Math 6410 • ODE

Let \(\R_+=(0,\infty)\). Consider the problem
\[
\begin{aligned}
  u_{tt}-u_{xx} &= 0, && \text{in } \R_+\times\{t>0\},\\
  u(x,0) &= g(x),\quad u_t(x,0)=h(x), && x\in\R_+
    && \text{(initial conditions)},\\
  u_x(0,t) &= 0, && t>0
    && \text{(boundary condition)}.
\end{aligned}
\]
Assume that \(g,h\in C^1(\overline{\R_+})\), and that \(g_x(0)=h_x(0)=0\).
\begin{enumerate}
\item[(a)] Find an explicit formula for the solution \(u(x,t)\) (d'Alembert's formula is a
good place to start).

\item[(b)] Suppose that for \(0\le t<\frac{1}{2}\) and \(x>0\), we know \(u(x,t)=g(x+t)\),
and assume \(\operatorname{supp}g\subset(1,2)\). Thus, initially a wave is travelling
toward the origin \(x=0\). What happens to the wave after it hits the origin, say for
\(t>2\)? Is it absorbed, reflected, damped, inverted? Calculate \(u(x,3)\).
\end{enumerate}