Fourier-Bessel series
/ BESS-uhl /
Strike a drumhead and the patterns you see are not sine waves — they are concentric rings set by Bessel functions. A Fourier-Bessel series expands a function on a disk's radius as a sum of Bessel functions scaled to vanish at the rim, the radial counterpart of an ordinary Fourier series. It is the right basis whenever your geometry is circular or cylindrical and the natural radial modes are Bessel functions.
Fix the order m. The functions J_m(alpha_k r / a), where alpha_k is the k-th positive zero of J_m, are the eigenfunctions of the Bessel Sturm-Liouville problem on [0, a] and are orthogonal with weight w = r: integral from 0 to a of J_m(alpha_j r/a) J_m(alpha_k r/a) r dr is zero unless j = k. So a radial function f(r) expands as f(r) = sum over k of c_k J_m(alpha_k r / a) with coefficients c_k obtained by projecting f against J_m with the weight r and dividing by the squared norm, which works out to (a^2/2) J_{m+1}(alpha_k)^2. The crucial detail, easy to forget, is the weight r dr — it comes from the area element of the disk and is exactly the weight the self-adjoint form demands.
Fourier-Bessel series are how separation of variables solves the heat or wave equation on a disk or a cylinder. After splitting off the angular part (which gives e^{i m theta}) and the time part, the radial equation is Bessel's, the boundary condition at the rim selects the zeros alpha_k, and the initial radial profile is expanded in a Fourier-Bessel series so each mode can decay or oscillate on its own. The series converges in mean square like other Sturm-Liouville expansions; near a discontinuity it exhibits the same Gibbs overshoot, and the zeros alpha_k are irrational, so coefficients are computed numerically.
Expand the constant f(r) = 1 on a unit disk in order-zero Bessel functions: 1 = sum over k of c_k J_0(alpha_k r), with c_k = 2 / (alpha_k J_1(alpha_k)). The first few alpha_k are 2.405, 5.520, 8.654, ... — the zeros of J_0 that fix the radial nodes of a vibrating circular membrane.
The Bessel zeros set the drumhead's radial nodes; the coefficients distribute a flat profile among those ring modes.
The expansion is over the zeros of a single fixed order m; the right boundary condition (function vanishes, or its derivative vanishes) changes which zeros you use, and forgetting the weight r dr in the coefficient integral is the most common mistake here.