= Radial vibrations of a circular membrane
{title2=$-(rR')'=\lambda rR,\quad\lambda=\omega^2/c^2$}
A fixed circular <elastic membrane> of radius $a$ has radial <normal modes> satisfying the displayed <Sturm-Liouville problem>, with weight $r$, $R(a)=0$ and regularity at the origin. The regular solutions are <Bessel functions of the first kind> $J_0(\sqrt\lambda r)$. Integration by parts gives the positive <Rayleigh quotient> $\lambda=\int_0^a r|R'|^2dr/\int_0^a r|R|^2dr$. Distinct modes are orthogonal in the <weighted inner product> $\int_0^a r\overline R S\,dr$.
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