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1. Shard Calculation

Query:

Response:

Calculated Shard: 152 (from laksa085)

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/Superposition_principle\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Superposition_principle\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Superposition_principle","crawl_time":1775773361,"first_indexed_time":1375978975,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Superposition_principle s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Superposition principle - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nSuperposition of almost\nplane waves\n(diagonal lines) from a distant source and waves from the\nwake\nof the\nducks\n.\nLinearity\nholds only approximately in water and only for waves with small amplitudes relative to their wavelengths.\nRolling\nmotion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions:\nrotation\nwithout\ntranslation\n, and translation without rotation.\nThe\nsuperposition principle\n,\n[\n1\n]\nalso known as\nsuperposition property\n, states that, for all\nlinear systems\n, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input\nA\nproduces response\nX\n, and input\nB\nproduces response\nY\n, then input (\nA\n+\nB\n) produces response (\nX\n+\nY\n).\nA\nfunction\nthat satisfies the superposition principle is called a\nlinear function\n. Superposition can be defined by two simpler properties:\nadditivity\nand\nhomogeneity\nfor\nscalar\na\n.\nThis principle has many applications in\nphysics\nand\nengineering\nbecause many physical systems can be modeled as linear systems. For example, a\nbeam\ncan be modeled as a linear system where the input stimulus is the\nload\non the beam and the output response is the\ndeflection\nof the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques,\nfrequency-domain\nlinear transform\nmethods such as\nFourier\nand\nLaplace\ntransforms, and\nlinear operator\ntheory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.\nThe superposition principle applies to\nany\nlinear system, including\nalgebraic equations\n,\nlinear differential equations\n, and\nsystems of equations\nof those forms. The stimuli and responses could be numbers, functions, vectors,\nvector fields\n, time-varying signals, or any other object that satisfies\ncertain axioms\n. Note that when vectors or vector fields are involved, a superposition is interpreted as a\nvector sum\n. If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).\nRelation to Fourier analysis and similar methods\n[\nedit\n]\nBy writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.\nFor example, in\nFourier analysis\n, the stimulus is written as the superposition of infinitely many\nsinusoids\n. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different\namplitude\nand\nphase\n.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.\nAs another common example, in\nGreen's function analysis\n, the stimulus is written as the superposition of infinitely many\nimpulse functions\n, and the response is then a superposition of\nimpulse responses\n.\nFourier analysis is particularly common for\nwaves\n. For example, in electromagnetic theory, ordinary\nlight\nis described as a superposition of\nplane waves\n(waves of fixed\nfrequency\n,\npolarization\n, and direction). As long as the superposition principle holds (which is often but not always; see\nnonlinear optics\n), the behavior of any light wave can be understood as a superposition of the behavior of these simpler\nplane waves\n.\nTwo waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a\nstanding wave\n.\nTwo waves permeate without influencing each other\nWaves are usually described by variations in some parameters through space and time—for example, height in a water wave,\npressure\nin a sound wave, or the\nelectromagnetic field\nin a light wave. The value of this parameter is called the\namplitude\nof the wave and the wave itself is a\nfunction\nspecifying the amplitude at each point.\nIn any system with waves, the waveform at a given time is a function of the\nsources\n(i.e., external forces, if any, that create or affect the wave) and\ninitial conditions\nof the system. In many cases (for example, in the classic\nwave equation\n), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)\nWave diffraction vs. wave interference\n[\nedit\n]\nWith regard to wave superposition,\nRichard Feynman\nwrote:\n[\n2\n]\nNo-one has ever been able to define the difference between\ninterference\nand diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.\nOther authors elaborate:\n[\n3\n]\nThe difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically, they are two limiting cases of superposition effects.\nYet another source concurs:\n[\n4\n]\nIn as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is, therefore, a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront.\nThe phenomenon of\ninterference\nbetween waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in\nnoise-canceling headphones\n, the summed variation has a smaller\namplitude\nthan the component variations; this is called\ndestructive interference\n. In other cases, such as in a\nline array\n, the summed variation will have a bigger amplitude than any of the components individually; this is called\nconstructive interference\n.\ngreen wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.\ncombined\nwaveform\nwave 1\nwave 2\nTwo waves in phase\nTwo waves 180° out\nof phase\nDepartures from linearity\n[\nedit\n]\nIn most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles\nnonlinear optics\nand\nnonlinear acoustics\n.\nQuantum superposition\n[\nedit\n]\nIn\nquantum mechanics\n, a principal task is to compute how a certain type of wave\npropagates\nand behaves. The wave is described by a\nwave function\n, and the equation governing its behavior is called the\nSchrödinger equation\n. A primary approach to computing the behavior of a wave function is to write it as a superposition (called \"\nquantum superposition\n\") of (possibly infinitely many) other wave functions of a certain type—\nstationary states\nwhose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.\n[\n5\n]\nThe projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a\nray\nin\nprojective Hilbert space\n, not a\nvector\n.\nAccording to\nDirac\n: \"\nif the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state\n[italics in original].\"\n[\n6\n]\nHowever, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself\nuses ket vector representations of states to decompose or split,\nfor example, a ket vector\ninto superposition of component ket vectors\nas:\nwhere the\n.\nThe equivalence class of the\nallows a well-defined meaning to be given to the relative phases of the\n.,\n[\n7\n]\nbut an absolute (same amount\nfor all the\n) phase change on the\ndoes not affect the equivalence class of the\n.\nThere are exact correspondences between the superposition presented in the main on this page and the quantum superposition.\nFor example, the\nBloch sphere\nto represent\npure state\nof a\ntwo-level quantum mechanical system\n(\nqubit\n) is also known as the\nPoincaré sphere\nrepresenting different types of classical\npure\npolarization\nstates.\nNevertheless, on the topic of quantum superposition,\nKramers\nwrites: \"The principle of [quantum] superposition ... has no analogy in classical physics\"\n[\ncitation needed\n]\n.\nAccording to\nDirac\n: \"\nthe superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory\n[italics in original].\"\n[\n8\n]\nThough reasoning by Dirac includes atomicity of observation, which is valid, as for phase,\nthey actually mean phase translation symmetry derived from\ntime translation symmetry\n, which is also\napplicable to classical states, as shown above with classical polarization states.\nBoundary-value problems\n[\nedit\n]\nA common type of boundary value problem is (to put it abstractly) finding a function\ny\nthat satisfies some equation\nwith some boundary specification\nFor example, in\nLaplace's equation\nwith\nDirichlet boundary conditions\n,\nF\nwould be the\nLaplacian\noperator in a region\nR\n,\nG\nwould be an operator that restricts\ny\nto the boundary of\nR\n, and\nz\nwould be the function that\ny\nis required to equal on the boundary of\nR\n.\nIn the case that\nF\nand\nG\nare both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation:\nwhile the boundary values superpose:\nUsing these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.\nAdditive state decomposition\n[\nedit\n]\nConsider a simple linear system:\nBy the superposition principle, the system can be decomposed into\nwith\nSuperposition principle is only available for linear systems. However, the\nadditive state decomposition\ncan be applied to both linear and nonlinear systems. Next, consider a nonlinear system\nwhere\nis a nonlinear function. By the additive state decomposition, the system can be additively decomposed into\nwith\nThis decomposition can help to simplify controller design.\nOther example applications\n[\nedit\n]\nIn\nelectrical engineering\n, in a\nlinear circuit\n, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses.\nIn\nphysics\n,\nMaxwell's equations\nimply that the (possibly time-varying) distributions of\ncharges\nand\ncurrents\nare related to the\nelectric\nand\nmagnetic fields\nby a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the\nheat equation\n.\nIn\nengineering\n, superposition is used to solve for beam and structure\ndeflections\nof combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).\n[\n9\n]\nMode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.\n[\n10\n]\nIn\nhydrogeology\n, the superposition principle is applied to the\ndrawdown\nof two or more\nwater wells\npumping in an ideal\naquifer\n. This principle is used in the\nanalytic element method\nto develop analytical elements capable of being combined in a single model.\nIn\nprocess control\n, the superposition principle is used in\nmodel predictive control\n.\nThe superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by\nlinearization\n.\nAccording to\nLéon Brillouin\n, the principle of superposition was first stated by\nDaniel Bernoulli\nin 1753: \"The general motion of a vibrating system is given by a superposition of its proper vibrations.\" The principle was rejected by\nLeonhard Euler\nand then by\nJoseph Lagrange\n. Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.\n[\n11\n]\nLater it became accepted, largely through the work of\nJoseph Fourier\n.\n[\n12\n]\nAdditive state decomposition\nBeat (acoustics)\nCoherence (physics)\nConvolution\nGreen's function\nImpulse response\nInterference\nQuantum superposition\n^\nThe Penguin Dictionary of Physics,  ed. Valerie Illingworth, 1991, Penguin Books, London.\n^\nLectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass\n[1]\n^\nN. K. VERMA,\nPhysics for Engineers\n, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361.\n[2]\n^\nTim Freegarde,\nIntroduction to the Physics of Waves\n, Cambridge University Press, Nov 8, 2012.\n[3]\n^\nQuantum Mechanics,\nKramers, H.A.\npublisher Dover, 1957, p. 62\nISBN\n \n978-0-486-66772-0\n^\nDirac, P. A. M.\n(1958).\nThe Principles of Quantum Mechanics\n, 4th edition, Oxford, UK: Oxford University Press, p. 17.\n^\nSolem, J. C.; Biedenharn, L. C. (1993). \"Understanding geometrical phases in quantum mechanics: An elementary example\".\nFoundations of Physics\n.\n23\n(2):\n185–\n195.\nBibcode\n:\n1993FoPh...23..185S\n.\ndoi\n:\n10.1007\/BF01883623\n.\nS2CID\n \n121930907\n.\n^\nDirac, P. A. M.\n(1958).\nThe Principles of Quantum Mechanics\n, 4th edition, Oxford, UK: Oxford University Press, p. 14.\n^\nMechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192\nISBN\n \n0-07-252036-1\n^\nFinite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785\nISBN\n \n0-13-301458-4\n^\nTopics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain,\np. 39\n^\nBrillouin, L.\n(1946).\nWave propagation in Periodic Structures: Electric Filters and Crystal Lattices\n, McGraw–Hill, New York, p. 2.\nHaberman, Richard (2004).\nApplied Partial Differential Equations\n. Prentice Hall.\nISBN\n \n978-0-13-065243-0\n.\nSuperposition of sound waves","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#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=Superposition+principle \"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=Superposition+principle \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle)\n- [1 Relation to Fourier analysis and similar methods](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Relation_to_Fourier_analysis_and_similar_methods)\n- [2 Wave superposition](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Wave_superposition)\n  Toggle Wave superposition subsection\n  - [2\\.1 Wave diffraction vs. wave interference](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Wave_diffraction_vs._wave_interference)\n  - [2\\.2 Wave interference](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Wave_interference)\n  - [2\\.3 Departures from linearity](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Departures_from_linearity)\n  - [2\\.4 Quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Quantum_superposition)\n- [3 Boundary-value problems](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Boundary-value_problems)\n- [4 Additive state decomposition](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Additive_state_decomposition)\n- [5 Other example applications](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Other_example_applications)\n- [6 History](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#History)\n- [7 See also](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#See_also)\n- [8 References](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#References)\n- [9 Further reading](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#Further_reading)\n- [10 External links](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#External_links)\n\nToggle the table of contents\n\n# Superposition principle\n40 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D9%85%D8%A8%D8%AF%D8%A3_%D8%A7%D9%84%D8%AA%D8%B1%D8%A7%D9%83%D8%A8 \"مبدأ التراكب – Arabic\")\n- [Беларуская (тарашкевіца)](https:\/\/be-tarask.wikipedia.org\/wiki\/%D0%9F%D1%80%D1%8B%D0%BD%D1%86%D1%8B%D0%BF_%D1%81%D1%83%D0%BF%D1%8D%D1%80%D0%BF%D0%B0%D0%B7%D1%8B%D1%86%D1%8B%D1%96 \"Прынцып супэрпазыцыі – Belarusian (Taraškievica orthography)\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Principi_de_superposici%C3%B3 \"Principi de superposició – Catalan\")\n- [کوردی](https:\/\/ckb.wikipedia.org\/wiki\/%D8%A8%D9%86%DB%95%D9%85%D8%A7%DB%8C_%D8%B3%DB%95%D8%B1%DB%8C%DB%95%DA%A9%DA%86%D9%88%D9%88%D9%86 \"بنەمای سەریەکچوون – Central Kurdish\")\n- [Чӑвашла](https:\/\/cv.wikipedia.org\/wiki\/%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8_%D0%BF%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF%C4%95 \"Суперпозици принципĕ – Chuvash\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Superposition_\\(Physik\\) \"Superposition (Physik) – German\")\n- [Ελληνικά](https:\/\/el.wikipedia.org\/wiki\/%CE%91%CF%81%CF%87%CE%AE_%CF%84%CE%B7%CF%82_%CE%B5%CF%80%CE%B1%CE%BB%CE%BB%CE%B7%CE%BB%CE%AF%CE%B1%CF%82 \"Αρχή της επαλληλίας – Greek\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Principio_de_superposici%C3%B3n \"Principio de superposición – Spanish\")\n- [Eesti](https:\/\/et.wikipedia.org\/wiki\/Superpositsiooniprintsiip \"Superpositsiooniprintsiip – Estonian\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Gainezarmen_printzipioa \"Gainezarmen printzipioa – Basque\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D8%A7%D8%B5%D9%84_%D8%A8%D8%B1%D9%87%D9%85%E2%80%8C%D9%86%D9%87%DB%8C \"اصل برهم‌نهی – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Superpositioperiaate \"Superpositioperiaate – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Principe_de_superposition \"Principe de superposition – French\")\n- [עברית](https:\/\/he.wikipedia.org\/wiki\/%D7%A1%D7%95%D7%A4%D7%A8%D7%A4%D7%95%D7%96%D7%99%D7%A6%D7%99%D7%94 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PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Superposition_principle&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikimedia Commons](https:\/\/commons.wikimedia.org\/wiki\/Category:Superposition_principle)\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q284885 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nFundamental principle of physics\n\nThis article is about the superposition principle in linear systems. For the geologic principle, see [Law of superposition](https:\/\/en.wikipedia.org\/wiki\/Law_of_superposition \"Law of superposition\"). For other uses, see [Superposition (disambiguation)](https:\/\/en.wikipedia.org\/wiki\/Superposition_\\(disambiguation\\) \"Superposition (disambiguation)\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/7f\/Anas_platyrhynchos_with_ducklings_reflecting_water.jpg\/250px-Anas_platyrhynchos_with_ducklings_reflecting_water.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Anas_platyrhynchos_with_ducklings_reflecting_water.jpg)\n\nSuperposition of almost [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") (diagonal lines) from a distant source and waves from the [wake](https:\/\/en.wikipedia.org\/wiki\/Wake_\\(physics\\) \"Wake (physics)\") of the [ducks](https:\/\/en.wikipedia.org\/wiki\/Duck \"Duck\"). [Linearity](https:\/\/en.wikipedia.org\/wiki\/Linearity \"Linearity\") holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/36\/Rolling_animation.gif\/250px-Rolling_animation.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Rolling_animation.gif)\n\n[Rolling](https:\/\/en.wikipedia.org\/wiki\/Rolling \"Rolling\") motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: [rotation](https:\/\/en.wikipedia.org\/wiki\/Rotation \"Rotation\") without [translation](https:\/\/en.wikipedia.org\/wiki\/Translation_\\(geometry\\) \"Translation (geometry)\"), and translation without rotation.\n\nThe **superposition principle**,[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-1) also known as **superposition property**, states that, for all [linear systems](https:\/\/en.wikipedia.org\/wiki\/Linear_system \"Linear system\"), the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X*, and input *B* produces response *Y*, then input (*A* + *B*) produces response (*X* + *Y*).\n\nA [function](https:\/\/en.wikipedia.org\/wiki\/Function_\\(mathematics\\) \"Function (mathematics)\") F ( x ) {\\\\displaystyle F(x)} ![{\\\\displaystyle F(x)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/71a82805d469cdfa7856c11d6ee756acd1dc7174) that satisfies the superposition principle is called a [linear function](https:\/\/en.wikipedia.org\/wiki\/Linear_function \"Linear function\"). Superposition can be defined by two simpler properties: [additivity](https:\/\/en.wikipedia.org\/wiki\/Additive_map \"Additive map\") F ( x 1 \\+ x 2 ) \\= F ( x 1 ) \\+ F ( x 2 ) {\\\\displaystyle F(x\\_{1}+x\\_{2})=F(x\\_{1})+F(x\\_{2})} ![{\\\\displaystyle F(x\\_{1}+x\\_{2})=F(x\\_{1})+F(x\\_{2})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fe048d07a4d57ffe738373ddafec4fb02faf7b6) and [homogeneity](https:\/\/en.wikipedia.org\/wiki\/Homogeneous_function \"Homogeneous function\") F ( a x ) \\= a F ( x ) {\\\\displaystyle F(ax)=aF(x)} ![{\\\\displaystyle F(ax)=aF(x)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/43e16187439ba65e17e0eddfc4d75a09d47cfb9c) for [scalar](https:\/\/en.wikipedia.org\/wiki\/Scalar_\\(mathematics\\) \"Scalar (mathematics)\") a.\n\nThis principle has many applications in [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\") and [engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering \"Engineering\") because many physical systems can be modeled as linear systems. For example, a [beam](https:\/\/en.wikipedia.org\/wiki\/Beam_\\(structure\\) \"Beam (structure)\") can be modeled as a linear system where the input stimulus is the [load](https:\/\/en.wikipedia.org\/wiki\/Structural_load \"Structural load\") on the beam and the output response is the [deflection](https:\/\/en.wikipedia.org\/wiki\/Deflection_\\(engineering\\) \"Deflection (engineering)\") of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, [frequency-domain](https:\/\/en.wikipedia.org\/wiki\/Frequency-domain \"Frequency-domain\") [linear transform](https:\/\/en.wikipedia.org\/wiki\/Linear_transform \"Linear transform\") methods such as [Fourier](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\") and [Laplace](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform \"Laplace transform\") transforms, and [linear operator](https:\/\/en.wikipedia.org\/wiki\/Linear_operator \"Linear operator\") theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.\n\nThe superposition principle applies to *any* linear system, including [algebraic equations](https:\/\/en.wikipedia.org\/wiki\/Algebraic_equation \"Algebraic equation\"), [linear differential equations](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equations \"Linear differential equations\"), and [systems of equations](https:\/\/en.wikipedia.org\/wiki\/System_of_equations \"System of equations\") of those forms. The stimuli and responses could be numbers, functions, vectors, [vector fields](https:\/\/en.wikipedia.org\/wiki\/Vector_field \"Vector field\"), time-varying signals, or any other object that satisfies [certain axioms](https:\/\/en.wikipedia.org\/wiki\/Vector_space \"Vector space\"). Note that when vectors or vector fields are involved, a superposition is interpreted as a [vector sum](https:\/\/en.wikipedia.org\/wiki\/Vector_sum \"Vector sum\"). If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).\n\n## Relation to Fourier analysis and similar methods\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=1 \"Edit section: Relation to Fourier analysis and similar methods\")\\]\n\nBy writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.\n\nFor example, in [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\"), the stimulus is written as the superposition of infinitely many [sinusoids](https:\/\/en.wikipedia.org\/wiki\/Sine_wave \"Sine wave\"). Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") and [phase](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\").) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.\n\nAs another common example, in [Green's function analysis](https:\/\/en.wikipedia.org\/wiki\/Green%27s_function \"Green's function\"), the stimulus is written as the superposition of infinitely many [impulse functions](https:\/\/en.wikipedia.org\/wiki\/Impulse_function \"Impulse function\"), and the response is then a superposition of [impulse responses](https:\/\/en.wikipedia.org\/wiki\/Impulse_response \"Impulse response\").\n\nFourier analysis is particularly common for [waves](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\"). For example, in electromagnetic theory, ordinary [light](https:\/\/en.wikipedia.org\/wiki\/Light \"Light\") is described as a superposition of [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") (waves of fixed [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\"), [polarization](https:\/\/en.wikipedia.org\/wiki\/Polarization_\\(waves\\) \"Polarization (waves)\"), and direction). As long as the superposition principle holds (which is often but not always; see [nonlinear optics](https:\/\/en.wikipedia.org\/wiki\/Nonlinear_optics \"Nonlinear optics\")), the behavior of any light wave can be understood as a superposition of the behavior of these simpler [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\").\n\n## Wave superposition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=2 \"Edit section: Wave superposition\")\\]\n\nFurther information: [Wave](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\") and [Wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\")\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/7d\/Standing_wave_2.gif\/250px-Standing_wave_2.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Standing_wave_2.gif)\n\nTwo waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a [standing wave](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/3\/3c\/Standing_waves1.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Standing_waves1.gif)\n\nTwo waves permeate without influencing each other\n\nWaves are usually described by variations in some parameters through space and time—for example, height in a water wave, [pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") in a sound wave, or the [electromagnetic field](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_field \"Electromagnetic field\") in a light wave. The value of this parameter is called the [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") of the wave and the wave itself is a [function](https:\/\/en.wikipedia.org\/wiki\/Function_\\(mathematics\\) \"Function (mathematics)\") specifying the amplitude at each point.\n\nIn any system with waves, the waveform at a given time is a function of the [sources](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\") (i.e., external forces, if any, that create or affect the wave) and [initial conditions](https:\/\/en.wikipedia.org\/wiki\/Initial_condition \"Initial condition\") of the system. In many cases (for example, in the classic [wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\")), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)\n\n### Wave diffraction vs. wave interference\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=3 \"Edit section: Wave diffraction vs. wave interference\")\\]\n\nWith regard to wave superposition, [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\") wrote:[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-2)\n\n> No-one has ever been able to define the difference between [interference](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\") and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.\n\nOther authors elaborate:[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-3)\n\n> The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is \\[a matter\\] of degree only, and basically, they are two limiting cases of superposition effects.\n\nYet another source concurs:[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-4)\n\n> In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter \\[Fraunhofer diffraction\\] is, therefore, a continuation of Chapter 8 \\[Interference\\]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront.\n\n### Wave interference\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=4 \"Edit section: Wave interference\")\\]\n\nMain article: [Interference (wave propagation)](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\")\n\nThe phenomenon of [interference](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\") between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in [noise-canceling headphones](https:\/\/en.wikipedia.org\/wiki\/Noise-canceling_headphones \"Noise-canceling headphones\"), the summed variation has a smaller [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") than the component variations; this is called *destructive interference*. In other cases, such as in a [line array](https:\/\/en.wikipedia.org\/wiki\/Line_array \"Line array\"), the summed variation will have a bigger amplitude than any of the components individually; this is called *constructive interference*.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/5d\/Waventerference.gif\/250px-Waventerference.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Waventerference.gif)\n\ngreen wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.\n\n|   |   |   |\n|---|---|---|\n| combined  waveform | [![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0f\/Interference_of_two_waves.svg\/500px-Interference_of_two_waves.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Interference_of_two_waves.svg) |   |\n| wave 1 |   |   |\n| wave 2 |   |   |\n|   | Two waves in phase | Two waves 180° out  of phase |\n\n### Departures from linearity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=5 \"Edit section: Departures from linearity\")\\]\n\nIn most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles [nonlinear optics](https:\/\/en.wikipedia.org\/wiki\/Nonlinear_optics \"Nonlinear optics\") and [nonlinear acoustics](https:\/\/en.wikipedia.org\/wiki\/Nonlinear_acoustics \"Nonlinear acoustics\").\n\n### Quantum superposition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=6 \"Edit section: Quantum superposition\")\\]\n\nMain article: [Quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\")\n\nIn [quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics\"), a principal task is to compute how a certain type of wave [propagates](https:\/\/en.wikipedia.org\/wiki\/Wave_propagation \"Wave propagation\") and behaves. The wave is described by a [wave function](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\"), and the equation governing its behavior is called the [Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\"). A primary approach to computing the behavior of a wave function is to write it as a superposition (called \"[quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\")\") of (possibly infinitely many) other wave functions of a certain type—[stationary states](https:\/\/en.wikipedia.org\/wiki\/Stationary_state \"Stationary state\") whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-QuaMech-5)\n\nThe projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a *ray* in [projective Hilbert space](https:\/\/en.wikipedia.org\/wiki\/Projective_Hilbert_space \"Projective Hilbert space\"), not a *vector*. According to [Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\"): \"*if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state* \\[italics in original\\].\"[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-6) However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself uses ket vector representations of states to decompose or split, for example, a ket vector \\| ψ i ⟩ {\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle } ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1284d734d01861923334d7ebfe1ddc855f25a66a) into superposition of component ket vectors \\| ϕ j ⟩ {\\\\displaystyle \\|\\\\phi \\_{j}\\\\rangle } ![{\\\\displaystyle \\|\\\\phi \\_{j}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/038f455c0dc2a38568eeac72e59131d3ad41d0a8) as: \\| ψ i ⟩ \\= ∑ j C j \\| ϕ j ⟩ , {\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle =\\\\sum \\_{j}{C\\_{j}}\\|\\\\phi \\_{j}\\\\rangle ,} ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle =\\\\sum \\_{j}{C\\_{j}}\\|\\\\phi \\_{j}\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5e6b0bdd3dcf5868472637fd364ef8131ceb5563) where the C j ∈ C {\\\\displaystyle C\\_{j}\\\\in {\\\\textbf {C}}} ![{\\\\displaystyle C\\_{j}\\\\in {\\\\textbf {C}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f05a8122fd2755f448f968a7765a3f71cb6c2ccb). The equivalence class of the \\| ψ i ⟩ {\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle } ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1284d734d01861923334d7ebfe1ddc855f25a66a) allows a well-defined meaning to be given to the relative phases of the C j {\\\\displaystyle C\\_{j}} ![{\\\\displaystyle C\\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4598eb9b4e1c079e0973de79383fc6f898744e18).,[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-7) but an absolute (same amount for all the C j {\\\\displaystyle C\\_{j}} ![{\\\\displaystyle C\\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4598eb9b4e1c079e0973de79383fc6f898744e18)) phase change on the C j {\\\\displaystyle C\\_{j}} ![{\\\\displaystyle C\\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4598eb9b4e1c079e0973de79383fc6f898744e18) does not affect the equivalence class of the \\| ψ i ⟩ {\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle } ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1284d734d01861923334d7ebfe1ddc855f25a66a).\n\nThere are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the [Bloch sphere](https:\/\/en.wikipedia.org\/wiki\/Bloch_sphere \"Bloch sphere\") to represent [pure state](https:\/\/en.wikipedia.org\/wiki\/Pure_state \"Pure state\") of a [two-level quantum mechanical system](https:\/\/en.wikipedia.org\/wiki\/Two-level_system \"Two-level system\") ([qubit](https:\/\/en.wikipedia.org\/wiki\/Qubit \"Qubit\")) is also known as the [Poincaré sphere](https:\/\/en.wikipedia.org\/wiki\/Bloch_sphere \"Bloch sphere\") representing different types of classical pure [polarization](https:\/\/en.wikipedia.org\/wiki\/Polarization_\\(waves\\) \"Polarization (waves)\") states.\n\nNevertheless, on the topic of quantum superposition, [Kramers](https:\/\/en.wikipedia.org\/wiki\/Hans_Kramers \"Hans Kramers\") writes: \"The principle of \\[quantum\\] superposition ... has no analogy in classical physics\"\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]. According to [Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\"): \"*the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory* \\[italics in original\\].\"[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-8) Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from [time translation symmetry](https:\/\/en.wikipedia.org\/wiki\/Time_translation_symmetry \"Time translation symmetry\"), which is also applicable to classical states, as shown above with classical polarization states.\n\n## Boundary-value problems\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=7 \"Edit section: Boundary-value problems\")\\]\n\nFurther information: [Boundary-value problem](https:\/\/en.wikipedia.org\/wiki\/Boundary-value_problem \"Boundary-value problem\")\n\nA common type of boundary value problem is (to put it abstractly) finding a function *y* that satisfies some equation F ( y ) \\= 0 {\\\\displaystyle F(y)=0} ![{\\\\displaystyle F(y)=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/937a4b209065cbbd3fe869f60f55cfa5fe9285c5) with some boundary specification G ( y ) \\= z . {\\\\displaystyle G(y)=z.} ![{\\\\displaystyle G(y)=z.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/adc40e235a4d43a4a07840d82de075f100e5f802) For example, in [Laplace's equation](https:\/\/en.wikipedia.org\/wiki\/Laplace%27s_equation \"Laplace's equation\") with [Dirichlet boundary conditions](https:\/\/en.wikipedia.org\/wiki\/Dirichlet_problem \"Dirichlet problem\"), *F* would be the [Laplacian](https:\/\/en.wikipedia.org\/wiki\/Laplacian \"Laplacian\") operator in a region *R*, *G* would be an operator that restricts *y* to the boundary of *R*, and *z* would be the function that *y* is required to equal on the boundary of *R*.\n\nIn the case that *F* and *G* are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: F ( y 1 ) \\= F ( y 2 ) \\= ⋯ \\= 0 ⇒ F ( y 1 \\+ y 2 \\+ ⋯ ) \\= 0 , {\\\\displaystyle F(y\\_{1})=F(y\\_{2})=\\\\cdots =0\\\\quad \\\\Rightarrow \\\\quad F(y\\_{1}+y\\_{2}+\\\\cdots )=0,} ![{\\\\displaystyle F(y\\_{1})=F(y\\_{2})=\\\\cdots =0\\\\quad \\\\Rightarrow \\\\quad F(y\\_{1}+y\\_{2}+\\\\cdots )=0,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9df7ad43bb0cb6488758147d3186a5cbcab69ba4) while the boundary values superpose: G ( y 1 ) \\+ G ( y 2 ) \\= G ( y 1 \\+ y 2 ) . {\\\\displaystyle G(y\\_{1})+G(y\\_{2})=G(y\\_{1}+y\\_{2}).} ![{\\\\displaystyle G(y\\_{1})+G(y\\_{2})=G(y\\_{1}+y\\_{2}).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6050e641d59e43363b8c1ea447ddbe165258b7f5) Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.\n\n## Additive state decomposition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=8 \"Edit section: Additive state decomposition\")\\]\n\nMain article: [Additive state decomposition](https:\/\/en.wikipedia.org\/wiki\/Additive_state_decomposition \"Additive state decomposition\")\n\nConsider a simple linear system: x ˙ \\= A x \\+ B ( u 1 \\+ u 2 ) , x ( 0 ) \\= x 0 . {\\\\displaystyle {\\\\dot {x}}=Ax+B(u\\_{1}+u\\_{2}),\\\\qquad x(0)=x\\_{0}.} ![{\\\\displaystyle {\\\\dot {x}}=Ax+B(u\\_{1}+u\\_{2}),\\\\qquad x(0)=x\\_{0}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a9571117cc3b1c0ddcfd61368718f0c8122bc4f0)\n\nBy the superposition principle, the system can be decomposed into x ˙ 1 \\= A x 1 \\+ B u 1 , x 1 ( 0 ) \\= x 0 , x ˙ 2 \\= A x 2 \\+ B u 2 , x 2 ( 0 ) \\= 0 {\\\\displaystyle {\\\\begin{aligned}{\\\\dot {x}}\\_{1}&=Ax\\_{1}+Bu\\_{1},&\\&x\\_{1}(0)=x\\_{0},\\\\\\\\{\\\\dot {x}}\\_{2}&=Ax\\_{2}+Bu\\_{2},&\\&x\\_{2}(0)=0\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}{\\\\dot {x}}\\_{1}&=Ax\\_{1}+Bu\\_{1},&\\&x\\_{1}(0)=x\\_{0},\\\\\\\\{\\\\dot {x}}\\_{2}&=Ax\\_{2}+Bu\\_{2},&\\&x\\_{2}(0)=0\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/33cd8d6f87b41a316638576cf0c25130907c9fee) with x \\= x 1 \\+ x 2 . {\\\\displaystyle x=x\\_{1}+x\\_{2}.} ![{\\\\displaystyle x=x\\_{1}+x\\_{2}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9ebf439216976f0058d8c371439dc49babe911cb)\n\nSuperposition principle is only available for linear systems. However, the [additive state decomposition](https:\/\/en.wikipedia.org\/wiki\/Additive_state_decomposition \"Additive state decomposition\") can be applied to both linear and nonlinear systems. Next, consider a nonlinear system x ˙ \\= A x \\+ B ( u 1 \\+ u 2 ) \\+ ϕ ( c T x ) , x ( 0 ) \\= x 0 , {\\\\displaystyle {\\\\dot {x}}=Ax+B(u\\_{1}+u\\_{2})+\\\\phi \\\\left(c^{\\\\mathsf {T}}x\\\\right),\\\\qquad x(0)=x\\_{0},} ![{\\\\displaystyle {\\\\dot {x}}=Ax+B(u\\_{1}+u\\_{2})+\\\\phi \\\\left(c^{\\\\mathsf {T}}x\\\\right),\\\\qquad x(0)=x\\_{0},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fa8951ec8b9f5d31d81ccf266a3510bb70973146) where ϕ {\\\\displaystyle \\\\phi } ![{\\\\displaystyle \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/72b1f30316670aee6270a28334bdf4f5072cdde4) is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into x ˙ 1 \\= A x 1 \\+ B u 1 \\+ ϕ ( y d ) , x 1 ( 0 ) \\= x 0 , x ˙ 2 \\= A x 2 \\+ B u 2 \\+ ϕ ( c T x 1 \\+ c T x 2 ) − ϕ ( y d ) , x 2 ( 0 ) \\= 0 {\\\\displaystyle {\\\\begin{aligned}{\\\\dot {x}}\\_{1}&=Ax\\_{1}+Bu\\_{1}+\\\\phi (y\\_{d}),&\\&x\\_{1}(0)=x\\_{0},\\\\\\\\{\\\\dot {x}}\\_{2}&=Ax\\_{2}+Bu\\_{2}+\\\\phi \\\\left(c^{\\\\mathsf {T}}x\\_{1}+c^{\\\\mathsf {T}}x\\_{2}\\\\right)-\\\\phi (y\\_{d}),&\\&x\\_{2}(0)=0\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}{\\\\dot {x}}\\_{1}&=Ax\\_{1}+Bu\\_{1}+\\\\phi (y\\_{d}),&\\&x\\_{1}(0)=x\\_{0},\\\\\\\\{\\\\dot {x}}\\_{2}&=Ax\\_{2}+Bu\\_{2}+\\\\phi \\\\left(c^{\\\\mathsf {T}}x\\_{1}+c^{\\\\mathsf {T}}x\\_{2}\\\\right)-\\\\phi (y\\_{d}),&\\&x\\_{2}(0)=0\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2561e25f8bb0b1d1116612026471b378e189ab83) with x \\= x 1 \\+ x 2 . {\\\\displaystyle x=x\\_{1}+x\\_{2}.} ![{\\\\displaystyle x=x\\_{1}+x\\_{2}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9ebf439216976f0058d8c371439dc49babe911cb)\n\nThis decomposition can help to simplify controller design.\n\n## Other example applications\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=9 \"Edit section: Other example applications\")\\]\n\n- In [electrical engineering](https:\/\/en.wikipedia.org\/wiki\/Electrical_engineering \"Electrical engineering\"), in a [linear circuit](https:\/\/en.wikipedia.org\/wiki\/Linear_circuit \"Linear circuit\"), the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses.\n- In [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\"), [Maxwell's equations](https:\/\/en.wikipedia.org\/wiki\/Maxwell%27s_equations \"Maxwell's equations\") imply that the (possibly time-varying) distributions of [charges](https:\/\/en.wikipedia.org\/wiki\/Electric_charge \"Electric charge\") and [currents](https:\/\/en.wikipedia.org\/wiki\/Electric_current \"Electric current\") are related to the [electric](https:\/\/en.wikipedia.org\/wiki\/Electric_field \"Electric field\") and [magnetic fields](https:\/\/en.wikipedia.org\/wiki\/Magnetic_field \"Magnetic field\") by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the [heat equation](https:\/\/en.wikipedia.org\/wiki\/Heat_equation \"Heat equation\").\n- In [engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering \"Engineering\"), superposition is used to solve for beam and structure [deflections](https:\/\/en.wikipedia.org\/wiki\/Deflection_\\(engineering\\) \"Deflection (engineering)\") of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-9) Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-10)\n- In [hydrogeology](https:\/\/en.wikipedia.org\/wiki\/Hydrogeology \"Hydrogeology\"), the superposition principle is applied to the [drawdown](https:\/\/en.wikipedia.org\/wiki\/Drawdown_\\(hydrology\\) \"Drawdown (hydrology)\") of two or more [water wells](https:\/\/en.wikipedia.org\/wiki\/Water_well \"Water well\") pumping in an ideal [aquifer](https:\/\/en.wikipedia.org\/wiki\/Aquifer \"Aquifer\"). This principle is used in the [analytic element method](https:\/\/en.wikipedia.org\/wiki\/Analytic_element_method \"Analytic element method\") to develop analytical elements capable of being combined in a single model.\n- In [process control](https:\/\/en.wikipedia.org\/wiki\/Process_control \"Process control\"), the superposition principle is used in [model predictive control](https:\/\/en.wikipedia.org\/wiki\/Model_predictive_control \"Model predictive control\").\n- The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by [linearization](https:\/\/en.wikipedia.org\/wiki\/Linearization \"Linearization\").\n\n## History\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=10 \"Edit section: History\")\\]\n\nAccording to [Léon Brillouin](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9on_Brillouin \"Léon Brillouin\"), the principle of superposition was first stated by [Daniel Bernoulli](https:\/\/en.wikipedia.org\/wiki\/Daniel_Bernoulli \"Daniel Bernoulli\") in 1753: \"The general motion of a vibrating system is given by a superposition of its proper vibrations.\" The principle was rejected by [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\") and then by [Joseph Lagrange](https:\/\/en.wikipedia.org\/wiki\/Joseph-Louis_Lagrange \"Joseph-Louis Lagrange\"). Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-11)\n\nLater it became accepted, largely through the work of [Joseph Fourier](https:\/\/en.wikipedia.org\/wiki\/Joseph_Fourier \"Joseph Fourier\").[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-12)\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=11 \"Edit section: See also\")\\]\n\n- [Additive state decomposition](https:\/\/en.wikipedia.org\/wiki\/Additive_state_decomposition \"Additive state decomposition\")\n- [Beat (acoustics)](https:\/\/en.wikipedia.org\/wiki\/Beat_\\(acoustics\\) \"Beat (acoustics)\")\n- [Coherence (physics)](https:\/\/en.wikipedia.org\/wiki\/Coherence_\\(physics\\) \"Coherence (physics)\")\n- [Convolution](https:\/\/en.wikipedia.org\/wiki\/Convolution \"Convolution\")\n- [Green's function](https:\/\/en.wikipedia.org\/wiki\/Green%27s_function \"Green's function\")\n- [Impulse response](https:\/\/en.wikipedia.org\/wiki\/Impulse_response \"Impulse response\")\n- [Interference](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\")\n- [Quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\")\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=12 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-1)** The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London.\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-2)** Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [\\[1\\]](https:\/\/books.google.com\/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1)\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-3)** N. K. VERMA, *Physics for Engineers*, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [\\[2\\]](https:\/\/books.google.com\/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361)\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-4)** Tim Freegarde, *Introduction to the Physics of Waves*, Cambridge University Press, Nov 8, 2012. [\\[3\\]](https:\/\/books.google.com\/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106)\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-QuaMech_5-0)**\n   Quantum Mechanics, [Kramers, H.A.](https:\/\/en.wikipedia.org\/wiki\/Hendrik_Anthony_Kramers \"Hendrik Anthony Kramers\") publisher Dover, 1957, p. 62 [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-486-66772-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-486-66772-0 \"Special:BookSources\/978-0-486-66772-0\")\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-6)** [Dirac, P. A. M.](https:\/\/en.wikipedia.org\/wiki\/Paul_Adrien_Maurice_Dirac \"Paul Adrien Maurice Dirac\") (1958). *The Principles of Quantum Mechanics*, 4th edition, Oxford, UK: Oxford University Press, p. 17.\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-7)**\n   Solem, J. C.; Biedenharn, L. C. (1993). \"Understanding geometrical phases in quantum mechanics: An elementary example\". *Foundations of Physics*. **23** (2): 185–195\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1993FoPh...23..185S](https:\/\/ui.adsabs.harvard.edu\/abs\/1993FoPh...23..185S). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF01883623](https:\/\/doi.org\/10.1007%2FBF01883623). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [121930907](https:\/\/api.semanticscholar.org\/CorpusID:121930907).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-8)** [Dirac, P. A. M.](https:\/\/en.wikipedia.org\/wiki\/Paul_Adrien_Maurice_Dirac \"Paul Adrien Maurice Dirac\") (1958). *The Principles of Quantum Mechanics*, 4th edition, Oxford, UK: Oxford University Press, p. 14.\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-9)**\n   Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192\n   \n   [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-07-252036-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-07-252036-1 \"Special:BookSources\/0-07-252036-1\")\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-10)**\n    Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785\n    \n    [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-13-301458-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-301458-4 \"Special:BookSources\/0-13-301458-4\")\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-11)** Topics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, [p. 39](http:\/\/www.bcamath.org\/documentos_public\/courses\/1_Course2012Chapter1WavesHistoryApplications.pdf)\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-12)** [Brillouin, L.](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9on_Brillouin \"Léon Brillouin\") (1946). *Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices*, McGraw–Hill, New York, p. 2.\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=13 \"Edit section: Further reading\")\\]\n\n- Haberman, Richard (2004). *Applied Partial Differential Equations*. Prentice Hall. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-13-065243-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-13-065243-0 \"Special:BookSources\/978-0-13-065243-0\")\n  \n  .\n- [Superposition of sound waves](http:\/\/www.acoustics.salford.ac.uk\/feschools\/waves\/super.htm)\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=14 \"Edit section: External links\")\\]\n\n- [![Wikimedia Commons logo](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/4\/4a\/Commons-logo.svg\/20px-Commons-logo.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Commons-logo.svg) Media related to [Superposition principle](https:\/\/commons.wikimedia.org\/wiki\/Category:Superposition_principle \"commons:Category:Superposition principle\") at Wikimedia Commons\n- [![Wiktionary logo](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/99\/Wiktionary-logo-en-v2.svg\/20px-Wiktionary-logo-en-v2.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Wiktionary-logo-en-v2.svg) The dictionary definition of [*interference*](https:\/\/en.wiktionary.org\/wiki\/interference \"wiktionary:interference\") at Wiktionary\n\n| [Authority control databases](https:\/\/en.wikipedia.org\/wiki\/Help:Authority_control \"Help:Authority control\") [![Edit this at Wikidata](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8a\/OOjs_UI_icon_edit-ltr-progressive.svg\/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https:\/\/www.wikidata.org\/wiki\/Q284885#identifiers \"Edit this at Wikidata\") |   |\n|---|---|\n| International | [GND](https:\/\/d-nb.info\/gnd\/4184121-9) |\n| National | [United States](https:\/\/id.loc.gov\/authorities\/sh85130645) [Israel](https:\/\/www.nli.org.il\/en\/authorities\/987007551031105171) |\n| Other | [Yale LUX](https:\/\/lux.collections.yale.edu\/view\/concept\/c1a93c1a-f26f-4d8c-9f03-f022bf9dec69) |\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=Superposition_principle&oldid=1344825758>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Mathematical physics](https:\/\/en.wikipedia.org\/wiki\/Category:Mathematical_physics \"Category:Mathematical physics\")\n- [Waves](https:\/\/en.wikipedia.org\/wiki\/Category:Waves \"Category:Waves\")\n- [Systems theory](https:\/\/en.wikipedia.org\/wiki\/Category:Systems_theory \"Category:Systems theory\")\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- [All articles with unsourced statements](https:\/\/en.wikipedia.org\/wiki\/Category:All_articles_with_unsourced_statements \"Category:All articles with unsourced statements\")\n- [Articles with unsourced statements from March 2023](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_unsourced_statements_from_March_2023 \"Category:Articles with unsourced statements from March 2023\")\n- [Commons category link is on Wikidata](https:\/\/en.wikipedia.org\/wiki\/Category:Commons_category_link_is_on_Wikidata \"Category:Commons category link is on Wikidata\")\n\n- This page was last edited on 22 March 2026, at 19:45 (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=Superposition_principle&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\nSuperposition principle\n\n40 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/7f\/Anas_platyrhynchos_with_ducklings_reflecting_water.jpg\/250px-Anas_platyrhynchos_with_ducklings_reflecting_water.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Anas_platyrhynchos_with_ducklings_reflecting_water.jpg)\n\nSuperposition of almost [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") (diagonal lines) from a distant source and waves from the [wake](https:\/\/en.wikipedia.org\/wiki\/Wake_\\(physics\\) \"Wake (physics)\") of the [ducks](https:\/\/en.wikipedia.org\/wiki\/Duck \"Duck\"). [Linearity](https:\/\/en.wikipedia.org\/wiki\/Linearity \"Linearity\") holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/36\/Rolling_animation.gif\/250px-Rolling_animation.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Rolling_animation.gif)\n\n[Rolling](https:\/\/en.wikipedia.org\/wiki\/Rolling \"Rolling\") motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: [rotation](https:\/\/en.wikipedia.org\/wiki\/Rotation \"Rotation\") without [translation](https:\/\/en.wikipedia.org\/wiki\/Translation_\\(geometry\\) \"Translation (geometry)\"), and translation without rotation.\n\nThe **superposition principle**,[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-1) also known as **superposition property**, states that, for all [linear systems](https:\/\/en.wikipedia.org\/wiki\/Linear_system \"Linear system\"), the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X*, and input *B* produces response *Y*, then input (*A* + *B*) produces response (*X* + *Y*).\n\nA [function](https:\/\/en.wikipedia.org\/wiki\/Function_\\(mathematics\\) \"Function (mathematics)\") ![{\\\\displaystyle F(x)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/71a82805d469cdfa7856c11d6ee756acd1dc7174) that satisfies the superposition principle is called a [linear function](https:\/\/en.wikipedia.org\/wiki\/Linear_function \"Linear function\"). Superposition can be defined by two simpler properties: [additivity](https:\/\/en.wikipedia.org\/wiki\/Additive_map \"Additive map\") ![{\\\\displaystyle F(x\\_{1}+x\\_{2})=F(x\\_{1})+F(x\\_{2})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fe048d07a4d57ffe738373ddafec4fb02faf7b6) and [homogeneity](https:\/\/en.wikipedia.org\/wiki\/Homogeneous_function \"Homogeneous function\") ![{\\\\displaystyle F(ax)=aF(x)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/43e16187439ba65e17e0eddfc4d75a09d47cfb9c) for [scalar](https:\/\/en.wikipedia.org\/wiki\/Scalar_\\(mathematics\\) \"Scalar (mathematics)\") a.\n\nThis principle has many applications in [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\") and [engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering \"Engineering\") because many physical systems can be modeled as linear systems. For example, a [beam](https:\/\/en.wikipedia.org\/wiki\/Beam_\\(structure\\) \"Beam (structure)\") can be modeled as a linear system where the input stimulus is the [load](https:\/\/en.wikipedia.org\/wiki\/Structural_load \"Structural load\") on the beam and the output response is the [deflection](https:\/\/en.wikipedia.org\/wiki\/Deflection_\\(engineering\\) \"Deflection (engineering)\") of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, [frequency-domain](https:\/\/en.wikipedia.org\/wiki\/Frequency-domain \"Frequency-domain\") [linear transform](https:\/\/en.wikipedia.org\/wiki\/Linear_transform \"Linear transform\") methods such as [Fourier](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform \"Fourier transform\") and [Laplace](https:\/\/en.wikipedia.org\/wiki\/Laplace_transform \"Laplace transform\") transforms, and [linear operator](https:\/\/en.wikipedia.org\/wiki\/Linear_operator \"Linear operator\") theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.\n\nThe superposition principle applies to *any* linear system, including [algebraic equations](https:\/\/en.wikipedia.org\/wiki\/Algebraic_equation \"Algebraic equation\"), [linear differential equations](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equations \"Linear differential equations\"), and [systems of equations](https:\/\/en.wikipedia.org\/wiki\/System_of_equations \"System of equations\") of those forms. The stimuli and responses could be numbers, functions, vectors, [vector fields](https:\/\/en.wikipedia.org\/wiki\/Vector_field \"Vector field\"), time-varying signals, or any other object that satisfies [certain axioms](https:\/\/en.wikipedia.org\/wiki\/Vector_space \"Vector space\"). Note that when vectors or vector fields are involved, a superposition is interpreted as a [vector sum](https:\/\/en.wikipedia.org\/wiki\/Vector_sum \"Vector sum\"). If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).\n\n## Relation to Fourier analysis and similar methods\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=1 \"Edit section: Relation to Fourier analysis and similar methods\")\\]\n\nBy writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.\n\nFor example, in [Fourier analysis](https:\/\/en.wikipedia.org\/wiki\/Fourier_analysis \"Fourier analysis\"), the stimulus is written as the superposition of infinitely many [sinusoids](https:\/\/en.wikipedia.org\/wiki\/Sine_wave \"Sine wave\"). Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") and [phase](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\").) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.\n\nAs another common example, in [Green's function analysis](https:\/\/en.wikipedia.org\/wiki\/Green%27s_function \"Green's function\"), the stimulus is written as the superposition of infinitely many [impulse functions](https:\/\/en.wikipedia.org\/wiki\/Impulse_function \"Impulse function\"), and the response is then a superposition of [impulse responses](https:\/\/en.wikipedia.org\/wiki\/Impulse_response \"Impulse response\").\n\nFourier analysis is particularly common for [waves](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\"). For example, in electromagnetic theory, ordinary [light](https:\/\/en.wikipedia.org\/wiki\/Light \"Light\") is described as a superposition of [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") (waves of fixed [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\"), [polarization](https:\/\/en.wikipedia.org\/wiki\/Polarization_\\(waves\\) \"Polarization (waves)\"), and direction). As long as the superposition principle holds (which is often but not always; see [nonlinear optics](https:\/\/en.wikipedia.org\/wiki\/Nonlinear_optics \"Nonlinear optics\")), the behavior of any light wave can be understood as a superposition of the behavior of these simpler [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/7d\/Standing_wave_2.gif\/250px-Standing_wave_2.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Standing_wave_2.gif)\n\nTwo waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a [standing wave](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/3\/3c\/Standing_waves1.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Standing_waves1.gif)\n\nTwo waves permeate without influencing each other\n\nWaves are usually described by variations in some parameters through space and time—for example, height in a water wave, [pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") in a sound wave, or the [electromagnetic field](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_field \"Electromagnetic field\") in a light wave. The value of this parameter is called the [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") of the wave and the wave itself is a [function](https:\/\/en.wikipedia.org\/wiki\/Function_\\(mathematics\\) \"Function (mathematics)\") specifying the amplitude at each point.\n\nIn any system with waves, the waveform at a given time is a function of the [sources](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\") (i.e., external forces, if any, that create or affect the wave) and [initial conditions](https:\/\/en.wikipedia.org\/wiki\/Initial_condition \"Initial condition\") of the system. In many cases (for example, in the classic [wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\")), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)\n\n### Wave diffraction vs. wave interference\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=3 \"Edit section: Wave diffraction vs. wave interference\")\\]\n\nWith regard to wave superposition, [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\") wrote:[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-2)\n\n> No-one has ever been able to define the difference between [interference](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\") and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.\n\nOther authors elaborate:[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-3)\n\n> The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is \\[a matter\\] of degree only, and basically, they are two limiting cases of superposition effects.\n\nYet another source concurs:[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-4)\n\n> In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter \\[Fraunhofer diffraction\\] is, therefore, a continuation of Chapter 8 \\[Interference\\]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront.\n\nThe phenomenon of [interference](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\") between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in [noise-canceling headphones](https:\/\/en.wikipedia.org\/wiki\/Noise-canceling_headphones \"Noise-canceling headphones\"), the summed variation has a smaller [amplitude](https:\/\/en.wikipedia.org\/wiki\/Amplitude \"Amplitude\") than the component variations; this is called *destructive interference*. In other cases, such as in a [line array](https:\/\/en.wikipedia.org\/wiki\/Line_array \"Line array\"), the summed variation will have a bigger amplitude than any of the components individually; this is called *constructive interference*.\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/5d\/Waventerference.gif\/250px-Waventerference.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Waventerference.gif)\n\ngreen wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.\n\n|   |   |   |\n|---|---|---|\n| combined  waveform | [![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0f\/Interference_of_two_waves.svg\/500px-Interference_of_two_waves.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Interference_of_two_waves.svg) |   |\n| wave 1 |   |   |\n| wave 2 |   |   |\n|   | Two waves in phase | Two waves 180° out  of phase |\n\n### Departures from linearity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=5 \"Edit section: Departures from linearity\")\\]\n\nIn most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles [nonlinear optics](https:\/\/en.wikipedia.org\/wiki\/Nonlinear_optics \"Nonlinear optics\") and [nonlinear acoustics](https:\/\/en.wikipedia.org\/wiki\/Nonlinear_acoustics \"Nonlinear acoustics\").\n\n### Quantum superposition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=6 \"Edit section: Quantum superposition\")\\]\n\nIn [quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics\"), a principal task is to compute how a certain type of wave [propagates](https:\/\/en.wikipedia.org\/wiki\/Wave_propagation \"Wave propagation\") and behaves. The wave is described by a [wave function](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\"), and the equation governing its behavior is called the [Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\"). A primary approach to computing the behavior of a wave function is to write it as a superposition (called \"[quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\")\") of (possibly infinitely many) other wave functions of a certain type—[stationary states](https:\/\/en.wikipedia.org\/wiki\/Stationary_state \"Stationary state\") whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-QuaMech-5)\n\nThe projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a *ray* in [projective Hilbert space](https:\/\/en.wikipedia.org\/wiki\/Projective_Hilbert_space \"Projective Hilbert space\"), not a *vector*. According to [Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\"): \"*if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state* \\[italics in original\\].\"[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-6) However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself uses ket vector representations of states to decompose or split, for example, a ket vector ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1284d734d01861923334d7ebfe1ddc855f25a66a) into superposition of component ket vectors ![{\\\\displaystyle \\|\\\\phi \\_{j}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/038f455c0dc2a38568eeac72e59131d3ad41d0a8) as: ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle =\\\\sum \\_{j}{C\\_{j}}\\|\\\\phi \\_{j}\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5e6b0bdd3dcf5868472637fd364ef8131ceb5563) where the ![{\\\\displaystyle C\\_{j}\\\\in {\\\\textbf {C}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f05a8122fd2755f448f968a7765a3f71cb6c2ccb). The equivalence class of the ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1284d734d01861923334d7ebfe1ddc855f25a66a) allows a well-defined meaning to be given to the relative phases of the ![{\\\\displaystyle C\\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4598eb9b4e1c079e0973de79383fc6f898744e18).,[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-7) but an absolute (same amount for all the ![{\\\\displaystyle C\\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4598eb9b4e1c079e0973de79383fc6f898744e18)) phase change on the ![{\\\\displaystyle C\\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4598eb9b4e1c079e0973de79383fc6f898744e18) does not affect the equivalence class of the ![{\\\\displaystyle \\|\\\\psi \\_{i}\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1284d734d01861923334d7ebfe1ddc855f25a66a).\n\nThere are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the [Bloch sphere](https:\/\/en.wikipedia.org\/wiki\/Bloch_sphere \"Bloch sphere\") to represent [pure state](https:\/\/en.wikipedia.org\/wiki\/Pure_state \"Pure state\") of a [two-level quantum mechanical system](https:\/\/en.wikipedia.org\/wiki\/Two-level_system \"Two-level system\") ([qubit](https:\/\/en.wikipedia.org\/wiki\/Qubit \"Qubit\")) is also known as the [Poincaré sphere](https:\/\/en.wikipedia.org\/wiki\/Bloch_sphere \"Bloch sphere\") representing different types of classical pure [polarization](https:\/\/en.wikipedia.org\/wiki\/Polarization_\\(waves\\) \"Polarization (waves)\") states.\n\nNevertheless, on the topic of quantum superposition, [Kramers](https:\/\/en.wikipedia.org\/wiki\/Hans_Kramers \"Hans Kramers\") writes: \"The principle of \\[quantum\\] superposition ... has no analogy in classical physics\"\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]. According to [Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\"): \"*the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory* \\[italics in original\\].\"[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-8) Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from [time translation symmetry](https:\/\/en.wikipedia.org\/wiki\/Time_translation_symmetry \"Time translation symmetry\"), which is also applicable to classical states, as shown above with classical polarization states.\n\n## Boundary-value problems\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=7 \"Edit section: Boundary-value problems\")\\]\n\nA common type of boundary value problem is (to put it abstractly) finding a function *y* that satisfies some equation ![{\\\\displaystyle F(y)=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/937a4b209065cbbd3fe869f60f55cfa5fe9285c5) with some boundary specification ![{\\\\displaystyle G(y)=z.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/adc40e235a4d43a4a07840d82de075f100e5f802) For example, in [Laplace's equation](https:\/\/en.wikipedia.org\/wiki\/Laplace%27s_equation \"Laplace's equation\") with [Dirichlet boundary conditions](https:\/\/en.wikipedia.org\/wiki\/Dirichlet_problem \"Dirichlet problem\"), *F* would be the [Laplacian](https:\/\/en.wikipedia.org\/wiki\/Laplacian \"Laplacian\") operator in a region *R*, *G* would be an operator that restricts *y* to the boundary of *R*, and *z* would be the function that *y* is required to equal on the boundary of *R*.\n\nIn the case that *F* and *G* are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: ![{\\\\displaystyle F(y\\_{1})=F(y\\_{2})=\\\\cdots =0\\\\quad \\\\Rightarrow \\\\quad F(y\\_{1}+y\\_{2}+\\\\cdots )=0,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9df7ad43bb0cb6488758147d3186a5cbcab69ba4) while the boundary values superpose: ![{\\\\displaystyle G(y\\_{1})+G(y\\_{2})=G(y\\_{1}+y\\_{2}).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6050e641d59e43363b8c1ea447ddbe165258b7f5) Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.\n\n## Additive state decomposition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=8 \"Edit section: Additive state decomposition\")\\]\n\nConsider a simple linear system: ![{\\\\displaystyle {\\\\dot {x}}=Ax+B(u\\_{1}+u\\_{2}),\\\\qquad x(0)=x\\_{0}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a9571117cc3b1c0ddcfd61368718f0c8122bc4f0)\n\nBy the superposition principle, the system can be decomposed into ![{\\\\displaystyle {\\\\begin{aligned}{\\\\dot {x}}\\_{1}&=Ax\\_{1}+Bu\\_{1},&\\&x\\_{1}(0)=x\\_{0},\\\\\\\\{\\\\dot {x}}\\_{2}&=Ax\\_{2}+Bu\\_{2},&\\&x\\_{2}(0)=0\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/33cd8d6f87b41a316638576cf0c25130907c9fee) with ![{\\\\displaystyle x=x\\_{1}+x\\_{2}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9ebf439216976f0058d8c371439dc49babe911cb)\n\nSuperposition principle is only available for linear systems. However, the [additive state decomposition](https:\/\/en.wikipedia.org\/wiki\/Additive_state_decomposition \"Additive state decomposition\") can be applied to both linear and nonlinear systems. Next, consider a nonlinear system ![{\\\\displaystyle {\\\\dot {x}}=Ax+B(u\\_{1}+u\\_{2})+\\\\phi \\\\left(c^{\\\\mathsf {T}}x\\\\right),\\\\qquad x(0)=x\\_{0},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fa8951ec8b9f5d31d81ccf266a3510bb70973146) where ![{\\\\displaystyle \\\\phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/72b1f30316670aee6270a28334bdf4f5072cdde4) is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into ![{\\\\displaystyle {\\\\begin{aligned}{\\\\dot {x}}\\_{1}&=Ax\\_{1}+Bu\\_{1}+\\\\phi (y\\_{d}),&\\&x\\_{1}(0)=x\\_{0},\\\\\\\\{\\\\dot {x}}\\_{2}&=Ax\\_{2}+Bu\\_{2}+\\\\phi \\\\left(c^{\\\\mathsf {T}}x\\_{1}+c^{\\\\mathsf {T}}x\\_{2}\\\\right)-\\\\phi (y\\_{d}),&\\&x\\_{2}(0)=0\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2561e25f8bb0b1d1116612026471b378e189ab83) with ![{\\\\displaystyle x=x\\_{1}+x\\_{2}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9ebf439216976f0058d8c371439dc49babe911cb)\n\nThis decomposition can help to simplify controller design.\n\n## Other example applications\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Superposition_principle&action=edit&section=9 \"Edit section: Other example applications\")\\]\n\n- In [electrical engineering](https:\/\/en.wikipedia.org\/wiki\/Electrical_engineering \"Electrical engineering\"), in a [linear circuit](https:\/\/en.wikipedia.org\/wiki\/Linear_circuit \"Linear circuit\"), the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses.\n- In [physics](https:\/\/en.wikipedia.org\/wiki\/Physics \"Physics\"), [Maxwell's equations](https:\/\/en.wikipedia.org\/wiki\/Maxwell%27s_equations \"Maxwell's equations\") imply that the (possibly time-varying) distributions of [charges](https:\/\/en.wikipedia.org\/wiki\/Electric_charge \"Electric charge\") and [currents](https:\/\/en.wikipedia.org\/wiki\/Electric_current \"Electric current\") are related to the [electric](https:\/\/en.wikipedia.org\/wiki\/Electric_field \"Electric field\") and [magnetic fields](https:\/\/en.wikipedia.org\/wiki\/Magnetic_field \"Magnetic field\") by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the [heat equation](https:\/\/en.wikipedia.org\/wiki\/Heat_equation \"Heat equation\").\n- In [engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering \"Engineering\"), superposition is used to solve for beam and structure [deflections](https:\/\/en.wikipedia.org\/wiki\/Deflection_\\(engineering\\) \"Deflection (engineering)\") of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-9) Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-10)\n- In [hydrogeology](https:\/\/en.wikipedia.org\/wiki\/Hydrogeology \"Hydrogeology\"), the superposition principle is applied to the [drawdown](https:\/\/en.wikipedia.org\/wiki\/Drawdown_\\(hydrology\\) \"Drawdown (hydrology)\") of two or more [water wells](https:\/\/en.wikipedia.org\/wiki\/Water_well \"Water well\") pumping in an ideal [aquifer](https:\/\/en.wikipedia.org\/wiki\/Aquifer \"Aquifer\"). This principle is used in the [analytic element method](https:\/\/en.wikipedia.org\/wiki\/Analytic_element_method \"Analytic element method\") to develop analytical elements capable of being combined in a single model.\n- In [process control](https:\/\/en.wikipedia.org\/wiki\/Process_control \"Process control\"), the superposition principle is used in [model predictive control](https:\/\/en.wikipedia.org\/wiki\/Model_predictive_control \"Model predictive control\").\n- The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by [linearization](https:\/\/en.wikipedia.org\/wiki\/Linearization \"Linearization\").\n\nAccording to [Léon Brillouin](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9on_Brillouin \"Léon Brillouin\"), the principle of superposition was first stated by [Daniel Bernoulli](https:\/\/en.wikipedia.org\/wiki\/Daniel_Bernoulli \"Daniel Bernoulli\") in 1753: \"The general motion of a vibrating system is given by a superposition of its proper vibrations.\" The principle was rejected by [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\") and then by [Joseph Lagrange](https:\/\/en.wikipedia.org\/wiki\/Joseph-Louis_Lagrange \"Joseph-Louis Lagrange\"). Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-11)\n\nLater it became accepted, largely through the work of [Joseph Fourier](https:\/\/en.wikipedia.org\/wiki\/Joseph_Fourier \"Joseph Fourier\").[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_note-12)\n\n- [Additive state decomposition](https:\/\/en.wikipedia.org\/wiki\/Additive_state_decomposition \"Additive state decomposition\")\n- [Beat (acoustics)](https:\/\/en.wikipedia.org\/wiki\/Beat_\\(acoustics\\) \"Beat (acoustics)\")\n- [Coherence (physics)](https:\/\/en.wikipedia.org\/wiki\/Coherence_\\(physics\\) \"Coherence (physics)\")\n- [Convolution](https:\/\/en.wikipedia.org\/wiki\/Convolution \"Convolution\")\n- [Green's function](https:\/\/en.wikipedia.org\/wiki\/Green%27s_function \"Green's function\")\n- [Impulse response](https:\/\/en.wikipedia.org\/wiki\/Impulse_response \"Impulse response\")\n- [Interference](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\")\n- [Quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-1)** The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London.\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-2)** Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [\\[1\\]](https:\/\/books.google.com\/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1)\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-3)** N. K. VERMA, *Physics for Engineers*, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [\\[2\\]](https:\/\/books.google.com\/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361)\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-4)** Tim Freegarde, *Introduction to the Physics of Waves*, Cambridge University Press, Nov 8, 2012. [\\[3\\]](https:\/\/books.google.com\/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106)\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-QuaMech_5-0)**\n   Quantum Mechanics, [Kramers, H.A.](https:\/\/en.wikipedia.org\/wiki\/Hendrik_Anthony_Kramers \"Hendrik Anthony Kramers\") publisher Dover, 1957, p. 62 [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-486-66772-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-486-66772-0 \"Special:BookSources\/978-0-486-66772-0\")\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-6)** [Dirac, P. A. M.](https:\/\/en.wikipedia.org\/wiki\/Paul_Adrien_Maurice_Dirac \"Paul Adrien Maurice Dirac\") (1958). *The Principles of Quantum Mechanics*, 4th edition, Oxford, UK: Oxford University Press, p. 17.\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-7)**\n   Solem, J. C.; Biedenharn, L. C. (1993). \"Understanding geometrical phases in quantum mechanics: An elementary example\". *Foundations of Physics*. **23** (2): 185–195\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1993FoPh...23..185S](https:\/\/ui.adsabs.harvard.edu\/abs\/1993FoPh...23..185S). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF01883623](https:\/\/doi.org\/10.1007%2FBF01883623). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [121930907](https:\/\/api.semanticscholar.org\/CorpusID:121930907).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-8)** [Dirac, P. A. M.](https:\/\/en.wikipedia.org\/wiki\/Paul_Adrien_Maurice_Dirac \"Paul Adrien Maurice Dirac\") (1958). *The Principles of Quantum Mechanics*, 4th edition, Oxford, UK: Oxford University Press, p. 14.\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-9)**\n   Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-07-252036-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-07-252036-1 \"Special:BookSources\/0-07-252036-1\")\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-10)**\n    Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-13-301458-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-301458-4 \"Special:BookSources\/0-13-301458-4\")\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-11)** Topics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, [p. 39](http:\/\/www.bcamath.org\/documentos_public\/courses\/1_Course2012Chapter1WavesHistoryApplications.pdf)\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle#cite_ref-12)** [Brillouin, L.](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9on_Brillouin \"Léon Brillouin\") (1946). *Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices*, McGraw–Hill, New York, p. 2.\n\n- Haberman, Richard (2004). *Applied Partial Differential Equations*. Prentice Hall. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-13-065243-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-13-065243-0 \"Special:BookSources\/978-0-13-065243-0\")\n  \n  .\n- [Superposition of sound waves](http:\/\/www.acoustics.salford.ac.uk\/feschools\/waves\/super.htm)","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks . Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths. Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: rotation without translation , and translation without rotation. The superposition principle , [ 1 ] also known as superposition property , states that, for all linear systems , the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X , and input B produces response Y , then input ( A + B ) produces response ( X + Y ). A function that satisfies the superposition principle is called a linear function . Superposition can be defined by two simpler properties: additivity and homogeneity for scalar a . This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency-domain linear transform methods such as Fourier and Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior. The superposition principle applies to any linear system, including algebraic equations , linear differential equations , and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields , time-varying signals, or any other object that satisfies certain axioms . Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum . If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist). Relation to Fourier analysis and similar methods [ edit ] By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute. For example, in Fourier analysis , the stimulus is written as the superposition of infinitely many sinusoids . Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase .) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses. As another common example, in Green's function analysis , the stimulus is written as the superposition of infinitely many impulse functions , and the response is then a superposition of impulse responses . Fourier analysis is particularly common for waves . For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency , polarization , and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics ), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves . Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave . Two waves permeate without influencing each other Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave and the wave itself is a function specifying the amplitude at each point. In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In many cases (for example, in the classic wave equation ), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.) Wave diffraction vs. wave interference [ edit ] With regard to wave superposition, Richard Feynman wrote: [ 2 ] No-one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used. Other authors elaborate: [ 3 ] The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically, they are two limiting cases of superposition effects. Yet another source concurs: [ 4 ] In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is, therefore, a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront. The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-canceling headphones , the summed variation has a smaller amplitude than the component variations; this is called destructive interference . In other cases, such as in a line array , the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference . green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. combined waveform wave 1 wave 2 Two waves in phase Two waves 180° out of phase Departures from linearity [ edit ] In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles nonlinear optics and nonlinear acoustics . Quantum superposition [ edit ] In quantum mechanics , a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function , and the equation governing its behavior is called the Schrödinger equation . A primary approach to computing the behavior of a wave function is to write it as a superposition (called " quantum superposition ") of (possibly infinitely many) other wave functions of a certain type— stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way. [ 5 ] The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a ray in projective Hilbert space , not a vector . According to Dirac : " if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state [italics in original]." [ 6 ] However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself uses ket vector representations of states to decompose or split, for example, a ket vector into superposition of component ket vectors as: where the . The equivalence class of the allows a well-defined meaning to be given to the relative phases of the ., [ 7 ] but an absolute (same amount for all the ) phase change on the does not affect the equivalence class of the . There are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the Bloch sphere to represent pure state of a two-level quantum mechanical system ( qubit ) is also known as the Poincaré sphere representing different types of classical pure polarization states. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics" [ citation needed ] . According to Dirac : " the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]." [ 8 ] Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from time translation symmetry , which is also applicable to classical states, as shown above with classical polarization states. Boundary-value problems [ edit ] A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation with some boundary specification For example, in Laplace's equation with Dirichlet boundary conditions , F would be the Laplacian operator in a region R , G would be an operator that restricts y to the boundary of R , and z would be the function that y is required to equal on the boundary of R . In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: while the boundary values superpose: Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems. Additive state decomposition [ edit ] Consider a simple linear system: By the superposition principle, the system can be decomposed into with Superposition principle is only available for linear systems. However, the additive state decomposition can be applied to both linear and nonlinear systems. Next, consider a nonlinear system where is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into with This decomposition can help to simplify controller design. Other example applications [ edit ] In electrical engineering , in a linear circuit , the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. In physics , Maxwell's equations imply that the (possibly time-varying) distributions of charges and currents are related to the electric and magnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the heat equation . In engineering , superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system). [ 9 ] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure. [ 10 ] In hydrogeology , the superposition principle is applied to the drawdown of two or more water wells pumping in an ideal aquifer . This principle is used in the analytic element method to develop analytical elements capable of being combined in a single model. In process control , the superposition principle is used in model predictive control . The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization . According to Léon Brillouin , the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by Leonhard Euler and then by Joseph Lagrange . Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution. [ 11 ] Later it became accepted, largely through the work of Joseph Fourier . [ 12 ] Additive state decomposition Beat (acoustics) Coherence (physics) Convolution Green's function Impulse response Interference Quantum superposition ^ The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London. ^ Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [1] ^ N. K. VERMA, Physics for Engineers , PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [2] ^ Tim Freegarde, Introduction to the Physics of Waves , Cambridge University Press, Nov 8, 2012. [3] ^ Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62 ISBN 978-0-486-66772-0 ^ Dirac, P. A. M. (1958). The Principles of Quantum Mechanics , 4th edition, Oxford, UK: Oxford University Press, p. 17. ^ Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics . 23 (2): 185– 195. Bibcode : 1993FoPh...23..185S . doi : 10.1007/BF01883623 . S2CID 121930907 . ^ Dirac, P. A. M. (1958). The Principles of Quantum Mechanics , 4th edition, Oxford, UK: Oxford University Press, p. 14. ^ Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 ISBN 0-07-252036-1 ^ Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 ISBN 0-13-301458-4 ^ Topics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, p. 39 ^ Brillouin, L. (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices , McGraw–Hill, New York, p. 2. Haberman, Richard (2004). Applied Partial Differential Equations . Prentice Hall. ISBN 978-0-13-065243-0 . Superposition of sound waves
Markdown	[Jump to content](https://en.wikipedia.org/wiki/Superposition_principle#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=Superposition+principle "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=Superposition+principle "You're encouraged to log in; however, it's not mandatory. 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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Superposition_principle) - [1 Relation to Fourier analysis and similar methods](https://en.wikipedia.org/wiki/Superposition_principle#Relation_to_Fourier_analysis_and_similar_methods) - [2 Wave superposition](https://en.wikipedia.org/wiki/Superposition_principle#Wave_superposition) Toggle Wave superposition subsection - [2\.1 Wave diffraction vs. wave interference](https://en.wikipedia.org/wiki/Superposition_principle#Wave_diffraction_vs._wave_interference) - [2\.2 Wave interference](https://en.wikipedia.org/wiki/Superposition_principle#Wave_interference) - [2\.3 Departures from linearity](https://en.wikipedia.org/wiki/Superposition_principle#Departures_from_linearity) - [2\.4 Quantum superposition](https://en.wikipedia.org/wiki/Superposition_principle#Quantum_superposition) - [3 Boundary-value problems](https://en.wikipedia.org/wiki/Superposition_principle#Boundary-value_problems) - [4 Additive state decomposition](https://en.wikipedia.org/wiki/Superposition_principle#Additive_state_decomposition) - [5 Other example applications](https://en.wikipedia.org/wiki/Superposition_principle#Other_example_applications) - [6 History](https://en.wikipedia.org/wiki/Superposition_principle#History) - [7 See also](https://en.wikipedia.org/wiki/Superposition_principle#See_also) - [8 References](https://en.wikipedia.org/wiki/Superposition_principle#References) - [9 Further reading](https://en.wikipedia.org/wiki/Superposition_principle#Further_reading) - [10 External links](https://en.wikipedia.org/wiki/Superposition_principle#External_links) Toggle the table of contents # Superposition principle 40 languages - [العربية](https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%AF%D8%A3_%D8%A7%D9%84%D8%AA%D8%B1%D8%A7%D9%83%D8%A8 "مبدأ التراكب – Arabic") - [Беларуская (тарашкевіца)](https://be-tarask.wikipedia.org/wiki/%D0%9F%D1%80%D1%8B%D0%BD%D1%86%D1%8B%D0%BF_%D1%81%D1%83%D0%BF%D1%8D%D1%80%D0%BF%D0%B0%D0%B7%D1%8B%D1%86%D1%8B%D1%96 "Прынцып супэрпазыцыі – Belarusian (Taraškievica orthography)") - [Català](https://ca.wikipedia.org/wiki/Principi_de_superposici%C3%B3 "Principi de superposició – Catalan") - 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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%2FSuperposition_principle) Print/export - [Download as PDF](https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=Superposition_principle&action=show-download-screen "Download this page as a PDF file") - [Printable version](https://en.wikipedia.org/w/index.php?title=Superposition_principle&printable=yes "Printable version of this page [p]") In other projects - [Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Superposition_principle) - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q284885 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Fundamental principle of physics This article is about the superposition principle in linear systems. For the geologic principle, see [Law of superposition](https://en.wikipedia.org/wiki/Law_of_superposition "Law of superposition"). For other uses, see [Superposition (disambiguation)](https://en.wikipedia.org/wiki/Superposition_\(disambiguation\) "Superposition (disambiguation)"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Anas_platyrhynchos_with_ducklings_reflecting_water.jpg/250px-Anas_platyrhynchos_with_ducklings_reflecting_water.jpg)](https://en.wikipedia.org/wiki/File:Anas_platyrhynchos_with_ducklings_reflecting_water.jpg) Superposition of almost [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") (diagonal lines) from a distant source and waves from the [wake](https://en.wikipedia.org/wiki/Wake_\(physics\) "Wake (physics)") of the [ducks](https://en.wikipedia.org/wiki/Duck "Duck"). [Linearity](https://en.wikipedia.org/wiki/Linearity "Linearity") holds only approximately in water and only for waves with small amplitudes relative to their wavelengths. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/36/Rolling_animation.gif/250px-Rolling_animation.gif)](https://en.wikipedia.org/wiki/File:Rolling_animation.gif) [Rolling](https://en.wikipedia.org/wiki/Rolling "Rolling") motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: [rotation](https://en.wikipedia.org/wiki/Rotation "Rotation") without [translation](https://en.wikipedia.org/wiki/Translation_\(geometry\) "Translation (geometry)"), and translation without rotation. The superposition principle,[\[1\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-1) also known as superposition property, states that, for all [linear systems](https://en.wikipedia.org/wiki/Linear_system "Linear system"), the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X, and input B produces response Y, then input (A + B) produces response (X + Y). A [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") F ( x ) {\\displaystyle F(x)} ![{\\displaystyle F(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174) that satisfies the superposition principle is called a [linear function](https://en.wikipedia.org/wiki/Linear_function "Linear function"). Superposition can be defined by two simpler properties: [additivity](https://en.wikipedia.org/wiki/Additive_map "Additive map") F ( x 1 \+ x 2 ) \= F ( x 1 ) \+ F ( x 2 ) {\\displaystyle F(x\_{1}+x\_{2})=F(x\_{1})+F(x\_{2})} ![{\\displaystyle F(x\_{1}+x\_{2})=F(x\_{1})+F(x\_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe048d07a4d57ffe738373ddafec4fb02faf7b6) and [homogeneity](https://en.wikipedia.org/wiki/Homogeneous_function "Homogeneous function") F ( a x ) \= a F ( x ) {\\displaystyle F(ax)=aF(x)} ![{\\displaystyle F(ax)=aF(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e16187439ba65e17e0eddfc4d75a09d47cfb9c) for [scalar](https://en.wikipedia.org/wiki/Scalar_\(mathematics\) "Scalar (mathematics)") a. This principle has many applications in [physics](https://en.wikipedia.org/wiki/Physics "Physics") and [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering") because many physical systems can be modeled as linear systems. For example, a [beam](https://en.wikipedia.org/wiki/Beam_\(structure\) "Beam (structure)") can be modeled as a linear system where the input stimulus is the [load](https://en.wikipedia.org/wiki/Structural_load "Structural load") on the beam and the output response is the [deflection](https://en.wikipedia.org/wiki/Deflection_\(engineering\) "Deflection (engineering)") of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, [frequency-domain](https://en.wikipedia.org/wiki/Frequency-domain "Frequency-domain") [linear transform](https://en.wikipedia.org/wiki/Linear_transform "Linear transform") methods such as [Fourier](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") and [Laplace](https://en.wikipedia.org/wiki/Laplace_transform "Laplace transform") transforms, and [linear operator](https://en.wikipedia.org/wiki/Linear_operator "Linear operator") theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior. The superposition principle applies to any linear system, including [algebraic equations](https://en.wikipedia.org/wiki/Algebraic_equation "Algebraic equation"), [linear differential equations](https://en.wikipedia.org/wiki/Linear_differential_equations "Linear differential equations"), and [systems of equations](https://en.wikipedia.org/wiki/System_of_equations "System of equations") of those forms. The stimuli and responses could be numbers, functions, vectors, [vector fields](https://en.wikipedia.org/wiki/Vector_field "Vector field"), time-varying signals, or any other object that satisfies [certain axioms](https://en.wikipedia.org/wiki/Vector_space "Vector space"). Note that when vectors or vector fields are involved, a superposition is interpreted as a [vector sum](https://en.wikipedia.org/wiki/Vector_sum "Vector sum"). If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist). ## Relation to Fourier analysis and similar methods \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=1 "Edit section: Relation to Fourier analysis and similar methods")\] By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute. For example, in [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"), the stimulus is written as the superposition of infinitely many [sinusoids](https://en.wikipedia.org/wiki/Sine_wave "Sine wave"). Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") and [phase](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)").) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses. As another common example, in [Green's function analysis](https://en.wikipedia.org/wiki/Green%27s_function "Green's function"), the stimulus is written as the superposition of infinitely many [impulse functions](https://en.wikipedia.org/wiki/Impulse_function "Impulse function"), and the response is then a superposition of [impulse responses](https://en.wikipedia.org/wiki/Impulse_response "Impulse response"). Fourier analysis is particularly common for [waves](https://en.wikipedia.org/wiki/Wave "Wave"). For example, in electromagnetic theory, ordinary [light](https://en.wikipedia.org/wiki/Light "Light") is described as a superposition of [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") (waves of fixed [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency"), [polarization](https://en.wikipedia.org/wiki/Polarization_\(waves\) "Polarization (waves)"), and direction). As long as the superposition principle holds (which is often but not always; see [nonlinear optics](https://en.wikipedia.org/wiki/Nonlinear_optics "Nonlinear optics")), the behavior of any light wave can be understood as a superposition of the behavior of these simpler [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave"). ## Wave superposition \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=2 "Edit section: Wave superposition")\] Further information: [Wave](https://en.wikipedia.org/wiki/Wave "Wave") and [Wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Standing_wave_2.gif/250px-Standing_wave_2.gif)](https://en.wikipedia.org/wiki/File:Standing_wave_2.gif) Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a [standing wave](https://en.wikipedia.org/wiki/Standing_wave "Standing wave"). [![](https://upload.wikimedia.org/wikipedia/commons/3/3c/Standing_waves1.gif)](https://en.wikipedia.org/wiki/File:Standing_waves1.gif) Two waves permeate without influencing each other Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") in a sound wave, or the [electromagnetic field](https://en.wikipedia.org/wiki/Electromagnetic_field "Electromagnetic field") in a light wave. The value of this parameter is called the [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") of the wave and the wave itself is a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") specifying the amplitude at each point. In any system with waves, the waveform at a given time is a function of the [sources](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") (i.e., external forces, if any, that create or affect the wave) and [initial conditions](https://en.wikipedia.org/wiki/Initial_condition "Initial condition") of the system. In many cases (for example, in the classic [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation")), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.) ### Wave diffraction vs. wave interference \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=3 "Edit section: Wave diffraction vs. wave interference")\] With regard to wave superposition, [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman") wrote:[\[2\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-2) > No-one has ever been able to define the difference between [interference](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used. Other authors elaborate:[\[3\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-3) > The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is \[a matter\] of degree only, and basically, they are two limiting cases of superposition effects. Yet another source concurs:[\[4\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-4) > In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter \[Fraunhofer diffraction\] is, therefore, a continuation of Chapter 8 \[Interference\]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront. ### Wave interference \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=4 "Edit section: Wave interference")\] Main article: [Interference (wave propagation)](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") The phenomenon of [interference](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in [noise-canceling headphones](https://en.wikipedia.org/wiki/Noise-canceling_headphones "Noise-canceling headphones"), the summed variation has a smaller [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") than the component variations; this is called destructive interference. In other cases, such as in a [line array](https://en.wikipedia.org/wiki/Line_array "Line array"), the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Waventerference.gif/250px-Waventerference.gif)](https://en.wikipedia.org/wiki/File:Waventerference.gif) green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. \| \| \| \| \|---\|---\|---\| \| combined waveform \| [![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Interference_of_two_waves.svg/500px-Interference_of_two_waves.svg.png)](https://en.wikipedia.org/wiki/File:Interference_of_two_waves.svg) \| \| \| wave 1 \| \| \| \| wave 2 \| \| \| \| \| Two waves in phase \| Two waves 180° out of phase \| ### Departures from linearity \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=5 "Edit section: Departures from linearity")\] In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles [nonlinear optics](https://en.wikipedia.org/wiki/Nonlinear_optics "Nonlinear optics") and [nonlinear acoustics](https://en.wikipedia.org/wiki/Nonlinear_acoustics "Nonlinear acoustics"). ### Quantum superposition \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=6 "Edit section: Quantum superposition")\] Main article: [Quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition") In [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics"), a principal task is to compute how a certain type of wave [propagates](https://en.wikipedia.org/wiki/Wave_propagation "Wave propagation") and behaves. The wave is described by a [wave function](https://en.wikipedia.org/wiki/Wave_function "Wave function"), and the equation governing its behavior is called the [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"). A primary approach to computing the behavior of a wave function is to write it as a superposition (called "[quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition")") of (possibly infinitely many) other wave functions of a certain type—[stationary states](https://en.wikipedia.org/wiki/Stationary_state "Stationary state") whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[\[5\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-QuaMech-5) The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a ray in [projective Hilbert space](https://en.wikipedia.org/wiki/Projective_Hilbert_space "Projective Hilbert space"), not a vector. According to [Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac"): "if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state \[italics in original\]."[\[6\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-6) However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself uses ket vector representations of states to decompose or split, for example, a ket vector \\| ψ i ⟩ {\\displaystyle \\|\\psi \_{i}\\rangle } ![{\\displaystyle \\|\\psi \_{i}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1284d734d01861923334d7ebfe1ddc855f25a66a) into superposition of component ket vectors \\| ϕ j ⟩ {\\displaystyle \\|\\phi \_{j}\\rangle } ![{\\displaystyle \\|\\phi \_{j}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/038f455c0dc2a38568eeac72e59131d3ad41d0a8) as: \\| ψ i ⟩ \= ∑ j C j \\| ϕ j ⟩ , {\\displaystyle \\|\\psi \_{i}\\rangle =\\sum \_{j}{C\_{j}}\\|\\phi \_{j}\\rangle ,} ![{\\displaystyle \\|\\psi \_{i}\\rangle =\\sum \_{j}{C\_{j}}\\|\\phi \_{j}\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6b0bdd3dcf5868472637fd364ef8131ceb5563) where the C j ∈ C {\\displaystyle C\_{j}\\in {\\textbf {C}}} ![{\\displaystyle C\_{j}\\in {\\textbf {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f05a8122fd2755f448f968a7765a3f71cb6c2ccb). The equivalence class of the \\| ψ i ⟩ {\\displaystyle \\|\\psi \_{i}\\rangle } ![{\\displaystyle \\|\\psi \_{i}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1284d734d01861923334d7ebfe1ddc855f25a66a) allows a well-defined meaning to be given to the relative phases of the C j {\\displaystyle C\_{j}} ![{\\displaystyle C\_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18).,[\[7\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-7) but an absolute (same amount for all the C j {\\displaystyle C\_{j}} ![{\\displaystyle C\_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18)) phase change on the C j {\\displaystyle C\_{j}} ![{\\displaystyle C\_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18) does not affect the equivalence class of the \\| ψ i ⟩ {\\displaystyle \\|\\psi \_{i}\\rangle } ![{\\displaystyle \\|\\psi \_{i}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1284d734d01861923334d7ebfe1ddc855f25a66a). There are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the [Bloch sphere](https://en.wikipedia.org/wiki/Bloch_sphere "Bloch sphere") to represent [pure state](https://en.wikipedia.org/wiki/Pure_state "Pure state") of a [two-level quantum mechanical system](https://en.wikipedia.org/wiki/Two-level_system "Two-level system") ([qubit](https://en.wikipedia.org/wiki/Qubit "Qubit")) is also known as the [Poincaré sphere](https://en.wikipedia.org/wiki/Bloch_sphere "Bloch sphere") representing different types of classical pure [polarization](https://en.wikipedia.org/wiki/Polarization_\(waves\) "Polarization (waves)") states. Nevertheless, on the topic of quantum superposition, [Kramers](https://en.wikipedia.org/wiki/Hans_Kramers "Hans Kramers") writes: "The principle of \[quantum\] superposition ... has no analogy in classical physics"\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\]. According to [Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac"): "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory \[italics in original\]."[\[8\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-8) Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from [time translation symmetry](https://en.wikipedia.org/wiki/Time_translation_symmetry "Time translation symmetry"), which is also applicable to classical states, as shown above with classical polarization states. ## Boundary-value problems \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=7 "Edit section: Boundary-value problems")\] Further information: [Boundary-value problem](https://en.wikipedia.org/wiki/Boundary-value_problem "Boundary-value problem") A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation F ( y ) \= 0 {\\displaystyle F(y)=0} ![{\\displaystyle F(y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/937a4b209065cbbd3fe869f60f55cfa5fe9285c5) with some boundary specification G ( y ) \= z . {\\displaystyle G(y)=z.} ![{\\displaystyle G(y)=z.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adc40e235a4d43a4a07840d82de075f100e5f802) For example, in [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") with [Dirichlet boundary conditions](https://en.wikipedia.org/wiki/Dirichlet_problem "Dirichlet problem"), F would be the [Laplacian](https://en.wikipedia.org/wiki/Laplacian "Laplacian") operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R. In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: F ( y 1 ) \= F ( y 2 ) \= ⋯ \= 0 ⇒ F ( y 1 \+ y 2 \+ ⋯ ) \= 0 , {\\displaystyle F(y\_{1})=F(y\_{2})=\\cdots =0\\quad \\Rightarrow \\quad F(y\_{1}+y\_{2}+\\cdots )=0,} ![{\\displaystyle F(y\_{1})=F(y\_{2})=\\cdots =0\\quad \\Rightarrow \\quad F(y\_{1}+y\_{2}+\\cdots )=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df7ad43bb0cb6488758147d3186a5cbcab69ba4) while the boundary values superpose: G ( y 1 ) \+ G ( y 2 ) \= G ( y 1 \+ y 2 ) . {\\displaystyle G(y\_{1})+G(y\_{2})=G(y\_{1}+y\_{2}).} ![{\\displaystyle G(y\_{1})+G(y\_{2})=G(y\_{1}+y\_{2}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6050e641d59e43363b8c1ea447ddbe165258b7f5) Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems. ## Additive state decomposition \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=8 "Edit section: Additive state decomposition")\] Main article: [Additive state decomposition](https://en.wikipedia.org/wiki/Additive_state_decomposition "Additive state decomposition") Consider a simple linear system: x ˙ \= A x \+ B ( u 1 \+ u 2 ) , x ( 0 ) \= x 0 . {\\displaystyle {\\dot {x}}=Ax+B(u\_{1}+u\_{2}),\\qquad x(0)=x\_{0}.} ![{\\displaystyle {\\dot {x}}=Ax+B(u\_{1}+u\_{2}),\\qquad x(0)=x\_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9571117cc3b1c0ddcfd61368718f0c8122bc4f0) By the superposition principle, the system can be decomposed into x ˙ 1 \= A x 1 \+ B u 1 , x 1 ( 0 ) \= x 0 , x ˙ 2 \= A x 2 \+ B u 2 , x 2 ( 0 ) \= 0 {\\displaystyle {\\begin{aligned}{\\dot {x}}\_{1}&=Ax\_{1}+Bu\_{1},&\&x\_{1}(0)=x\_{0},\\\\{\\dot {x}}\_{2}&=Ax\_{2}+Bu\_{2},&\&x\_{2}(0)=0\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}{\\dot {x}}\_{1}&=Ax\_{1}+Bu\_{1},&\&x\_{1}(0)=x\_{0},\\\\{\\dot {x}}\_{2}&=Ax\_{2}+Bu\_{2},&\&x\_{2}(0)=0\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33cd8d6f87b41a316638576cf0c25130907c9fee) with x \= x 1 \+ x 2 . {\\displaystyle x=x\_{1}+x\_{2}.} ![{\\displaystyle x=x\_{1}+x\_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebf439216976f0058d8c371439dc49babe911cb) Superposition principle is only available for linear systems. However, the [additive state decomposition](https://en.wikipedia.org/wiki/Additive_state_decomposition "Additive state decomposition") can be applied to both linear and nonlinear systems. Next, consider a nonlinear system x ˙ \= A x \+ B ( u 1 \+ u 2 ) \+ ϕ ( c T x ) , x ( 0 ) \= x 0 , {\\displaystyle {\\dot {x}}=Ax+B(u\_{1}+u\_{2})+\\phi \\left(c^{\\mathsf {T}}x\\right),\\qquad x(0)=x\_{0},} ![{\\displaystyle {\\dot {x}}=Ax+B(u\_{1}+u\_{2})+\\phi \\left(c^{\\mathsf {T}}x\\right),\\qquad x(0)=x\_{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8951ec8b9f5d31d81ccf266a3510bb70973146) where ϕ {\\displaystyle \\phi } ![{\\displaystyle \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4) is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into x ˙ 1 \= A x 1 \+ B u 1 \+ ϕ ( y d ) , x 1 ( 0 ) \= x 0 , x ˙ 2 \= A x 2 \+ B u 2 \+ ϕ ( c T x 1 \+ c T x 2 ) − ϕ ( y d ) , x 2 ( 0 ) \= 0 {\\displaystyle {\\begin{aligned}{\\dot {x}}\_{1}&=Ax\_{1}+Bu\_{1}+\\phi (y\_{d}),&\&x\_{1}(0)=x\_{0},\\\\{\\dot {x}}\_{2}&=Ax\_{2}+Bu\_{2}+\\phi \\left(c^{\\mathsf {T}}x\_{1}+c^{\\mathsf {T}}x\_{2}\\right)-\\phi (y\_{d}),&\&x\_{2}(0)=0\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}{\\dot {x}}\_{1}&=Ax\_{1}+Bu\_{1}+\\phi (y\_{d}),&\&x\_{1}(0)=x\_{0},\\\\{\\dot {x}}\_{2}&=Ax\_{2}+Bu\_{2}+\\phi \\left(c^{\\mathsf {T}}x\_{1}+c^{\\mathsf {T}}x\_{2}\\right)-\\phi (y\_{d}),&\&x\_{2}(0)=0\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2561e25f8bb0b1d1116612026471b378e189ab83) with x \= x 1 \+ x 2 . {\\displaystyle x=x\_{1}+x\_{2}.} ![{\\displaystyle x=x\_{1}+x\_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebf439216976f0058d8c371439dc49babe911cb) This decomposition can help to simplify controller design. ## Other example applications \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=9 "Edit section: Other example applications")\] - In [electrical engineering](https://en.wikipedia.org/wiki/Electrical_engineering "Electrical engineering"), in a [linear circuit](https://en.wikipedia.org/wiki/Linear_circuit "Linear circuit"), the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. - In [physics](https://en.wikipedia.org/wiki/Physics "Physics"), [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations") imply that the (possibly time-varying) distributions of [charges](https://en.wikipedia.org/wiki/Electric_charge "Electric charge") and [currents](https://en.wikipedia.org/wiki/Electric_current "Electric current") are related to the [electric](https://en.wikipedia.org/wiki/Electric_field "Electric field") and [magnetic fields](https://en.wikipedia.org/wiki/Magnetic_field "Magnetic field") by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"). - In [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"), superposition is used to solve for beam and structure [deflections](https://en.wikipedia.org/wiki/Deflection_\(engineering\) "Deflection (engineering)") of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[\[9\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-9) Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[\[10\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-10) - In [hydrogeology](https://en.wikipedia.org/wiki/Hydrogeology "Hydrogeology"), the superposition principle is applied to the [drawdown](https://en.wikipedia.org/wiki/Drawdown_\(hydrology\) "Drawdown (hydrology)") of two or more [water wells](https://en.wikipedia.org/wiki/Water_well "Water well") pumping in an ideal [aquifer](https://en.wikipedia.org/wiki/Aquifer "Aquifer"). This principle is used in the [analytic element method](https://en.wikipedia.org/wiki/Analytic_element_method "Analytic element method") to develop analytical elements capable of being combined in a single model. - In [process control](https://en.wikipedia.org/wiki/Process_control "Process control"), the superposition principle is used in [model predictive control](https://en.wikipedia.org/wiki/Model_predictive_control "Model predictive control"). - The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by [linearization](https://en.wikipedia.org/wiki/Linearization "Linearization"). ## History \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=10 "Edit section: History")\] According to [Léon Brillouin](https://en.wikipedia.org/wiki/L%C3%A9on_Brillouin "Léon Brillouin"), the principle of superposition was first stated by [Daniel Bernoulli](https://en.wikipedia.org/wiki/Daniel_Bernoulli "Daniel Bernoulli") in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") and then by [Joseph Lagrange](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange "Joseph-Louis Lagrange"). Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.[\[11\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-11) Later it became accepted, largely through the work of [Joseph Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier").[\[12\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-12) ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=11 "Edit section: See also")\] - [Additive state decomposition](https://en.wikipedia.org/wiki/Additive_state_decomposition "Additive state decomposition") - [Beat (acoustics)](https://en.wikipedia.org/wiki/Beat_\(acoustics\) "Beat (acoustics)") - [Coherence (physics)](https://en.wikipedia.org/wiki/Coherence_\(physics\) "Coherence (physics)") - [Convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") - [Green's function](https://en.wikipedia.org/wiki/Green%27s_function "Green's function") - [Impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") - [Interference](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") - [Quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition") ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=12 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-1) The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London. 2. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-2) Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [\[1\]](https://books.google.com/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1) 3. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-3) N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [\[2\]](https://books.google.com/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361) 4. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-4) Tim Freegarde, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [\[3\]](https://books.google.com/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106) 5. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-QuaMech_5-0) Quantum Mechanics, [Kramers, H.A.](https://en.wikipedia.org/wiki/Hendrik_Anthony_Kramers "Hendrik Anthony Kramers") publisher Dover, 1957, p. 62 [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-66772-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-66772-0 "Special:BookSources/978-0-486-66772-0") 6. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-6) [Dirac, P. A. M.](https://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac "Paul Adrien Maurice Dirac") (1958). The Principles of Quantum Mechanics, 4th edition, Oxford, UK: Oxford University Press, p. 17. 7. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-7) Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1993FoPh...23..185S](https://ui.adsabs.harvard.edu/abs/1993FoPh...23..185S). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01883623](https://doi.org/10.1007%2FBF01883623). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [121930907](https://api.semanticscholar.org/CorpusID:121930907). 8. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-8) [Dirac, P. A. M.](https://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac "Paul Adrien Maurice Dirac") (1958). The Principles of Quantum Mechanics, 4th edition, Oxford, UK: Oxford University Press, p. 14. 9. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-9) Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-07-252036-1](https://en.wikipedia.org/wiki/Special:BookSources/0-07-252036-1 "Special:BookSources/0-07-252036-1") 10. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-10) Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-301458-4](https://en.wikipedia.org/wiki/Special:BookSources/0-13-301458-4 "Special:BookSources/0-13-301458-4") 11. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-11) Topics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, [p. 39](http://www.bcamath.org/documentos_public/courses/1_Course2012Chapter1WavesHistoryApplications.pdf) 12. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-12) [Brillouin, L.](https://en.wikipedia.org/wiki/L%C3%A9on_Brillouin "Léon Brillouin") (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2. ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=13 "Edit section: Further reading")\] - Haberman, Richard (2004). Applied Partial Differential Equations. Prentice Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-13-065243-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-065243-0 "Special:BookSources/978-0-13-065243-0") . - [Superposition of sound waves](http://www.acoustics.salford.ac.uk/feschools/waves/super.htm) ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=14 "Edit section: External links")\] - [![Wikimedia Commons logo](https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png)](https://en.wikipedia.org/wiki/File:Commons-logo.svg) Media related to [Superposition principle](https://commons.wikimedia.org/wiki/Category:Superposition_principle "commons:Category:Superposition principle") at Wikimedia Commons - [![Wiktionary logo](https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/20px-Wiktionary-logo-en-v2.svg.png)](https://en.wikipedia.org/wiki/File:Wiktionary-logo-en-v2.svg) The dictionary definition of [interference](https://en.wiktionary.org/wiki/interference "wiktionary:interference") at Wiktionary \| [Authority control databases](https://en.wikipedia.org/wiki/Help:Authority_control "Help:Authority control") [![Edit this at Wikidata](https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https://www.wikidata.org/wiki/Q284885#identifiers "Edit this at Wikidata") \| \| \|---\|---\| \| International \| [GND](https://d-nb.info/gnd/4184121-9) \| \| National \| [United States](https://id.loc.gov/authorities/sh85130645) [Israel](https://www.nli.org.il/en/authorities/987007551031105171) \| \| Other \| [Yale LUX](https://lux.collections.yale.edu/view/concept/c1a93c1a-f26f-4d8c-9f03-f022bf9dec69) \| ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Superposition_principle&oldid=1344825758>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Mathematical physics](https://en.wikipedia.org/wiki/Category:Mathematical_physics "Category:Mathematical physics") - [Waves](https://en.wikipedia.org/wiki/Category:Waves "Category:Waves") - [Systems theory](https://en.wikipedia.org/wiki/Category:Systems_theory "Category:Systems theory") Hidden categories: - [Articles with short description](https://en.wikipedia.org/wiki/Category:Articles_with_short_description "Category:Articles with short description") - [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") - [All articles with unsourced statements](https://en.wikipedia.org/wiki/Category:All_articles_with_unsourced_statements "Category:All articles with unsourced statements") - [Articles with unsourced statements from March 2023](https://en.wikipedia.org/wiki/Category:Articles_with_unsourced_statements_from_March_2023 "Category:Articles with unsourced statements from March 2023") - [Commons category link is on Wikidata](https://en.wikipedia.org/wiki/Category:Commons_category_link_is_on_Wikidata "Category:Commons category link is on Wikidata") - This page was last edited on 22 March 2026, at 19:45 (UTC). - 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. 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Readable Markdown	From Wikipedia, the free encyclopedia [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Anas_platyrhynchos_with_ducklings_reflecting_water.jpg/250px-Anas_platyrhynchos_with_ducklings_reflecting_water.jpg)](https://en.wikipedia.org/wiki/File:Anas_platyrhynchos_with_ducklings_reflecting_water.jpg) Superposition of almost [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") (diagonal lines) from a distant source and waves from the [wake](https://en.wikipedia.org/wiki/Wake_\(physics\) "Wake (physics)") of the [ducks](https://en.wikipedia.org/wiki/Duck "Duck"). [Linearity](https://en.wikipedia.org/wiki/Linearity "Linearity") holds only approximately in water and only for waves with small amplitudes relative to their wavelengths. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/36/Rolling_animation.gif/250px-Rolling_animation.gif)](https://en.wikipedia.org/wiki/File:Rolling_animation.gif) [Rolling](https://en.wikipedia.org/wiki/Rolling "Rolling") motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: [rotation](https://en.wikipedia.org/wiki/Rotation "Rotation") without [translation](https://en.wikipedia.org/wiki/Translation_\(geometry\) "Translation (geometry)"), and translation without rotation. The superposition principle,[\[1\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-1) also known as superposition property, states that, for all [linear systems](https://en.wikipedia.org/wiki/Linear_system "Linear system"), the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X, and input B produces response Y, then input (A + B) produces response (X + Y). A [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") ![{\\displaystyle F(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174) that satisfies the superposition principle is called a [linear function](https://en.wikipedia.org/wiki/Linear_function "Linear function"). Superposition can be defined by two simpler properties: [additivity](https://en.wikipedia.org/wiki/Additive_map "Additive map") ![{\\displaystyle F(x\_{1}+x\_{2})=F(x\_{1})+F(x\_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe048d07a4d57ffe738373ddafec4fb02faf7b6) and [homogeneity](https://en.wikipedia.org/wiki/Homogeneous_function "Homogeneous function") ![{\\displaystyle F(ax)=aF(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e16187439ba65e17e0eddfc4d75a09d47cfb9c) for [scalar](https://en.wikipedia.org/wiki/Scalar_\(mathematics\) "Scalar (mathematics)") a. This principle has many applications in [physics](https://en.wikipedia.org/wiki/Physics "Physics") and [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering") because many physical systems can be modeled as linear systems. For example, a [beam](https://en.wikipedia.org/wiki/Beam_\(structure\) "Beam (structure)") can be modeled as a linear system where the input stimulus is the [load](https://en.wikipedia.org/wiki/Structural_load "Structural load") on the beam and the output response is the [deflection](https://en.wikipedia.org/wiki/Deflection_\(engineering\) "Deflection (engineering)") of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, [frequency-domain](https://en.wikipedia.org/wiki/Frequency-domain "Frequency-domain") [linear transform](https://en.wikipedia.org/wiki/Linear_transform "Linear transform") methods such as [Fourier](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") and [Laplace](https://en.wikipedia.org/wiki/Laplace_transform "Laplace transform") transforms, and [linear operator](https://en.wikipedia.org/wiki/Linear_operator "Linear operator") theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior. The superposition principle applies to any linear system, including [algebraic equations](https://en.wikipedia.org/wiki/Algebraic_equation "Algebraic equation"), [linear differential equations](https://en.wikipedia.org/wiki/Linear_differential_equations "Linear differential equations"), and [systems of equations](https://en.wikipedia.org/wiki/System_of_equations "System of equations") of those forms. The stimuli and responses could be numbers, functions, vectors, [vector fields](https://en.wikipedia.org/wiki/Vector_field "Vector field"), time-varying signals, or any other object that satisfies [certain axioms](https://en.wikipedia.org/wiki/Vector_space "Vector space"). Note that when vectors or vector fields are involved, a superposition is interpreted as a [vector sum](https://en.wikipedia.org/wiki/Vector_sum "Vector sum"). If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist). ## Relation to Fourier analysis and similar methods \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=1 "Edit section: Relation to Fourier analysis and similar methods")\] By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute. For example, in [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"), the stimulus is written as the superposition of infinitely many [sinusoids](https://en.wikipedia.org/wiki/Sine_wave "Sine wave"). Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") and [phase](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)").) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses. As another common example, in [Green's function analysis](https://en.wikipedia.org/wiki/Green%27s_function "Green's function"), the stimulus is written as the superposition of infinitely many [impulse functions](https://en.wikipedia.org/wiki/Impulse_function "Impulse function"), and the response is then a superposition of [impulse responses](https://en.wikipedia.org/wiki/Impulse_response "Impulse response"). Fourier analysis is particularly common for [waves](https://en.wikipedia.org/wiki/Wave "Wave"). For example, in electromagnetic theory, ordinary [light](https://en.wikipedia.org/wiki/Light "Light") is described as a superposition of [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") (waves of fixed [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency"), [polarization](https://en.wikipedia.org/wiki/Polarization_\(waves\) "Polarization (waves)"), and direction). As long as the superposition principle holds (which is often but not always; see [nonlinear optics](https://en.wikipedia.org/wiki/Nonlinear_optics "Nonlinear optics")), the behavior of any light wave can be understood as a superposition of the behavior of these simpler [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Standing_wave_2.gif/250px-Standing_wave_2.gif)](https://en.wikipedia.org/wiki/File:Standing_wave_2.gif) Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a [standing wave](https://en.wikipedia.org/wiki/Standing_wave "Standing wave"). [![](https://upload.wikimedia.org/wikipedia/commons/3/3c/Standing_waves1.gif)](https://en.wikipedia.org/wiki/File:Standing_waves1.gif) Two waves permeate without influencing each other Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") in a sound wave, or the [electromagnetic field](https://en.wikipedia.org/wiki/Electromagnetic_field "Electromagnetic field") in a light wave. The value of this parameter is called the [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") of the wave and the wave itself is a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") specifying the amplitude at each point. In any system with waves, the waveform at a given time is a function of the [sources](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") (i.e., external forces, if any, that create or affect the wave) and [initial conditions](https://en.wikipedia.org/wiki/Initial_condition "Initial condition") of the system. In many cases (for example, in the classic [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation")), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.) ### Wave diffraction vs. wave interference \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=3 "Edit section: Wave diffraction vs. wave interference")\] With regard to wave superposition, [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman") wrote:[\[2\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-2) > No-one has ever been able to define the difference between [interference](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used. Other authors elaborate:[\[3\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-3) > The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is \[a matter\] of degree only, and basically, they are two limiting cases of superposition effects. Yet another source concurs:[\[4\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-4) > In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter \[Fraunhofer diffraction\] is, therefore, a continuation of Chapter 8 \[Interference\]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront. The phenomenon of [interference](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in [noise-canceling headphones](https://en.wikipedia.org/wiki/Noise-canceling_headphones "Noise-canceling headphones"), the summed variation has a smaller [amplitude](https://en.wikipedia.org/wiki/Amplitude "Amplitude") than the component variations; this is called destructive interference. In other cases, such as in a [line array](https://en.wikipedia.org/wiki/Line_array "Line array"), the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference. [![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Waventerference.gif/250px-Waventerference.gif)](https://en.wikipedia.org/wiki/File:Waventerference.gif) green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. \| \| \| \| \|---\|---\|---\| \| combined waveform \| [![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Interference_of_two_waves.svg/500px-Interference_of_two_waves.svg.png)](https://en.wikipedia.org/wiki/File:Interference_of_two_waves.svg) \| \| \| wave 1 \| \| \| \| wave 2 \| \| \| \| \| Two waves in phase \| Two waves 180° out of phase \| ### Departures from linearity \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=5 "Edit section: Departures from linearity")\] In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles [nonlinear optics](https://en.wikipedia.org/wiki/Nonlinear_optics "Nonlinear optics") and [nonlinear acoustics](https://en.wikipedia.org/wiki/Nonlinear_acoustics "Nonlinear acoustics"). ### Quantum superposition \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=6 "Edit section: Quantum superposition")\] In [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics"), a principal task is to compute how a certain type of wave [propagates](https://en.wikipedia.org/wiki/Wave_propagation "Wave propagation") and behaves. The wave is described by a [wave function](https://en.wikipedia.org/wiki/Wave_function "Wave function"), and the equation governing its behavior is called the [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"). A primary approach to computing the behavior of a wave function is to write it as a superposition (called "[quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition")") of (possibly infinitely many) other wave functions of a certain type—[stationary states](https://en.wikipedia.org/wiki/Stationary_state "Stationary state") whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[\[5\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-QuaMech-5) The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a ray in [projective Hilbert space](https://en.wikipedia.org/wiki/Projective_Hilbert_space "Projective Hilbert space"), not a vector. According to [Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac"): "if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state \[italics in original\]."[\[6\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-6) However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself uses ket vector representations of states to decompose or split, for example, a ket vector ![{\\displaystyle \\|\\psi \_{i}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1284d734d01861923334d7ebfe1ddc855f25a66a) into superposition of component ket vectors ![{\\displaystyle \\|\\phi \_{j}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/038f455c0dc2a38568eeac72e59131d3ad41d0a8) as: ![{\\displaystyle \\|\\psi \_{i}\\rangle =\\sum \_{j}{C\_{j}}\\|\\phi \_{j}\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6b0bdd3dcf5868472637fd364ef8131ceb5563) where the ![{\\displaystyle C\_{j}\\in {\\textbf {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f05a8122fd2755f448f968a7765a3f71cb6c2ccb). The equivalence class of the ![{\\displaystyle \\|\\psi \_{i}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1284d734d01861923334d7ebfe1ddc855f25a66a) allows a well-defined meaning to be given to the relative phases of the ![{\\displaystyle C\_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18).,[\[7\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-7) but an absolute (same amount for all the ![{\\displaystyle C\_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18)) phase change on the ![{\\displaystyle C\_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18) does not affect the equivalence class of the ![{\\displaystyle \\|\\psi \_{i}\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1284d734d01861923334d7ebfe1ddc855f25a66a). There are exact correspondences between the superposition presented in the main on this page and the quantum superposition. For example, the [Bloch sphere](https://en.wikipedia.org/wiki/Bloch_sphere "Bloch sphere") to represent [pure state](https://en.wikipedia.org/wiki/Pure_state "Pure state") of a [two-level quantum mechanical system](https://en.wikipedia.org/wiki/Two-level_system "Two-level system") ([qubit](https://en.wikipedia.org/wiki/Qubit "Qubit")) is also known as the [Poincaré sphere](https://en.wikipedia.org/wiki/Bloch_sphere "Bloch sphere") representing different types of classical pure [polarization](https://en.wikipedia.org/wiki/Polarization_\(waves\) "Polarization (waves)") states. Nevertheless, on the topic of quantum superposition, [Kramers](https://en.wikipedia.org/wiki/Hans_Kramers "Hans Kramers") writes: "The principle of \[quantum\] superposition ... has no analogy in classical physics"\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\]. According to [Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac"): "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory \[italics in original\]."[\[8\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-8) Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase, they actually mean phase translation symmetry derived from [time translation symmetry](https://en.wikipedia.org/wiki/Time_translation_symmetry "Time translation symmetry"), which is also applicable to classical states, as shown above with classical polarization states. ## Boundary-value problems \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=7 "Edit section: Boundary-value problems")\] A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation ![{\\displaystyle F(y)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/937a4b209065cbbd3fe869f60f55cfa5fe9285c5) with some boundary specification ![{\\displaystyle G(y)=z.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adc40e235a4d43a4a07840d82de075f100e5f802) For example, in [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") with [Dirichlet boundary conditions](https://en.wikipedia.org/wiki/Dirichlet_problem "Dirichlet problem"), F would be the [Laplacian](https://en.wikipedia.org/wiki/Laplacian "Laplacian") operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R. In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation: ![{\\displaystyle F(y\_{1})=F(y\_{2})=\\cdots =0\\quad \\Rightarrow \\quad F(y\_{1}+y\_{2}+\\cdots )=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df7ad43bb0cb6488758147d3186a5cbcab69ba4) while the boundary values superpose: ![{\\displaystyle G(y\_{1})+G(y\_{2})=G(y\_{1}+y\_{2}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6050e641d59e43363b8c1ea447ddbe165258b7f5) Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems. ## Additive state decomposition \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=8 "Edit section: Additive state decomposition")\] Consider a simple linear system: ![{\\displaystyle {\\dot {x}}=Ax+B(u\_{1}+u\_{2}),\\qquad x(0)=x\_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9571117cc3b1c0ddcfd61368718f0c8122bc4f0) By the superposition principle, the system can be decomposed into ![{\\displaystyle {\\begin{aligned}{\\dot {x}}\_{1}&=Ax\_{1}+Bu\_{1},&\&x\_{1}(0)=x\_{0},\\\\{\\dot {x}}\_{2}&=Ax\_{2}+Bu\_{2},&\&x\_{2}(0)=0\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33cd8d6f87b41a316638576cf0c25130907c9fee) with ![{\\displaystyle x=x\_{1}+x\_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebf439216976f0058d8c371439dc49babe911cb) Superposition principle is only available for linear systems. However, the [additive state decomposition](https://en.wikipedia.org/wiki/Additive_state_decomposition "Additive state decomposition") can be applied to both linear and nonlinear systems. Next, consider a nonlinear system ![{\\displaystyle {\\dot {x}}=Ax+B(u\_{1}+u\_{2})+\\phi \\left(c^{\\mathsf {T}}x\\right),\\qquad x(0)=x\_{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8951ec8b9f5d31d81ccf266a3510bb70973146) where ![{\\displaystyle \\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4) is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into ![{\\displaystyle {\\begin{aligned}{\\dot {x}}\_{1}&=Ax\_{1}+Bu\_{1}+\\phi (y\_{d}),&\&x\_{1}(0)=x\_{0},\\\\{\\dot {x}}\_{2}&=Ax\_{2}+Bu\_{2}+\\phi \\left(c^{\\mathsf {T}}x\_{1}+c^{\\mathsf {T}}x\_{2}\\right)-\\phi (y\_{d}),&\&x\_{2}(0)=0\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2561e25f8bb0b1d1116612026471b378e189ab83) with ![{\\displaystyle x=x\_{1}+x\_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebf439216976f0058d8c371439dc49babe911cb) This decomposition can help to simplify controller design. ## Other example applications \[[edit](https://en.wikipedia.org/w/index.php?title=Superposition_principle&action=edit&section=9 "Edit section: Other example applications")\] - In [electrical engineering](https://en.wikipedia.org/wiki/Electrical_engineering "Electrical engineering"), in a [linear circuit](https://en.wikipedia.org/wiki/Linear_circuit "Linear circuit"), the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. - In [physics](https://en.wikipedia.org/wiki/Physics "Physics"), [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations") imply that the (possibly time-varying) distributions of [charges](https://en.wikipedia.org/wiki/Electric_charge "Electric charge") and [currents](https://en.wikipedia.org/wiki/Electric_current "Electric current") are related to the [electric](https://en.wikipedia.org/wiki/Electric_field "Electric field") and [magnetic fields](https://en.wikipedia.org/wiki/Magnetic_field "Magnetic field") by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"). - In [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"), superposition is used to solve for beam and structure [deflections](https://en.wikipedia.org/wiki/Deflection_\(engineering\) "Deflection (engineering)") of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[\[9\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-9) Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[\[10\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-10) - In [hydrogeology](https://en.wikipedia.org/wiki/Hydrogeology "Hydrogeology"), the superposition principle is applied to the [drawdown](https://en.wikipedia.org/wiki/Drawdown_\(hydrology\) "Drawdown (hydrology)") of two or more [water wells](https://en.wikipedia.org/wiki/Water_well "Water well") pumping in an ideal [aquifer](https://en.wikipedia.org/wiki/Aquifer "Aquifer"). This principle is used in the [analytic element method](https://en.wikipedia.org/wiki/Analytic_element_method "Analytic element method") to develop analytical elements capable of being combined in a single model. - In [process control](https://en.wikipedia.org/wiki/Process_control "Process control"), the superposition principle is used in [model predictive control](https://en.wikipedia.org/wiki/Model_predictive_control "Model predictive control"). - The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by [linearization](https://en.wikipedia.org/wiki/Linearization "Linearization"). According to [Léon Brillouin](https://en.wikipedia.org/wiki/L%C3%A9on_Brillouin "Léon Brillouin"), the principle of superposition was first stated by [Daniel Bernoulli](https://en.wikipedia.org/wiki/Daniel_Bernoulli "Daniel Bernoulli") in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") and then by [Joseph Lagrange](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange "Joseph-Louis Lagrange"). Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.[\[11\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-11) Later it became accepted, largely through the work of [Joseph Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier").[\[12\]](https://en.wikipedia.org/wiki/Superposition_principle#cite_note-12) - [Additive state decomposition](https://en.wikipedia.org/wiki/Additive_state_decomposition "Additive state decomposition") - [Beat (acoustics)](https://en.wikipedia.org/wiki/Beat_\(acoustics\) "Beat (acoustics)") - [Coherence (physics)](https://en.wikipedia.org/wiki/Coherence_\(physics\) "Coherence (physics)") - [Convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") - [Green's function](https://en.wikipedia.org/wiki/Green%27s_function "Green's function") - [Impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") - [Interference](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)") - [Quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition") 1. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-1) The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London. 2. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-2) Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [\[1\]](https://books.google.com/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1) 3. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-3) N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [\[2\]](https://books.google.com/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361) 4. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-4) Tim Freegarde, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [\[3\]](https://books.google.com/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106) 5. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-QuaMech_5-0) Quantum Mechanics, [Kramers, H.A.](https://en.wikipedia.org/wiki/Hendrik_Anthony_Kramers "Hendrik Anthony Kramers") publisher Dover, 1957, p. 62 [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-66772-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-66772-0 "Special:BookSources/978-0-486-66772-0") 6. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-6) [Dirac, P. A. M.](https://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac "Paul Adrien Maurice Dirac") (1958). The Principles of Quantum Mechanics, 4th edition, Oxford, UK: Oxford University Press, p. 17. 7. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-7) Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1993FoPh...23..185S](https://ui.adsabs.harvard.edu/abs/1993FoPh...23..185S). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01883623](https://doi.org/10.1007%2FBF01883623). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [121930907](https://api.semanticscholar.org/CorpusID:121930907). 8. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-8) [Dirac, P. A. M.](https://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac "Paul Adrien Maurice Dirac") (1958). The Principles of Quantum Mechanics, 4th edition, Oxford, UK: Oxford University Press, p. 14. 9. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-9) Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192 [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-07-252036-1](https://en.wikipedia.org/wiki/Special:BookSources/0-07-252036-1 "Special:BookSources/0-07-252036-1") 10. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-10) Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785 [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-301458-4](https://en.wikipedia.org/wiki/Special:BookSources/0-13-301458-4 "Special:BookSources/0-13-301458-4") 11. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-11) Topics on Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, [p. 39](http://www.bcamath.org/documentos_public/courses/1_Course2012Chapter1WavesHistoryApplications.pdf) 12. [^](https://en.wikipedia.org/wiki/Superposition_principle#cite_ref-12) [Brillouin, L.](https://en.wikipedia.org/wiki/L%C3%A9on_Brillouin "Léon Brillouin") (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2. - Haberman, Richard (2004). Applied Partial Differential Equations. Prentice Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-13-065243-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-065243-0 "Special:BookSources/978-0-13-065243-0") . - [Superposition of sound waves](http://www.acoustics.salford.ac.uk/feschools/waves/super.htm)
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