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 . It is named after the mathematician Jean le Rond d'Alembert.

The characteristics of the PDE are $xpm\; ct=mathrm\{const\},$, so use the change of variables $mu=x+ct,\; eta=x-ct,$ to transform the PDE to $u\_\{mueta\}=0,$. 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 $cF\text{'}(x)-cG\text{'}(x)=h(x),$.

Integrate the last equation to get

:$cF(x)-cG(x)=int\_\{-infty\}^x\; h(xi)\; dxi\; +\; c\_1,$

Now solve this system of equations to get

:$F(x)\; =\; frac\{-1\}\{2c\}left(-cg(x)-left(int\_\{-infty\}^x\; h(xi)\; dxi\; +c\_1\; ight)\; ight),$

:$G(x)\; =\; frac\{-1\}\{2c\}left(-cg(x)+left(int\_\{-infty\}^x\; h(xi)\; dxi\; +c\_1\; ight)\; ight),$

Now, using

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

d´Alembert's formula becomes:

:$u(x,t)\; =\; frac\{1\}\{2\}left\; [g(x-ct)\; +\; g(x+ct)\; ight]\; +\; frac\{1\}\{2c\}\; int\_\{x-ct\}^\{x+ct\}\; h(xi)\; dxi$

External links

* [http://www.exampleproblems.com/wiki/index.php/PDE27 An example] of solving a nonhomogeneous wave equation from www.exampleproblems.com