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 Incoming links: Radial vibrations of a circular membrane

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 10 / ii / Solution 2026-10-18

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^{'})^{'} = λ r R

by

\overline{R}

and integrate. Integration by parts gives

λ \int_{0}^{a} r ∣ R ∣^{2} d r = \int_{0}^{a} r ∣ R^{'} ∣^{2} d r - [r \overline{R} R^{'}]_{0}^{a} = \int_{0}^{a} r ∣ R^{'} ∣^{2} d r .

The 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. Consequently

λ = \frac{\int _{0}^{a} r ∣ R ^{'} ∣ ^{2} d r}{\int _{0}^{a} r ∣ R ∣ ^{2} d r} > 0, ν = \frac{c λ}{2 π} \in R .

This also proves reality without presuming real-valued eigenfunctions. Explicitly, regular solutions are Bessel functions of the first kind

R = J_{0} (λ r)

, and the positive zeros of

J_{0} (λ a)

give the discrete spectrum.

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