= Solution
The time dependence gives $u_{tt}=-\omega^2u$, and the angular derivative vanishes for these radial <normal modes>. The <Laplacian> in <polar coordinates> therefore yields
$$
R''+\frac1rR'+\frac{\omega^2}{c^2}R=0.
$$
Multiplication by $r$ puts the <Sturm-Liouville problem> in the form
$$
\boxed{-(rR')'=\lambda rR,\qquad p(r)=r,\quad q(r)=0,\quad w(r)=r,\quad\lambda=\omega^2/c^2.}
$$
The relevant <self-adjoint operator> is $\mathcal A R=-r^{-1}(rR')'$ 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 $rR'\to0$ there; regular eigenfunctions also have $R'(0)=0$. This origin condition is needed in addition to the fixed outer boundary.
Back to article page
