Wave equation: d'Alembert's formula

In 1747, Jean le Rond d'Alembert (1717-1783) published a paper on vibrating strings that included his famous solution to the one-dimensional wave equation:

\[ u_{tt}(x,t)=a^2u_{xx}(x,t). \]

Consider the initial value problem for the wave equation on the entire real line

\begin{eqnarray*} u_{tt}(x,t)&=&a^2u_{xx}(x,t),\quad x\in \mathbb R,\quad t>0,\\ u(x,0) &=& f(x),\quad x\in \mathbb R,\\ u_t(x, 0) &=& g(x),\quad x\in \mathbb R. \end{eqnarray*}

The solution of this initial value problem is given by

\begin{eqnarray}\label{solution} u(x,t)=\frac{1}{2}\big(f(x+at)+f(x-at) \big)+\frac{1}{2a}\int_{x-at}^{x+at}g(s)ds. \end{eqnarray}

The form of $u(x, t)$ given in (\ref{solution}) is called d'Alembert's formula.

The wave equation on a half line

Consider the initial boundary value problem

\begin{eqnarray*} u_{tt}(x,t)&=&a^2u_{xx}(x,t),\quad x>0,\quad t>0,\\ u(x,0) &=& f(x),\quad x>0,\\ u_t(x, 0) &=& g(x),\quad x>0,\\ u(0,t)&=&0\quad t>0. \end{eqnarray*}

The boundary condition specifying $u(0, t)$ means that the end of the string is held in place.

For $x > at$, we have d'Alembert's formula (\ref{solution}):

\[ u(x,t)=\frac{1}{2}\big(f(x+at)+f(x-at) \big)+\frac{1}{2a}\int_{x-at}^{x+at}g(s)ds \]

with $f$ and $g$ evaluated only at positive values of their arguments. For $0 < x < at$, we have:

\[ u(x,t)=\frac{1}{2}\big(f(x+at)-f(at-x) \big)+\frac{1}{2a}\int_{at-x}^{x+at}g(s)ds \]

Interact with the following simulation of a semi-infinite string.

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Simulation

The following simulation shows a dynamic representation of the solution on the entire real line of the wave equation using d'Alembert's formula (\ref{solution}).

Things to try:

Change the initial conditions $u(x,0)$ and $u_t(x,0)$.
Change the value of $a$.
Click on Play button to start simulation.

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Reference

[1] D'Alembert (1747). Recherches sur la courbe que forme une corde tenduë mise en vibration (Researches on the curve that a tense cord forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214-219.