Math 6410 • ODE

Consider the wave equation
\[
  u_{tt}-c^2u_{xx}=0,\qquad t>0,\quad x\in\R.
\]
along with the intial conditions
\[
  u(x,0)=g(x),\qquad u_t(x,0)=h(x).
\]
\begin{enumerate}
\item[(a)] Assuming \(c\) is constant, derive d'Alembert's formula
\[
  u(x,t)=\frac12\bigl(g(x+ct)+g(x-ct)\bigr)
  +\frac{1}{2c}\int_{x-ct}^{x+ct}h(\xi)\dd\xi.
\]
\item[(b)] Briefly explain the concepts of \textit{domain of dependence}, and
\textit{finite propagation speed}, and prove that these properties follow from
d'Alembert's formula.
\item[(c)] Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary
condition \(u(0,t)\) for \(t\ge0\). If the initial conditions satisfy \(g(0)=h(0)=0\),
using d'Alembert's formula as a starting point, find a closed-form solution for
\(u(x,t)\), with \(x,t>0\).
\end{enumerate}