Math 6420 • PDE

Find the first four eigenvalues for a ``half drum'', that is,
for the Laplacian on a semicircular domain, \(0\le r\le R\), \(0\le\theta\le\pi\), with
homogeneous Dirichlet boundary conditions. Sketch the nodal lines for these eigenfunctions.

Hint: The zeros of Bessel functions are \(\lambda_{nm}\) where \(J_n(\lambda_{nm})=0\). They
are ordered as
\[
  \lambda_{01}<\lambda_{11}<\lambda_{21}<\lambda_{02}<\lambda_{31}<\lambda_{12}<\lambda_{41}.
\]
The Laplacian in polar coordinates is
\[
  \nabla^2u=\frac{1}{r}\frac{\partial}{\partial r}
  \left(r\frac{\partial u}{\partial r}\right)
  +\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}.
\]
\end{enumerate}