NationMaster - Encyclopedia: 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: . It is named after the mathematician Jean le Rond d'Alembert. Euclid, detail from The School of Athens by Raphael. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean le Rond dAlembert (November 16, 1717 â€“ October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ...

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, The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ...

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

$u,$ 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

$cF(x)-cG(x)=int_{-infty}^x h(&# 0; d&# 0;+ c_1,$

Now solve this system of equations to get

$F(x) = frac{-1}{2c}left(-cg(x)-left(int_{-infty}^x h(&# 0; d&# 0;+c_1 right)right),$

$G(x) = frac{-1}{2c}left(-cg(x)+left(int_{-infty}^x h(&# 0; d&# 0;+c_1 right)right),$

Now, using

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

d´Alembert's formula becomes:

$u(x,t) = frac{1}{2}left[g(x-ct) + g(x+ct)right] + frac{1}{2c} int_{x-ct}^{x+ct} h(&# 0; d&# 0; /></p> <p><br/><a name=$

External links

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

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

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