# 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

\(\displaystyle \map u {x, 0}\) | \(=\) | \(\displaystyle \map \phi x\) | |||||||||||

\(\displaystyle \map {u_t} {x, 0}\) | \(=\) | \(\displaystyle \map \psi x\) |

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:

\(\displaystyle \map \phi x\) | \(=\) | \(\, \displaystyle \map u {x, 0} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle \map f x + \map g x\) | |||||||||

\(\displaystyle \map \psi x\) | \(=\) | \(\, \displaystyle \map {u_t} {x, 0} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle c \map {f'} x - c \map {g'} x\) | Chain Rule for Partial Derivatives |

So we have:

\(\displaystyle \map {\phi'} x\) | \(=\) | \(\displaystyle \map {f'} x + \map {g'} x\) | Sum Rule for Derivatives | ||||||||||

\(\displaystyle \dfrac {\map \psi x} c\) | \(=\) | \(\displaystyle \map {f'} x - \map {g'} x\) |

Solving the equations give:

\(\displaystyle \map {f'} x\) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {\map {\phi'} x + \dfrac {\map \psi x} c}\) | |||||||||||

\(\displaystyle \map {g'} x\) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {\map {\phi'} x - \dfrac {\map \psi x} c}\) |

Integrating both equations and using Fundamental Theorem of Calculus:

\(\displaystyle \map f x\) | \(=\) | \(\displaystyle \dfrac 1 2 \map \phi x + \dfrac 1 {2c} \int_0^x \map \psi s \rd s + A\) | |||||||||||

\(\displaystyle \map g x\) | \(=\) | \(\displaystyle \dfrac 1 2 \map \phi x - \dfrac 1 {2c} \int_0^x \map \psi s \rd s + B\) |

for some constants $A,B$.

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

Therefore:

\(\displaystyle \map u {x,t}\) | \(=\) | \(\displaystyle \map f {x + c t} + \map g {x - c t}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \map \phi {x + c t} + \dfrac 1 {2c} \int_0^{x + c t} \map \psi s \rd s + A + \dfrac 1 2 \map \phi {x - c t} - \dfrac 1 {2c} \int_0^{x - c t} \map \psi s \rd s + B\) | substitution | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {\map \phi {x + c t} + \map \phi {x - c t} } + \dfrac 1 {2 c} \int_{x - c t}^{x + c t} \map \psi s \rd s\) | simplification |

$\blacksquare$

## Source of Name

This entry was named for Jean le Rond d'Alembert.

## Historical Note

Jean le Rond d'Alembert devised this solution to the $1$-dimensional wave equation in $1746$.

## Sources

- 2008: Walter A. Strauss:
*Partial Differential Equations: An Introduction*(2nd ed.): Chapter $2$: Waves and Diffusions