Math 6410 • ODE

\begin{enumerate}
\item[(a)] Consider the velocity potential \(\phi\) for linear water waves. Let \(h_0\)
denote the depth of water in the absence of waves, and \(g\) denote the gravitational
constant. The corresponding boundary value problem on \(\R\times[-h_0,0]\) is given
by Laplace's equation
\[
  \frac{\partial^2\phi}{\partial x^2}
  +\frac{\partial^2\phi}{\partial y^2}=0,
\]
with
\[
  \frac{\partial^2\phi}{\partial t^2}
  +g\frac{\partial\phi}{\partial y}=0
  \qquad\text{on } y=0,
\]
and
\[
  \frac{\partial\phi}{\partial y}=0
  \qquad\text{on } y=-h_0.
\]
The linear wave solution takes the form
\[
  \phi(x,y,t)=Y(y)e^{ikx-i\omega t}.
\]
Derive the transcendental dispersion relation
\[
  \omega^2=gk\tanh(kh_0).
\]
Determine the group and phase velocities for (i) deep water waves (large \(h_0\)) and
(ii) shallow water waves (small \(h_0\)). Which case is non-dispersive?
\item[(b)] Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain
\(\Omega\subset\R^n\), so \(\Delta u=0\) in \(\Omega\). Prove the mean value property:
if \(x\in\Omega\) and \(r>0\) is chosen such that \(B_r(x)\subset\Omega\) (ball of radius
\(r\) centered at \(x\)) then
\[
  u(x)=\frac{1}{\omega_nr^{n-1}}\int_{\partial B_r(x)}u(s)\dd s,
\]
where \(\omega_n\) is the measure of \(\partial B_1\). Hence show that
\[
  u(x)\le\frac{n}{\omega_nr^n}\int_{B_r(x)}u(y)\dd y.
\]
\end{enumerate}