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| {{Lowercase|title=d'Alembert's Solution of the wave equation}}
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| {{Refimprove|date=September 2009}}
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| {{Textbook|date=June 2010}}
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| In [[mathematics]], and specifically [[partial differential equations]], '''d´Alembert's formula''' is the general solution to the one-dimensional [[wave equation]]:
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| :<math>u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),</math> | |
| for <math>-\infty < x<\infty,\,\, t>0</math>. It is named after the mathematician [[Jean le Rond d'Alembert]].<ref>D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA214#v=onepage&q&f=false "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 214-219. See also: D'Alembert (1747) [http://books.google.com/books?id=lJQDAAAAMAAJ&pg=PA220#v=onepage&q&f=false "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration"] (Further researches on the curve that a tense cord forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 220-249. See also: D'Alembert (1750) [http://books.google.com/books?id=m5UDAAAAMAAJ&pg=PA355#v=onepage&q&f=false "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"] ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 6, pages 355-360.</ref>
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| The [[method of characteristics|characteristics]] of the PDE are <math>x\pm ct=\mathrm{const}\,</math>, so use the change of variables <math>\mu=x+ct, \eta=x-ct\,</math> to transform the PDE to <math>u_{\mu\eta}=0\,</math>. The general solution of this PDE is <math>u(\mu,\eta) = F(\mu) + G(\eta)\,</math> where <math>F\,</math> and <math>G\,</math> are <math>C^1\,</math> functions. Back in <math>x,t\,</math> coordinates,
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| :<math>u(x,t)=F(x+ct)+G(x-ct)\,</math>
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| :<math>u\,</math> is <math>C^2\,</math> if <math>F\,</math> and <math>G\,</math> are <math>C^2\,</math>. | |
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| This solution <math>u\,</math> can be interpreted as two waves with constant velocity <math>c\,</math> moving in opposite directions along the x-axis.
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| Now consider this solution with the [[Cauchy data]] <math>u(x,0)=g(x), u_t(x,0)=h(x)\,</math>.
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| Using <math>u(x,0)=g(x)\,</math> we get <math>F(x)+G(x)=g(x)\,</math>.
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| Using <math>u_t(x,0)=h(x)\,</math> we get <math>cF'(x)-cG'(x)=h(x)\,</math>.
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| Integrate the last equation to get
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| :<math>cF(x)-cG(x)=\int_{-\infty}^x h(\xi) \, d\xi + c_1.\,</math>
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| Now solve this system of equations to get
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| :<math>F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) \, d\xi +c_1 \right)\right)\,</math>
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| :<math>G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right).\,</math>
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| Now, using
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| :<math>u(x,t) = F(x+ct)+G(x-ct)\,</math>
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| d´Alembert's formula becomes:
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| :<math>u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) \, d\xi.</math>
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| ==See also==
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| *[[D'Alembert operator]]
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| == Notes ==
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| {{reflist}}
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| ==External links==
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| *[http://www.exampleproblems.com/wiki/index.php/PDE27 An example] of solving a nonhomogeneous wave equation from www.exampleproblems.com
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| [[Category:Partial differential equations]]
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