aufirst=George+M.&rft.au=Schweinsberg%2C+Aaron&rft.au=Barsi%2C+Christopher&rft.au=Kostinski%2C+Natalie&rft.au=Boyd%2C+Robert+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></span> </li> <li id="cite_note-SLBBJ05-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-SLBBJ05_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchweinsbergLepeshkinBigelowBoyd2005" class="citation cs2">Schweinsberg, A.; Lepeshkin, N. N.; Bigelow, M.S.; Boyd, R. W.; Jarabo, S. (2005), <a rel="nofollow" class="external text" href="http://www.hep.princeton.edu/%7Emcdonald/examples/optics/schweinsberg_epl_73_218_06.pdf">"Observation of superluminal and slow light propagation in erbium-doped optical fiber"</a> <span class="cs1-format">(PDF)</span>, <i>Europhysics Letters</i>, <b>73</b> (2): 218–224, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006EL.....73..218S">2006EL.....73..218S</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.5564">10.1.1.205.5564</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1209%2Fepl%2Fi2005-10371-0">10.1209/epl/i2005-10371-0</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250852270">250852270</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Europhysics+Letters&rft.atitle=Observation+of+superluminal+and+slow+light+propagation+in+erbium-doped+optical+fiber&rft.volume=73&rft.issue=2&rft.pages=218-224&rft.date=2005&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.205.5564%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250852270%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1209%2Fepl%2Fi2005-10371-0&rft_id=info%3Abibcode%2F2006EL.....73..218S&rft.aulast=Schweinsberg&rft.aufirst=A.&rft.au=Lepeshkin%2C+N.+N.&rft.au=Bigelow%2C+M.S.&rft.au=Boyd%2C+R.+W.&rft.au=Jarabo%2C+S.&rft_id=http%3A%2F%2Fwww.hep.princeton.edu%2F%257Emcdonald%2Fexamples%2Foptics%2Fschweinsberg_epl_73_218_06.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged June 2024">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_velocity&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Crawford jr., Frank S. (1968). <i>Waves (Berkeley Physics Course, Vol. 3)</i>, McGraw-Hill, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070048607" title="Special:BookSources/978-0070048607">978-0070048607</a> <a rel="nofollow" class="external text" href="https://archive.org/details/Waves_371">Free online version</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTiplerLlewellyn2003" class="citation cs2">Tipler, Paul A.; Llewellyn, Ralph A. (2003), <i>Modern Physics</i> (4th ed.), New York: W. H. Freeman and Company, p. 223, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-4345-3" title="Special:BookSources/978-0-7167-4345-3"><bdi>978-0-7167-4345-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Physics&rft.place=New+York&rft.pages=223&rft.edition=4th&rft.pub=W.+H.+Freeman+and+Company&rft.date=2003&rft.isbn=978-0-7167-4345-3&rft.aulast=Tipler&rft.aufirst=Paul+A.&rft.au=Llewellyn%2C+Ralph+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiot1957" class="citation cs2">Biot, M. A. (1957), "General theorems on the equivalence of group velocity and energy transport", <i>Physical Review</i>, <b>105</b> (4): 1129–1137, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957PhRv..105.1129B">1957PhRv..105.1129B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.105.1129">10.1103/PhysRev.105.1129</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=General+theorems+on+the+equivalence+of+group+velocity+and+energy+transport&rft.volume=105&rft.issue=4&rft.pages=1129-1137&rft.date=1957&rft_id=info%3Adoi%2F10.1103%2FPhysRev.105.1129&rft_id=info%3Abibcode%2F1957PhRv..105.1129B&rft.aulast=Biot&rft.aufirst=M.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitham1961" class="citation cs2">Whitham, G. B. (1961), "Group velocity and energy propagation for three-dimensional waves", <i>Communications on Pure and Applied Mathematics</i>, <b>14</b> (3): 675–691, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.7999">10.1.1.205.7999</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160140337">10.1002/cpa.3160140337</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&rft.atitle=Group+velocity+and+energy+propagation+for+three-dimensional+waves&rft.volume=14&rft.issue=3&rft.pages=675-691&rft.date=1961&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.205.7999%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160140337&rft.aulast=Whitham&rft.aufirst=G.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLighthill1965" class="citation cs2">Lighthill, M. J. (1965), "Group velocity", <i>IMA Journal of Applied Mathematics</i>, <b>1</b> (1): 1–28, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimamat%2F1.1.1">10.1093/imamat/1.1.1</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IMA+Journal+of+Applied+Mathematics&rft.atitle=Group+velocity&rft.volume=1&rft.issue=1&rft.pages=1-28&rft.date=1965&rft_id=info%3Adoi%2F10.1093%2Fimamat%2F1.1.1&rft.aulast=Lighthill&rft.aufirst=M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrethertonGarrett1968" class="citation cs2"><a href="/wiki/Francis_Patton_Bretherton" class="mw-redirect" title="Francis Patton Bretherton">Bretherton, F. P.</a>; Garrett, C. J. R. (1968), "Wavetrains in inhomogeneous moving media", <i>Proceedings of the Royal Society of London</i>, Series A, Mathematical and Physical Sciences, <b>302</b> (1471): 529–554, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1968RSPSA.302..529B">1968RSPSA.302..529B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1968.0034">10.1098/rspa.1968.0034</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:202575349">202575349</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London&rft.atitle=Wavetrains+in+inhomogeneous+moving+media&rft.volume=302&rft.issue=1471&rft.pages=529-554&rft.date=1968&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A202575349%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1968.0034&rft_id=info%3Abibcode%2F1968RSPSA.302..529B&rft.aulast=Bretherton&rft.aufirst=F.+P.&rft.au=Garrett%2C+C.+J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayes1973" class="citation cs2">Hayes, W. D. (1973), "Group velocity and nonlinear dispersive wave propagation", <i>Proceedings of the Royal Society of London</i>, Series A, Mathematical and Physical Sciences, <b>332</b> (1589): 199–221, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973RSPSA.332..199H">1973RSPSA.332..199H</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1973.0021">10.1098/rspa.1973.0021</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121521673">121521673</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London&rft.atitle=Group+velocity+and+nonlinear+dispersive+wave+propagation&rft.volume=332&rft.issue=1589&rft.pages=199-221&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121521673%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1973.0021&rft_id=info%3Abibcode%2F1973RSPSA.332..199H&rft.aulast=Hayes&rft.aufirst=W.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitham1974" class="citation cs2">Whitham, G. B. (1974), <i>Linear and nonlinear waves</i>, Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471940906" title="Special:BookSources/978-0471940906"><bdi>978-0471940906</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+and+nonlinear+waves&rft.pub=Wiley&rft.date=1974&rft.isbn=978-0471940906&rft.aulast=Whitham&rft.aufirst=G.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+velocity" class="Z3988"></span></li></ul> </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=Group_velocity&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Greg_Egan" title="Greg Egan">Greg Egan</a> has an excellent Java applet on <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050615174625/http://gregegan.customer.netspace.net.au/APPLETS/20/20.html">his web site</a> that illustrates the apparent difference in group velocity from <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>.</li> <li>Maarten Ambaum has a <a rel="nofollow" class="external text" href="http://www.met.rdg.ac.uk/~sws97mha/Downstream/">webpage with movie</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190504020931/http://www.met.rdg.ac.uk/~sws97mha/Downstream/">Archived</a> 2019-05-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> demonstrating the importance of group velocity to downstream development of weather systems.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200418070125/http://www.engineeredmiracle.com/fady/GroupAndPhaseVelocities/">Phase vs. Group Velocity </a> – Various Phase- and Group-velocity relations (animation)</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 li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Velocities_of_waves" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Velocities_of_Waves" title="Template:Velocities of Waves"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Velocities_of_Waves" title="Template talk:Velocities of Waves"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Velocities_of_Waves" title="Special:EditPage/Template:Velocities of Waves"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Velocities_of_waves" style="font-size:114%;margin:0 4em">Velocities of <a href="/wiki/Wave" title="Wave">waves</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Phase_velocity" title="Phase velocity">Phase</a></li> <li><a class="mw-selflink selflink">Group</a></li> <li><a href="/wiki/Front_velocity" title="Front velocity">Front</a></li> <li><a href="/wiki/Signal_velocity" title="Signal velocity">Signal</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q217361#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4158446-6">Germany</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Group_velocity&oldid=1246175866">https://en.wikipedia.org/w/index.php?title=Group_velocity&oldid=1246175866</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Radio_frequency_propagation" title="Category:Radio frequency propagation">Radio frequency propagation</a></li><li><a href="/wiki/Category:Optical_quantities" title="Category:Optical quantities">Optical quantities</a></li><li><a href="/wiki/Category:Wave_mechanics" title="Category:Wave mechanics">Wave mechanics</a></li><li><a href="/wiki/Category:Physical_quantities" title="Category:Physical quantities">Physical quantities</a></li><li><a href="/wiki/Category:Mathematical_physics" title="Category:Mathematical physics">Mathematical physics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:All_articles_with_dead_external_links" title="Category:All articles with dead external links">All articles with dead external links</a></li><li><a href="/wiki/Category:Articles_with_dead_external_links_from_June_2024" title="Category:Articles with dead external links from June 2024">Articles with dead external links from June 2024</a></li><li><a href="/wiki/Category:Articles_with_permanently_dead_external_links" title="Category:Articles with permanently dead external links">Articles with permanently dead external links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_with_excerpts" title="Category:Articles with excerpts">Articles with excerpts</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 17 September 2024, at 11:31<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Group_velocity&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b8d669998-7btdt","wgBackendResponseTime":137,"wgPageParseReport":{"limitreport":{"cputime":"0.725","walltime":"0.946","ppvisitednodes":{"value":4539,"limit":1000000},"postexpandincludesize":{"value":84322,"limit":2097152},"templateargumentsize":{"value":5590,"limit":2097152},"expansiondepth":{"value":17,"limit":100},"expensivefunctioncount":{"value":7,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":108674,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 762.857 1 -total"," 41.16% 313.997 1 Template:Reflist"," 16.42% 125.241 2 Template:Excerpt"," 13.39% 102.146 5 Template:Cite_journal"," 10.79% 82.300 15 Template:Citation"," 9.76% 74.460 1 Template:Velocities_of_Waves"," 9.39% 71.595 1 Template:Navbox"," 8.62% 65.782 1 Template:Short_description"," 7.54% 57.520 1 Template:Dead_link"," 7.30% 55.696 1 Template:Fix"]},"scribunto":{"limitreport-timeusage":{"value":"0.478","limit":"10.000"},"limitreport-memusage":{"value":7541384,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBigelowLepeshkinShinBoyd2006\"] = 1,\n [\"CITEREFBiot1957\"] = 1,\n [\"CITEREFBoydGauthier2009\"] = 1,\n [\"CITEREFBrethertonGarrett1968\"] = 1,\n [\"CITEREFBrillouin,_L.1946\"] = 1,\n [\"CITEREFBrillouin1960\"] = 1,\n [\"CITEREFBrillouin2003\"] = 1,\n [\"CITEREFDavid_K._Ferry2001\"] = 1,\n [\"CITEREFDollingEnkrichWegenerSoukoulis2006\"] = 1,\n [\"CITEREFGehringSchweinsbergBarsiKostinski2006\"] = 1,\n [\"CITEREFGerasikStastna2010\"] = 1,\n [\"CITEREFGriffiths,_David_J.1995\"] = 1,\n [\"CITEREFHayes1973\"] = 1,\n [\"CITEREFLighthill1965\"] = 1,\n [\"CITEREFLighthill2001\"] = 1,\n [\"CITEREFLoudon,_R.1973\"] = 1,\n [\"CITEREFMorin2009\"] = 1,\n [\"CITEREFMuschiettiDum1993\"] = 1,\n [\"CITEREFNemirovskyRechtsman,_Mikael_CSegev,_Mordechai2012\"] = 1,\n [\"CITEREFSchweinsbergLepeshkinBigelowBoyd2005\"] = 1,\n [\"CITEREFTiplerLlewellyn2003\"] = 1,\n [\"CITEREFWhitham1961\"] = 1,\n [\"CITEREFWhitham1974\"] = 1,\n [\"CITEREFWithayachumnankulFischerFergusonDavis2010\"] = 1,\n [\"CITEREFWorrall,_G.2012\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 6,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 15,\n [\"Cite book\"] = 4,\n [\"Cite journal\"] = 5,\n [\"Cite web\"] = 1,\n [\"Colorbox\"] = 1,\n [\"DEFAULTSORT:Group Velocity\"] = 1,\n [\"Dead link\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Excerpt\"] = 2,\n [\"Further\"] = 1,\n [\"Harvtxt\"] = 2,\n [\"ISBN\"] = 1,\n [\"K\"] = 1,\n [\"Legend-line\"] = 2,\n [\"Main\"] = 1,\n [\"Math\"] = 36,\n [\"Mvar\"] = 8,\n [\"Nowrap\"] = 1,\n [\"Paragraph\"] = 4,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"See also\"] = 1,\n [\"Sfrac\"] = 1,\n [\"Short description\"] = 1,\n [\"Velocities of Waves\"] = 1,\n [\"Webarchive\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-ext.codfw.main-67468f59cb-rsrnl","timestamp":"20241127132834","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Group velocity","url":"https:\/\/en.wikipedia.org\/wiki\/Group_velocity","sameAs":"http:\/\/www.wikidata.org\/entity\/Q217361","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q217361","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-10-27T13:21:22Z","dateModified":"2024-09-17T11:31:30Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d8\/Wave_packet_propagation_%28phase_faster_than_group%2C_nondispersive%29.gif","headline":"physical quantity"}</script> </body> </html>

CINXE.COM

Group velocity - Wikipedia