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.

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:

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

See also

• D'Alembert operator

External links

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