🕷️ Crawler Inspector

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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 152 (from laksa071)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Sine_wave\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Sine_wave\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Sine_wave","crawl_time":1775715657,"first_indexed_time":1375984497,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Sine_wave s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Sine wave - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nTracing the y component of a\ncircle\nwhile going around the circle results in a sine wave (red). Tracing the x component results in a\ncosine\nwave (blue). Both waves are sinusoids of the same frequency but different phases.\nA\nsine wave\n,\nsinusoidal wave\n, or\nsinusoid\n(symbol:\n∿\n) is a\nperiodic wave\nwhose\nwaveform\n(shape) is the\ntrigonometric\nsine function\n. In\nmechanics\n, as a linear\nmotion\nover time, this is\nsimple harmonic motion\n; as\nrotation\n, it corresponds to\nuniform circular motion\n. Sine waves occur often in\nphysics\n, including\nwind waves\n,\nsound\nwaves, and\nlight\nwaves, such as\nmonochromatic radiation\n. In\nengineering\n,\nsignal processing\n, and\nmathematics\n,\nFourier analysis\ndecomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.\nWhen any two sine waves of the same\nfrequency\n(but arbitrary\nphase\n) are\nlinearly combined\n, the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the\nsine\nand\ncosine\ncomponents\n, respectively.\nFive seconds of a 220 Hz sine wave. This is the\nsound wave\ndescribed by a sine function with\nf\n= 220 oscillations per second.\nA sine wave represents a single frequency with no\nharmonics\nand is considered an\nacoustically\npure tone\n. Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the\nfundamental\ncauses variation in the\ntimbre\n, which is the reason why the same\nmusical pitch\nplayed on different instruments sounds different.\nSine waves of arbitrary phase and amplitude are called\nsinusoids\nand have the general form:\n[\n1\n]\n[\n2\n]\nwhere:\n,\namplitude\n, the peak deviation of the function from zero.\n, the\nreal\nindependent variable\n, usually representing\ntime\nin\nseconds\n.\n,\nangular frequency\n, the rate of change of the function argument in units of\nradians per second\n.\n,\nordinary frequency\n, the\nnumber\nof oscillations (\ncycles\n) that occur each second of time.\n,\nphase\n, specifies (in\nradians\n) where in its cycle the oscillation is at\nt\n= 0.\nAs a function of both position and time\n[\nedit\n]\nThe displacement of an undamped\nspring-mass system\noscillating around the equilibrium over time is a sine wave.\nSinusoids that exist in both position and time also have:\nDepending on their direction of travel, they can take the form:\n, if the wave is moving to the right, or\n, if the wave is moving to the left.\nSince sine waves propagate without changing form in\ndistributed linear systems\n,\n[\ndefinition needed\n]\nthey are often used to analyze\nwave propagation\n.\nWhen two waves with the same\namplitude\nand\nfrequency\ntraveling in opposite directions\nsuperpose\neach other, then a\nstanding wave\npattern is created.\nOn a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's\nresonant\nfrequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the\nfundamental frequency\n) and integer divisions of that (corresponding to higher harmonics).\nMultiple spatial dimensions\n[\nedit\n]\nThe earlier equation gives the displacement\nof the wave at a position\nat time\nalong a single line. This could, for example, be considered the value of a wave along a wire.\nIn two or three spatial dimensions, the same equation describes a travelling\nplane wave\nif position\nand wavenumber\nare interpreted as vectors, and their product as a\ndot product\n. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.\nSinusoidal plane wave\n[\nedit\n]\nFrench mathematician\nJoseph Fourier\ndiscovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including\nsquare waves\n. These\nFourier series\nare frequently used in\nsignal processing\nand the statistical analysis of\ntime series\n. The\nFourier transform\nthen extended Fourier series to handle general functions, and birthed the field of\nFourier analysis\n.\nDifferentiation and integration\n[\nedit\n]\nDifferentiating\nany sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:\nA\ndifferentiator\nhas a\nzero\nat the origin of the\ncomplex frequency\nplane. The\ngain\nof its\nfrequency response\nincreases at a rate of +20 \ndB\nper\ndecade\nof frequency (for\nroot-power\nquantities), the same positive slope as a 1\nst\norder\nhigh-pass filter\n's\nstopband\n, although a differentiator does not have a\ncutoff frequency\nor a flat\npassband\n. A n\nth\n-order high-pass filter approximately applies the n\nth\ntime derivative of\nsignals\nwhose frequency band is significantly lower than the filter's cutoff frequency.\nIntegrating\nany sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle:\nThe\nconstant of integration\nwill be zero if the\nbounds of integration\nis an integer multiple of the sinusoid's period.\nAn\nintegrator\nhas a\npole\nat the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1\nst\norder\nlow-pass filter\n's stopband, although an integrator does not have a cutoff frequency or a flat passband. A n\nth\n-order low-pass filter approximately performs the n\nth\ntime integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.\nCrest (physics)\nComplex exponential\nDamped sine wave\nEuler's formula\nFourier transform\nHarmonic analysis\nHarmonic series (mathematics)\nHarmonic series (music)\nHelmholtz equation\nInstantaneous phase\nIn-phase and quadrature components\nLeast-squares spectral analysis\nOscilloscope\nPhasor\nPure tone\nSimple harmonic motion\nSinusoidal model\nWave (physics)\nWave equation\n∿\nthe sine wave symbol (U+223F)\n^\nSmith, Julius Orion.\n\"Sinusoids\"\n.\nccrma.stanford.edu\n. Retrieved\n2024-01-05\n.\n^\n\"1.2: Sine Waves\"\n.\nPhysics LibreTexts\n. 2021-02-21\n. Retrieved\n2026-03-16\n.\n\"Sine Wave\"\n.\nMathematical Mysteries\n. 2021-11-17\n. Retrieved\n2022-09-30\n.","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Sine+wave \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Sine+wave \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\nPersonal tools\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Sine+wave \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Sine+wave \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Sine_wave)\n- [1 Audio example](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Audio_example)\n- [2 Sinusoid form](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Sinusoid_form)\n- [3 As a function of both position and time](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#As_a_function_of_both_position_and_time)\n  Toggle As a function of both position and time subsection\n  - [3\\.1 Standing waves](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Standing_waves)\n  - [3\\.2 Multiple spatial dimensions](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Multiple_spatial_dimensions)\n    - [3\\.2.1 Sinusoidal plane wave](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Sinusoidal_plane_wave)\n- [4 Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Fourier_analysis)\n- [5 Differentiation and integration](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Differentiation_and_integration)\n  Toggle Differentiation and integration subsection\n  - [5\\.1 Differentiation](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Differentiation)\n  - [5\\.2 Integration](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#Integration)\n- [6 See also](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#See_also)\n- [7 References](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#References)\n- [8 External links](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#External_links)\n\nToggle the table of contents\n\n# Sine wave\n53 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D9%85%D9%88%D8%AC%D8%A9_%D8%AC%D9%8A%D8%A8%D9%8A%D8%A9 \"موجة جيبية – Arabic\")\n- [বাংলা](https:\/\/bn.wikipedia.org\/wiki\/%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%A4%E0%A6%B0%E0%A6%99%E0%A7%8D%E0%A6%97 \"সাইন তরঙ্গ – Bangla\")\n- [Bosanski](https:\/\/bs.wikipedia.org\/wiki\/Sinusoida \"Sinusoida – Bosnian\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Sinusoide \"Sinusoide – Catalan\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Sinusoida \"Sinusoida – Czech\")\n- [Чӑвашла](https:\/\/cv.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 \"Синусоида – Chuvash\")\n- [Dansk](https:\/\/da.wikipedia.org\/wiki\/Sinusb%C3%B8lge \"Sinusbølge – Danish\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Sinusoid \"Sinusoid – German\")\n- [Ελληνικά](https:\/\/el.wikipedia.org\/wiki\/%CE%97%CE%BC%CE%B9%CF%84%CE%BF%CE%BD%CE%BF%CE%B5%CE%B9%CE%B4%CE%AE%CF%82_%CE%BA%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B7 \"Ημιτονοειδής καμπύλη – Greek\")\n- [Esperanto](https:\/\/eo.wikipedia.org\/wiki\/Sinusoido \"Sinusoido – Esperanto\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Sinusoide \"Sinusoide – Spanish\")\n- [Eesti](https:\/\/et.wikipedia.org\/wiki\/Sinusoid \"Sinusoid – Estonian\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Sinusoide \"Sinusoide – Basque\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D9%85%D9%88%D8%AC_%D8%B3%DB%8C%D9%86%D9%88%D8%B3%DB%8C \"موج سینوسی – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Siniaalto \"Siniaalto – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Signal_sinuso%C3%AFdal \"Signal sinusoïdal – French\")\n- [हिन्दी](https:\/\/hi.wikipedia.org\/wiki\/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97 \"ज्या तरंग – Hindi\")\n- [Kreyòl ayisyen](https:\/\/ht.wikipedia.org\/wiki\/Ond_sinisoyidal \"Ond sinisoyidal – Haitian Creole\")\n- [Հայերեն](https:\/\/hy.wikipedia.org\/wiki\/%D5%8D%D5%AB%D5%B6%D5%B8%D6%82%D5%BD%D5%B8%D5%AB%D5%A4 \"Սինուսոիդ – Armenian\")\n- [Bahasa Indonesia](https:\/\/id.wikipedia.org\/wiki\/Gelombang_sinus \"Gelombang sinus – Indonesian\")\n- [Ido](https:\/\/io.wikipedia.org\/wiki\/Sinusoido \"Sinusoido – Ido\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Onda_sinusoidale \"Onda sinusoidale – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E6%AD%A3%E5%BC%A6%E6%B3%A2 \"正弦波 – Japanese\")\n- [ქართული](https:\/\/ka.wikipedia.org\/wiki\/%E1%83%A1%E1%83%98%E1%83%9C%E1%83%A3%E1%83%A1%E1%83%9D%E1%83%98%E1%83%93%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A2%E1%83%90%E1%83%9A%E1%83%A6%E1%83%90 \"სინუსოიდალური ტალღა – Georgian\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%82%AC%EC%9D%B8%ED%8C%8C \"사인파 – Korean\")\n- [Лезги](https:\/\/lez.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 \"Синусоида – Lezghian\")\n- [Lietuvių](https:\/\/lt.wikipedia.org\/wiki\/Sinusoid%C4%97 \"Sinusoidė – Lithuanian\")\n- [Мокшень](https:\/\/mdf.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0%D1%81%D1%8C \"Синусоидась – Moksha\")\n- [Македонски](https:\/\/mk.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%B5%D0%BD_%D0%B1%D1%80%D0%B0%D0%BD \"Синусен бран – Macedonian\")\n- [मराठी](https:\/\/mr.wikipedia.org\/wiki\/%E0%A4%B8%E0%A4%BE%E0%A4%87%E0%A4%A8_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97 \"साइन तरंग – Marathi\")\n- [Bahasa Melayu](https:\/\/ms.wikipedia.org\/wiki\/Gelombang_sinus \"Gelombang sinus – Malay\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Sinuso%C3%AFde \"Sinusoïde – Dutch\")\n- [Norsk nynorsk](https:\/\/nn.wikipedia.org\/wiki\/Sinuskurve \"Sinuskurve – Norwegian Nynorsk\")\n- [Norsk bokmål](https:\/\/no.wikipedia.org\/wiki\/Sinuskurve \"Sinuskurve – Norwegian Bokmål\")\n- [ਪੰਜਾਬੀ](https:\/\/pa.wikipedia.org\/wiki\/%E0%A8%B8%E0%A8%BE%E0%A8%88%E0%A8%A8_%E0%A8%B5%E0%A9%87%E0%A8%B5 \"ਸਾਈਨ ਵੇਵ – Punjabi\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Fala_sinusoidalna \"Fala sinusoidalna – Polish\")\n- [پنجابی](https:\/\/pnb.wikipedia.org\/wiki\/%D8%B3%D8%A7%D8%A6%DB%8C%D9%86_%D9%88%DB%8C%D9%88 \"سائین ویو – Western Punjabi\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Senoide \"Senoide – Portuguese\")\n- [Română](https:\/\/ro.wikipedia.org\/wiki\/Sinusoid%C4%83 \"Sinusoidă – Romanian\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 \"Синусоида – Russian\")\n- [Srpskohrvatski \/ српскохрватски](https:\/\/sh.wikipedia.org\/wiki\/Sinusoida \"Sinusoida – Serbo-Croatian\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Sine_wave \"Sine wave – Simple English\")\n- [Shqip](https:\/\/sq.wikipedia.org\/wiki\/Vala_sinusoidale \"Vala sinusoidale – Albanian\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 \"Синусоида – Serbian\")\n- [Sunda](https:\/\/su.wikipedia.org\/wiki\/Gelombang_sinus \"Gelombang sinus – Sundanese\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Sinusv%C3%A5g \"Sinusvåg – Swedish\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/Sin%C3%BCs_dalgas%C4%B1 \"Sinüs dalgası – Turkish\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D1%97%D0%B4%D0%B0 \"Синусоїда – Ukrainian\")\n- [اردو](https:\/\/ur.wikipedia.org\/wiki\/%D8%AC%DB%8C%D8%A8_%D9%85%D9%88%D8%AC \"جیب موج – Urdu\")\n- [Tiếng Việt](https:\/\/vi.wikipedia.org\/wiki\/S%C3%B3ng_sin \"Sóng sin – Vietnamese\")\n- [Wolof](https:\/\/wo.wikipedia.org\/wiki\/Duusub_sin \"Duusub sin – Wolof\")\n- [粵語](https:\/\/zh-yue.wikipedia.org\/wiki\/%E6%AD%A3%E5%BC%A6%E6%B3%A2 \"正弦波 – Cantonese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E6%AD%A3%E5%BC%A6%E6%9B%B2%E7%B7%9A \"正弦曲線 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q207527#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Sine_wave \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Sine_wave \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Sine_wave)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit \"Edit this page [e]\")\n- [View 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[Page information](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=info \"More information about this page\")\n- [Cite this page](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CiteThisPage&page=Sine_wave&id=1346599128&wpFormIdentifier=titleform \"Information on how to cite this page\")\n- [Get shortened URL](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSine_wave)\n\nPrint\/export\n\n- [Download as PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Sine_wave&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikimedia Commons](https:\/\/commons.wikimedia.org\/wiki\/Category:Sine_waves)\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q207527 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nWave shaped like the sine function\n\n\"Sinusoid\" redirects here; not to be confused with [Sinusoid (blood vessel)](https:\/\/en.wikipedia.org\/wiki\/Sinusoid_\\(blood_vessel\\) \"Sinusoid (blood vessel)\").\n\n|   |   |\n|---|---|\n| [![icon](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/99\/Question_book-new.svg\/60px-Question_book-new.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Question_book-new.svg) | This article **relies largely or entirely on a [single source](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Articles_with_a_single_source \"Wikipedia:Articles with a single source\")**. Relevant discussion may be found on the [talk page](https:\/\/en.wikipedia.org\/wiki\/Talk:Sine_wave \"Talk:Sine wave\"). Please help [improve this article](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit) by [introducing citations to additional sources](https:\/\/en.wikipedia.org\/wiki\/Help:Referencing_for_beginners \"Help:Referencing for beginners\"). *Find sources:* [\"Sine wave\"](https:\/\/www.google.com\/search?as_eq=wikipedia&q=%22Sine+wave%22) – [news](https:\/\/www.google.com\/search?tbm=nws&q=%22Sine+wave%22+-wikipedia&tbs=ar:1) **·** [newspapers](https:\/\/www.google.com\/search?&q=%22Sine+wave%22&tbs=bkt:s&tbm=bks) **·** [books](https:\/\/www.google.com\/search?tbs=bks:1&q=%22Sine+wave%22+-wikipedia) **·** [scholar](https:\/\/scholar.google.com\/scholar?q=%22Sine+wave%22) **·** [JSTOR](https:\/\/www.jstor.org\/action\/doBasicSearch?Query=%22Sine+wave%22&acc=on&wc=on) *(January 2024)* |\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/bd\/Sine_and_cosine_animation.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Sine_and_cosine_animation.gif)\n\nTracing the y component of a [circle](https:\/\/en.wikipedia.org\/wiki\/Circle \"Circle\") while going around the circle results in a sine wave (red). Tracing the x component results in a [cosine](https:\/\/en.wikipedia.org\/wiki\/Cosine \"Cosine\") wave (blue). Both waves are sinusoids of the same frequency but different phases.\n\nA **sine wave**, **sinusoidal wave**, or **sinusoid** (symbol: **∿**) is a [periodic wave](https:\/\/en.wikipedia.org\/wiki\/Periodic_function \"Periodic function\") whose [waveform](https:\/\/en.wikipedia.org\/wiki\/Waveform \"Waveform\") (shape) is the [trigonometric](https:\/\/en.wikipedia.org\/wiki\/Trigonometric_function \"Trigonometric function\") [sine function](https:\/\/en.wikipedia.org\/wiki\/Sine \"Sine\"). In [mechanics](https:\/\/en.wikipedia.org\/wiki\/Mechanics \"Mechanics\"), as a linear [motion](https:\/\/en.wikipedia.org\/wiki\/Motion \"Motion\") over time, this is *[simple harmonic motion](https:\/\/en.wikipedia.org\/wiki\/Simple_harmonic_motion \"Simple harmonic motion\")*; as [rotation](https:\/\/en.wikipedia.org\/wiki\/Rotation \"Rotation\"), it corresponds to *[uniform circular motion](https:\/\/en.wikipedia.org\/wiki\/Uniform_circular_motion \"Uniform circular motion\")*. Sine waves occur often in [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\"), including [wind waves](https:\/\/en.wikipedia.org\/wiki\/Wind_wave \"Wind wave\"), [sound](https:\/\/en.wikipedia.org\/wiki\/Sound \"Sound\") waves, and [light](https:\/\/en.wikipedia.org\/wiki\/Light \"Light\") waves, such as [monochromatic radiation](https:\/\/en.wikipedia.org\/wiki\/Monochromatic_radiation \"Monochromatic radiation\"). In [engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering \"Engineering\"), [signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\"), and [mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics \"Mathematics\"), [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\") decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.\n\nWhen any two sine waves of the same [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") (but arbitrary [phase](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\")) are [linearly combined](https:\/\/en.wikipedia.org\/wiki\/Linear_combination \"Linear combination\"), the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the *sine* and *cosine* [components](https:\/\/en.wikipedia.org\/wiki\/Vector_component \"Vector component\"), respectively.\n\n## Audio example\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=1 \"Edit section: Audio example\")\\]\n\n![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/87\/Gnome-mime-sound-openclipart.svg\/60px-Gnome-mime-sound-openclipart.svg.png)\n\n[Sine wave](https:\/\/en.wikipedia.org\/wiki\/File:220_Hz_sine_wave.ogg \"File:220 Hz sine wave.ogg\")\n\nFive seconds of a 220 Hz sine wave. This is the [sound wave](https:\/\/en.wikipedia.org\/wiki\/Sound#Waves \"Sound\") described by a sine function with *f* = 220 oscillations per second.\n\n***\n*Problems playing this file? See [media help](https:\/\/en.wikipedia.org\/wiki\/Help:Media \"Help:Media\").*\n\nA sine wave represents a single frequency with no [harmonics](https:\/\/en.wikipedia.org\/wiki\/Harmonic \"Harmonic\") and is considered an [acoustically](https:\/\/en.wikipedia.org\/wiki\/Acoustics \"Acoustics\") [pure tone](https:\/\/en.wikipedia.org\/wiki\/Pure_tone \"Pure tone\"). Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the [fundamental](https:\/\/en.wikipedia.org\/wiki\/Fundamental_frequency \"Fundamental frequency\") causes variation in the [timbre](https:\/\/en.wikipedia.org\/wiki\/Timbre \"Timbre\"), which is the reason why the same [musical pitch](https:\/\/en.wikipedia.org\/wiki\/Pitch_\\(music\\) \"Pitch (music)\") played on different instruments sounds different.\n\n## Sinusoid form\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=2 \"Edit section: Sinusoid form\")\\]\n\nSine waves of arbitrary phase and amplitude are called *sinusoids* and have the general form:[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_note-1) y ( t ) \\= A sin ⁡ ( ω t \\+ φ ) \\= A sin ⁡ ( 2 π f t \\+ φ ) {\\\\displaystyle y(t)=A\\\\sin(\\\\omega t+\\\\varphi )=A\\\\sin(2\\\\pi ft+\\\\varphi )} ![{\\\\displaystyle y(t)=A\\\\sin(\\\\omega t+\\\\varphi )=A\\\\sin(2\\\\pi ft+\\\\varphi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/11f7f3ec4cdc5ad694b390c5adfcdc0a858f1457)[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_note-2) where:\n\n- *A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3)*, *[amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\")*, the peak deviation of the function from zero.\n- t\n  \n  {\\\\displaystyle t}\n  \n  ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560)\n  , the [real](https:\/\/en.wikipedia.org\/wiki\/Real_number \"Real number\") [independent variable](https:\/\/en.wikipedia.org\/wiki\/Independent_variable \"Independent variable\"), usually representing [time](https:\/\/en.wikipedia.org\/wiki\/Time \"Time\") in [seconds](https:\/\/en.wikipedia.org\/wiki\/Seconds \"Seconds\").\n- ω\n  \n  {\\\\displaystyle \\\\omega }\n  \n  ![{\\\\displaystyle \\\\omega }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48eff443f9de7a985bb94ca3bde20813ea737be8)\n  , *[angular frequency](https:\/\/en.wikipedia.org\/wiki\/Angular_frequency \"Angular frequency\")*, the rate of change of the function argument in units of [radians per second](https:\/\/en.wikipedia.org\/wiki\/Radians_per_second \"Radians per second\").\n- *f {\\\\displaystyle f} ![{\\\\displaystyle f}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/132e57acb643253e7810ee9702d9581f159a1c61)*, *[ordinary frequency](https:\/\/en.wikipedia.org\/wiki\/Ordinary_frequency \"Ordinary frequency\")*, the *[number](https:\/\/en.wikipedia.org\/wiki\/Real_number \"Real number\")* of oscillations ([cycles](https:\/\/en.wikipedia.org\/wiki\/Turn_\\(angle\\) \"Turn (angle)\")) that occur each second of time.\n- φ\n  \n  {\\\\displaystyle \\\\varphi }\n  \n  ![{\\\\displaystyle \\\\varphi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)\n  , *[phase](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\")*, specifies (in [radians](https:\/\/en.wikipedia.org\/wiki\/Radian \"Radian\")) where in its cycle the oscillation is at *t* = 0.\n  - When\n    φ\n    \n    {\\\\displaystyle \\\\varphi }\n    \n    ![{\\\\displaystyle \\\\varphi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)\n    is non-zero, the entire waveform appears to be shifted backwards in time by the amount\n    φ\n    \n    ω\n    \n    {\\\\displaystyle {\\\\tfrac {\\\\varphi }{\\\\omega }}}\n    \n    ![{\\\\displaystyle {\\\\tfrac {\\\\varphi }{\\\\omega }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7c84c658594464cc4246e11afcfba081cdac5ac2)\n    seconds. A negative value represents a delay, and a positive value represents an advance.\n  - Adding or subtracting\n    2\n    \n    π\n    \n    {\\\\displaystyle 2\\\\pi }\n    \n    ![{\\\\displaystyle 2\\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/73efd1f6493490b058097060a572606d2c550a06)\n    (one cycle) to the phase results in an equivalent wave.\n\n## As a function of both position and time\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=3 \"Edit section: As a function of both position and time\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/2\/25\/Animated-mass-spring.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Animated-mass-spring.gif)\n\nThe displacement of an undamped [spring-mass system](https:\/\/en.wikipedia.org\/wiki\/Spring_mass_system \"Spring mass system\") oscillating around the equilibrium over time is a sine wave.\n\nSinusoids that exist in both position and time also have:\n\n- a spatial variable\n  x\n  \n  {\\\\displaystyle x}\n  \n  ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4)\n  that represents the *position* on the dimension on which the wave propagates.\n- a [wave number](https:\/\/en.wikipedia.org\/wiki\/Wave_number \"Wave number\") (or angular wave number)\n  k\n  \n  {\\\\displaystyle k}\n  \n  ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40)\n  , which represents the proportionality between the [angular frequency](https:\/\/en.wikipedia.org\/wiki\/Angular_frequency \"Angular frequency\")\n  ω\n  \n  {\\\\displaystyle \\\\omega }\n  \n  ![{\\\\displaystyle \\\\omega }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48eff443f9de7a985bb94ca3bde20813ea737be8)\n  and the linear speed ([speed of propagation](https:\/\/en.wikipedia.org\/wiki\/Phase_velocity \"Phase velocity\"))\n  v\n  \n  {\\\\displaystyle v}\n  \n  ![{\\\\displaystyle v}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e07b00e7fc0847fbd16391c778d65bc25c452597)\n  :\n  - wavenumber is related to the angular frequency by\n    k\n    \n    \\=\n    \n    ω\n    \n    v\n    \n    \\=\n    \n    2\n    \n    π\n    \n    f\n    \n    v\n    \n    \\=\n    \n    2\n    \n    π\n    \n    λ\n    \n    {\\\\textstyle k{=}{\\\\frac {\\\\omega }{v}}{=}{\\\\frac {2\\\\pi f}{v}}{=}{\\\\frac {2\\\\pi }{\\\\lambda }}}\n    \n    ![{\\\\textstyle k{=}{\\\\frac {\\\\omega }{v}}{=}{\\\\frac {2\\\\pi f}{v}}{=}{\\\\frac {2\\\\pi }{\\\\lambda }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8bb85eecafc3508df95d29528d5b7c620541d918)\n    where\n    λ\n    \n    {\\\\displaystyle \\\\lambda }\n    \n    ![{\\\\displaystyle \\\\lambda }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)\n    ([lambda](https:\/\/en.wikipedia.org\/wiki\/Lambda \"Lambda\")) is the [wavelength](https:\/\/en.wikipedia.org\/wiki\/Wavelength \"Wavelength\").\n\nDepending on their direction of travel, they can take the form:\n\n- y\n  \n  (\n  \n  x\n  \n  ,\n  \n  t\n  \n  )\n  \n  \\=\n  \n  A\n  \n  sin\n  \n  ⁡\n  \n  (\n  \n  k\n  \n  x\n  \n  −\n  \n  ω\n  \n  t\n  \n  \\+\n  \n  φ\n  \n  )\n  \n  {\\\\displaystyle y(x,t)=A\\\\sin(kx-\\\\omega t+\\\\varphi )}\n  \n  ![{\\\\displaystyle y(x,t)=A\\\\sin(kx-\\\\omega t+\\\\varphi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aab3502270349ff4b9c819170eb370c7d35a440a)\n  , if the wave is moving to the right, or\n- y\n  \n  (\n  \n  x\n  \n  ,\n  \n  t\n  \n  )\n  \n  \\=\n  \n  A\n  \n  sin\n  \n  ⁡\n  \n  (\n  \n  k\n  \n  x\n  \n  \\+\n  \n  ω\n  \n  t\n  \n  \\+\n  \n  φ\n  \n  )\n  \n  {\\\\displaystyle y(x,t)=A\\\\sin(kx+\\\\omega t+\\\\varphi )}\n  \n  ![{\\\\displaystyle y(x,t)=A\\\\sin(kx+\\\\omega t+\\\\varphi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/42b125c02254130171f1ae41b4ea0abe04843b93)\n  , if the wave is moving to the left.\n\nSince sine waves propagate without changing form in *distributed linear systems*,\\[*[definition needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Please_clarify \"Wikipedia:Please clarify\")*\\] they are often used to analyze [wave propagation](https:\/\/en.wikipedia.org\/wiki\/Wave_propagation \"Wave propagation\").\n\n### Standing waves\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=4 \"Edit section: Standing waves\")\\]\n\nMain article: [Standing wave](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\")\n\nWhen two waves with the same [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") and [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") traveling in opposite directions [superpose](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle \"Superposition principle\") each other, then a [standing wave](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\") pattern is created.\n\nOn a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's [resonant](https:\/\/en.wikipedia.org\/wiki\/Resonant \"Resonant\") frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the [fundamental frequency](https:\/\/en.wikipedia.org\/wiki\/Fundamental_frequency \"Fundamental frequency\")) and integer divisions of that (corresponding to higher harmonics).\n\n### Multiple spatial dimensions\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=5 \"Edit section: Multiple spatial dimensions\")\\]\n\nThe earlier equation gives the displacement y {\\\\displaystyle y} ![{\\\\displaystyle y}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b8a6208ec717213d4317e666f1ae872e00620a0d) of the wave at a position x {\\\\displaystyle x} ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4) at time t {\\\\displaystyle t} ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560) along a single line. This could, for example, be considered the value of a wave along a wire.\n\nIn two or three spatial dimensions, the same equation describes a travelling [plane wave](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") if position x {\\\\displaystyle x} ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4) and wavenumber k {\\\\displaystyle k} ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40) are interpreted as vectors, and their product as a [dot product](https:\/\/en.wikipedia.org\/wiki\/Dot_product \"Dot product\"). For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.\n\n#### Sinusoidal plane wave\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=6 \"Edit section: Sinusoidal plane wave\")\\]\n\nThis section is an excerpt from [Sinusoidal plane wave](https:\/\/en.wikipedia.org\/wiki\/Sinusoidal_plane_wave \"Sinusoidal plane wave\").\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sinusoidal_plane_wave&action=edit)\\]\n\nIn [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\"), a [sinusoidal plane wave](https:\/\/en.wikipedia.org\/wiki\/Sinusoidal_plane_wave \"Sinusoidal plane wave\") is a special case of [plane wave](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\"): a [field](https:\/\/en.wikipedia.org\/wiki\/Field_\\(physics\\) \"Field (physics)\") whose value varies as a [sinusoidal function](https:\/\/en.wikipedia.org\/wiki\/Sinusoidal_function \"Sinusoidal function\") of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constant [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") (as in [monochromatic radiation](https:\/\/en.wikipedia.org\/wiki\/Monochromatic_radiation \"Monochromatic radiation\")).\n\n## Fourier analysis\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=7 \"Edit section: Fourier analysis\")\\]\n\nMain articles: [Fourier series](https:\/\/en.wikipedia.org\/wiki\/Fourier_series \"Fourier series\"), [Fourier transform](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\"), and [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\")\n\nFrench mathematician [Joseph Fourier](https:\/\/en.wikipedia.org\/wiki\/Joseph_Fourier \"Joseph Fourier\") discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including [square waves](https:\/\/en.wikipedia.org\/wiki\/Square_wave_\\(waveform\\) \"Square wave (waveform)\"). These [Fourier series](https:\/\/en.wikipedia.org\/wiki\/Fourier_series \"Fourier series\") are frequently used in [signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\") and the statistical analysis of [time series](https:\/\/en.wikipedia.org\/wiki\/Time_series \"Time series\"). The [Fourier transform](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\") then extended Fourier series to handle general functions, and birthed the field of [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\").\n\n## Differentiation and integration\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=8 \"Edit section: Differentiation and integration\")\\]\n\nSee also: [Phasor § Differentiation and integration](https:\/\/en.wikipedia.org\/wiki\/Phasor#Differentiation_and_integration \"Phasor\")\n\n### Differentiation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=9 \"Edit section: Differentiation\")\\]\n\n[Differentiating](https:\/\/en.wikipedia.org\/wiki\/Derivative \"Derivative\") any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:\n\nd d t \\[ A sin ⁡ ( ω t \\+ φ ) \\] \\= A ω cos ⁡ ( ω t \\+ φ ) \\= A ω sin ⁡ ( ω t \\+ φ \\+ π 2 ) . {\\\\displaystyle {\\\\begin{aligned}{\\\\frac {d}{dt}}\\[A\\\\sin(\\\\omega t+\\\\varphi )\\]&=A\\\\omega \\\\cos(\\\\omega t+\\\\varphi )\\\\\\\\&=A\\\\omega \\\\sin(\\\\omega t+\\\\varphi +{\\\\tfrac {\\\\pi }{2}})\\\\,.\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}{\\\\frac {d}{dt}}\\[A\\\\sin(\\\\omega t+\\\\varphi )\\]&=A\\\\omega \\\\cos(\\\\omega t+\\\\varphi )\\\\\\\\&=A\\\\omega \\\\sin(\\\\omega t+\\\\varphi +{\\\\tfrac {\\\\pi }{2}})\\\\,.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ef1d774c1775716120ab0ded159150b63b8c4a2c)\n\nA [differentiator](https:\/\/en.wikipedia.org\/wiki\/Differentiator \"Differentiator\") has a [zero](https:\/\/en.wikipedia.org\/wiki\/Zeros_and_poles \"Zeros and poles\") at the origin of the [complex frequency](https:\/\/en.wikipedia.org\/wiki\/Complex_frequency \"Complex frequency\") plane. The [gain](https:\/\/en.wikipedia.org\/wiki\/Gain_\\(electronics\\) \"Gain (electronics)\") of its [frequency response](https:\/\/en.wikipedia.org\/wiki\/Frequency_response \"Frequency response\") increases at a rate of +20 [dB](https:\/\/en.wikipedia.org\/wiki\/Decibel \"Decibel\") per [decade](https:\/\/en.wikipedia.org\/wiki\/Decade_\\(log_scale\\) \"Decade (log scale)\") of frequency (for [root-power](https:\/\/en.wikipedia.org\/wiki\/Root-power \"Root-power\") quantities), the same positive slope as a 1st order [high-pass filter](https:\/\/en.wikipedia.org\/wiki\/High-pass_filter \"High-pass filter\")'s [stopband](https:\/\/en.wikipedia.org\/wiki\/Stopband \"Stopband\"), although a differentiator does not have a [cutoff frequency](https:\/\/en.wikipedia.org\/wiki\/Cutoff_frequency \"Cutoff frequency\") or a flat [passband](https:\/\/en.wikipedia.org\/wiki\/Passband \"Passband\"). A nth\\-order high-pass filter approximately applies the nth time derivative of [signals](https:\/\/en.wikipedia.org\/wiki\/Signals \"Signals\") whose frequency band is significantly lower than the filter's cutoff frequency.\n\n### Integration\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=10 \"Edit section: Integration\")\\]\n\n[Integrating](https:\/\/en.wikipedia.org\/wiki\/Integral \"Integral\") any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle:\n\n∫ A sin ⁡ ( ω t \\+ φ ) d t \\= − A ω cos ⁡ ( ω t \\+ φ ) \\+ C \\= − A ω sin ⁡ ( ω t \\+ φ \\+ π 2 ) \\+ C \\= A ω sin ⁡ ( ω t \\+ φ − π 2 ) \\+ C . {\\\\displaystyle {\\\\begin{aligned}\\\\int A\\\\sin(\\\\omega t+\\\\varphi )dt&=-{\\\\frac {A}{\\\\omega }}\\\\cos(\\\\omega t+\\\\varphi )+C\\\\\\\\&=-{\\\\frac {A}{\\\\omega }}\\\\sin(\\\\omega t+\\\\varphi +{\\\\tfrac {\\\\pi }{2}})+C\\\\\\\\&={\\\\frac {A}{\\\\omega }}\\\\sin(\\\\omega t+\\\\varphi -{\\\\tfrac {\\\\pi }{2}})+C\\\\,.\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}\\\\int A\\\\sin(\\\\omega t+\\\\varphi )dt&=-{\\\\frac {A}{\\\\omega }}\\\\cos(\\\\omega t+\\\\varphi )+C\\\\\\\\&=-{\\\\frac {A}{\\\\omega }}\\\\sin(\\\\omega t+\\\\varphi +{\\\\tfrac {\\\\pi }{2}})+C\\\\\\\\&={\\\\frac {A}{\\\\omega }}\\\\sin(\\\\omega t+\\\\varphi -{\\\\tfrac {\\\\pi }{2}})+C\\\\,.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5f27fb900539e12f857c9c575510c3f84c6b4d8e)\n\nThe [constant of integration](https:\/\/en.wikipedia.org\/wiki\/Constant_of_integration \"Constant of integration\") C {\\\\displaystyle C} ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) will be zero if the [bounds of integration](https:\/\/en.wikipedia.org\/wiki\/Bounds_of_integration \"Bounds of integration\") is an integer multiple of the sinusoid's period.\n\nAn [integrator](https:\/\/en.wikipedia.org\/wiki\/Integrator \"Integrator\") has a [pole](https:\/\/en.wikipedia.org\/wiki\/Zeros_and_poles \"Zeros and poles\") at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1st order [low-pass filter](https:\/\/en.wikipedia.org\/wiki\/Low-pass_filter \"Low-pass filter\")'s stopband, although an integrator does not have a cutoff frequency or a flat passband. A nth\\-order low-pass filter approximately performs the nth time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=11 \"Edit section: See also\")\\]\n\n- [Crest (physics)](https:\/\/en.wikipedia.org\/wiki\/Crest_\\(physics\\) \"Crest (physics)\")\n- [Complex exponential](https:\/\/en.wikipedia.org\/wiki\/Complex_exponential \"Complex exponential\")\n- [Damped sine wave](https:\/\/en.wikipedia.org\/wiki\/Damped_sine_wave \"Damped sine wave\")\n- [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\")\n- [Fourier transform](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\")\n- [Harmonic analysis](https:\/\/en.wikipedia.org\/wiki\/Harmonic_analysis \"Harmonic analysis\")\n- [Harmonic series (mathematics)](https:\/\/en.wikipedia.org\/wiki\/Harmonic_series_\\(mathematics\\) \"Harmonic series (mathematics)\")\n- [Harmonic series (music)](https:\/\/en.wikipedia.org\/wiki\/Harmonic_series_\\(music\\) \"Harmonic series (music)\")\n- [Helmholtz equation](https:\/\/en.wikipedia.org\/wiki\/Helmholtz_equation \"Helmholtz equation\")\n- [Instantaneous phase](https:\/\/en.wikipedia.org\/wiki\/Instantaneous_phase \"Instantaneous phase\")\n- [In-phase and quadrature components](https:\/\/en.wikipedia.org\/wiki\/In-phase_and_quadrature_components \"In-phase and quadrature components\")\n- [Least-squares spectral analysis](https:\/\/en.wikipedia.org\/wiki\/Least-squares_spectral_analysis \"Least-squares spectral analysis\")\n- [Oscilloscope](https:\/\/en.wikipedia.org\/wiki\/Oscilloscope \"Oscilloscope\")\n- [Phasor](https:\/\/en.wikipedia.org\/wiki\/Phasor \"Phasor\")\n- [Pure tone](https:\/\/en.wikipedia.org\/wiki\/Pure_tone \"Pure tone\")\n- [Simple harmonic motion](https:\/\/en.wikipedia.org\/wiki\/Simple_harmonic_motion \"Simple harmonic motion\")\n- [Sinusoidal model](https:\/\/en.wikipedia.org\/wiki\/Sinusoidal_model \"Sinusoidal model\")\n- [Wave (physics)](https:\/\/en.wikipedia.org\/wiki\/Wave_\\(physics\\) \"Wave (physics)\")\n- [Wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\")\n- [∿](https:\/\/en.wikipedia.org\/wiki\/Tilde#Electronics \"Tilde\") the sine wave symbol (U+223F)\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=12 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_ref-1)**\n   Smith, Julius Orion. [\"Sinusoids\"](https:\/\/ccrma.stanford.edu\/~jos\/st\/Sinusoids.html). *ccrma.stanford.edu*. Retrieved 2024-01-05.\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_ref-2)**\n   [\"1.2: Sine Waves\"](https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Radically_Modern_Introductory_Physics_Text_I_\\(Raymond\\)\/01%3A_Waves_in_One_Dimension\/1.02%3A_Sine_Waves). *Physics LibreTexts*. 2021-02-21. Retrieved 2026-03-16.\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=13 \"Edit section: External links\")\\]\n\n- [\"Sine Wave\"](https:\/\/mathematicalmysteries.org\/sine-wave\/). *Mathematical Mysteries*. 2021-11-17. Retrieved 2022-09-30.\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Waveforms \"Template:Waveforms\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Waveforms \"Template talk:Waveforms\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Waveforms \"Special:EditPage\/Template:Waveforms\")[Waveforms](https:\/\/en.wikipedia.org\/wiki\/Waveform \"Waveform\") |\n|---|\n| [Sine wave]() [Non-sinusoidal](https:\/\/en.wikipedia.org\/wiki\/Non-sinusoidal_waveform \"Non-sinusoidal waveform\") [Rectangular wave](https:\/\/en.wikipedia.org\/wiki\/Pulse_wave \"Pulse wave\") [Sawtooth wave](https:\/\/en.wikipedia.org\/wiki\/Sawtooth_wave \"Sawtooth wave\") [Square wave](https:\/\/en.wikipedia.org\/wiki\/Square_wave_\\(waveform\\) \"Square wave (waveform)\") [Triangle wave](https:\/\/en.wikipedia.org\/wiki\/Triangle_wave \"Triangle wave\") |\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&oldid=1346599128>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Trigonometry](https:\/\/en.wikipedia.org\/wiki\/Category:Trigonometry \"Category:Trigonometry\")\n- [Wave mechanics](https:\/\/en.wikipedia.org\/wiki\/Category:Wave_mechanics \"Category:Wave mechanics\")\n- [Waves](https:\/\/en.wikipedia.org\/wiki\/Category:Waves \"Category:Waves\")\n- [Waveforms](https:\/\/en.wikipedia.org\/wiki\/Category:Waveforms \"Category:Waveforms\")\n- [Sound](https:\/\/en.wikipedia.org\/wiki\/Category:Sound \"Category:Sound\")\n- [Acoustics](https:\/\/en.wikipedia.org\/wiki\/Category:Acoustics \"Category:Acoustics\")\n\nHidden categories:\n\n- [Articles with short description](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_short_description \"Category:Articles with short description\")\n- [Short description is different from Wikidata](https:\/\/en.wikipedia.org\/wiki\/Category:Short_description_is_different_from_Wikidata \"Category:Short description is different from Wikidata\")\n- [Articles needing additional references from January 2024](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_needing_additional_references_from_January_2024 \"Category:Articles needing additional references from January 2024\")\n- [All articles needing additional references](https:\/\/en.wikipedia.org\/wiki\/Category:All_articles_needing_additional_references \"Category:All articles needing additional references\")\n- [Articles with hAudio microformats](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_hAudio_microformats \"Category:Articles with hAudio microformats\")\n- [Wikipedia articles needing clarification from August 2019](https:\/\/en.wikipedia.org\/wiki\/Category:Wikipedia_articles_needing_clarification_from_August_2019 \"Category:Wikipedia articles needing clarification from August 2019\")\n- [Articles with excerpts](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_excerpts \"Category:Articles with excerpts\")\n\n- This page was last edited on 1 April 2026, at 19:37 (UTC).\n- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License \"Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License\"); additional terms may apply. By using this site, you agree to the [Terms of Use](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Terms_of_Use \"foundation:Special:MyLanguage\/Policy:Terms of Use\") and [Privacy Policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy \"foundation:Special:MyLanguage\/Policy:Privacy policy\"). Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nSine wave\n\n53 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Sine_wave)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/bd\/Sine_and_cosine_animation.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Sine_and_cosine_animation.gif)\n\nTracing the y component of a [circle](https:\/\/en.wikipedia.org\/wiki\/Circle \"Circle\") while going around the circle results in a sine wave (red). Tracing the x component results in a [cosine](https:\/\/en.wikipedia.org\/wiki\/Cosine \"Cosine\") wave (blue). Both waves are sinusoids of the same frequency but different phases.\n\nA **sine wave**, **sinusoidal wave**, or **sinusoid** (symbol: **∿**) is a [periodic wave](https:\/\/en.wikipedia.org\/wiki\/Periodic_function \"Periodic function\") whose [waveform](https:\/\/en.wikipedia.org\/wiki\/Waveform \"Waveform\") (shape) is the [trigonometric](https:\/\/en.wikipedia.org\/wiki\/Trigonometric_function \"Trigonometric function\") [sine function](https:\/\/en.wikipedia.org\/wiki\/Sine \"Sine\"). In [mechanics](https:\/\/en.wikipedia.org\/wiki\/Mechanics \"Mechanics\"), as a linear [motion](https:\/\/en.wikipedia.org\/wiki\/Motion \"Motion\") over time, this is *[simple harmonic motion](https:\/\/en.wikipedia.org\/wiki\/Simple_harmonic_motion \"Simple harmonic motion\")*; as [rotation](https:\/\/en.wikipedia.org\/wiki\/Rotation \"Rotation\"), it corresponds to *[uniform circular motion](https:\/\/en.wikipedia.org\/wiki\/Uniform_circular_motion \"Uniform circular motion\")*. Sine waves occur often in [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\"), including [wind waves](https:\/\/en.wikipedia.org\/wiki\/Wind_wave \"Wind wave\"), [sound](https:\/\/en.wikipedia.org\/wiki\/Sound \"Sound\") waves, and [light](https:\/\/en.wikipedia.org\/wiki\/Light \"Light\") waves, such as [monochromatic radiation](https:\/\/en.wikipedia.org\/wiki\/Monochromatic_radiation \"Monochromatic radiation\"). In [engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering \"Engineering\"), [signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\"), and [mathematics](https:\/\/en.wikipedia.org\/wiki\/Mathematics \"Mathematics\"), [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\") decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.\n\nWhen any two sine waves of the same [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") (but arbitrary [phase](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\")) are [linearly combined](https:\/\/en.wikipedia.org\/wiki\/Linear_combination \"Linear combination\"), the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the *sine* and *cosine* [components](https:\/\/en.wikipedia.org\/wiki\/Vector_component \"Vector component\"), respectively.\n\n![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/87\/Gnome-mime-sound-openclipart.svg\/60px-Gnome-mime-sound-openclipart.svg.png)\n\nFive seconds of a 220 Hz sine wave. This is the [sound wave](https:\/\/en.wikipedia.org\/wiki\/Sound#Waves \"Sound\") described by a sine function with *f* = 220 oscillations per second.\n\nA sine wave represents a single frequency with no [harmonics](https:\/\/en.wikipedia.org\/wiki\/Harmonic \"Harmonic\") and is considered an [acoustically](https:\/\/en.wikipedia.org\/wiki\/Acoustics \"Acoustics\") [pure tone](https:\/\/en.wikipedia.org\/wiki\/Pure_tone \"Pure tone\"). Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the [fundamental](https:\/\/en.wikipedia.org\/wiki\/Fundamental_frequency \"Fundamental frequency\") causes variation in the [timbre](https:\/\/en.wikipedia.org\/wiki\/Timbre \"Timbre\"), which is the reason why the same [musical pitch](https:\/\/en.wikipedia.org\/wiki\/Pitch_\\(music\\) \"Pitch (music)\") played on different instruments sounds different.\n\nSine waves of arbitrary phase and amplitude are called *sinusoids* and have the general form:[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_note-1) ![{\\\\displaystyle y(t)=A\\\\sin(\\\\omega t+\\\\varphi )=A\\\\sin(2\\\\pi ft+\\\\varphi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/11f7f3ec4cdc5ad694b390c5adfcdc0a858f1457)[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_note-2) where:\n\n- *![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3)*, *[amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\")*, the peak deviation of the function from zero.\n- ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560), the [real](https:\/\/en.wikipedia.org\/wiki\/Real_number \"Real number\") [independent variable](https:\/\/en.wikipedia.org\/wiki\/Independent_variable \"Independent variable\"), usually representing [time](https:\/\/en.wikipedia.org\/wiki\/Time \"Time\") in [seconds](https:\/\/en.wikipedia.org\/wiki\/Seconds \"Seconds\").\n- ![{\\\\displaystyle \\\\omega }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48eff443f9de7a985bb94ca3bde20813ea737be8), *[angular frequency](https:\/\/en.wikipedia.org\/wiki\/Angular_frequency \"Angular frequency\")*, the rate of change of the function argument in units of [radians per second](https:\/\/en.wikipedia.org\/wiki\/Radians_per_second \"Radians per second\").\n- *![{\\\\displaystyle f}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/132e57acb643253e7810ee9702d9581f159a1c61)*, *[ordinary frequency](https:\/\/en.wikipedia.org\/wiki\/Ordinary_frequency \"Ordinary frequency\")*, the *[number](https:\/\/en.wikipedia.org\/wiki\/Real_number \"Real number\")* of oscillations ([cycles](https:\/\/en.wikipedia.org\/wiki\/Turn_\\(angle\\) \"Turn (angle)\")) that occur each second of time.\n- ![{\\\\displaystyle \\\\varphi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e), *[phase](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\")*, specifies (in [radians](https:\/\/en.wikipedia.org\/wiki\/Radian \"Radian\")) where in its cycle the oscillation is at *t* = 0.\n\n## As a function of both position and time\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=3 \"Edit section: As a function of both position and time\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/2\/25\/Animated-mass-spring.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Animated-mass-spring.gif)\n\nThe displacement of an undamped [spring-mass system](https:\/\/en.wikipedia.org\/wiki\/Spring_mass_system \"Spring mass system\") oscillating around the equilibrium over time is a sine wave.\n\nSinusoids that exist in both position and time also have:\n\nDepending on their direction of travel, they can take the form:\n\n- ![{\\\\displaystyle y(x,t)=A\\\\sin(kx-\\\\omega t+\\\\varphi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aab3502270349ff4b9c819170eb370c7d35a440a), if the wave is moving to the right, or\n- ![{\\\\displaystyle y(x,t)=A\\\\sin(kx+\\\\omega t+\\\\varphi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/42b125c02254130171f1ae41b4ea0abe04843b93), if the wave is moving to the left.\n\nSince sine waves propagate without changing form in *distributed linear systems*,\\[*[definition needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Please_clarify \"Wikipedia:Please clarify\")*\\] they are often used to analyze [wave propagation](https:\/\/en.wikipedia.org\/wiki\/Wave_propagation \"Wave propagation\").\n\nWhen two waves with the same [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") and [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") traveling in opposite directions [superpose](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle \"Superposition principle\") each other, then a [standing wave](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\") pattern is created.\n\nOn a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's [resonant](https:\/\/en.wikipedia.org\/wiki\/Resonant \"Resonant\") frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the [fundamental frequency](https:\/\/en.wikipedia.org\/wiki\/Fundamental_frequency \"Fundamental frequency\")) and integer divisions of that (corresponding to higher harmonics).\n\n### Multiple spatial dimensions\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=5 \"Edit section: Multiple spatial dimensions\")\\]\n\nThe earlier equation gives the displacement ![{\\\\displaystyle y}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b8a6208ec717213d4317e666f1ae872e00620a0d) of the wave at a position ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4) at time ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560) along a single line. This could, for example, be considered the value of a wave along a wire.\n\nIn two or three spatial dimensions, the same equation describes a travelling [plane wave](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") if position ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4) and wavenumber ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40) are interpreted as vectors, and their product as a [dot product](https:\/\/en.wikipedia.org\/wiki\/Dot_product \"Dot product\"). For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.\n\n#### Sinusoidal plane wave\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=6 \"Edit section: Sinusoidal plane wave\")\\]\n\nFrench mathematician [Joseph Fourier](https:\/\/en.wikipedia.org\/wiki\/Joseph_Fourier \"Joseph Fourier\") discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including [square waves](https:\/\/en.wikipedia.org\/wiki\/Square_wave_\\(waveform\\) \"Square wave (waveform)\"). These [Fourier series](https:\/\/en.wikipedia.org\/wiki\/Fourier_series \"Fourier series\") are frequently used in [signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\") and the statistical analysis of [time series](https:\/\/en.wikipedia.org\/wiki\/Time_series \"Time series\"). The [Fourier transform](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\") then extended Fourier series to handle general functions, and birthed the field of [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\").\n\n## Differentiation and integration\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Sine_wave&action=edit&section=8 \"Edit section: Differentiation and integration\")\\]\n\n[Differentiating](https:\/\/en.wikipedia.org\/wiki\/Derivative \"Derivative\") any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:\n\n![{\\\\displaystyle {\\\\begin{aligned}{\\\\frac {d}{dt}}\\[A\\\\sin(\\\\omega t+\\\\varphi )\\]&=A\\\\omega \\\\cos(\\\\omega t+\\\\varphi )\\\\\\\\&=A\\\\omega \\\\sin(\\\\omega t+\\\\varphi +{\\\\tfrac {\\\\pi }{2}})\\\\,.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ef1d774c1775716120ab0ded159150b63b8c4a2c)\n\nA [differentiator](https:\/\/en.wikipedia.org\/wiki\/Differentiator \"Differentiator\") has a [zero](https:\/\/en.wikipedia.org\/wiki\/Zeros_and_poles \"Zeros and poles\") at the origin of the [complex frequency](https:\/\/en.wikipedia.org\/wiki\/Complex_frequency \"Complex frequency\") plane. The [gain](https:\/\/en.wikipedia.org\/wiki\/Gain_\\(electronics\\) \"Gain (electronics)\") of its [frequency response](https:\/\/en.wikipedia.org\/wiki\/Frequency_response \"Frequency response\") increases at a rate of +20 [dB](https:\/\/en.wikipedia.org\/wiki\/Decibel \"Decibel\") per [decade](https:\/\/en.wikipedia.org\/wiki\/Decade_\\(log_scale\\) \"Decade (log scale)\") of frequency (for [root-power](https:\/\/en.wikipedia.org\/wiki\/Root-power \"Root-power\") quantities), the same positive slope as a 1st order [high-pass filter](https:\/\/en.wikipedia.org\/wiki\/High-pass_filter \"High-pass filter\")'s [stopband](https:\/\/en.wikipedia.org\/wiki\/Stopband \"Stopband\"), although a differentiator does not have a [cutoff frequency](https:\/\/en.wikipedia.org\/wiki\/Cutoff_frequency \"Cutoff frequency\") or a flat [passband](https:\/\/en.wikipedia.org\/wiki\/Passband \"Passband\"). A nth\\-order high-pass filter approximately applies the nth time derivative of [signals](https:\/\/en.wikipedia.org\/wiki\/Signals \"Signals\") whose frequency band is significantly lower than the filter's cutoff frequency.\n\n[Integrating](https:\/\/en.wikipedia.org\/wiki\/Integral \"Integral\") any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle:\n\n![{\\\\displaystyle {\\\\begin{aligned}\\\\int A\\\\sin(\\\\omega t+\\\\varphi )dt&=-{\\\\frac {A}{\\\\omega }}\\\\cos(\\\\omega t+\\\\varphi )+C\\\\\\\\&=-{\\\\frac {A}{\\\\omega }}\\\\sin(\\\\omega t+\\\\varphi +{\\\\tfrac {\\\\pi }{2}})+C\\\\\\\\&={\\\\frac {A}{\\\\omega }}\\\\sin(\\\\omega t+\\\\varphi -{\\\\tfrac {\\\\pi }{2}})+C\\\\,.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5f27fb900539e12f857c9c575510c3f84c6b4d8e)\n\nThe [constant of integration](https:\/\/en.wikipedia.org\/wiki\/Constant_of_integration \"Constant of integration\") ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) will be zero if the [bounds of integration](https:\/\/en.wikipedia.org\/wiki\/Bounds_of_integration \"Bounds of integration\") is an integer multiple of the sinusoid's period.\n\nAn [integrator](https:\/\/en.wikipedia.org\/wiki\/Integrator \"Integrator\") has a [pole](https:\/\/en.wikipedia.org\/wiki\/Zeros_and_poles \"Zeros and poles\") at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1st order [low-pass filter](https:\/\/en.wikipedia.org\/wiki\/Low-pass_filter \"Low-pass filter\")'s stopband, although an integrator does not have a cutoff frequency or a flat passband. A nth\\-order low-pass filter approximately performs the nth time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.\n\n- [Crest (physics)](https:\/\/en.wikipedia.org\/wiki\/Crest_\\(physics\\) \"Crest (physics)\")\n- [Complex exponential](https:\/\/en.wikipedia.org\/wiki\/Complex_exponential \"Complex exponential\")\n- [Damped sine wave](https:\/\/en.wikipedia.org\/wiki\/Damped_sine_wave \"Damped sine wave\")\n- [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\")\n- [Fourier transform](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\")\n- [Harmonic analysis](https:\/\/en.wikipedia.org\/wiki\/Harmonic_analysis \"Harmonic analysis\")\n- [Harmonic series (mathematics)](https:\/\/en.wikipedia.org\/wiki\/Harmonic_series_\\(mathematics\\) \"Harmonic series (mathematics)\")\n- [Harmonic series (music)](https:\/\/en.wikipedia.org\/wiki\/Harmonic_series_\\(music\\) \"Harmonic series (music)\")\n- [Helmholtz equation](https:\/\/en.wikipedia.org\/wiki\/Helmholtz_equation \"Helmholtz equation\")\n- [Instantaneous phase](https:\/\/en.wikipedia.org\/wiki\/Instantaneous_phase \"Instantaneous phase\")\n- [In-phase and quadrature components](https:\/\/en.wikipedia.org\/wiki\/In-phase_and_quadrature_components \"In-phase and quadrature components\")\n- [Least-squares spectral analysis](https:\/\/en.wikipedia.org\/wiki\/Least-squares_spectral_analysis \"Least-squares spectral analysis\")\n- [Oscilloscope](https:\/\/en.wikipedia.org\/wiki\/Oscilloscope \"Oscilloscope\")\n- [Phasor](https:\/\/en.wikipedia.org\/wiki\/Phasor \"Phasor\")\n- [Pure tone](https:\/\/en.wikipedia.org\/wiki\/Pure_tone \"Pure tone\")\n- [Simple harmonic motion](https:\/\/en.wikipedia.org\/wiki\/Simple_harmonic_motion \"Simple harmonic motion\")\n- [Sinusoidal model](https:\/\/en.wikipedia.org\/wiki\/Sinusoidal_model \"Sinusoidal model\")\n- [Wave (physics)](https:\/\/en.wikipedia.org\/wiki\/Wave_\\(physics\\) \"Wave (physics)\")\n- [Wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\")\n- [∿](https:\/\/en.wikipedia.org\/wiki\/Tilde#Electronics \"Tilde\") the sine wave symbol (U+223F)\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_ref-1)**\n   Smith, Julius Orion. [\"Sinusoids\"](https:\/\/ccrma.stanford.edu\/~jos\/st\/Sinusoids.html). *ccrma.stanford.edu*. Retrieved 2024-01-05.\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Sine_wave#cite_ref-2)**\n   [\"1.2: Sine Waves\"](https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Radically_Modern_Introductory_Physics_Text_I_\\(Raymond\\)\/01%3A_Waves_in_One_Dimension\/1.02%3A_Sine_Waves). *Physics LibreTexts*. 2021-02-21. Retrieved 2026-03-16.\n\n- [\"Sine Wave\"](https:\/\/mathematicalmysteries.org\/sine-wave\/). *Mathematical Mysteries*. 2021-11-17. Retrieved 2022-09-30.","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia Tracing the y component of a circle while going around the circle results in a sine wave (red). Tracing the x component results in a cosine wave (blue). Both waves are sinusoids of the same frequency but different phases. A sine wave , sinusoidal wave , or sinusoid (symbol: ∿ ) is a periodic wave whose waveform (shape) is the trigonometric sine function . In mechanics , as a linear motion over time, this is simple harmonic motion ; as rotation , it corresponds to uniform circular motion . Sine waves occur often in physics , including wind waves , sound waves, and light waves, such as monochromatic radiation . In engineering , signal processing , and mathematics , Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same frequency (but arbitrary phase ) are linearly combined , the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine components , respectively. Five seconds of a 220 Hz sine wave. This is the sound wave described by a sine function with f = 220 oscillations per second. A sine wave represents a single frequency with no harmonics and is considered an acoustically pure tone . Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the fundamental causes variation in the timbre , which is the reason why the same musical pitch played on different instruments sounds different. Sine waves of arbitrary phase and amplitude are called sinusoids and have the general form: [ 1 ] [ 2 ] where: , amplitude , the peak deviation of the function from zero. , the real independent variable , usually representing time in seconds . , angular frequency , the rate of change of the function argument in units of radians per second . , ordinary frequency , the number of oscillations ( cycles ) that occur each second of time. , phase , specifies (in radians ) where in its cycle the oscillation is at t = 0. As a function of both position and time [ edit ] The displacement of an undamped spring-mass system oscillating around the equilibrium over time is a sine wave. Sinusoids that exist in both position and time also have: Depending on their direction of travel, they can take the form: , if the wave is moving to the right, or , if the wave is moving to the left. Since sine waves propagate without changing form in distributed linear systems , [ definition needed ] they are often used to analyze wave propagation . When two waves with the same amplitude and frequency traveling in opposite directions superpose each other, then a standing wave pattern is created. On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's resonant frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the fundamental frequency ) and integer divisions of that (corresponding to higher harmonics). Multiple spatial dimensions [ edit ] The earlier equation gives the displacement of the wave at a position at time along a single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling plane wave if position and wavenumber are interpreted as vectors, and their product as a dot product . For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. Sinusoidal plane wave [ edit ] French mathematician Joseph Fourier discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including square waves . These Fourier series are frequently used in signal processing and the statistical analysis of time series . The Fourier transform then extended Fourier series to handle general functions, and birthed the field of Fourier analysis . Differentiation and integration [ edit ] Differentiating any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: A differentiator has a zero at the origin of the complex frequency plane. The gain of its frequency response increases at a rate of +20 dB per decade of frequency (for root-power quantities), the same positive slope as a 1 st order high-pass filter 's stopband , although a differentiator does not have a cutoff frequency or a flat passband . A n th -order high-pass filter approximately applies the n th time derivative of signals whose frequency band is significantly lower than the filter's cutoff frequency. Integrating any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: The constant of integration will be zero if the bounds of integration is an integer multiple of the sinusoid's period. An integrator has a pole at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1 st order low-pass filter 's stopband, although an integrator does not have a cutoff frequency or a flat passband. A n th -order low-pass filter approximately performs the n th time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency. Crest (physics) Complex exponential Damped sine wave Euler's formula Fourier transform Harmonic analysis Harmonic series (mathematics) Harmonic series (music) Helmholtz equation Instantaneous phase In-phase and quadrature components Least-squares spectral analysis Oscilloscope Phasor Pure tone Simple harmonic motion Sinusoidal model Wave (physics) Wave equation ∿ the sine wave symbol (U+223F) ^ Smith, Julius Orion. "Sinusoids" . ccrma.stanford.edu . Retrieved 2024-01-05 . ^ "1.2: Sine Waves" . Physics LibreTexts . 2021-02-21 . Retrieved 2026-03-16 . "Sine Wave" . Mathematical Mysteries . 2021-11-17 . Retrieved 2022-09-30 .
Markdown	[Jump to content](https://en.wikipedia.org/wiki/Sine_wave#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages "A list of all special pages [q]") [![](https://en.wikipedia.org/static/images/icons/enwiki-25.svg) ![Wikipedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en-25.svg)](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Sine+wave "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Sine+wave "You're encouraged to log in; however, it's not mandatory. [o]") Personal tools - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Sine+wave "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Sine+wave "You're encouraged to log in; however, it's not mandatory. [o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Sine_wave) - [1 Audio example](https://en.wikipedia.org/wiki/Sine_wave#Audio_example) - [2 Sinusoid form](https://en.wikipedia.org/wiki/Sine_wave#Sinusoid_form) - [3 As a function of both position and time](https://en.wikipedia.org/wiki/Sine_wave#As_a_function_of_both_position_and_time) Toggle As a function of both position and time subsection - [3\.1 Standing waves](https://en.wikipedia.org/wiki/Sine_wave#Standing_waves) - [3\.2 Multiple spatial dimensions](https://en.wikipedia.org/wiki/Sine_wave#Multiple_spatial_dimensions) - [3\.2.1 Sinusoidal plane wave](https://en.wikipedia.org/wiki/Sine_wave#Sinusoidal_plane_wave) - [4 Fourier analysis](https://en.wikipedia.org/wiki/Sine_wave#Fourier_analysis) - [5 Differentiation and integration](https://en.wikipedia.org/wiki/Sine_wave#Differentiation_and_integration) Toggle Differentiation and integration subsection - [5\.1 Differentiation](https://en.wikipedia.org/wiki/Sine_wave#Differentiation) - [5\.2 Integration](https://en.wikipedia.org/wiki/Sine_wave#Integration) - [6 See also](https://en.wikipedia.org/wiki/Sine_wave#See_also) - [7 References](https://en.wikipedia.org/wiki/Sine_wave#References) - [8 External links](https://en.wikipedia.org/wiki/Sine_wave#External_links) Toggle the table of contents # Sine wave 53 languages - [العربية](https://ar.wikipedia.org/wiki/%D9%85%D9%88%D8%AC%D8%A9_%D8%AC%D9%8A%D8%A8%D9%8A%D8%A9 "موجة جيبية – Arabic") - [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%A4%E0%A6%B0%E0%A6%99%E0%A7%8D%E0%A6%97 "সাইন তরঙ্গ – Bangla") - [Bosanski](https://bs.wikipedia.org/wiki/Sinusoida "Sinusoida – Bosnian") - [Català](https://ca.wikipedia.org/wiki/Sinusoide "Sinusoide – Catalan") - [Čeština](https://cs.wikipedia.org/wiki/Sinusoida "Sinusoida – Czech") - [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 "Синусоида – Chuvash") - [Dansk](https://da.wikipedia.org/wiki/Sinusb%C3%B8lge "Sinusbølge – Danish") - [Deutsch](https://de.wikipedia.org/wiki/Sinusoid "Sinusoid – German") - [Ελληνικά](https://el.wikipedia.org/wiki/%CE%97%CE%BC%CE%B9%CF%84%CE%BF%CE%BD%CE%BF%CE%B5%CE%B9%CE%B4%CE%AE%CF%82_%CE%BA%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B7 "Ημιτονοειδής καμπύλη – Greek") - [Esperanto](https://eo.wikipedia.org/wiki/Sinusoido "Sinusoido – Esperanto") - [Español](https://es.wikipedia.org/wiki/Sinusoide "Sinusoide – Spanish") - [Eesti](https://et.wikipedia.org/wiki/Sinusoid "Sinusoid – Estonian") - [Euskara](https://eu.wikipedia.org/wiki/Sinusoide "Sinusoide – Basque") - [فارسی](https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AC_%D8%B3%DB%8C%D9%86%D9%88%D8%B3%DB%8C "موج سینوسی – Persian") - [Suomi](https://fi.wikipedia.org/wiki/Siniaalto "Siniaalto – Finnish") - [Français](https://fr.wikipedia.org/wiki/Signal_sinuso%C3%AFdal "Signal sinusoïdal – French") - [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97 "ज्या तरंग – Hindi") - [Kreyòl ayisyen](https://ht.wikipedia.org/wiki/Ond_sinisoyidal "Ond sinisoyidal – Haitian Creole") - [Հայերեն](https://hy.wikipedia.org/wiki/%D5%8D%D5%AB%D5%B6%D5%B8%D6%82%D5%BD%D5%B8%D5%AB%D5%A4 "Սինուսոիդ – Armenian") - [Bahasa Indonesia](https://id.wikipedia.org/wiki/Gelombang_sinus "Gelombang sinus – Indonesian") - [Ido](https://io.wikipedia.org/wiki/Sinusoido "Sinusoido – Ido") - [Italiano](https://it.wikipedia.org/wiki/Onda_sinusoidale "Onda sinusoidale – Italian") - [日本語](https://ja.wikipedia.org/wiki/%E6%AD%A3%E5%BC%A6%E6%B3%A2 "正弦波 – Japanese") - [ქართული](https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%9C%E1%83%A3%E1%83%A1%E1%83%9D%E1%83%98%E1%83%93%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A2%E1%83%90%E1%83%9A%E1%83%A6%E1%83%90 "სინუსოიდალური ტალღა – Georgian") - [한국어](https://ko.wikipedia.org/wiki/%EC%82%AC%EC%9D%B8%ED%8C%8C "사인파 – Korean") - [Лезги](https://lez.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 "Синусоида – Lezghian") - [Lietuvių](https://lt.wikipedia.org/wiki/Sinusoid%C4%97 "Sinusoidė – Lithuanian") - [Мокшень](https://mdf.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0%D1%81%D1%8C "Синусоидась – Moksha") - [Македонски](https://mk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%B5%D0%BD_%D0%B1%D1%80%D0%B0%D0%BD "Синусен бран – Macedonian") - [मराठी](https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%87%E0%A4%A8_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97 "साइन तरंग – Marathi") - [Bahasa Melayu](https://ms.wikipedia.org/wiki/Gelombang_sinus "Gelombang sinus – Malay") - [Nederlands](https://nl.wikipedia.org/wiki/Sinuso%C3%AFde "Sinusoïde – Dutch") - [Norsk nynorsk](https://nn.wikipedia.org/wiki/Sinuskurve "Sinuskurve – Norwegian Nynorsk") - [Norsk bokmål](https://no.wikipedia.org/wiki/Sinuskurve "Sinuskurve – Norwegian Bokmål") - [ਪੰਜਾਬੀ](https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%BE%E0%A8%88%E0%A8%A8_%E0%A8%B5%E0%A9%87%E0%A8%B5 "ਸਾਈਨ ਵੇਵ – Punjabi") - [Polski](https://pl.wikipedia.org/wiki/Fala_sinusoidalna "Fala sinusoidalna – Polish") - [پنجابی](https://pnb.wikipedia.org/wiki/%D8%B3%D8%A7%D8%A6%DB%8C%D9%86_%D9%88%DB%8C%D9%88 "سائین ویو – Western Punjabi") - [Português](https://pt.wikipedia.org/wiki/Senoide "Senoide – Portuguese") - [Română](https://ro.wikipedia.org/wiki/Sinusoid%C4%83 "Sinusoidă – Romanian") - [Русский](https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 "Синусоида – Russian") - [Srpskohrvatski / српскохрватски](https://sh.wikipedia.org/wiki/Sinusoida "Sinusoida – Serbo-Croatian") - [Simple English](https://simple.wikipedia.org/wiki/Sine_wave "Sine wave – Simple English") - [Shqip](https://sq.wikipedia.org/wiki/Vala_sinusoidale "Vala sinusoidale – Albanian") - [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D0%B8%D0%B4%D0%B0 "Синусоида – Serbian") - [Sunda](https://su.wikipedia.org/wiki/Gelombang_sinus "Gelombang sinus – Sundanese") - [Svenska](https://sv.wikipedia.org/wiki/Sinusv%C3%A5g "Sinusvåg – Swedish") - [Türkçe](https://tr.wikipedia.org/wiki/Sin%C3%BCs_dalgas%C4%B1 "Sinüs dalgası – Turkish") - [Українська](https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81%D0%BE%D1%97%D0%B4%D0%B0 "Синусоїда – Ukrainian") - [اردو](https://ur.wikipedia.org/wiki/%D8%AC%DB%8C%D8%A8_%D9%85%D9%88%D8%AC "جیب موج – Urdu") - [Tiếng Việt](https://vi.wikipedia.org/wiki/S%C3%B3ng_sin "Sóng sin – Vietnamese") - [Wolof](https://wo.wikipedia.org/wiki/Duusub_sin "Duusub sin – Wolof") - [粵語](https://zh-yue.wikipedia.org/wiki/%E6%AD%A3%E5%BC%A6%E6%B3%A2 "正弦波 – Cantonese") - [中文](https://zh.wikipedia.org/wiki/%E6%AD%A3%E5%BC%A6%E6%9B%B2%E7%B7%9A "正弦曲線 – Chinese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q207527#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/Sine_wave "View the content page [c]") - [Talk](https://en.wikipedia.org/wiki/Talk:Sine_wave "Discuss improvements to the content page [t]") English - [Read](https://en.wikipedia.org/wiki/Sine_wave) - [Edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=history "Past revisions of this page [h]") Tools Tools move to sidebar hide Actions - [Read](https://en.wikipedia.org/wiki/Sine_wave) - 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[Get shortened URL](https://en.wikipedia.org/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSine_wave) Print/export - [Download as PDF](https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=Sine_wave&action=show-download-screen "Download this page as a PDF file") - [Printable version](https://en.wikipedia.org/w/index.php?title=Sine_wave&printable=yes "Printable version of this page [p]") In other projects - [Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Sine_waves) - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q207527 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Wave shaped like the sine function "Sinusoid" redirects here; not to be confused with [Sinusoid (blood vessel)](https://en.wikipedia.org/wiki/Sinusoid_\(blood_vessel\) "Sinusoid (blood vessel)"). \| \| \| \|---\|---\| \| [![icon](https://upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/60px-Question_book-new.svg.png)](https://en.wikipedia.org/wiki/File:Question_book-new.svg) \| This article relies largely or entirely on a [single source](https://en.wikipedia.org/wiki/Wikipedia:Articles_with_a_single_source "Wikipedia:Articles with a single source"). Relevant discussion may be found on the [talk page](https://en.wikipedia.org/wiki/Talk:Sine_wave "Talk:Sine wave"). Please help [improve this article](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit) by [introducing citations to additional sources](https://en.wikipedia.org/wiki/Help:Referencing_for_beginners "Help:Referencing for beginners"). Find sources: ["Sine wave"](https://www.google.com/search?as_eq=wikipedia&q=%22Sine+wave%22) – [news](https://www.google.com/search?tbm=nws&q=%22Sine+wave%22+-wikipedia&tbs=ar:1) · [newspapers](https://www.google.com/search?&q=%22Sine+wave%22&tbs=bkt:s&tbm=bks) · [books](https://www.google.com/search?tbs=bks:1&q=%22Sine+wave%22+-wikipedia) · [scholar](https://scholar.google.com/scholar?q=%22Sine+wave%22) · [JSTOR](https://www.jstor.org/action/doBasicSearch?Query=%22Sine+wave%22&acc=on&wc=on) (January 2024) \| [![](https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and_cosine_animation.gif)](https://en.wikipedia.org/wiki/File:Sine_and_cosine_animation.gif) Tracing the y component of a [circle](https://en.wikipedia.org/wiki/Circle "Circle") while going around the circle results in a sine wave (red). Tracing the x component results in a [cosine](https://en.wikipedia.org/wiki/Cosine "Cosine") wave (blue). Both waves are sinusoids of the same frequency but different phases. A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a [periodic wave](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") whose [waveform](https://en.wikipedia.org/wiki/Waveform "Waveform") (shape) is the [trigonometric](https://en.wikipedia.org/wiki/Trigonometric_function "Trigonometric function") [sine function](https://en.wikipedia.org/wiki/Sine "Sine"). In [mechanics](https://en.wikipedia.org/wiki/Mechanics "Mechanics"), as a linear [motion](https://en.wikipedia.org/wiki/Motion "Motion") over time, this is [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion "Simple harmonic motion"); as [rotation](https://en.wikipedia.org/wiki/Rotation "Rotation"), it corresponds to [uniform circular motion](https://en.wikipedia.org/wiki/Uniform_circular_motion "Uniform circular motion"). Sine waves occur often in [physics](https://en.wikipedia.org/wiki/Physics "Physics"), including [wind waves](https://en.wikipedia.org/wiki/Wind_wave "Wind wave"), [sound](https://en.wikipedia.org/wiki/Sound "Sound") waves, and [light](https://en.wikipedia.org/wiki/Light "Light") waves, such as [monochromatic radiation](https://en.wikipedia.org/wiki/Monochromatic_radiation "Monochromatic radiation"). In [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"), [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing"), and [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis") decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") (but arbitrary [phase](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)")) are [linearly combined](https://en.wikipedia.org/wiki/Linear_combination "Linear combination"), the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine [components](https://en.wikipedia.org/wiki/Vector_component "Vector component"), respectively. ## Audio example \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=1 "Edit section: Audio example")\] ![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/60px-Gnome-mime-sound-openclipart.svg.png) [Sine wave](https://en.wikipedia.org/wiki/File:220_Hz_sine_wave.ogg "File:220 Hz sine wave.ogg") Five seconds of a 220 Hz sine wave. This is the [sound wave](https://en.wikipedia.org/wiki/Sound#Waves "Sound") described by a sine function with f = 220 oscillations per second. *** Problems playing this file? See [media help](https://en.wikipedia.org/wiki/Help:Media "Help:Media"). A sine wave represents a single frequency with no [harmonics](https://en.wikipedia.org/wiki/Harmonic "Harmonic") and is considered an [acoustically](https://en.wikipedia.org/wiki/Acoustics "Acoustics") [pure tone](https://en.wikipedia.org/wiki/Pure_tone "Pure tone"). Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the [fundamental](https://en.wikipedia.org/wiki/Fundamental_frequency "Fundamental frequency") causes variation in the [timbre](https://en.wikipedia.org/wiki/Timbre "Timbre"), which is the reason why the same [musical pitch](https://en.wikipedia.org/wiki/Pitch_\(music\) "Pitch (music)") played on different instruments sounds different. ## Sinusoid form \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=2 "Edit section: Sinusoid form")\] Sine waves of arbitrary phase and amplitude are called sinusoids and have the general form:[\[1\]](https://en.wikipedia.org/wiki/Sine_wave#cite_note-1) y ( t ) \= A sin ⁡ ( ω t \+ φ ) \= A sin ⁡ ( 2 π f t \+ φ ) {\\displaystyle y(t)=A\\sin(\\omega t+\\varphi )=A\\sin(2\\pi ft+\\varphi )} ![{\\displaystyle y(t)=A\\sin(\\omega t+\\varphi )=A\\sin(2\\pi ft+\\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f7f3ec4cdc5ad694b390c5adfcdc0a858f1457)[\[2\]](https://en.wikipedia.org/wiki/Sine_wave#cite_note-2) where: - A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude"), the peak deviation of the function from zero. - t {\\displaystyle t} ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) , the [real](https://en.wikipedia.org/wiki/Real_number "Real number") [independent variable](https://en.wikipedia.org/wiki/Independent_variable "Independent variable"), usually representing [time](https://en.wikipedia.org/wiki/Time "Time") in [seconds](https://en.wikipedia.org/wiki/Seconds "Seconds"). - ω {\\displaystyle \\omega } ![{\\displaystyle \\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8) , [angular frequency](https://en.wikipedia.org/wiki/Angular_frequency "Angular frequency"), the rate of change of the function argument in units of [radians per second](https://en.wikipedia.org/wiki/Radians_per_second "Radians per second"). - f {\\displaystyle f} ![{\\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61), [ordinary frequency](https://en.wikipedia.org/wiki/Ordinary_frequency "Ordinary frequency"), the [number](https://en.wikipedia.org/wiki/Real_number "Real number") of oscillations ([cycles](https://en.wikipedia.org/wiki/Turn_\(angle\) "Turn (angle)")) that occur each second of time. - φ {\\displaystyle \\varphi } ![{\\displaystyle \\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e) , [phase](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)"), specifies (in [radians](https://en.wikipedia.org/wiki/Radian "Radian")) where in its cycle the oscillation is at t = 0. - When φ {\\displaystyle \\varphi } ![{\\displaystyle \\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e) is non-zero, the entire waveform appears to be shifted backwards in time by the amount φ ω {\\displaystyle {\\tfrac {\\varphi }{\\omega }}} ![{\\displaystyle {\\tfrac {\\varphi }{\\omega }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c84c658594464cc4246e11afcfba081cdac5ac2) seconds. A negative value represents a delay, and a positive value represents an advance. - Adding or subtracting 2 π {\\displaystyle 2\\pi } ![{\\displaystyle 2\\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06) (one cycle) to the phase results in an equivalent wave. ## As a function of both position and time \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=3 "Edit section: As a function of both position and time")\] [![](https://upload.wikimedia.org/wikipedia/commons/2/25/Animated-mass-spring.gif)](https://en.wikipedia.org/wiki/File:Animated-mass-spring.gif) The displacement of an undamped [spring-mass system](https://en.wikipedia.org/wiki/Spring_mass_system "Spring mass system") oscillating around the equilibrium over time is a sine wave. Sinusoids that exist in both position and time also have: - a spatial variable x {\\displaystyle x} ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) that represents the position on the dimension on which the wave propagates. - a [wave number](https://en.wikipedia.org/wiki/Wave_number "Wave number") (or angular wave number) k {\\displaystyle k} ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) , which represents the proportionality between the [angular frequency](https://en.wikipedia.org/wiki/Angular_frequency "Angular frequency") ω {\\displaystyle \\omega } ![{\\displaystyle \\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8) and the linear speed ([speed of propagation](https://en.wikipedia.org/wiki/Phase_velocity "Phase velocity")) v {\\displaystyle v} ![{\\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597) : - wavenumber is related to the angular frequency by k \= ω v \= 2 π f v \= 2 π λ {\\textstyle k{=}{\\frac {\\omega }{v}}{=}{\\frac {2\\pi f}{v}}{=}{\\frac {2\\pi }{\\lambda }}} ![{\\textstyle k{=}{\\frac {\\omega }{v}}{=}{\\frac {2\\pi f}{v}}{=}{\\frac {2\\pi }{\\lambda }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb85eecafc3508df95d29528d5b7c620541d918) where λ {\\displaystyle \\lambda } ![{\\displaystyle \\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) ([lambda](https://en.wikipedia.org/wiki/Lambda "Lambda")) is the [wavelength](https://en.wikipedia.org/wiki/Wavelength "Wavelength"). Depending on their direction of travel, they can take the form: - y ( x , t ) \= A sin ⁡ ( k x − ω t \+ φ ) {\\displaystyle y(x,t)=A\\sin(kx-\\omega t+\\varphi )} ![{\\displaystyle y(x,t)=A\\sin(kx-\\omega t+\\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab3502270349ff4b9c819170eb370c7d35a440a) , if the wave is moving to the right, or - y ( x , t ) \= A sin ⁡ ( k x \+ ω t \+ φ ) {\\displaystyle y(x,t)=A\\sin(kx+\\omega t+\\varphi )} ![{\\displaystyle y(x,t)=A\\sin(kx+\\omega t+\\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42b125c02254130171f1ae41b4ea0abe04843b93) , if the wave is moving to the left. Since sine waves propagate without changing form in distributed linear systems,\[[definition needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")\] they are often used to analyze [wave propagation](https://en.wikipedia.org/wiki/Wave_propagation "Wave propagation"). ### Standing waves \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=4 "Edit section: Standing waves")\] Main article: [Standing wave](https://en.wikipedia.org/wiki/Standing_wave "Standing wave") When two waves with the same [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") and [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") traveling in opposite directions [superpose](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle") each other, then a [standing wave](https://en.wikipedia.org/wiki/Standing_wave "Standing wave") pattern is created. On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's [resonant](https://en.wikipedia.org/wiki/Resonant "Resonant") frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the [fundamental frequency](https://en.wikipedia.org/wiki/Fundamental_frequency "Fundamental frequency")) and integer divisions of that (corresponding to higher harmonics). ### Multiple spatial dimensions \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=5 "Edit section: Multiple spatial dimensions")\] The earlier equation gives the displacement y {\\displaystyle y} ![{\\displaystyle y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d) of the wave at a position x {\\displaystyle x} ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) at time t {\\displaystyle t} ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) along a single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling [plane wave](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") if position x {\\displaystyle x} ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) and wavenumber k {\\displaystyle k} ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) are interpreted as vectors, and their product as a [dot product](https://en.wikipedia.org/wiki/Dot_product "Dot product"). For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. #### Sinusoidal plane wave \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=6 "Edit section: Sinusoidal plane wave")\] This section is an excerpt from [Sinusoidal plane wave](https://en.wikipedia.org/wiki/Sinusoidal_plane_wave "Sinusoidal plane wave").\[[edit](https://en.wikipedia.org/w/index.php?title=Sinusoidal_plane_wave&action=edit)\] In [physics](https://en.wikipedia.org/wiki/Physics "Physics"), a [sinusoidal plane wave](https://en.wikipedia.org/wiki/Sinusoidal_plane_wave "Sinusoidal plane wave") is a special case of [plane wave](https://en.wikipedia.org/wiki/Plane_wave "Plane wave"): a [field](https://en.wikipedia.org/wiki/Field_\(physics\) "Field (physics)") whose value varies as a [sinusoidal function](https://en.wikipedia.org/wiki/Sinusoidal_function "Sinusoidal function") of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constant [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") (as in [monochromatic radiation](https://en.wikipedia.org/wiki/Monochromatic_radiation "Monochromatic radiation")). ## Fourier analysis \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=7 "Edit section: Fourier analysis")\] Main articles: [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series"), [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform"), and [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis") French mathematician [Joseph Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier") discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including [square waves](https://en.wikipedia.org/wiki/Square_wave_\(waveform\) "Square wave (waveform)"). These [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") are frequently used in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") and the statistical analysis of [time series](https://en.wikipedia.org/wiki/Time_series "Time series"). The [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") then extended Fourier series to handle general functions, and birthed the field of [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"). ## Differentiation and integration \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=8 "Edit section: Differentiation and integration")\] See also: [Phasor § Differentiation and integration](https://en.wikipedia.org/wiki/Phasor#Differentiation_and_integration "Phasor") ### Differentiation \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=9 "Edit section: Differentiation")\] [Differentiating](https://en.wikipedia.org/wiki/Derivative "Derivative") any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: d d t \[ A sin ⁡ ( ω t \+ φ ) \] \= A ω cos ⁡ ( ω t \+ φ ) \= A ω sin ⁡ ( ω t \+ φ \+ π 2 ) . {\\displaystyle {\\begin{aligned}{\\frac {d}{dt}}\[A\\sin(\\omega t+\\varphi )\]&=A\\omega \\cos(\\omega t+\\varphi )\\\\&=A\\omega \\sin(\\omega t+\\varphi +{\\tfrac {\\pi }{2}})\\,.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}{\\frac {d}{dt}}\[A\\sin(\\omega t+\\varphi )\]&=A\\omega \\cos(\\omega t+\\varphi )\\\\&=A\\omega \\sin(\\omega t+\\varphi +{\\tfrac {\\pi }{2}})\\,.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1d774c1775716120ab0ded159150b63b8c4a2c) A [differentiator](https://en.wikipedia.org/wiki/Differentiator "Differentiator") has a [zero](https://en.wikipedia.org/wiki/Zeros_and_poles "Zeros and poles") at the origin of the [complex frequency](https://en.wikipedia.org/wiki/Complex_frequency "Complex frequency") plane. The [gain](https://en.wikipedia.org/wiki/Gain_\(electronics\) "Gain (electronics)") of its [frequency response](https://en.wikipedia.org/wiki/Frequency_response "Frequency response") increases at a rate of +20 [dB](https://en.wikipedia.org/wiki/Decibel "Decibel") per [decade](https://en.wikipedia.org/wiki/Decade_\(log_scale\) "Decade (log scale)") of frequency (for [root-power](https://en.wikipedia.org/wiki/Root-power "Root-power") quantities), the same positive slope as a 1st order [high-pass filter](https://en.wikipedia.org/wiki/High-pass_filter "High-pass filter")'s [stopband](https://en.wikipedia.org/wiki/Stopband "Stopband"), although a differentiator does not have a [cutoff frequency](https://en.wikipedia.org/wiki/Cutoff_frequency "Cutoff frequency") or a flat [passband](https://en.wikipedia.org/wiki/Passband "Passband"). A nth\-order high-pass filter approximately applies the nth time derivative of [signals](https://en.wikipedia.org/wiki/Signals "Signals") whose frequency band is significantly lower than the filter's cutoff frequency. ### Integration \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=10 "Edit section: Integration")\] [Integrating](https://en.wikipedia.org/wiki/Integral "Integral") any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: ∫ A sin ⁡ ( ω t \+ φ ) d t \= − A ω cos ⁡ ( ω t \+ φ ) \+ C \= − A ω sin ⁡ ( ω t \+ φ \+ π 2 ) \+ C \= A ω sin ⁡ ( ω t \+ φ − π 2 ) \+ C . {\\displaystyle {\\begin{aligned}\\int A\\sin(\\omega t+\\varphi )dt&=-{\\frac {A}{\\omega }}\\cos(\\omega t+\\varphi )+C\\\\&=-{\\frac {A}{\\omega }}\\sin(\\omega t+\\varphi +{\\tfrac {\\pi }{2}})+C\\\\&={\\frac {A}{\\omega }}\\sin(\\omega t+\\varphi -{\\tfrac {\\pi }{2}})+C\\,.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\int A\\sin(\\omega t+\\varphi )dt&=-{\\frac {A}{\\omega }}\\cos(\\omega t+\\varphi )+C\\\\&=-{\\frac {A}{\\omega }}\\sin(\\omega t+\\varphi +{\\tfrac {\\pi }{2}})+C\\\\&={\\frac {A}{\\omega }}\\sin(\\omega t+\\varphi -{\\tfrac {\\pi }{2}})+C\\,.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f27fb900539e12f857c9c575510c3f84c6b4d8e) The [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration") C {\\displaystyle C} ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) will be zero if the [bounds of integration](https://en.wikipedia.org/wiki/Bounds_of_integration "Bounds of integration") is an integer multiple of the sinusoid's period. An [integrator](https://en.wikipedia.org/wiki/Integrator "Integrator") has a [pole](https://en.wikipedia.org/wiki/Zeros_and_poles "Zeros and poles") at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1st order [low-pass filter](https://en.wikipedia.org/wiki/Low-pass_filter "Low-pass filter")'s stopband, although an integrator does not have a cutoff frequency or a flat passband. A nth\-order low-pass filter approximately performs the nth time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency. ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=11 "Edit section: See also")\] - [Crest (physics)](https://en.wikipedia.org/wiki/Crest_\(physics\) "Crest (physics)") - [Complex exponential](https://en.wikipedia.org/wiki/Complex_exponential "Complex exponential") - [Damped sine wave](https://en.wikipedia.org/wiki/Damped_sine_wave "Damped sine wave") - [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") - [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") - [Harmonic analysis](https://en.wikipedia.org/wiki/Harmonic_analysis "Harmonic analysis") - [Harmonic series (mathematics)](https://en.wikipedia.org/wiki/Harmonic_series_\(mathematics\) "Harmonic series (mathematics)") - [Harmonic series (music)](https://en.wikipedia.org/wiki/Harmonic_series_\(music\) "Harmonic series (music)") - [Helmholtz equation](https://en.wikipedia.org/wiki/Helmholtz_equation "Helmholtz equation") - [Instantaneous phase](https://en.wikipedia.org/wiki/Instantaneous_phase "Instantaneous phase") - [In-phase and quadrature components](https://en.wikipedia.org/wiki/In-phase_and_quadrature_components "In-phase and quadrature components") - [Least-squares spectral analysis](https://en.wikipedia.org/wiki/Least-squares_spectral_analysis "Least-squares spectral analysis") - [Oscilloscope](https://en.wikipedia.org/wiki/Oscilloscope "Oscilloscope") - [Phasor](https://en.wikipedia.org/wiki/Phasor "Phasor") - [Pure tone](https://en.wikipedia.org/wiki/Pure_tone "Pure tone") - [Simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion "Simple harmonic motion") - [Sinusoidal model](https://en.wikipedia.org/wiki/Sinusoidal_model "Sinusoidal model") - [Wave (physics)](https://en.wikipedia.org/wiki/Wave_\(physics\) "Wave (physics)") - [Wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") - [∿](https://en.wikipedia.org/wiki/Tilde#Electronics "Tilde") the sine wave symbol (U+223F) ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=12 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Sine_wave#cite_ref-1) Smith, Julius Orion. ["Sinusoids"](https://ccrma.stanford.edu/~jos/st/Sinusoids.html). ccrma.stanford.edu. Retrieved 2024-01-05. 2. [^](https://en.wikipedia.org/wiki/Sine_wave#cite_ref-2) ["1.2: Sine Waves"](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_\(Raymond\)/01%3A_Waves_in_One_Dimension/1.02%3A_Sine_Waves). Physics LibreTexts. 2021-02-21. Retrieved 2026-03-16. ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=13 "Edit section: External links")\] - ["Sine Wave"](https://mathematicalmysteries.org/sine-wave/). Mathematical Mysteries. 2021-11-17. 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Readable Markdown	From Wikipedia, the free encyclopedia [![](https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and_cosine_animation.gif)](https://en.wikipedia.org/wiki/File:Sine_and_cosine_animation.gif) Tracing the y component of a [circle](https://en.wikipedia.org/wiki/Circle "Circle") while going around the circle results in a sine wave (red). Tracing the x component results in a [cosine](https://en.wikipedia.org/wiki/Cosine "Cosine") wave (blue). Both waves are sinusoids of the same frequency but different phases. A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a [periodic wave](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") whose [waveform](https://en.wikipedia.org/wiki/Waveform "Waveform") (shape) is the [trigonometric](https://en.wikipedia.org/wiki/Trigonometric_function "Trigonometric function") [sine function](https://en.wikipedia.org/wiki/Sine "Sine"). In [mechanics](https://en.wikipedia.org/wiki/Mechanics "Mechanics"), as a linear [motion](https://en.wikipedia.org/wiki/Motion "Motion") over time, this is [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion "Simple harmonic motion"); as [rotation](https://en.wikipedia.org/wiki/Rotation "Rotation"), it corresponds to [uniform circular motion](https://en.wikipedia.org/wiki/Uniform_circular_motion "Uniform circular motion"). Sine waves occur often in [physics](https://en.wikipedia.org/wiki/Physics "Physics"), including [wind waves](https://en.wikipedia.org/wiki/Wind_wave "Wind wave"), [sound](https://en.wikipedia.org/wiki/Sound "Sound") waves, and [light](https://en.wikipedia.org/wiki/Light "Light") waves, such as [monochromatic radiation](https://en.wikipedia.org/wiki/Monochromatic_radiation "Monochromatic radiation"). In [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"), [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing"), and [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis") decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") (but arbitrary [phase](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)")) are [linearly combined](https://en.wikipedia.org/wiki/Linear_combination "Linear combination"), the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine [components](https://en.wikipedia.org/wiki/Vector_component "Vector component"), respectively. ![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/60px-Gnome-mime-sound-openclipart.svg.png) Five seconds of a 220 Hz sine wave. This is the [sound wave](https://en.wikipedia.org/wiki/Sound#Waves "Sound") described by a sine function with f = 220 oscillations per second. A sine wave represents a single frequency with no [harmonics](https://en.wikipedia.org/wiki/Harmonic "Harmonic") and is considered an [acoustically](https://en.wikipedia.org/wiki/Acoustics "Acoustics") [pure tone](https://en.wikipedia.org/wiki/Pure_tone "Pure tone"). Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the [fundamental](https://en.wikipedia.org/wiki/Fundamental_frequency "Fundamental frequency") causes variation in the [timbre](https://en.wikipedia.org/wiki/Timbre "Timbre"), which is the reason why the same [musical pitch](https://en.wikipedia.org/wiki/Pitch_\(music\) "Pitch (music)") played on different instruments sounds different. Sine waves of arbitrary phase and amplitude are called sinusoids and have the general form:[\[1\]](https://en.wikipedia.org/wiki/Sine_wave#cite_note-1) ![{\\displaystyle y(t)=A\\sin(\\omega t+\\varphi )=A\\sin(2\\pi ft+\\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f7f3ec4cdc5ad694b390c5adfcdc0a858f1457)[\[2\]](https://en.wikipedia.org/wiki/Sine_wave#cite_note-2) where: - ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude"), the peak deviation of the function from zero. - ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560), the [real](https://en.wikipedia.org/wiki/Real_number "Real number") [independent variable](https://en.wikipedia.org/wiki/Independent_variable "Independent variable"), usually representing [time](https://en.wikipedia.org/wiki/Time "Time") in [seconds](https://en.wikipedia.org/wiki/Seconds "Seconds"). - ![{\\displaystyle \\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8), [angular frequency](https://en.wikipedia.org/wiki/Angular_frequency "Angular frequency"), the rate of change of the function argument in units of [radians per second](https://en.wikipedia.org/wiki/Radians_per_second "Radians per second"). - ![{\\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61), [ordinary frequency](https://en.wikipedia.org/wiki/Ordinary_frequency "Ordinary frequency"), the [number](https://en.wikipedia.org/wiki/Real_number "Real number") of oscillations ([cycles](https://en.wikipedia.org/wiki/Turn_\(angle\) "Turn (angle)")) that occur each second of time. - ![{\\displaystyle \\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e), [phase](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)"), specifies (in [radians](https://en.wikipedia.org/wiki/Radian "Radian")) where in its cycle the oscillation is at t = 0. ## As a function of both position and time \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=3 "Edit section: As a function of both position and time")\] [![](https://upload.wikimedia.org/wikipedia/commons/2/25/Animated-mass-spring.gif)](https://en.wikipedia.org/wiki/File:Animated-mass-spring.gif) The displacement of an undamped [spring-mass system](https://en.wikipedia.org/wiki/Spring_mass_system "Spring mass system") oscillating around the equilibrium over time is a sine wave. Sinusoids that exist in both position and time also have: Depending on their direction of travel, they can take the form: - ![{\\displaystyle y(x,t)=A\\sin(kx-\\omega t+\\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab3502270349ff4b9c819170eb370c7d35a440a), if the wave is moving to the right, or - ![{\\displaystyle y(x,t)=A\\sin(kx+\\omega t+\\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42b125c02254130171f1ae41b4ea0abe04843b93), if the wave is moving to the left. Since sine waves propagate without changing form in distributed linear systems,\[[definition needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")\] they are often used to analyze [wave propagation](https://en.wikipedia.org/wiki/Wave_propagation "Wave propagation"). When two waves with the same [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") and [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") traveling in opposite directions [superpose](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle") each other, then a [standing wave](https://en.wikipedia.org/wiki/Standing_wave "Standing wave") pattern is created. On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's [resonant](https://en.wikipedia.org/wiki/Resonant "Resonant") frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the [fundamental frequency](https://en.wikipedia.org/wiki/Fundamental_frequency "Fundamental frequency")) and integer divisions of that (corresponding to higher harmonics). ### Multiple spatial dimensions \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=5 "Edit section: Multiple spatial dimensions")\] The earlier equation gives the displacement ![{\\displaystyle y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d) of the wave at a position ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) at time ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) along a single line. This could, for example, be considered the value of a wave along a wire. In two or three spatial dimensions, the same equation describes a travelling [plane wave](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") if position ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) and wavenumber ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) are interpreted as vectors, and their product as a [dot product](https://en.wikipedia.org/wiki/Dot_product "Dot product"). For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. #### Sinusoidal plane wave \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=6 "Edit section: Sinusoidal plane wave")\] French mathematician [Joseph Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier") discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including [square waves](https://en.wikipedia.org/wiki/Square_wave_\(waveform\) "Square wave (waveform)"). These [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") are frequently used in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") and the statistical analysis of [time series](https://en.wikipedia.org/wiki/Time_series "Time series"). The [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") then extended Fourier series to handle general functions, and birthed the field of [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"). ## Differentiation and integration \[[edit](https://en.wikipedia.org/w/index.php?title=Sine_wave&action=edit&section=8 "Edit section: Differentiation and integration")\] [Differentiating](https://en.wikipedia.org/wiki/Derivative "Derivative") any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: ![{\\displaystyle {\\begin{aligned}{\\frac {d}{dt}}\[A\\sin(\\omega t+\\varphi )\]&=A\\omega \\cos(\\omega t+\\varphi )\\\\&=A\\omega \\sin(\\omega t+\\varphi +{\\tfrac {\\pi }{2}})\\,.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1d774c1775716120ab0ded159150b63b8c4a2c) A [differentiator](https://en.wikipedia.org/wiki/Differentiator "Differentiator") has a [zero](https://en.wikipedia.org/wiki/Zeros_and_poles "Zeros and poles") at the origin of the [complex frequency](https://en.wikipedia.org/wiki/Complex_frequency "Complex frequency") plane. The [gain](https://en.wikipedia.org/wiki/Gain_\(electronics\) "Gain (electronics)") of its [frequency response](https://en.wikipedia.org/wiki/Frequency_response "Frequency response") increases at a rate of +20 [dB](https://en.wikipedia.org/wiki/Decibel "Decibel") per [decade](https://en.wikipedia.org/wiki/Decade_\(log_scale\) "Decade (log scale)") of frequency (for [root-power](https://en.wikipedia.org/wiki/Root-power "Root-power") quantities), the same positive slope as a 1st order [high-pass filter](https://en.wikipedia.org/wiki/High-pass_filter "High-pass filter")'s [stopband](https://en.wikipedia.org/wiki/Stopband "Stopband"), although a differentiator does not have a [cutoff frequency](https://en.wikipedia.org/wiki/Cutoff_frequency "Cutoff frequency") or a flat [passband](https://en.wikipedia.org/wiki/Passband "Passband"). A nth\-order high-pass filter approximately applies the nth time derivative of [signals](https://en.wikipedia.org/wiki/Signals "Signals") whose frequency band is significantly lower than the filter's cutoff frequency. [Integrating](https://en.wikipedia.org/wiki/Integral "Integral") any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle: ![{\\displaystyle {\\begin{aligned}\\int A\\sin(\\omega t+\\varphi )dt&=-{\\frac {A}{\\omega }}\\cos(\\omega t+\\varphi )+C\\\\&=-{\\frac {A}{\\omega }}\\sin(\\omega t+\\varphi +{\\tfrac {\\pi }{2}})+C\\\\&={\\frac {A}{\\omega }}\\sin(\\omega t+\\varphi -{\\tfrac {\\pi }{2}})+C\\,.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f27fb900539e12f857c9c575510c3f84c6b4d8e) The [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration") ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) will be zero if the [bounds of integration](https://en.wikipedia.org/wiki/Bounds_of_integration "Bounds of integration") is an integer multiple of the sinusoid's period. An [integrator](https://en.wikipedia.org/wiki/Integrator "Integrator") has a [pole](https://en.wikipedia.org/wiki/Zeros_and_poles "Zeros and poles") at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1st order [low-pass filter](https://en.wikipedia.org/wiki/Low-pass_filter "Low-pass filter")'s stopband, although an integrator does not have a cutoff frequency or a flat passband. A nth\-order low-pass filter approximately performs the nth time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency. - [Crest (physics)](https://en.wikipedia.org/wiki/Crest_\(physics\) "Crest (physics)") - [Complex exponential](https://en.wikipedia.org/wiki/Complex_exponential "Complex exponential") - [Damped sine wave](https://en.wikipedia.org/wiki/Damped_sine_wave "Damped sine wave") - [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") - [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") - [Harmonic analysis](https://en.wikipedia.org/wiki/Harmonic_analysis "Harmonic analysis") - [Harmonic series (mathematics)](https://en.wikipedia.org/wiki/Harmonic_series_\(mathematics\) "Harmonic series (mathematics)") - [Harmonic series (music)](https://en.wikipedia.org/wiki/Harmonic_series_\(music\) "Harmonic series (music)") - [Helmholtz equation](https://en.wikipedia.org/wiki/Helmholtz_equation "Helmholtz equation") - [Instantaneous phase](https://en.wikipedia.org/wiki/Instantaneous_phase "Instantaneous phase") - [In-phase and quadrature components](https://en.wikipedia.org/wiki/In-phase_and_quadrature_components "In-phase and quadrature components") - [Least-squares spectral analysis](https://en.wikipedia.org/wiki/Least-squares_spectral_analysis "Least-squares spectral analysis") - [Oscilloscope](https://en.wikipedia.org/wiki/Oscilloscope "Oscilloscope") - [Phasor](https://en.wikipedia.org/wiki/Phasor "Phasor") - [Pure tone](https://en.wikipedia.org/wiki/Pure_tone "Pure tone") - [Simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion "Simple harmonic motion") - [Sinusoidal model](https://en.wikipedia.org/wiki/Sinusoidal_model "Sinusoidal model") - [Wave (physics)](https://en.wikipedia.org/wiki/Wave_\(physics\) "Wave (physics)") - [Wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") - [∿](https://en.wikipedia.org/wiki/Tilde#Electronics "Tilde") the sine wave symbol (U+223F) 1. [^](https://en.wikipedia.org/wiki/Sine_wave#cite_ref-1) Smith, Julius Orion. ["Sinusoids"](https://ccrma.stanford.edu/~jos/st/Sinusoids.html). ccrma.stanford.edu. Retrieved 2024-01-05. 2. [^](https://en.wikipedia.org/wiki/Sine_wave#cite_ref-2) ["1.2: Sine Waves"](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_\(Raymond\)/01%3A_Waves_in_One_Dimension/1.02%3A_Sine_Waves). Physics LibreTexts. 2021-02-21. Retrieved 2026-03-16. - ["Sine Wave"](https://mathematicalmysteries.org/sine-wave/). Mathematical Mysteries. 2021-11-17. Retrieved 2022-09-30.
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