D'Alembert's Formula
Theorem
Let $t$ be time.
Let $x$ be position.
Let $\tuple {t, x} \stackrel u {\longrightarrow} \map u {t, x}: \R^2 \to \R$ be a twice-differentiable function in both variables.
Let $x \stackrel \phi {\longrightarrow} \map \phi x: \R \to \R$ be a differentiable function.
Let $x \stackrel \psi {\longrightarrow} \map \psi x: \R \to \R$ be a Riemann integrable function.
Let $c \in \R_{> 0}$ be a constant.
Then the solution to the partial differential equation:
- $\dfrac {\partial^2} {\partial t^2} \map u {x, t} = c^2 \dfrac {\partial^2} {\partial x^2} \map u {x, t}$
with initial conditions:
\(\ds \map u {x, 0}\) | \(=\) | \(\ds \map \phi x\) | ||||||||||||
\(\ds \valueat {\dfrac {\partial} {\partial t} \map u {x, t} } {t \mathop = 0}\) | \(=\) | \(\ds \map \psi x\) |
is given by:
- $\ds \map u {x, t} = \dfrac 1 2 \paren {\map \phi {x + c t} + \map \phi {x - c t} } + \dfrac 1 {2 c} \int_{x - c t}^{x + c t} \map \psi s \rd s$
The above solution formula is called d'Alembert's formula.
Proof
The general solution to the $1$-D wave equation:
- $\dfrac {\partial^2} {\partial t^2} \map u {x, t} = c^2 \dfrac {\partial^2} {\partial x^2} \map u {x, t}$
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:
\(\ds \map \phi x\) | \(=\) | \(\, \ds \map u {x, 0} \, \) | \(\, \ds = \, \) | \(\ds \map f x + \map g x\) | ||||||||||
\(\ds \map \psi x\) | \(=\) | \(\, \ds \valueat {\dfrac \partial {\partial t} \map u {x, t} } {t \mathop = 0} \, \) | \(\, \ds = \, \) | \(\ds c \map {f'} x - c \map {g'} x\) | Chain Rule for Partial Derivatives |
So we have:
\(\ds \map {\phi'} x\) | \(=\) | \(\ds \map {f'} x + \map {g'} x\) | Sum Rule for Derivatives | |||||||||||
\(\ds \dfrac {\map \psi x} c\) | \(=\) | \(\ds \map {f'} x - \map {g'} x\) |
Solving the equations give:
\(\ds \map {f'} x\) | \(=\) | \(\ds \dfrac 1 2 \paren {\map {\phi'} x + \dfrac {\map \psi x} c}\) | ||||||||||||
\(\ds \map {g'} x\) | \(=\) | \(\ds \dfrac 1 2 \paren {\map {\phi'} x - \dfrac {\map \psi x} c}\) |
Integrating both equations and using Fundamental Theorem of Calculus:
\(\ds \map f x\) | \(=\) | \(\ds \dfrac 1 2 \map \phi x + \dfrac 1 {2 c} \int_0^x \map \psi s \rd s + A\) | ||||||||||||
\(\ds \map g x\) | \(=\) | \(\ds \dfrac 1 2 \map \phi x - \dfrac 1 {2 c} \int_0^x \map \psi s \rd s + B\) |
for some constants $A, B$.
From $\map \phi x = \map f x + \map g x$, we have $A + B = 0$.
Therefore:
\(\ds \map u {x, t}\) | \(=\) | \(\ds \map f {x + c t} + \map g {x - c t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \map \phi {x + c t} + \dfrac 1 {2 c} \int_0^{x + c t} \map \psi s \rd s + A + \dfrac 1 2 \map \phi {x - c t} - \dfrac 1 {2 c} \int_0^{x - c t} \map \psi s \rd s + B\) | substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {\map \phi {x + c t} + \map \phi {x - c t} } + \dfrac 1 {2 c} \int_{x - c t}^{x + c t} \map \psi s \rd s\) | 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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations