Partial differential equations Hyperbolic equations Theorem

d'Alembert's formula

Statement

Let $c > 0$ , let $g \in C^2(\R)$ and let $h \in C^1(\R)$ . Define $u \colon \R \times [0, \infty) \to \R$ by $u(x, t) = \frac{g(x + ct) + g(x - ct)}{2} + \frac{1}{2c} \int_{x - ct}^{x + ct} h(s)\, ds.$ Then $u \in C^2(\R \times [0, \infty))$ , $u$ solves the wave equation $u_{tt} = c^2 u_{xx}$ on $\R \times (0, \infty)$ , and $u(x, 0) = g(x)$ , $u_t(x, 0) = h(x)$ for all $x \in \R$ . Moreover, $u$ is the only $C^2$ solution of this initial value problem.

Remarks

The change of variables $\xi = x + ct$ , $\eta = x - ct$ turns the equation into $u_{\xi\eta} = 0$ . Hence every $C^2$ solution has the form $u = F(x + ct) + G(x - ct)$ , and the two initial conditions determine $F$ and $G$ up to a constant that cancels. This argument also proves uniqueness. Example: with $g = 0$ and $h = \cos$ , the formula gives $u(x, t) = \frac{1}{2c}\big(\sin(x + ct) - \sin(x - ct)\big) = \frac{1}{c}\cos x \sin(ct)$ , a standing wave. The value $u(x, t)$ depends only on the data in the interval $[x - ct, x + ct]$ , which shows finite speed of propagation in one dimension. If $g$ is only continuous, the formula still defines a function, but it is a weak solution rather than a classical one.

Going deeper

For a student

d'Alembert's formula is the first explicit solution formula in a PDE course and the basis for the other dimensions. By the method of spherical means, it gives Kirchhoff's formula in dimension $3$ , and then by descent Poisson's formula in dimension $2$ . On the half line, the formula combined with an odd or even reflection of the data solves the problems with Dirichlet or Neumann boundary condition. For the inhomogeneous equation $u_{tt} - c^2 u_{xx} = f$ , Duhamel's principle adds the term $\frac{1}{2c}\int_0^t \int_{x - c(t - s)}^{x + c(t - s)} f(y, s)\, dy\, ds$ .

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Wave equation concept
d'Alembert's formula statement

Tag	`YVQRAPSF77`
URI	`m2://YVQRAPSF77`
Kind	statement, theorem
Writing	W0 stub
MSC	35L05, 35C05, 35L15
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About	Wave equation
Uses	Wave equation
Same as	Wikidata Q2388753 Wikipedia D'Alembert's formula