D'Alembert's Formula

Theorem
Let $u: \R^2 \to \R$ be a twice-differentiable function in two variables.

Let $\phi: \R \to \R$ be a differentiable function in $x$.

Let $\psi: \R \to \R$ be an integrable function in $x$.

Let $c \in \R_{> 0}$ be a constant.

Then the solution to the partial differential equation:


 * $u_{tt} = c^2 u_{xx}$

with initial conditions

is given by:


 * $\displaystyle \map u {x, t} = \dfrac 1 2 \paren {\map \phi {x + c t} + \map \phi {x - c t} } + \dfrac 1 {2 c} \int_{x - c t}^{x + c t} \map \psi s \rd s$

The above solution formula is called d'Alembert's Formula.

Proof
The general solution to the 1-D wave equation:


 * $u_{tt} = c^2 u_{xx} \quad \text{for } - \infty < x < \infty$

is given by


 * $\map u {x,t} = \map f {x + c t} + \map g {x - c t}$

where $f,g$ are arbitrary twice-differentiable functions.

From initial conditions we have:

So we have:

Solving the equations give:

Integrating both equations and using Fundamental Theorem of Calculus:

for some constants $A,B$.

From $\map \phi x = \map f x + \map g x$, we have $A + B = 0$.

Therefore: