Math 6410 • ODE

\textbf{(a)} Consider the three--dimensional wave equation
\[
  \frac{\partial^2u}{\partial t^2}=c^2\nabla^2u
\]
with Cauchy data
\[
  u(\mathbf{x},0)=0,\qquad \frac{\partial u}{\partial t}(\mathbf{x},0)=g(\mathbf{x})
\]
\begin{enumerate}
\item[(a)] Show that radially symmetric solutions are in the form of outgoing and
incoming waves
\[
  u(r,t)=\frac{1}{r}F(r\pm ct).
\]
[Hint: Perform the substitution \(v=ur\).]

\item[(b)] Writing the general solution to the Cauchy problem as a superposition of
outgoing waves
\[
  u(\mathbf{x},t)=\int_{\R^3}\frac{\delta(r-ct)}{r}f(x',y',z')\dd x'\dd y'\dd z',
\]
where \(r^2=(x-x')^2+(y-y')^2+(z-z')^2\), derive the retarded potential solution
\[
\begin{aligned}
  u(\mathbf{x},t)=ct\int_0^{2\pi}\int_0^\pi
  &f(x+ct\sin\theta\cos\phi,\ y+ct\sin\theta\sin\phi,\\
  &z+ct\cos\theta)\sin\theta\dd\theta\dd\phi
\end{aligned}
\]
and show that \(f=g/(4\pi c)\).

\item[(c)] Assuming that the initial data \(g\) is only nonzero in a bounded domain \(D\),
use a graphical construction to derive Huygen's principle.
\end{enumerate}