11.2 The vibrating drum

A circular membrane of radius \(a\), fixed at its rim, obeys \(u_{tt} = c^2\Delta u\) in \(r < a\) with \(u = 0\) at \(r = a\). Separating \(u = R(r)\Theta (\theta )T(t)\) gives \(T'' = -c^2k^2T\), \(\Theta '' = -m^2\Theta \) with \(\Theta \) \(2\pi \)-periodic, so \(m = 0, 1, 2, \ldots \) and \(\Theta = \cos m\theta \) or \(\sin m\theta \), and \[ r^2R'' + rR' + (k^2r^2 - m^2)R = 0 , \] Bessel’s equation in the variable \(kr\). Its solution bounded at the origin is \(J_m(kr)\); the other, \(Y_m\), is singular there. The rim condition \(J_m(ka) = 0\) makes \(ka\) a zero of \(J_m\). Writing \(j_{mn}\) for the \(n\)th positive zero, the normal modes and their angular frequencies are \[ u_{mn} = J_m\Bigl (\frac {j_{mn}r}a\Bigr )\cos m\theta \,\cos \omega _{mn}t, \qquad \omega _{mn} = \frac {c\,j_{mn}}a , \] together with the \(\sin m\theta \) and \(\sin \omega _{mn}t\) variants. Mode \((m, n)\) has \(m\) nodal diameters and \(n - 1\) nodal circles (Figure 14). Unlike those of a string, the frequencies are not integer multiples of the fundamental: \(\omega _{11}/\omega _{01} = 1.593\), \(\omega _{21}/\omega _{01} = 2.136\) (Software 11.6.1), which is why a drum has a less definite pitch than a string.

(1(1(2(2(2(20.01.52.10.23.61.9,10,19,13,29,15,21)0)3)6)5)3)7 — Figure 14: The first six normal modes of a circular drum, labelled \((m, n)\), with their frequencies relative to the fundamental. Red and blue regions move in opposite directions; the boundaries between them are the nodal lines, \(m\) diameters and \(n - 1\) circles at the radii \(j_{mk}/j_{mn}\), \(k < n\). The frequency ratios are irregularly spaced, not whole numbers.

To expand initial data in these modes one needs orthogonality.

Theorem 11.2.1. If \(\alpha \) and \(\beta \) are distinct positive zeros of \(J_m\), then \[ \int _0^1J_m(\alpha r)J_m(\beta r)\,r\,dr = 0, \qquad \int _0^1J_m(\alpha r)^2\,r\,dr = \tfrac 12J_m'(\alpha )^2 = \tfrac 12J_{m + 1}(\alpha )^2 . \] Moreover, the functions \(J_m(j_{mn}r)\), \(n = 1, 2, \ldots \), are complete: every \(f\) with \(\int _0^1f^2r\,dr < \infty \) is the mean-square limit of its expansion \(\sum c_nJ_m(j_{mn}r)\), with \(c_n = \int _0^1fJ_m(j_{mn}r)\,r\,dr\big /\frac 12J_{m + 1}(j_{mn})^2\).

Proof. \(U = J_m(\alpha r)\) and \(V = J_m(\beta r)\) satisfy \((rU')' + (\alpha ^2r - m^2/r)U = 0\) and the same with \(\beta \). Multiplying crosswise and subtracting, \(\bigl (r(U'V - UV')\bigr )' = (\beta ^2 - \alpha ^2)rUV\). Integrating from \(0\) to \(1\), for any \(\beta \) and a zero \(\alpha \), \[ (\beta ^2 - \alpha ^2)\int _0^1UV\,r\,dr = \alpha J_m'(\alpha )J_m(\beta ) , \] which vanishes when \(\beta \) is also a zero. Letting \(\beta \to \alpha \), \(J_m(\beta ) \approx J_m'(\alpha )(\beta - \alpha )\) and \(\beta ^2 - \alpha ^2 \approx 2\alpha (\beta - \alpha )\), giving the norm; at a zero, the recurrence \(J_m' = \frac mxJ_m - J_{m + 1}\) gives \(J_m'(\alpha ) = -J_{m + 1}(\alpha )\). Completeness is the Sturm–Liouville expansion theorem for this singular problem, which we quote. □

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