Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 10

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.

The conditions of regularity at $r=0$ and $R(a)=0$ select the radial vibrations of a circular membrane as an eigenvalue problem. Multiply $-(r{R}^{\prime }{)}^{\prime }=\lambda rR$ by $\overline{R}$ and integrate. Integration by parts givesThe boundary term vanishes by regularity and $R(a)=0$. For a nonzero eigenfunction the denominator is positive; the numerator could vanish only for a constant function, and the outer condition would then make that function zero. ConsequentlyThis also proves reality without presuming real-valued eigenfunctions. Explicitly, regular solutions are Bessel functions of the first kind $R=J_0(\sqrt{\lambda }r)$, and the positive zeros of $J_0(\sqrt{\lambda }a)$ give the discrete spectrum.

Use the weighted inner productMultiply the equation for $R_n$ by ${\overline{R}}_m$, subtract the complex conjugate equation for $R_m$ times $R_n$, and integrate. Since the eigenvalues are real,The boundary term vanishes as before. Thus eigenfunctions with distinct eigenvalues are orthogonal. The full disk inner product differs only by the angular factor $2\pi$.

The Rayleigh-Ritz variational principle bounds the lowest eigenvalue by the Rayleigh quotient of any nonzero admissible trial function. For $f=1-(r/a{)}^p$, take $p>0$, which ensures boundedness, the outer Dirichlet boundary condition, and finite radial Dirichlet energy. Direct integration givesThereforeThe derivative is $1-2/p^2$, so the unique minimizing positive exponent is $p=\sqrt{2}$. This yields the power-law variational bound for a circular membraneThe optimizing function has zero derivative at the origin and is admissible for the energy principle despite its unbounded second derivative there. For comparison, the exact radial fundamental has $\sqrt{{\lambda }_1}a\simeq 2.404826$, giving ${\nu }_1\simeq 0.382740c/a$. The bound is about $0.39\%$ above this value.