# d'Alembert's formula

In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation ${\displaystyle u_{tt}(x,t)=c^{2}u_{xx}(x,t)}$ (where subscript indices indicate partial differentiation, using the d'Alembert operator, the PDE becomes: ${\displaystyle \Box u=0}$).

The solution depends on the boundary conditions at ${\displaystyle t=0}$: ${\displaystyle u(x,0)}$ and ${\displaystyle u_{t}(x,0)}$. It consists of separate terms for the boundary conditions ${\displaystyle u(x,0)}$ and ${\displaystyle u_{t}(x,0)}$:

${\displaystyle u(x,t)={\frac {1}{2}}\left[u(x-ct,0)+u(x+ct,0)\right]+{\frac {1}{2c}}\int _{x-ct}^{x+ct}u_{t}(\xi ,0)\,d\xi .}$

It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a vibrating string.[1]

## Details

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

${\displaystyle u(x,t)=F(x+ct)+G(x-ct)\,}$
${\displaystyle u\,}$  is ${\displaystyle C^{2}\,}$  if ${\displaystyle F\,}$  and ${\displaystyle G\,}$  are ${\displaystyle C^{2}\,}$ .

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

Now consider this solution with the Cauchy data ${\displaystyle u(x,0)=g(x),u_{t}(x,0)=h(x)\,}$ .

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

Using ${\displaystyle u_{t}(x,0)=h(x)\,}$  we get ${\displaystyle cF'(x)-cG'(x)=h(x)\,}$ .

We can integrate the last equation to get

${\displaystyle cF(x)-cG(x)=\int _{-\infty }^{x}h(\xi )\,d\xi +c_{1}.\,}$

Now we can solve this system of equations to get

${\displaystyle F(x)={\frac {-1}{2c}}\left(-cg(x)-\left(\int _{-\infty }^{x}h(\xi )\,d\xi +c_{1}\right)\right)\,}$
${\displaystyle G(x)={\frac {-1}{2c}}\left(-cg(x)+\left(\int _{-\infty }^{x}h(\xi )d\xi +c_{1}\right)\right).\,}$

Now, using

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

d'Alembert's formula becomes:

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

## Generalization for inhomogeneous canonical hyperbolic differential equations

The general form of an inhomogeneous canonical hyperbolic type differential equation takes the form of:

${\displaystyle u_{tt}-u_{xx}=f(x,t),\,u(x,0)=g(x),\,u_{t}(x,0)=h(x),}$

for ${\displaystyle -\infty 0,f\in C^{2}(\mathbb {R} ^{2},\mathbb {R} )}$ .

All second order differential equations with constant coefficients can be transformed into their respective canonic forms. This equation is one of these three cases: Elliptic partial differential equation, Parabolic partial differential equation and Hyperbolic partial differential equation.

The only difference between a homogeneous and an inhomogeneous (partial) differential equation is that in the homogeneous form we only allow 0 to stand on the right side ( ${\displaystyle f(x,t)=0}$  ), while the inhomogeneous one is much more general, as in ${\displaystyle f(x,t)}$  could be any function as long as it's continuous and can be continuously differentiated twice.

The solution of the above equation is given by the formula:

${\displaystyle u(x,t)={\frac {1}{2}}{\Bigl (}g(x+t)+g(x-t){\biggr )}+{\frac {1}{2}}{\biggl (}\int \limits _{x-t}^{x+t}h(s)ds{\biggr )}+{\frac {1}{2}}{\biggl (}\int \limits _{0}^{t}\int \limits _{x-t+\tau }^{x+t-\tau }f(s,\tau )dsd\tau {\Bigr )}}$ .

If ${\displaystyle g(x)=0}$ , the first part disappears, if ${\displaystyle h(x)=0}$ , the second part disappears, and if ${\displaystyle f(x)=0}$ , the third part disappears from the solution, since integrating the 0-function between any two bounds always results in 0.

This means, that the homogeneous equation ( ${\displaystyle f(x,t)=0}$  ) gives back our original formula for the case of ${\displaystyle c=1}$ .

## 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 [string] 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.