 Solution

The time dependence gives $u_{tt}=-{\omega }^{2}u$, and the angular derivative vanishes for these radial normal modes. The Laplacian in polar coordinates therefore yieldsMultiplication by $r$ puts the Sturm-Liouville problem in the formThe relevant self-adjoint operator is $\mathcal{A}R=-r^{-1}(r{R}^{\prime }{)}^{\prime }$ in the weighted inner product with weight $r$. A physical displacement is regular at the origin, excluding the logarithmic singular solution. In particular $R$ is bounded and $r{R}^{\prime }\to 0$ there; regular eigenfunctions also have $R^{\prime }(0)=0$. This origin condition is needed in addition to the fixed outer boundary.