Radial vibrations of a circular membrane ($-(r{R}^{\prime }{)}^{\prime }=\lambda rR,\lambda ={\omega }^2/c^2$)

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 $\lambda ={\int }_0^{a}r|R^{\prime }{|}^{2}dr/{\int }_0^{a}r|R{|}^{2}dr$. Distinct modes are orthogonal in the weighted inner product ${\int }_0^{a}r\overline{R}Sdr$.

For a fixed circular elastic membrane, the Rayleigh-Ritz variational principle applied to $f(r)=1-(r/a{)}^p$, $p>0$, gives ${\lambda }_1\le (p+3+2/p)/a^2$. The numerator of its Rayleigh quotient is $p/2$, and the denominator is $a^2{p}^2/[2(p+1)(p+2)]$. Minimization gives $p=\sqrt{2}$, ${\lambda }_1\le (3+2\sqrt{2})/a^2$, and the displayed upper bound for the ordinary frequency ${\nu }_1=c\sqrt{{\lambda }_1}/(2\pi )$. Every $p>0$ has finite radial Dirichlet energy, which is the required variational admissibility condition.