D'Alembert's formula

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The correct title of this article is d'Alembert's formula. The initial letter is capitalized due to technical restrictions.

In mathematics, and specifically partial differential equations, d´Alembert's formula is the general solution to the one-dimensional wave equation: $u_{tt}-c^2u_{xx}=0, u(x,0)=g(x), u_t(x,0)=h(x)\,$ . It is named after the mathematician Jean le Rond d'Alembert.

The characteristics of the PDE are $x\pm ct=\mathrm{const}\,$ , so use the change of variables $\mu=x+ct, \eta=x-ct\,$ to transform the PDE to $u_{\mu\eta}=0\,$ . The general solution of this PDE is $u(\mu,\eta) = F(\mu) + G(\eta)\,$ where $F\,$ and $G\,$ are $C^1\,$ functions. Back in $x,t\,$ coordinates,

$u(x,t)=F(x+ct)+G(x-ct)\,$

$u\,$ is $C^2\,$ if $F\,$ and $G\,$ are $C^2\,$ .

This solution $u\,$ can be interpreted as two waves with constant velocity $c\,$ moving in opposite directions along the x-axis.

Now consider this solution with the Cauchy data $u(x,0)=g(x), u_t(x,0)=h(x)\,$ .

Using $u(x,0)=g(x)\,$ we get $F(x)+G(x)=g(x)\,$ .

Using $u_t(x,0)=h(x)\,$ we get $cF'(x)-cG'(x)=h(x)\,$ .

Integrate the last equation to get

$cF(x)-cG(x)=\int_{-\infty}^x h(\xi) d\xi + c_1\,$

Now solve this system of equations to get

$F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right)\,$

$G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right)\,$

Now, using

$u(x,t) = F(x+ct)+G(x-ct)\,$

d´Alembert's formula becomes:

$u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) d\xi$

External links

An example of solving a nonhomogenous wave equation from www.exampleproblems.com

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Categories: Article titles with lowercase initial letters | Partial differential equations

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