CINXE.COM

<!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>Waveguide (radio frequency) - 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":"51214b85-1cd6-4208-a396-85414268ead4","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Waveguide_(radio_frequency)","wgTitle":"Waveguide (radio frequency)","wgCurRevisionId":1245315675,"wgRevisionId":1245315675,"wgArticleId":3854225,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1: long volume value","Articles with short description","Short description is different from Wikidata","All articles with unsourced statements","Articles with unsourced statements from September 2015","Commons category link is locally defined","Telecommunications engineering","Electrodynamics","Microwave technology","Wave mechanics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName": "Waveguide_(radio_frequency)","wgRelevantArticleId":3854225,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"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":40000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q354724","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","mediawiki.page.gallery.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=["ext.cite.ux-enhancements","mediawiki.page.media","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.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%7Cmediawiki.page.gallery.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/1/16/Waveguide_collection.jpg"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="865"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/1/16/Waveguide_collection.jpg/800px-Waveguide_collection.jpg"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="577"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/1/16/Waveguide_collection.jpg/640px-Waveguide_collection.jpg"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="462"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Waveguide (radio frequency) - 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/Waveguide_(radio_frequency)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Waveguide_(radio_frequency)&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/Waveguide_(radio_frequency)"> <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-Waveguide_radio_frequency rootpage-Waveguide_radio_frequency 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=Waveguide+%28radio+frequency%29" 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=Waveguide+%28radio+frequency%29" 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=Waveguide+%28radio+frequency%29" 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=Waveguide+%28radio+frequency%29" 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-Principle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Principle</span> </div> </a> <ul id="toc-Principle-sublist" class="vector-toc-list"> </ul> </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">2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Description" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Description"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Description</span> </div> </a> <ul id="toc-Description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_practice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_practice"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In practice</span> </div> </a> <ul id="toc-In_practice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mathematical analysis</span> </div> </a> <ul id="toc-Mathematical_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dielectric_waveguides" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dielectric_waveguides"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Dielectric waveguides</span> </div> </a> <ul id="toc-Dielectric_waveguides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">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> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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">Waveguide (radio frequency)</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 31 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-31" 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">31 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Golfleier" title="Golfleier – Afrikaans" lang="af" hreflang="af" data-title="Golfleier" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%84%D9%8A%D9%84_%D9%85%D9%88%D8%AC%D8%A9" title="دليل موجة – Arabic" lang="ar" hreflang="ar" data-title="دليل موجة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Radio_dal%C4%9Fa%C3%B6t%C3%BCr%C9%99n" title="Radio dalğaötürən – Azerbaijani" lang="az" hreflang="az" data-title="Radio dalğaötürən" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%8A%D0%BB%D0%BD%D0%BE%D0%B2%D0%BE%D0%B4" title="Вълновод – Bulgarian" lang="bg" hreflang="bg" data-title="Вълновод" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Guia_d%27ones_(electromagnetisme)" title="Guia d'ones (electromagnetisme) – Catalan" lang="ca" hreflang="ca" data-title="Guia d'ones (electromagnetisme)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%BE%D1%85%D1%83%D0%BC_%D1%8F%D1%80%C4%83%D1%88" title="Радиохум ярăш – Chuvash" lang="cv" hreflang="cv" data-title="Радиохум ярăш" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlnovod" title="Vlnovod – Czech" lang="cs" hreflang="cs" data-title="Vlnovod" 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/Hohlleiter" title="Hohlleiter – German" lang="de" hreflang="de" data-title="Hohlleiter" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%85%CE%BC%CE%B1%CF%84%CE%BF%CE%B4%CE%B7%CE%B3%CF%8C%CF%82" title="Κυματοδηγός – Greek" lang="el" hreflang="el" data-title="Κυματοδηγός" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ondogvidilo" title="Ondogvidilo – Esperanto" lang="eo" hreflang="eo" data-title="Ondogvidilo" 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/%D9%85%D9%88%D8%AC%E2%80%8C%D8%A8%D8%B1_(%D9%81%D8%B1%DA%A9%D8%A7%D9%86%D8%B3_%D8%B1%D8%A7%D8%AF%DB%8C%D9%88%DB%8C%DB%8C)" title="موج‌بر (فرکانس رادیویی) – Persian" lang="fa" hreflang="fa" data-title="موج‌بر (فرکانس رادیویی)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Guide_d%27ondes" title="Guide d'ondes – French" lang="fr" hreflang="fr" data-title="Guide d'ondes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97%E0%A4%AA%E0%A4%A5%E0%A4%95_(%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4%E0%A4%9A%E0%A5%81%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%95%E0%A4%A4%E0%A5%8D%E0%A4%B5)" title="तरंगपथक (विद्युतचुम्बकत्व) – Hindi" lang="hi" hreflang="hi" data-title="तरंगपथक (विद्युतचुम्बकत्व)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Guida_d%27onda" title="Guida d'onda – Italian" lang="it" hreflang="it" data-title="Guida d'onda" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%BE%D1%82%D0%BE%D0%BB%D2%9B%D1%8B%D0%BD%D0%B6%D0%BE%D0%BB" title="Радиотолқынжол – Kazakh" lang="kk" hreflang="kk" data-title="Радиотолқынжол" 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/Bangolaidis" title="Bangolaidis – Lithuanian" lang="lt" hreflang="lt" data-title="Bangolaidis" 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%91%D1%80%D0%B0%D0%BD%D0%BE%D0%B2%D0%BE%D0%B4" title="Брановод – Macedonian" lang="mk" hreflang="mk" data-title="Брановод" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pandu_gelombang" title="Pandu gelombang – Malay" lang="ms" hreflang="ms" data-title="Pandu gelombang" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Golfpijp" title="Golfpijp – Dutch" lang="nl" hreflang="nl" data-title="Golfpijp" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B0%8E%E6%B3%A2%E7%AE%A1_(%E9%9B%BB%E7%A3%81%E6%B0%97)" title="導波管 (電磁気) – Japanese" lang="ja" hreflang="ja" data-title="導波管 (電磁気)" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Falow%C3%B3d_mikrofalowy" title="Falowód mikrofalowy – Polish" lang="pl" hreflang="pl" data-title="Falowód mikrofalowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Guia_de_onda_circular" title="Guia de onda circular – Portuguese" lang="pt" hreflang="pt" data-title="Guia de onda circular" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D0%B8%D0%BE%D0%B2%D0%BE%D0%BB%D0%BD%D0%BE%D0%B2%D0%BE%D0%B4" title="Радиоволновод – Russian" lang="ru" hreflang="ru" data-title="Радиоволновод" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aaltoputki" title="Aaltoputki – Finnish" lang="fi" hreflang="fi" data-title="Aaltoputki" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/V%C3%A5gledare" title="Vågledare – Swedish" lang="sv" hreflang="sv" data-title="Vågledare" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B2%E0%AF%88%E0%AE%A8%E0%AE%9F%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF" title="அலைநடத்தி – Tamil" lang="ta" hreflang="ta" data-title="அலைநடத்தி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dalga_rehberi" title="Dalga rehberi – Turkish" lang="tr" hreflang="tr" data-title="Dalga rehberi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B4%D1%96%D0%BE%D1%85%D0%B2%D0%B8%D0%BB%D0%B5%D0%B2%D1%96%D0%B4" title="Радіохвилевід – Ukrainian" lang="uk" hreflang="uk" data-title="Радіохвилевід" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%A7%D8%A6%D8%AF_%D8%A7%D9%84%D9%85%D9%88%D8%AC" title="قائد الموج – Urdu" lang="ur" hreflang="ur" data-title="قائد الموج" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%E1%BB%90ng_d%E1%BA%ABn_s%C3%B3ng_%C4%91i%E1%BB%87n_t%E1%BB%AB" title="Ống dẫn sóng điện từ – Vietnamese" lang="vi" hreflang="vi" data-title="Ống dẫn sóng điện từ" 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%94%B5%E7%A3%81%E6%B3%A2%E5%AF%BC" title="电磁波导 – Chinese" lang="zh" hreflang="zh" data-title="电磁波导" 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/Q354724#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/Waveguide_(radio_frequency)" 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:Waveguide_(radio_frequency)" 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/Waveguide_(radio_frequency)"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Waveguide_(radio_frequency)&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=Waveguide_(radio_frequency)&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/Waveguide_(radio_frequency)"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Waveguide_(radio_frequency)&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=Waveguide_(radio_frequency)&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/Waveguide_(radio_frequency)" 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/Waveguide_(radio_frequency)" 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=Waveguide_(radio_frequency)&oldid=1245315675" 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=Waveguide_(radio_frequency)&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=Waveguide_%28radio_frequency%29&id=1245315675&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%2FWaveguide_%28radio_frequency%29"><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%2FWaveguide_%28radio_frequency%29"><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=Waveguide_%28radio_frequency%29&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=Waveguide_(radio_frequency)&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 id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q354724" 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"></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">Hollow metal pipe used to carry radio waves</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Waveguide (electromagnetism)" redirects here. For optical waveguides, see <a href="/wiki/Waveguide_(optics)" title="Waveguide (optics)">Waveguide (optics)</a>. For other types of waveguide, see <a href="/wiki/Waveguide" title="Waveguide">Waveguide</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Waveguide_collection.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Waveguide_collection.jpg/370px-Waveguide_collection.jpg" decoding="async" width="370" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Waveguide_collection.jpg/555px-Waveguide_collection.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Waveguide_collection.jpg/740px-Waveguide_collection.jpg 2x" data-file-width="857" data-file-height="618" /></a><figcaption>Collection of standard waveguide components.</figcaption></figure> <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:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File/Frameless"><a href="/wiki/File:VFPt_Solenoid_correct2.svg" class="mw-file-description" title="Solenoid"><img alt="Solenoid" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/190px-VFPt_Solenoid_correct2.svg.png" decoding="async" width="190" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/285px-VFPt_Solenoid_correct2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/380px-VFPt_Solenoid_correct2.svg.png 2x" data-file-width="490" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Magnetism" title="Magnetism">Magnetism</a></li> <li><a href="/wiki/Optics" title="Optics">Optics</a></li> <li><a href="/wiki/History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li> <li><a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li> <li><a href="/wiki/List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li> <li><a href="/wiki/List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Electrostatics" title="Electrostatics">Electrostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Charge_density" title="Charge density">Charge density</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb law</a></li> <li><a href="/wiki/Electret" title="Electret">Electret</a></li> <li><a href="/wiki/Electric_charge" title="Electric charge">Electric charge</a></li> <li><a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li> <li><a href="/wiki/Electric_field" title="Electric field">Electric field</a></li> <li><a href="/wiki/Electric_flux" title="Electric flux">Electric flux</a></li> <li><a href="/wiki/Electric_potential" title="Electric potential">Electric potential</a></li> <li><a href="/wiki/Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li> <li><a href="/wiki/Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li> <li><a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss law</a></li> <li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Permittivity" title="Permittivity">Permittivity</a></li> <li><a href="/wiki/Polarization_density" title="Polarization density">Polarization</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Potential energy</a></li> <li><a href="/wiki/Static_electricity" title="Static electricity">Static electricity</a></li> <li><a href="/wiki/Triboelectric_effect" title="Triboelectric effect">Triboelectricity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Magnetostatics" title="Magnetostatics">Magnetostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère law</a></li> <li><a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li> <li><a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss magnetic law</a></li> <li><a href="/wiki/Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li> <li><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></li> <li><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li> <li><a href="/wiki/Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li> <li><a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li> <li><a href="/wiki/Magnetization" title="Magnetization">Magnetization</a></li> <li><a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li> <li><a href="/wiki/Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li> <li><a href="/wiki/Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li> <li><a href="/wiki/Displacement_current" title="Displacement current">Displacement current</a></li> <li><a href="/wiki/Eddy_current" title="Eddy current">Eddy current</a></li> <li><a href="/wiki/Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li> <li><a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li> <li><a href="/wiki/Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li> <li><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li> <li><a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday law</a></li> <li><a href="/wiki/Jefimenko%27s_equations" title="Jefimenko's equations">Jefimenko equations</a></li> <li><a href="/wiki/Larmor_formula" title="Larmor formula">Larmor formula</a></li> <li><a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz law</a></li> <li><a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li> <li><a href="/wiki/London_equations" title="London equations">London equations</a></li> <li><a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell equations</a></li> <li><a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li> <li><a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a></li> <li><a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Electrical_network" title="Electrical network">Electrical network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Alternating_current" title="Alternating current">Alternating current</a></li> <li><a href="/wiki/Capacitance" title="Capacitance">Capacitance</a></li> <li><a href="/wiki/Current_density" title="Current density">Current density</a></li> <li><a href="/wiki/Direct_current" title="Direct current">Direct current</a></li> <li><a href="/wiki/Electric_current" title="Electric current">Electric current</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electrolysis" title="Electrolysis">Electrolysis</a></li> <li><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a></li> <li><a href="/wiki/Electrical_impedance" title="Electrical impedance">Impedance</a></li> <li><a href="/wiki/Inductance" title="Inductance">Inductance</a></li> <li><a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a></li> <li><a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff laws</a></li> <li><a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li> <li><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm law</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li> <li><a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li> <li><a href="/wiki/Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li> <li><a href="/wiki/Voltage" title="Voltage">Voltage</a></li> <li><a href="/wiki/Watt" title="Watt">Watt</a></li> <li><a class="mw-selflink selflink">Waveguides</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/AC_motor" title="AC motor">AC motor</a></li> <li><a href="/wiki/DC_motor" title="DC motor">DC motor</a></li> <li><a href="/wiki/Electric_machine" title="Electric machine">Electric machine</a></li> <li><a href="/wiki/Electric_motor" title="Electric motor">Electric motor</a></li> <li><a href="/wiki/Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li> <li><a href="/wiki/Induction_motor" title="Induction motor">Induction motor</a></li> <li><a href="/wiki/Linear_motor" title="Linear motor">Linear motor</a></li> <li><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li> <li><a href="/wiki/Permeance" title="Permeance">Permeance</a></li> <li><a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li> <li><a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li> <li><a href="/wiki/Rotor_(electric)" title="Rotor (electric)">Rotor</a></li> <li><a href="/wiki/Stator" title="Stator">Stator</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li> <li><a href="/wiki/Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li> <li><a href="/wiki/Four-current" title="Four-current">Four-current</a></li> <li><a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li> <li><a href="/wiki/Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li> <li><a href="/wiki/Maxwell%27s_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li> <li><a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li> <li><a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li> <li><a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li> <li><a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li> <li><a href="/wiki/Humphry_Davy" title="Humphry Davy">Davy</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li> <li><a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a></li> <li><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li> <li><a href="/wiki/John_Hopkinson" title="John Hopkinson">Hopkinson</a></li> <li><a href="/wiki/Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li> <li><a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Larmor</a></li> <li><a href="/wiki/Emil_Lenz" title="Emil Lenz">Lenz</a></li> <li><a href="/wiki/Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li> <li><a href="/wiki/Georg_Ohm" title="Georg Ohm">Ohm</a></li> <li><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li> <li><a href="/wiki/William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li> <li><a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li> <li><a href="/wiki/George_Singer" title="George Singer">Singer</a></li> <li><a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a></li> <li><a href="/wiki/Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li> <li><a href="/wiki/Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><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:Electromagnetism" title="Template:Electromagnetism"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electromagnetism" title="Template talk:Electromagnetism"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electromagnetism" title="Special:EditPage/Template:Electromagnetism"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Radio-frequency_engineering" title="Radio-frequency engineering">radio-frequency engineering</a> and <a href="/wiki/Communications_system_engineering" class="mw-redirect" title="Communications system engineering">communications engineering</a>, a <b>waveguide</b> is a hollow metal pipe used to carry <a href="/wiki/Radio_wave" title="Radio wave">radio waves</a>.<sup id="cite_ref-IEEEdict1997_1-0" class="reference"><a href="#cite_note-IEEEdict1997-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This type of <a href="/wiki/Waveguide" title="Waveguide">waveguide</a> is used as a <a href="/wiki/Transmission_line" title="Transmission line">transmission line</a> mostly at <a href="/wiki/Microwave" title="Microwave">microwave</a> frequencies, for such purposes as connecting microwave <a href="/wiki/Transmitter" title="Transmitter">transmitters</a> and <a href="/wiki/Radio_receiver" title="Radio receiver">receivers</a> to their <a href="/wiki/Antenna_(radio)" title="Antenna (radio)">antennas</a>, in equipment such as <a href="/wiki/Microwave_oven" title="Microwave oven">microwave ovens</a>, <a href="/wiki/Radar" title="Radar">radar</a> sets, <a href="/wiki/Satellite_communications" class="mw-redirect" title="Satellite communications">satellite communications</a>, and microwave radio links. </p><p>The electromagnetic waves in a (metal-pipe) waveguide may be imagined as travelling down the guide in a zig-zag path, being repeatedly reflected between opposite walls of the guide. For the particular case of <b>rectangular waveguide</b>, it is possible to base an exact analysis on this view. Propagation in a dielectric waveguide may be viewed in the same way, with the waves confined to the dielectric by <a href="/wiki/Total_internal_reflection" title="Total internal reflection">total internal reflection</a> at its surface. Some structures, such as <a href="/wiki/Non-radiative_dielectric_waveguide" title="Non-radiative dielectric waveguide">non-radiative dielectric waveguides</a> and the <a href="/wiki/Goubau_line" title="Goubau line">Goubau line</a>, use both metal walls and dielectric surfaces to confine the wave. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Principle">Principle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=1" title="Edit section: Principle"><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:Diplexer1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Diplexer1.jpg/220px-Diplexer1.jpg" decoding="async" width="220" height="277" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Diplexer1.jpg/330px-Diplexer1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Diplexer1.jpg/440px-Diplexer1.jpg 2x" data-file-width="590" data-file-height="744" /></a><figcaption>Example of waveguides and a <a href="/wiki/Diplexer" title="Diplexer">diplexer</a> in an air traffic control radar</figcaption></figure> <p>Depending on the frequency, waveguides can be constructed from either conductive or <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> materials. Generally, the lower the frequency to be passed the larger the waveguide is. For example, the natural waveguide the earth forms given by the dimensions between the conductive ionosphere and the ground as well as the circumference at the median altitude of the Earth is resonant at 7.83 Hz. This is known as <a href="/wiki/Schumann_resonance" class="mw-redirect" title="Schumann resonance">Schumann resonance</a>. On the other hand, waveguides used in <a href="/wiki/Extremely_high_frequency" title="Extremely high frequency">extremely high frequency</a> (EHF) communications can be less than a millimeter in width. </p> <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=Waveguide_(radio_frequency)&action=edit&section=2" title="Edit section: History"><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:George_Southworth_%26_waveguides.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/George_Southworth_%26_waveguides.jpg/220px-George_Southworth_%26_waveguides.jpg" decoding="async" width="220" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/George_Southworth_%26_waveguides.jpg/330px-George_Southworth_%26_waveguides.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/George_Southworth_%26_waveguides.jpg/440px-George_Southworth_%26_waveguides.jpg 2x" data-file-width="640" data-file-height="623" /></a><figcaption><a href="/wiki/George_C._Southworth" class="mw-redirect" title="George C. Southworth">George C. Southworth</a> who developed waveguides in the early 1930s, in front of mile-long experimental waveguide run at Bell Labs, Holmdel, New Jersey, used in his research<sup id="cite_ref-Southworth_2-0" class="reference"><a href="#cite_note-Southworth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Southworth_demonstrating_waveguide.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Southworth_demonstrating_waveguide.jpg/220px-Southworth_demonstrating_waveguide.jpg" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Southworth_demonstrating_waveguide.jpg/330px-Southworth_demonstrating_waveguide.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Southworth_demonstrating_waveguide.jpg/440px-Southworth_demonstrating_waveguide.jpg 2x" data-file-width="810" data-file-height="620" /></a><figcaption>Southworth <i>(at left)</i> demonstrating waveguide at <a href="/wiki/Institute_of_Radio_Engineers" title="Institute of Radio Engineers">IRE</a> meeting in 1938,<sup id="cite_ref-Southworth_2-1" class="reference"><a href="#cite_note-Southworth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> showing 1.5 GHz microwaves passing through the 7.5 m flexible metal hose registering on a diode detector.</figcaption></figure> <p>During the 1890s theorists did the first analyses of electromagnetic waves in ducts.<sup id="cite_ref-Packard_3-0" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Around 1893 <a href="/wiki/J._J._Thomson" title="J. J. Thomson">J. J. Thomson</a> derived the electromagnetic modes inside a cylindrical metal cavity.<sup id="cite_ref-Packard_3-1" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In 1897 <a href="/wiki/Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a> did a definitive analysis of waveguides; he solved the <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary value problem</a> of electromagnetic waves propagating through both conducting tubes and dielectric rods of arbitrary shape.<sup id="cite_ref-Packard_3-2" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Rayleigh_4-0" class="reference"><a href="#cite_note-Rayleigh-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kizer_5-0" class="reference"><a href="#cite_note-Kizer-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lee_6-0" class="reference"><a href="#cite_note-Lee-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> He showed that the waves could travel without attenuation only in specific <a href="/wiki/Normal_mode" title="Normal mode">normal modes</a> with either the <a href="/wiki/Electric_field" title="Electric field">electric field</a> (<a href="/wiki/TE_mode" class="mw-redirect" title="TE mode">TE modes</a>) or <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> (<a href="/wiki/TM_mode" class="mw-redirect" title="TM mode">TM modes</a>), perpendicular to the direction of propagation. He also showed each mode had a <a href="/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequency</a> below which waves would not propagate. Since the cutoff wavelength for a given tube was of the same order as its width, it was clear that a hollow conducting tube could not carry radio wavelengths much larger than its diameter. In 1902 R. H. Weber observed that electromagnetic waves travel at a slower speed in tubes than in free space, and deduced the reason; that the waves travel in a "zigzag" path as they reflect from the walls.<sup id="cite_ref-Packard_3-3" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kizer_5-1" class="reference"><a href="#cite_note-Kizer-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Weber_7-0" class="reference"><a href="#cite_note-Weber-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Prior to the 1920s, practical work on radio waves concentrated on the low frequency end of the radio spectrum, as these frequencies were better for long-range communication.<sup id="cite_ref-Packard_3-4" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> These were far below the frequencies that could propagate in even large waveguides, so there was little experimental work on waveguides during this period, although a few experiments were done. In a June 1, 1894 lecture, "The work of Hertz", before the <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a>, <a href="/wiki/Oliver_Lodge" title="Oliver Lodge">Oliver Lodge</a> demonstrated the transmission of 3 inch radio waves from a <a href="/wiki/Spark_gap" title="Spark gap">spark gap</a> through a short cylindrical copper duct.<sup id="cite_ref-Packard_3-5" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lodge_8-0" class="reference"><a href="#cite_note-Lodge-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In his pioneering 1894-1900 research on microwaves, <a href="/wiki/Jagadish_Chandra_Bose" title="Jagadish Chandra Bose">Jagadish Chandra Bose</a> used short lengths of pipe to conduct the waves, so some sources credit him with inventing the waveguide.<sup id="cite_ref-Emerson_9-0" class="reference"><a href="#cite_note-Emerson-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> However, after this, the concept of radio waves being carried by a tube or duct passed out of engineering knowledge.<sup id="cite_ref-Packard_3-6" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>During the 1920s the first continuous sources of high frequency radio waves were developed: the <a href="/wiki/Barkhausen%E2%80%93Kurz_tube" title="Barkhausen–Kurz tube">Barkhausen–Kurz tube</a>,<sup id="cite_ref-Brown_10-0" class="reference"><a href="#cite_note-Brown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> the first oscillator which could produce power at <a href="/wiki/Ultrahigh_frequency" class="mw-redirect" title="Ultrahigh frequency">UHF</a> frequencies; and the <a href="/wiki/Cavity_magnetron" title="Cavity magnetron">split-anode magnetron</a> which by the 1930s had generated radio waves at up to 10 GHz.<sup id="cite_ref-Packard_3-7" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> These made possible the first systematic research on microwaves in the 1930s. It was discovered that <a href="/wiki/Transmission_line" title="Transmission line">transmission lines</a> used to carry lower frequency radio waves, <a href="/wiki/Twin_lead" class="mw-redirect" title="Twin lead">parallel line</a> and <a href="/wiki/Coaxial_cable" title="Coaxial cable">coaxial cable</a>, had excessive power losses at microwave frequencies, creating a need for a new transmission method.<sup id="cite_ref-Packard_3-8" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brown_10-1" class="reference"><a href="#cite_note-Brown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The waveguide was developed independently between 1932 and 1936 by <a href="/wiki/George_C._Southworth" class="mw-redirect" title="George C. Southworth">George C. Southworth</a> at <a href="/wiki/Bell_Telephone_Laboratories" class="mw-redirect" title="Bell Telephone Laboratories">Bell Telephone Laboratories</a><sup id="cite_ref-Southworth_2-2" class="reference"><a href="#cite_note-Southworth-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Wilmer_L._Barrow" title="Wilmer L. Barrow">Wilmer L. Barrow</a> at the <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a>, who worked without knowledge of one another.<sup id="cite_ref-Packard_3-9" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kizer_5-2" class="reference"><a href="#cite_note-Kizer-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lee_6-1" class="reference"><a href="#cite_note-Lee-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brown_10-2" class="reference"><a href="#cite_note-Brown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Southworth's interest was sparked during his 1920s doctoral work in which he measured the <a href="/wiki/Dielectric_constant" class="mw-redirect" title="Dielectric constant">dielectric constant</a> of water with a radio frequency <a href="/wiki/Lecher_line" title="Lecher line">Lecher line</a> in a long tank of water. He found that if he removed the Lecher line, the tank of water still showed resonance peaks, indicating it was acting as a <a href="/wiki/Dielectric_waveguide" class="mw-redirect" title="Dielectric waveguide">dielectric waveguide</a>.<sup id="cite_ref-Packard_3-10" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> At Bell Labs in 1931 he resumed work in dielectric waveguides. By March 1932 he observed waves in water-filled copper pipes. Rayleigh's previous work had been forgotten, and <a href="/wiki/Sergei_A._Schelkunoff" class="mw-redirect" title="Sergei A. Schelkunoff">Sergei A. Schelkunoff</a>, a Bell Labs mathematician, did theoretical analyses of waveguides<sup id="cite_ref-Packard_3-11" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schelkunoff_11-0" class="reference"><a href="#cite_note-Schelkunoff-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and rediscovered waveguide modes. In December 1933 it was realized that with a metal sheath the dielectric is superfluous and attention shifted to metal waveguides. </p><p>Barrow had become interested in high frequencies in 1930 studying under <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> in Germany.<sup id="cite_ref-Packard_3-12" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> At MIT beginning in 1932 he worked on high frequency antennas to generate narrow beams of radio waves to locate aircraft in fog. He invented a <a href="/wiki/Horn_antenna" title="Horn antenna">horn antenna</a> and hit on the idea of using a hollow pipe as a feedline to feed radio waves to the antenna.<sup id="cite_ref-Packard_3-13" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> By March 1936 he had derived the propagation modes and cutoff frequency in a rectangular waveguide.<sup id="cite_ref-Brown_10-3" class="reference"><a href="#cite_note-Brown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The source he was using had a large wavelength of 40 cm, so for his first successful waveguide experiments he used a 16-foot section of air duct, 18 inches in diameter.<sup id="cite_ref-Packard_3-14" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Barrow and Southworth became aware of each other's work a few weeks before both were scheduled to present papers on waveguides to a combined meeting of the <a href="/wiki/American_Physical_Society" title="American Physical Society">American Physical Society</a> and the <a href="/wiki/Institute_of_Radio_Engineers" title="Institute of Radio Engineers">Institute of Radio Engineers</a> in May 1936.<sup id="cite_ref-Packard_3-15" class="reference"><a href="#cite_note-Packard-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brown_10-4" class="reference"><a href="#cite_note-Brown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> They amicably worked out credit sharing and patent division arrangements. </p><p>The development of centimeter <a href="/wiki/Radar" title="Radar">radar</a> during World War 2 and the first high power microwave tubes, the <a href="/wiki/Klystron" title="Klystron">klystron</a> (1938) and cavity magnetron (1940), resulted in the first widespread use of waveguide.<sup id="cite_ref-Brown_10-5" class="reference"><a href="#cite_note-Brown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Standard waveguide "plumbing" components were manufactured, with flanges on the end which could be bolted together. After the war in the 1950s and 60s waveguides became common in commercial microwave systems, such as airport radar and <a href="/wiki/Microwave_relay" class="mw-redirect" title="Microwave relay">microwave relay</a> networks which were built to transmit telephone calls and television programs between cities. </p> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=3" title="Edit section: Description"><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:Waveguide.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Waveguide.svg/220px-Waveguide.svg.png" decoding="async" width="220" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Waveguide.svg/330px-Waveguide.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Waveguide.svg/440px-Waveguide.svg.png 2x" data-file-width="512" data-file-height="303" /></a><figcaption>Rectangular hollow Waveguide</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:WaveguideJ-Band.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/WaveguideJ-Band.png/200px-WaveguideJ-Band.png" decoding="async" width="200" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/WaveguideJ-Band.png/300px-WaveguideJ-Band.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/WaveguideJ-Band.png/400px-WaveguideJ-Band.png 2x" data-file-width="1592" data-file-height="1216" /></a><figcaption>Flexible waveguide from a J-Band radar</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Radar_antenna_feed_waveguide.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Radar_antenna_feed_waveguide.png/290px-Radar_antenna_feed_waveguide.png" decoding="async" width="290" height="315" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Radar_antenna_feed_waveguide.png/435px-Radar_antenna_feed_waveguide.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e1/Radar_antenna_feed_waveguide.png 2x" data-file-width="522" data-file-height="567" /></a><figcaption>Typical waveguide application: <a href="/wiki/Antenna_feed" title="Antenna feed">antenna feed</a> for military radar. </figcaption></figure> <p>In the <a href="/wiki/Microwave" title="Microwave">microwave</a> region of the <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a>, a waveguide normally consists of a hollow metallic conductor. These waveguides can take the form of single conductors with or without a dielectric coating, e.g. the <a href="/wiki/Goubau_line" title="Goubau line">Goubau line</a> and helical waveguides. Hollow waveguides must be one-half wavelength or more in diameter in order to support one or more transverse wave modes. </p><p>Waveguides may be filled with pressurized gas to inhibit arcing and prevent <a href="/wiki/Multipaction" class="mw-redirect" title="Multipaction">multipaction</a>, allowing higher power transmission. Conversely, waveguides may be required to be evacuated as part of evacuated systems (e.g. electron beam systems). </p><p>A <a href="/wiki/Slotted_waveguide" class="mw-redirect" title="Slotted waveguide">slotted waveguide</a> is generally used for radar and other similar applications. The waveguide serves as a feed path, and each slot is a separate radiator, thus forming an antenna. This structure has the capability of generating a radiation pattern to launch an <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a> in a specific relatively narrow and controllable direction. </p><p>A <b>closed waveguide</b> is an electromagnetic waveguide (a) that is tubular, usually with a circular or rectangular cross section, (b) that has electrically conducting walls, (c) that may be hollow or filled with a dielectric material, (d) that can support a large number of discrete propagating modes, though only a few may be practical, (e) in which each discrete mode defines the <a href="/wiki/Propagation_constant" title="Propagation constant">propagation constant</a> for that mode, (f) in which the <a href="/wiki/Field_(physics)" title="Field (physics)">field</a> at any point is describable in terms of the supported modes, (g) in which there is no <a href="/wiki/Radiation" title="Radiation">radiation</a> field, and (h) in which discontinuities and bends may cause mode conversion but not radiation.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2015)">citation needed</span></a></i>]</sup> </p><p>The dimensions of a hollow metallic waveguide determine which wavelengths it can support, and in which modes. Typically the waveguide is operated so that only a single mode is present. The lowest order mode possible is generally selected. Frequencies below the guide's cutoff frequency will not propagate. It is possible to operate waveguides at higher order modes, or with multiple modes present, but this is usually impractical. </p><p>Waveguides are almost exclusively made of metal and mostly rigid structures. There are certain types of "corrugated" waveguides that have the ability to flex and bend but only used where essential since they degrade propagation properties. Due to propagation of energy in mostly air or space within the waveguide, it is one of the lowest loss transmission line types and highly preferred for high frequency applications where most other types of transmission structures introduce large losses. Due to the <a href="/wiki/Skin_effect" title="Skin effect">skin effect</a> at high frequencies, electric current along the walls penetrates typically only a few <a href="/wiki/Micrometre" title="Micrometre">micrometers</a> into the metal of the inner surface. Since this is where most of the resistive loss occurs, it is important that the conductivity of interior surface be kept as high as possible. For this reason, most waveguide interior surfaces are plated with <a href="/wiki/Copper" title="Copper">copper</a>, <a href="/wiki/Silver" title="Silver">silver</a>, or <a href="/wiki/Gold" title="Gold">gold</a>. </p><p>Voltage standing wave ratio (<a href="/wiki/VSWR" class="mw-redirect" title="VSWR">VSWR</a>) measurements may be taken to ensure that a waveguide is contiguous and has no leaks or sharp bends. If such bends or holes in the waveguide surface are present, this may diminish the performance of both transmitter and receiver equipment connected at either end. Poor transmission through the waveguide may also occur as a result of moisture build up which corrodes and degrades conductivity of the inner surfaces, which is crucial for low loss propagation. For this reason, waveguides are nominally fitted with <a href="/w/index.php?title=Microwave_windows&action=edit&redlink=1" class="new" title="Microwave windows (page does not exist)">microwave windows</a> at the outer end that will not interfere with propagation but keep the elements out. Moisture can also cause <a href="/wiki/Fungus" title="Fungus">fungus</a> build up or arcing in high power systems such as radio or radar transmitters. Moisture in waveguides can typically be prevented with <a href="/wiki/Silica_gel" title="Silica gel">silica gel</a>, a <a href="/wiki/Desiccant" title="Desiccant">desiccant</a>, or slight pressurization of the waveguide cavities with dry <a href="/wiki/Nitrogen" title="Nitrogen">nitrogen</a> or <a href="/wiki/Argon" title="Argon">argon</a>. Desiccant silica gel canisters may be attached with screw-on nibs and higher power systems will have pressurized tanks for maintaining pressure including leakage monitors. Arcing may also occur if there is a hole, tear or bump in the conducting walls, if transmitting at high power (usually 200 watts or more). Waveguide plumbing<sup id="cite_ref-PBP_12-0" class="reference"><a href="#cite_note-PBP-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> is crucial for proper waveguide performance. Voltage standing waves occur when impedance mismatches in the waveguide cause energy to reflect back in the opposite direction of propagation. In addition to limiting the effective transfer of energy, these reflections can cause higher voltages in the waveguide and damage equipment. </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:752px;max-width:752px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Waveguide17-with-UBR120-flanges.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Waveguide17-with-UBR120-flanges.svg/200px-Waveguide17-with-UBR120-flanges.svg.png" decoding="async" width="200" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Waveguide17-with-UBR120-flanges.svg/300px-Waveguide17-with-UBR120-flanges.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Waveguide17-with-UBR120-flanges.svg/400px-Waveguide17-with-UBR120-flanges.svg.png 2x" data-file-width="645" data-file-height="502" /></a></span></div><div class="thumbcaption">Short length of rectangular waveguide (WG17 with <a href="/wiki/Waveguide_flange" title="Waveguide flange">UBR120 connection-flanges</a>)</div></div><div class="tsingle" style="width:132px;max-width:132px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG/130px-J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG" decoding="async" width="130" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG/195px-J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG/260px-J_M_Briscoe24_07_200713_14_56IMG2128_WAVEGUIDE.JPG 2x" data-file-width="2200" data-file-height="2632" /></a></span></div><div class="thumbcaption">Section of the flexible waveguide</div></div><div class="tsingle" style="width:412px;max-width:412px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Waveguide900MHz.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Waveguide900MHz.jpg/410px-Waveguide900MHz.jpg" decoding="async" width="410" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Waveguide900MHz.jpg/615px-Waveguide900MHz.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Waveguide900MHz.jpg/820px-Waveguide900MHz.jpg 2x" data-file-width="4320" data-file-height="1620" /></a></span></div><div class="thumbcaption">Waveguide (ankle piece 900MHz)</div></div></div></div></div> <div class="mw-heading mw-heading2"><h2 id="In_practice">In practice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=4" title="Edit section: In practice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In practice, waveguides act as the equivalent of cables for <a href="/wiki/Super_high_frequency" title="Super high frequency">super high frequency</a> (SHF) systems. For such applications, it is desired to operate waveguides with only one mode propagating through the waveguide. With rectangular waveguides, it is possible to design the waveguide such that the frequency band over which only one mode propagates is as high as 2:1 (i.e. the ratio of the upper band edge to lower band edge is two). The relation between the waveguide dimensions and the lowest frequency is simple: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>W</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fceeaecf17a9338c2af11eb016af513e36679f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.722ex; height:1.676ex;" alt="{\displaystyle \scriptstyle W}"></span> is the greater of its two dimensions, then the longest wavelength that will propagate is <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 \;=\;2W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mn>2</mn> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \;=\;2W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326875e406d4eaa5207f825c860274e4b6dcf3c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.342ex; height:2.176ex;" alt="{\displaystyle \lambda \;=\;2W}"></span> and the lowest frequency is thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\;=\;c/\lambda \;=\;c/2W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\;=\;c/\lambda \;=\;c/2W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b722396ca1939eeabfdd372af6747265dada08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.348ex; height:2.843ex;" alt="{\displaystyle f\;=\;c/\lambda \;=\;c/2W}"></span> </p><p>With circular waveguides, the highest possible bandwidth allowing only a single mode to propagate is only 1.3601:1.<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>Because rectangular waveguides have a much larger bandwidth over which only a single mode can propagate, standards exist for rectangular waveguides, but not for circular waveguides. In general (but not always), standard waveguides are designed such that </p> <ul><li>one band starts where another band ends, with another band that overlaps the two bands<sup id="cite_ref-Harvey_14-0" class="reference"><a href="#cite_note-Harvey-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li>the lower edge of the band is approximately 30% higher than the waveguide's <a href="/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequency</a></li> <li>the upper edge of the band is approximately 5% lower than the cutoff frequency of the next higher order mode</li> <li>the waveguide height is half the waveguide width</li></ul> <p>The first condition is to allow for applications near band edges. The second condition limits <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a>, a phenomenon in which the velocity of propagation is a function of frequency. It also limits the loss per unit length. The third condition is to avoid <a href="/wiki/Evanescent_wave#Evanescent-wave_coupling" class="mw-redirect" title="Evanescent wave">evanescent-wave coupling</a> via higher order modes. The fourth condition is that which allows a 2:1 operation bandwidth. Although it is possible to have a 2:1 operating bandwidth when the height is less than half the width, having the height exactly half the width maximizes the power that can propagate inside the waveguide before <a href="/wiki/Dielectric_breakdown" class="mw-redirect" title="Dielectric breakdown">dielectric breakdown</a> occurs. </p><p>Below is a table of standard waveguides. The waveguide name <i>WR</i> stands for <i>waveguide rectangular</i>, and the number is the inner dimension width of the waveguide in hundredths of an <a href="/wiki/Inch" title="Inch">inch</a> (0.01 inch = 0.254 mm) rounded to the nearest hundredth of an inch. </p> <table class="wikitable collapsible" style="text-align: center; max-width:880px;"> <tbody><tr> <th colspan="9">Standard sizes of rectangular waveguide </th></tr> <tr> <th colspan="3" rowspan="2">Waveguide name </th> <th rowspan="3">Frequency<br />band name </th> <th rowspan="3">Recommended<br />frequency band<br />of operation (GHz) </th> <th colspan="2">Cutoff frequency (GHz) of </th> <th colspan="2" rowspan="2">Inner dimensions of waveguide opening </th></tr> <tr> <th rowspan="2">lowest order mode </th> <th rowspan="2">next mode </th></tr> <tr> <th><a href="/wiki/Electronic_Industries_Alliance" title="Electronic Industries Alliance">EIA</a> </th> <th>RCSC <sup>*</sup> </th> <th><a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a> </th> <th>(inch)</th> <th>(mm) </th></tr> <tr style="white-space:nowrap;"> <td>WR2300 </td> <td>WG0.0 </td> <td>R3 </td> <td> </td> <td>0.32 — 0.45 </td> <td>0.257 </td> <td>0.513 </td> <td>23.000 × 11.500</td> <td>584.20 × 292.10 </td></tr> <tr style="white-space:nowrap;"> <td>WR2100 </td> <td>WG0 </td> <td>R4 </td> <td> </td> <td>0.35 — 0.50 </td> <td>0.281 </td> <td>0.562 </td> <td>21.000 × 10.500</td> <td>533.40 × 266.7 </td></tr> <tr> <td>WR1800 </td> <td>WG1 </td> <td>R5 </td> <td> </td> <td>0.45 — 0.63 </td> <td>0.328 </td> <td>0.656 </td> <td>18.000 × 9.000</td> <td>457.20 × 228.6 </td></tr> <tr> <td>WR1500 </td> <td>WG2 </td> <td>R6 </td> <td> </td> <td>0.50 — 0.75 </td> <td>0.393 </td> <td>0.787 </td> <td>15.000 × 7.500</td> <td>381.00 × 190.5 </td></tr> <tr> <td>WR1150 </td> <td>WG3 </td> <td>R8 </td> <td> </td> <td>0.63 — 0.97 </td> <td>0.513 </td> <td>1.026 </td> <td>11.500 × 5.750</td> <td>292.10 × 146.5 </td></tr> <tr> <td>WR975 </td> <td>WG4 </td> <td>R9 </td> <td> </td> <td>0.75 — 1.15 </td> <td>0.605 </td> <td>1.211 </td> <td>9.750 × 4.875</td> <td>247.7 × 123.8 </td></tr> <tr> <td>WR770 </td> <td>WG5 </td> <td>R12 </td> <td> </td> <td>0.97 — 1.45 </td> <td>0.766 </td> <td>1.533 </td> <td>7.700 × 3.850</td> <td>195,6 × 97.79 </td></tr> <tr> <td>WR650 </td> <td>WG6 </td> <td>R14 </td> <td><a href="/wiki/L_band" title="L band">L band</a> (part) </td> <td>1.15 — 1.72 </td> <td>0.908 </td> <td>1.816 </td> <td>6.500 × 3.250</td> <td>165.1 × 82.55 </td></tr> <tr> <td>WR510 </td> <td>WG7 </td> <td>R18 </td> <td> </td> <td>1.45 — 2.20 </td> <td>1.157 </td> <td>2.314 </td> <td>5.100 × 2.550</td> <td>129.5 × 64.77 </td></tr> <tr> <td>WR430 </td> <td>WG8 </td> <td>R22 </td> <td> </td> <td>1.72 — 2.60 </td> <td>1.372 </td> <td>2.745 </td> <td>4.300 × 2.150</td> <td>109.2 × 54.61 </td></tr> <tr> <td>WR340 </td> <td>WG9A </td> <td>R26 </td> <td><a href="/wiki/S_band" title="S band">S band</a> (part) </td> <td>2.20 — 3.30 </td> <td>1.736 </td> <td>3.471 </td> <td>3.400 × 1.700</td> <td>86.36 × 43.18 </td></tr> <tr> <td>WR284 </td> <td>WG10 </td> <td>R32 </td> <td><a href="/wiki/S_band" title="S band">S band</a> (part) </td> <td>2.60 — 3.95 </td> <td>2.078 </td> <td>4.156 </td> <td>2.840 × 1.340 <sup>†</sup></td> <td>72.14 × 34.94 </td></tr> <tr style="white-space:nowrap;"> <td>WR229 </td> <td>WG11A </td> <td>R40 </td> <td><a href="/wiki/C_band_(IEEE)" title="C band (IEEE)">C band</a> (part) </td> <td>3.30 — 4.90 </td> <td>2.577 </td> <td>5.154 </td> <td>2.290 × 1.145</td> <td>58.17 × 29.08 </td></tr> <tr> <td>WR187 </td> <td>WG12 </td> <td>R48 </td> <td><a href="/wiki/C_band_(IEEE)" title="C band (IEEE)">C band</a> (part) </td> <td>3.95 — 5.85 </td> <td>3.153 </td> <td>6.305 </td> <td>1.872 × 0.872 <sup>†</sup></td> <td>47.55 × 22.2 </td></tr> <tr> <td>WR159 </td> <td>WG13 </td> <td>R58 </td> <td><a href="/wiki/C_band_(IEEE)" title="C band (IEEE)">C band</a> (part) </td> <td>4.90 — 7.05 </td> <td>3.712 </td> <td>7.423 </td> <td>1.590 × 0.795</td> <td>40.38 × 20.2 </td></tr> <tr> <td>WR137 </td> <td>WG14 </td> <td>R70 </td> <td><a href="/wiki/C_band_(IEEE)" title="C band (IEEE)">C band</a> (part) </td> <td>5.85 — 8.20 </td> <td>4.301 </td> <td>8.603 </td> <td>1.372 × 0.622 <sup>†</sup></td> <td>34.90 × 15.8 </td></tr> <tr> <td>WR112 </td> <td>WG15 </td> <td>R84 </td> <td>— </td> <td>7.05 — 10.0 </td> <td>5.260 </td> <td>10.520 </td> <td>1.122 × 0.497 <sup>†</sup></td> <td>28.50 × 12.6 </td></tr> <tr> <td>WR90 </td> <td>WG16 </td> <td>R100 </td> <td><a href="/wiki/X_band" title="X band">X band</a> </td> <td><span style="visibility:hidden;color:transparent;">0</span>8.2 — 12.4 </td> <td>6.557 </td> <td>13.114 </td> <td>0.900 × 0.400 <sup>†</sup></td> <td>22.9 × 10.2 </td></tr> <tr> <td>WR75 </td> <td>WG17 </td> <td>R120 </td> <td>— </td> <td>10.0 — 15.0 </td> <td>7.869 </td> <td>15.737 </td> <td>0.750 × 0.375</td> <td>19.1 × 9.53 </td></tr> <tr> <td>WR62 </td> <td>WG18 </td> <td>R140 </td> <td><a href="/wiki/Ku_band" title="Ku band">K<sub>u</sub> band</a> </td> <td>12.4 — 18.0 </td> <td>9.488 </td> <td>18.976 </td> <td>0.622 × 0.311</td> <td>15.8 × 7.90 </td></tr> <tr> <td>WR51 </td> <td>WG19 </td> <td>R180 </td> <td>— </td> <td>15 — 22 </td> <td>11.572 </td> <td>23.143 </td> <td>0.510 × 0.255</td> <td>13.0 × 6.48 </td></tr> <tr> <td>WR42 </td> <td>WG20 </td> <td>R220 </td> <td><a href="/wiki/K_band_(IEEE)" title="K band (IEEE)">K band</a> </td> <td><span style="visibility:hidden;color:transparent;">0.</span>18 — 26.5 </td> <td>14.051 </td> <td>28.102 </td> <td>0.420 × 0.170 <sup>†</sup></td> <td>10.7 × 4.32 </td></tr> <tr> <td>WR34 </td> <td>WG21 </td> <td>R260 </td> <td>— </td> <td>22 — 33 </td> <td>17.357 </td> <td>34.715 </td> <td>0.340 × 0.170</td> <td>8.64 × 4.32 </td></tr> <tr> <td>WR28 </td> <td>WG22 </td> <td>R320 </td> <td><a href="/wiki/Ka_band" title="Ka band">K<sub>a</sub> band</a> </td> <td>26.5 — 40<span style="visibility:hidden;color:transparent;">.0</span> </td> <td>21.077 </td> <td>42.154 </td> <td>0.280 × 0.140</td> <td>7.11 × 3.56 </td></tr> <tr> <td>WR22 </td> <td>WG23 </td> <td>R400 </td> <td><a href="/wiki/Q_band" title="Q band">Q band</a> </td> <td>33 — 50 </td> <td>26.346 </td> <td>52.692 </td> <td>0.224 × 0.112</td> <td>5.68 × 2.84 </td></tr> <tr> <td>WR19 </td> <td>WG24 </td> <td>R500 </td> <td><a href="/wiki/U_band" title="U band">U band</a> </td> <td>40 — 60 </td> <td>31.391 </td> <td>62.782 </td> <td>0.188 × 0.094</td> <td>4.78 × 2.39 </td></tr> <tr> <td>WR15 </td> <td>WG25 </td> <td>R620 </td> <td><a href="/wiki/V_band" title="V band">V band</a> </td> <td>50 — 75 </td> <td>39.875 </td> <td>79.750 </td> <td>0.148 × 0.074</td> <td>3.76 × 1.88 </td></tr> <tr> <td>WR12 </td> <td>WG26 </td> <td>R740 </td> <td><a href="/wiki/E_band_(waveguide)" title="E band (waveguide)">E band</a> </td> <td>60 — 90 </td> <td>48.373 </td> <td>96.746 </td> <td>0.122 × 0.061</td> <td>3.10 × 1.55 </td></tr> <tr> <td>WR10 </td> <td>WG27 </td> <td>R900 </td> <td><a href="/wiki/W_band" title="W band">W band</a> </td> <td><span style="visibility:hidden;color:transparent;">0</span>75 — 110 </td> <td>59.015 </td> <td>118.030 </td> <td>0.100 × 0.050</td> <td>2.54 × 1.27 </td></tr> <tr> <td>WR8 </td> <td>WG28 </td> <td>R1200 </td> <td><a href="/wiki/F_band_(waveguide)" title="F band (waveguide)">F band</a> </td> <td><span style="visibility:hidden;color:transparent;">0</span>90 — 140 </td> <td>73.768 </td> <td>147.536 </td> <td>0.080 × 0.040</td> <td>2.03 × 1.02 </td></tr> <tr> <td>WR6<br />WR7<br />WR6.5 </td> <td>WG29 </td> <td>R1400 </td> <td><a href="/wiki/D_band_(waveguide)" title="D band (waveguide)">D band</a> </td> <td>110 — 170 </td> <td>90.791 </td> <td>181.583 </td> <td>0.0650 × 0.0325</td> <td>1.65 × 0.826 </td></tr> <tr> <td>WR5 </td> <td>WG30 </td> <td>R1800 </td> <td><a href="/wiki/G_band_(IEEE)" class="mw-redirect" title="G band (IEEE)">G band</a> </td> <td>140 — 220 </td> <td>115.714 </td> <td>231.429 </td> <td>0.0510 × 0.0255</td> <td>1.30 × 0.648 </td></tr> <tr> <td>WR4 </td> <td>WG31 </td> <td>R2200 </td> <td> </td> <td>170 — 260 </td> <td>137.243 </td> <td>274.485 </td> <td>0.0430 × 0.0215</td> <td>1.09 × 0.546 </td></tr> <tr style="white-space:nowrap;"> <td>WR3 </td> <td>WG32 </td> <td>R2600 </td> <td> </td> <td>220 — 330 </td> <td>173.571 </td> <td>347.143 </td> <td>0.0340 × 0.0170</td> <td>0.864 × 0.432 </td></tr> <tr> <td>WR2.8 </td> <td> </td> <td> </td> <td> </td> <td>260 — 400 </td> <td>211.121 </td> <td>422.243 </td> <td>0.02795 × 0.01398</td> <td>0.71 × 0.355 </td></tr> <tr> <td>WR2.2 </td> <td> </td> <td> </td> <td> </td> <td>325 — 500 </td> <td>262.975 </td> <td>525.951 </td> <td>0.02244 × 0.01122</td> <td>0.57 × 0.285 </td></tr> <tr> <td>WR1.9 </td> <td> </td> <td> </td> <td> </td> <td>400 — 600 </td> <td>318.928 </td> <td>637.856 </td> <td>0.01850 × 0.009252</td> <td>0.47 × 0.235 </td></tr> <tr> <td>WR1.5 </td> <td> </td> <td> </td> <td> </td> <td>500 — 750 </td> <td>394.463 </td> <td>788.927 </td> <td>0.01496 × 0.007480</td> <td>0.38 × 0.19 </td></tr> <tr> <td>WR1.2 </td> <td> </td> <td> </td> <td> </td> <td>600 — 900 </td> <td>483.536 </td> <td>967.072 </td> <td>0.01220 × 0.006102</td> <td>0.31 × 0.155 </td></tr> <tr> <td>WR1 </td> <td> </td> <td> </td> <td> </td> <td>750 — 1100 </td> <td>599.584 </td> <td>1199.2 </td> <td>0.009843 × 0.004921</td> <td>0.25 × 0.125 </td></tr></tbody></table> <dl><dd><sup>*</sup> Radio Components Standardization Committee</dd></dl> <dl><dd><sup>†</sup> For historical reasons the outside rather than the inside dimensions of these waveguides are 2:1 (with wall thickness WG6–WG10: 0.08" (2.0 mm), WG11A–WG15: 0.064" (1.6 mm), WG16–WG17: 0.05" (1.3 mm), WG18–WG28: 0.04" (1.0 mm))<sup id="cite_ref-Baden-Fuller_15-0" class="reference"><a href="#cite_note-Baden-Fuller-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></dd></dl> <p>For the frequencies in the table above, the main advantage of waveguides over <a href="/wiki/Coax_cable" class="mw-redirect" title="Coax cable">coaxial cables</a> is that waveguides support propagation with lower loss. For lower frequencies, the waveguide dimensions become impractically large, and for higher frequencies the dimensions become impractically small (the manufacturing tolerance becomes a significant portion of the waveguide size). </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_analysis">Mathematical analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=5" title="Edit section: Mathematical analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Electromagnetic waveguides are analyzed by solving <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, or their reduced form, the <a href="/wiki/Electromagnetic_wave_equation" title="Electromagnetic wave equation">electromagnetic wave equation</a>, with <a href="/wiki/Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a> determined by the properties of the materials and their interfaces. These equations have multiple solutions, or modes, which are <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the equation system. Each mode is characterized by a cutoff frequency below which the mode cannot exist in the guide. Waveguide propagation modes depend on the operating <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> and <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> and the shape and size of the guide. The <a href="/wiki/Longitudinal_mode" title="Longitudinal mode">longitudinal mode</a> of a waveguide is a particular <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> pattern formed by waves confined in the cavity. The <a href="/wiki/Transverse_mode" title="Transverse mode">transverse modes</a> are classified into different types: </p> <ul><li>TE modes (transverse electric) have no electric field in the direction of propagation.</li> <li>TM modes (transverse magnetic) have no magnetic field in the direction of propagation.</li> <li>TEM modes (transverse electromagnetic) have no electric nor magnetic field in the direction of propagation.</li> <li>Hybrid modes have both electric and magnetic field components in the direction of propagation.</li></ul> <p>Waveguides with certain symmetries may be solved using the method of <a href="/wiki/Separation_of_variables" title="Separation of variables">separation of variables</a>. Rectangular wave guides may be solved in rectangular coordinates.<sup id="cite_ref-Harrington_16-0" class="reference"><a href="#cite_note-Harrington-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 143">: 143 </span></sup> Round waveguides may be solved in cylindrical coordinates.<sup id="cite_ref-Harrington_16-1" class="reference"><a href="#cite_note-Harrington-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 198">: 198 </span></sup> </p><p>In hollow, single conductor waveguides, TEM waves are not possible. This contrasts with two-conductor <a href="/wiki/Transmission_line" title="Transmission line">transmission lines</a> used at lower frequencies; <a href="/wiki/Coaxial_cable" title="Coaxial cable">coaxial cable</a>, <a href="/wiki/Twin_lead" class="mw-redirect" title="Twin lead">parallel wire line</a> and <a href="/wiki/Stripline" title="Stripline">stripline</a>, in which TEM mode is possible. Additionally, the propagating modes (i.e. TE and TM) inside the waveguide can be mathematically expressed as the superposition of two TEM waves.<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> </p><p>The mode with the lowest cutoff frequency is termed the <b>dominant mode</b> of the guide. It is common to choose the size of the guide such that only this one mode can exist in the frequency band of operation. In rectangular and circular (hollow pipe) waveguides, the dominant modes are designated the TE<sub>1,0</sub> mode and TE<sub>1,1</sub> modes respectively.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:TE11.svg" class="mw-file-description" title="TE1,1 mode of a circular hollow metallic waveguide."><img alt="TE1,1 mode of a circular hollow metallic waveguide." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/TE11.svg/120px-TE11.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/TE11.svg/180px-TE11.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/TE11.svg/240px-TE11.svg.png 2x" data-file-width="155" data-file-height="155" /></a></span></div> <div class="gallerytext">TE<sub>1,1</sub> mode of a circular hollow metallic waveguide.</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Dielectric_waveguides">Dielectric waveguides</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=6" title="Edit section: Dielectric waveguides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>dielectric waveguide</i> employs a solid dielectric rod rather than a hollow pipe. An <a href="/wiki/Optical_fibre" class="mw-redirect" title="Optical fibre">optical fibre</a> is a dielectric guide designed to work at optical frequencies. Transmission lines such as <a href="/wiki/Microstrip" title="Microstrip">microstrip</a>, <a href="/wiki/Coplanar_waveguide" title="Coplanar waveguide">coplanar waveguide</a>, <a href="/wiki/Stripline" title="Stripline">stripline</a> or <a href="/wiki/Coaxial_cable" title="Coaxial cable">coaxial cable</a> may also be considered to be waveguides. </p><p>Dielectric rod and slab waveguides are used to conduct radio waves, mostly at <a href="/wiki/Millimeter_wave" class="mw-redirect" title="Millimeter wave">millimeter wave</a> frequencies and above.<sup id="cite_ref-Lioubtchenko_19-0" class="reference"><a href="#cite_note-Lioubtchenko-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Shevgaonkar_20-0" class="reference"><a href="#cite_note-Shevgaonkar-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> These confine the radio waves by <a href="/wiki/Total_internal_reflection" title="Total internal reflection">total internal reflection</a> from the step in <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a> due to the change in dielectric constant at the material surface.<sup id="cite_ref-Rana_21-0" class="reference"><a href="#cite_note-Rana-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> At millimeter wave frequencies and above, metal is not a good conductor, so metal waveguides can have increasing attenuation. At these wavelengths dielectric waveguides can have lower losses than metal waveguides. Optical fibre is a form of dielectric waveguide used at optical wavelengths. </p><p>One difference between dielectric and metal waveguides is that at a metal surface the electromagnetic waves are tightly confined; at high frequencies the electric and magnetic fields penetrate a very short distance into the metal. In contrast, the surface of the dielectric waveguide is an interface between two dielectrics, so the fields of the wave penetrate outside the dielectric in the form of an <a href="/wiki/Evanescent_wave" class="mw-redirect" title="Evanescent wave">evanescent</a> (non-propagating) wave.<sup id="cite_ref-Rana_21-1" class="reference"><a href="#cite_note-Rana-21"><span class="cite-bracket">[</span>21<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=Waveguide_(radio_frequency)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Angular_misalignment_loss" title="Angular misalignment loss">Angular misalignment loss</a></li> <li><a href="/wiki/Cantenna" title="Cantenna">Cantenna</a></li> <li><a href="/wiki/Cavity_resonator" class="mw-redirect" title="Cavity resonator">Cavity resonator</a></li> <li><a href="/wiki/Cutoff_frequency" title="Cutoff frequency">Cutoff frequency</a></li> <li><a href="/wiki/Feed_horn" title="Feed horn">Feed horn</a></li> <li><a href="/wiki/Filled_cable" title="Filled cable">Filled cable</a></li> <li><a href="/wiki/Leaky_mode" title="Leaky mode">Leaky mode</a></li> <li><a href="/wiki/Magic_tee" title="Magic tee">Magic tee</a></li> <li><a href="/wiki/Optical_waveguide" class="mw-redirect" title="Optical waveguide">Optical waveguide</a></li> <li><a href="/wiki/Radiation_mode" title="Radiation mode">Radiation mode</a></li> <li><a href="/wiki/Radio_propagation" title="Radio propagation">Radio propagation</a></li> <li><a href="/wiki/Radio_wave" title="Radio wave">Radio wave</a></li> <li><a href="/wiki/Substrate-integrated_waveguide" title="Substrate-integrated waveguide">Substrate-integrated waveguide</a></li> <li><a href="/wiki/Transmission_medium" title="Transmission medium">Transmission medium</a></li> <li><a href="/wiki/Waveguide_filter" title="Waveguide filter">Waveguide filter</a></li> <li><a href="/wiki/Waveguide_flange" title="Waveguide flange">Waveguide flange</a></li> <li><a href="/wiki/Waveguide_rotary_joint" title="Waveguide rotary joint">Waveguide rotary joint</a></li> <li><a href="/wiki/Flap_attenuator" title="Flap attenuator">Flap attenuator</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=Waveguide_(radio_frequency)&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-IEEEdict1997-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-IEEEdict1997_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="CITEREFRadatz" class="citation book cs1">Radatz, Jane. <a rel="nofollow" class="external text" href="https://standards.ieee.org/ieee/100/256/"><i>The IEEE Standard Dictionary of Electrical and Electronics Terms</i></a> (6 ed.). IEEE Standards Association. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1559378336" title="Special:BookSources/1559378336"><bdi>1559378336</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+IEEE+Standard+Dictionary+of+Electrical+and+Electronics+Terms&rft.edition=6&rft.pub=IEEE+Standards+Association&rft.isbn=1559378336&rft.aulast=Radatz&rft.aufirst=Jane&rft_id=https%3A%2F%2Fstandards.ieee.org%2Fieee%2F100%2F256%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Southworth-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Southworth_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Southworth_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Southworth_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSouthworth1936" class="citation journal cs1">Southworth, G. C. (August 1936). <a rel="nofollow" class="external text" href="http://www.americanradiohistory.com/Archive-Short-Wave-Television/30s/SW-TV-1936-08.pdf">"Electric Wave Guides"</a> <span class="cs1-format">(PDF)</span>. <i>Short Wave Craft</i>. <b>7</b> (1): 198, 233<span class="reference-accessdate">. Retrieved <span class="nowrap">March 27,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Short+Wave+Craft&rft.atitle=Electric+Wave+Guides&rft.volume=7&rft.issue=1&rft.pages=198%2C+233&rft.date=1936-08&rft.aulast=Southworth&rft.aufirst=G.+C.&rft_id=http%3A%2F%2Fwww.americanradiohistory.com%2FArchive-Short-Wave-Television%2F30s%2FSW-TV-1936-08.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Packard-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Packard_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Packard_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Packard_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Packard_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Packard_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Packard_3-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Packard_3-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Packard_3-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Packard_3-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Packard_3-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-Packard_3-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-Packard_3-11"><sup><i><b>l</b></i></sup></a> <a href="#cite_ref-Packard_3-12"><sup><i><b>m</b></i></sup></a> <a href="#cite_ref-Packard_3-13"><sup><i><b>n</b></i></sup></a> <a href="#cite_ref-Packard_3-14"><sup><i><b>o</b></i></sup></a> <a href="#cite_ref-Packard_3-15"><sup><i><b>p</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPackard1984" class="citation journal cs1 cs1-prop-long-vol">Packard, Karle S. (September 1984). <a rel="nofollow" class="external text" href="http://www.ieeeghn.org/wiki/images/8/86/MTT_Waveguide_History.pdf">"The Origin of Waveguides: A Case of Multiple Rediscovery"</a> <span class="cs1-format">(PDF)</span>. <i>IEEE Transactions on Microwave Theory and Techniques</i>. MTT-32 (9): 961–969. <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/1984ITMTT..32..961P">1984ITMTT..32..961P</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=10.1.1.532.8921">10.1.1.532.8921</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/10.1109%2Ftmtt.1984.1132809">10.1109/tmtt.1984.1132809</a><span class="reference-accessdate">. Retrieved <span class="nowrap">March 24,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Microwave+Theory+and+Techniques&rft.atitle=The+Origin+of+Waveguides%3A+A+Case+of+Multiple+Rediscovery&rft.volume=MTT-32&rft.issue=9&rft.pages=961-969&rft.date=1984-09&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.532.8921%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1109%2Ftmtt.1984.1132809&rft_id=info%3Abibcode%2F1984ITMTT..32..961P&rft.aulast=Packard&rft.aufirst=Karle+S.&rft_id=http%3A%2F%2Fwww.ieeeghn.org%2Fwiki%2Fimages%2F8%2F86%2FMTT_Waveguide_History.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Rayleigh-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rayleigh_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrutt1897" class="citation journal cs1">Strutt, William (Lord Rayleigh) (February 1897). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1431225">"On the passage of electric waves through tubes, or the vibrations of dielectric cylinders"</a>. <i>Philosophical Magazine</i>. <b>43</b> (261): 125–132. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786449708620969">10.1080/14786449708620969</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+passage+of+electric+waves+through+tubes%2C+or+the+vibrations+of+dielectric+cylinders&rft.volume=43&rft.issue=261&rft.pages=125-132&rft.date=1897-02&rft_id=info%3Adoi%2F10.1080%2F14786449708620969&rft.aulast=Strutt&rft.aufirst=William+%28Lord+Rayleigh%29&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1431225&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Kizer-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kizer_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kizer_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Kizer_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKizer2013" class="citation book cs1">Kizer, George (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JVhGmjQ8TyoC&q=southworth+bose+lodge+waveguide"><i>Digital Microwave Communication: Engineering Point-to-Point Microwave Systems</i></a>. John Wiley and Sons. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1118636800" title="Special:BookSources/978-1118636800"><bdi>978-1118636800</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Microwave+Communication%3A+Engineering+Point-to-Point+Microwave+Systems&rft.pages=7&rft.pub=John+Wiley+and+Sons&rft.date=2013&rft.isbn=978-1118636800&rft.aulast=Kizer&rft.aufirst=George&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJVhGmjQ8TyoC%26q%3Dsouthworth%2Bbose%2Blodge%2Bwaveguide&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Lee-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lee_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lee_6-1"><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="CITEREFLee2004" class="citation book cs1">Lee, Thomas H. (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uoj3IWFxbVYC&pg=PA18"><i>Planar Microwave Engineering: A Practical Guide to Theory, Measurement, and Circuits, Vol. 1</i></a>. Cambridge University Press. pp. 18, 118. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521835268" title="Special:BookSources/9780521835268"><bdi>9780521835268</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Planar+Microwave+Engineering%3A+A+Practical+Guide+to+Theory%2C+Measurement%2C+and+Circuits%2C+Vol.+1&rft.pages=18%2C+118&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=9780521835268&rft.aulast=Lee&rft.aufirst=Thomas+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Duoj3IWFxbVYC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Weber-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Weber_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeber1902" class="citation journal cs1">Weber, R. H. (1902). "Elektromagnetische Schwingungen in Metallrohren". <i>Annalen der Physik</i>. <b>8</b> (4): 721–751. <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/1902AnP...313..721W">1902AnP...313..721W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19023130802">10.1002/andp.19023130802</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuc1.%24b24304">2027/uc1.$b24304</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Elektromagnetische+Schwingungen+in+Metallrohren&rft.volume=8&rft.issue=4&rft.pages=721-751&rft.date=1902&rft_id=info%3Ahdl%2F2027%2Fuc1.%24b24304&rft_id=info%3Adoi%2F10.1002%2Fandp.19023130802&rft_id=info%3Abibcode%2F1902AnP...313..721W&rft.aulast=Weber&rft.aufirst=R.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Lodge-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lodge_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLodge1984" class="citation journal cs1">Lodge, Oliver (June 1, 1984). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JNINAQAAIAAJ&q=%22hollow+cylinder%22&pg=PA331">"The Work of Hertz"</a>. <i>Proc. Of the Royal Institution</i>. <b>14</b> (88): 331–332<span class="reference-accessdate">. Retrieved <span class="nowrap">April 11,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Of+the+Royal+Institution&rft.atitle=The+Work+of+Hertz&rft.volume=14&rft.issue=88&rft.pages=331-332&rft.date=1984-06-01&rft.aulast=Lodge&rft.aufirst=Oliver&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJNINAQAAIAAJ%26q%3D%2522hollow%2Bcylinder%2522%26pg%3DPA331&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Emerson-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Emerson_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmerson1998" class="citation web cs1">Emerson, Darrel T. (1998). <a rel="nofollow" class="external text" href="https://www.cv.nrao.edu/~demerson/bose/emerson_delhi.pdf">"Jagadish Chandra Bose: Millimeter-wave research in the 19th century"</a> <span class="cs1-format">(PDF)</span>. US National Radio Astronomy Observatory<span class="reference-accessdate">. Retrieved <span class="nowrap">April 11,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Jagadish+Chandra+Bose%3A+Millimeter-wave+research+in+the+19th+century&rft.pub=US+National+Radio+Astronomy+Observatory&rft.date=1998&rft.aulast=Emerson&rft.aufirst=Darrel+T.&rft_id=https%3A%2F%2Fwww.cv.nrao.edu%2F~demerson%2Fbose%2Femerson_delhi.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Brown-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Brown_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Brown_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Brown_10-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Brown_10-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Brown_10-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Brown_10-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown1999" class="citation book cs1">Brown, Louis (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uYgsr3exvS4C&q=barrow+southworth+waveguide&pg=PA147"><i>Technical and Military Imperatives: A Radar History of World War 2</i></a>. CRC Press. pp. 146–148. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1420050660" title="Special:BookSources/978-1420050660"><bdi>978-1420050660</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Technical+and+Military+Imperatives%3A+A+Radar+History+of+World+War+2&rft.pages=146-148&rft.pub=CRC+Press&rft.date=1999&rft.isbn=978-1420050660&rft.aulast=Brown&rft.aufirst=Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuYgsr3exvS4C%26q%3Dbarrow%2Bsouthworth%2Bwaveguide%26pg%3DPA147&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Schelkunoff-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schelkunoff_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchelkunoff1937" class="citation journal cs1">Schelkunoff, Sergei A. (November 1937). "Electromagnetic Waves in Conducting Tubes". <i>Physical Review</i>. <b>52</b> (10): 1078. <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/1937PhRv...52.1078S">1937PhRv...52.1078S</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.52.1078">10.1103/PhysRev.52.1078</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=Electromagnetic+Waves+in+Conducting+Tubes&rft.volume=52&rft.issue=10&rft.pages=1078&rft.date=1937-11&rft_id=info%3Adoi%2F10.1103%2FPhysRev.52.1078&rft_id=info%3Abibcode%2F1937PhRv...52.1078S&rft.aulast=Schelkunoff&rft.aufirst=Sergei+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-PBP-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-PBP_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.fnrf.science.cmu.ac.th/waveguide/Waveguide%20theory%2012.html">"Module 12: Waveguide Plumbing"</a>. <i>Introduction to Waveguides</i>. Plasma and Beam Physics Research Facility, Dept. of Physics and Materials Science, Chiang Mai University, Thailand. 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">September 21,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Introduction+to+Waveguides&rft.atitle=Module+12%3A+Waveguide+Plumbing&rft.date=2012&rft_id=http%3A%2F%2Fwww.fnrf.science.cmu.ac.th%2Fwaveguide%2FWaveguide%2520theory%252012.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" 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">For bandwidths lower than 2:1 it is more common to express them as a percentage of the center frequency, which in the case of 1.360:1 is 26.55 %. For reference, a 2:1 bandwidth corresponds to a 66.67 % bandwidth. The reason for expressing bandwidths as a ratio of upper to lower band edges for bandwidths greater than 66.67 % is that in the limiting case that the lower edge goes to zero (or the upper edge goes to infinity), the bandwidth approaches 200 %, which means that the entire range of 3:1 to infinity:1 map into the range 100 % to 200 %.</span> </li> <li id="cite_note-Harvey-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Harvey_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarvey1955" class="citation journal cs1">Harvey, A. F. (July 1955). "Standard waveguides and couplings for microwave equipment". <i>Proceedings of the IEE - Part B: Radio and Electronic Engineering</i>. <b>102</b> (4): 493–499. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fpi-b-1.1955.0095">10.1049/pi-b-1.1955.0095</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+IEE+-+Part+B%3A+Radio+and+Electronic+Engineering&rft.atitle=Standard+waveguides+and+couplings+for+microwave+equipment&rft.volume=102&rft.issue=4&rft.pages=493-499&rft.date=1955-07&rft_id=info%3Adoi%2F10.1049%2Fpi-b-1.1955.0095&rft.aulast=Harvey&rft.aufirst=A.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Baden-Fuller-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Baden-Fuller_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaden_Fuller1969" class="citation book cs1">Baden Fuller, A. J. (1969). <i>Microwaves</i> (1 ed.). Pergamon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-006616-5" title="Special:BookSources/978-0-08-006616-5"><bdi>978-0-08-006616-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microwaves&rft.edition=1&rft.pub=Pergamon+Press&rft.date=1969&rft.isbn=978-0-08-006616-5&rft.aulast=Baden+Fuller&rft.aufirst=A.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Harrington-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Harrington_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Harrington_16-1"><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="CITEREFHarrington1961" class="citation cs2"><a href="/wiki/Roger_F._Harrington" title="Roger F. Harrington">Harrington, Roger F.</a> (1961), <i>Time-Harmonic Electromagnetic Fields</i>, McGraw-Hill, pp. 7–8, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.39015002091489">2027/mdp.39015002091489</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-026745-6" title="Special:BookSources/0-07-026745-6"><bdi>0-07-026745-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time-Harmonic+Electromagnetic+Fields&rft.pages=7-8&rft.pub=McGraw-Hill&rft.date=1961&rft_id=info%3Ahdl%2F2027%2Fmdp.39015002091489&rft.isbn=0-07-026745-6&rft.aulast=Harrington&rft.aufirst=Roger+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" 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="CITEREFSomeda1998" class="citation book cs1">Someda, Carlo G. (1998). <i>Electromagnetic Waves</i>. CRC Press. pp. 257–258. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0412578700" title="Special:BookSources/0412578700"><bdi>0412578700</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Waves&rft.pages=257-258&rft.pub=CRC+Press&rft.date=1998&rft.isbn=0412578700&rft.aulast=Someda&rft.aufirst=Carlo+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFModiBalanis2016" class="citation journal cs1">Modi, Anuj Y.; Balanis, Constantine A. (2016). "PEC-PMC Baffle Inside Circular Cross Section Waveguide for Reduction of Cut-Off Frequency". <i>IEEE Microwave and Wireless Components Letters</i>. <b>26</b> (3): 171–173. <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%2FLMWC.2016.2524529">10.1109/LMWC.2016.2524529</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:9594124">9594124</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Microwave+and+Wireless+Components+Letters&rft.atitle=PEC-PMC+Baffle+Inside+Circular+Cross+Section+Waveguide+for+Reduction+of+Cut-Off+Frequency&rft.volume=26&rft.issue=3&rft.pages=171-173&rft.date=2016&rft_id=info%3Adoi%2F10.1109%2FLMWC.2016.2524529&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9594124%23id-name%3DS2CID&rft.aulast=Modi&rft.aufirst=Anuj+Y.&rft.au=Balanis%2C+Constantine+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Lioubtchenko-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lioubtchenko_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLioubtchenkoSergei_TretyakovSergey_Dudorov2003" class="citation book cs1">Lioubtchenko, Dmitri; Sergei Tretyakov; Sergey Dudorov (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CoihHs1Ctp4C&q=%22dielectric+waveguide&pg=PA149"><i>Millimeter-Wave Waveguides</i></a>. Springer. p. 149. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1402075315" title="Special:BookSources/978-1402075315"><bdi>978-1402075315</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Millimeter-Wave+Waveguides&rft.pages=149&rft.pub=Springer&rft.date=2003&rft.isbn=978-1402075315&rft.aulast=Lioubtchenko&rft.aufirst=Dmitri&rft.au=Sergei+Tretyakov&rft.au=Sergey+Dudorov&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCoihHs1Ctp4C%26q%3D%2522dielectric%2Bwaveguide%26pg%3DPA149&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Shevgaonkar-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shevgaonkar_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShevgaonkar2005" class="citation book cs1">Shevgaonkar, R. K. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VRMGaHDuaFQC&q=%22dielectric+slab%22+waveguide&pg=PA327"><i>Electromagnetic Waves</i></a>. Tata McGraw-Hill Education. p. 327. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070591165" title="Special:BookSources/978-0070591165"><bdi>978-0070591165</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Waves&rft.pages=327&rft.pub=Tata+McGraw-Hill+Education&rft.date=2005&rft.isbn=978-0070591165&rft.aulast=Shevgaonkar&rft.aufirst=R.+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVRMGaHDuaFQC%26q%3D%2522dielectric%2Bslab%2522%2Bwaveguide%26pg%3DPA327&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> <li id="cite_note-Rana-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rana_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rana_21-1"><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="CITEREFRana2005" class="citation web cs1">Rana, Farhan (Fall 2005). <a rel="nofollow" class="external text" href="https://courses.cit.cornell.edu/ece303/Lectures/lecture26.pdf">"Lecture 26: Dielectric slab waveguides"</a> <span class="cs1-format">(PDF)</span>. <i>Class notes ECE 303: Electromagnetic Fields and Waves</i>. Electrical Engineering Dept. Cornell Univ. pp. 2–3, 10<span class="reference-accessdate">. Retrieved <span class="nowrap">June 21,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Class+notes+ECE+303%3A+Electromagnetic+Fields+and+Waves&rft.atitle=Lecture+26%3A+Dielectric+slab+waveguides&rft.ssn=fall&rft.pages=2-3%2C+10&rft.date=2005&rft.aulast=Rana&rft.aufirst=Farhan&rft_id=https%3A%2F%2Fcourses.cit.cornell.edu%2Fece303%2FLectures%2Flecture26.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWaveguide+%28radio+frequency%29" class="Z3988"></span></span> </li> </ol></div> <ul><li>This article is based in part on material from <a href="/wiki/Federal_Standard_1037C" title="Federal Standard 1037C">Federal Standard 1037C</a> and from <a href="/wiki/MIL-STD-188" title="MIL-STD-188">MIL-STD-188</a>, and <a href="/wiki/Alliance_for_Telecommunications_Industry_Solutions" title="Alliance for Telecommunications Industry Solutions">ATIS</a></li> <li>J. J. Thomson, <i>Recent Researches</i> (1893).</li> <li>O. J. Lodge, <i>Proc. Roy. Inst.</i> <b>14</b>, p. 321 (1894).</li> <li>Lord Rayleigh, <i>Phil. Mag.</i> <b>43</b>, p. 125 (1897).</li> <li>N. W. McLachlan, <i>Theory and Applications of Mathieu Functions</i>, p. 8 (1947) (reprinted by Dover: New York, 1964).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/George_Clark_Southworth" title="George Clark Southworth">George Clark Southworth</a>, "<i>Principles and applications of wave-guide transmission</i>". New York, Van Nostrand [1950], xi, 689 p. illus. 24 cm. Bell Telephone Laboratories series. LCCN 50009834</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Waveguide_(radio_frequency)&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Waveguides" class="extiw" title="commons:Category:Waveguides">Waveguides</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.feynmanlectures.caltech.edu/II_24.html">The Feynman Lectures on Physics Vol. II Ch. 24: Waveguides</a></li> <li><a rel="nofollow" class="external text" href="http://www.antenna-theory.com/tutorial/waveguides/waveguides2.php">Derivation of Fields Within a Rectangular Waveguide</a> antenna-theory.com</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><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><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></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/Q354724#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/4160442-8">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00561566">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="vlnovody"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph195847&CON_LNG=ENG">Czech Republic</a></span></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=Waveguide_(radio_frequency)&oldid=1245315675">https://en.wikipedia.org/w/index.php?title=Waveguide_(radio_frequency)&oldid=1245315675</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:Telecommunications_engineering" title="Category:Telecommunications engineering">Telecommunications engineering</a></li><li><a href="/wiki/Category:Electrodynamics" title="Category:Electrodynamics">Electrodynamics</a></li><li><a href="/wiki/Category:Microwave_technology" title="Category:Microwave technology">Microwave technology</a></li><li><a href="/wiki/Category:Wave_mechanics" title="Category:Wave mechanics">Wave mechanics</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:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</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:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2015" title="Category:Articles with unsourced statements from September 2015">Articles with unsourced statements from September 2015</a></li><li><a href="/wiki/Category:Commons_category_link_is_locally_defined" title="Category:Commons category link is locally defined">Commons category link is locally defined</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 12 September 2024, at 08:39<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=Waveguide_(radio_frequency)&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-f69cdc8f6-qr4c8","wgBackendResponseTime":140,"wgPageParseReport":{"limitreport":{"cputime":"0.510","walltime":"0.679","ppvisitednodes":{"value":2725,"limit":1000000},"postexpandincludesize":{"value":73707,"limit":2097152},"templateargumentsize":{"value":1914,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":98257,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 519.964 1 -total"," 19.25% 100.103 8 Template:Cite_book"," 14.23% 74.014 1 Template:Electromagnetism"," 13.92% 72.354 1 Template:Sidebar_with_collapsible_lists"," 10.46% 54.401 1 Template:Short_description"," 9.51% 49.429 1 Template:Commons_category"," 9.20% 47.829 1 Template:Sister_project"," 8.95% 46.515 1 Template:Side_box"," 8.87% 46.097 1 Template:Authority_control"," 6.46% 33.583 8 Template:Cite_journal"]},"scribunto":{"limitreport-timeusage":{"value":"0.303","limit":"10.000"},"limitreport-memusage":{"value":7140333,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-r5tzf","timestamp":"20241122141534","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Waveguide (radio frequency)","url":"https:\/\/en.wikipedia.org\/wiki\/Waveguide_(radio_frequency)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q354724","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q354724","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":"2006-01-26T18:10:48Z","dateModified":"2024-09-12T08:39:44Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/1\/16\/Waveguide_collection.jpg","headline":"structure used to propagate electromagnetic waves"}</script> </body> </html>