FormulaProved

d'Alembert's formula for the wave equation

Statement

For the wave equation $u_{tt} = c^2 u_{xx}$ on $\mathbb{R} \times (0,\infty)$ with initial data $u(x,0)=\varphi(x)$ , $u_t(x,0)=\psi(x)$ , the unique solution is $u(x,t) = \tfrac{1}{2}\big(\varphi(x-ct)+\varphi(x+ct)\big) + \tfrac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds$ .

Why is it true?

A wave on an infinite string is just two copies of the initial shape, one sliding left and one sliding right at speed $c$ , plus a correction that accounts for the initial velocity. Anything that happens at a point $(x,t)$ can only depend on the initial data in the interval $[x-ct, x+ct]$ — information cannot travel faster than $c$ .

Proof sketch

Change variables to characteristic coordinates $\xi = x - ct$ , $\eta = x + ct$ ; the wave equation becomes $u_{\xi\eta} = 0$ , so $u = F(\xi) + G(\eta)$ for some functions $F, G$ . Impose the initial conditions $u(x,0)=\varphi(x)$ and $u_t(x,0)=\psi(x)$ to solve for $F$ and $G$ in terms of $\varphi$ and an antiderivative of $\psi$ , and substitute back.

Topics that use this theorem

The wave equation

Related theorems

Fundamental theorem of calculus

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Lawrence C. Evans (2010). Partial Differential Equations · DOI:10.1090/gsm/019