d'Alembert's formula

In mathematics, and specifically partial differential equations, d´Alembert's formula is the general solution to the one-dimensional wave equation:

$u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),$

for $-\infty < x<\infty,\,\, t>0$. It is named after the mathematician Jean le Rond d'Alembert.^[1]

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:

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

Notes

1. ^ D'Alembert (1747) "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) "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) "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.

External links

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