a Bessel-function expansion

Separation of variables is not tied to straight lines and rectangles. Run it on a circular drumhead — a disk — and the round geometry refuses to give you neat sines and cosines for the radial part. Instead the radial factor obeys an equation whose natural solutions are Bessel functions: oscillating curves that look a bit like decaying sines but with spacing and amplitude that drift, the special functions tailored to circular and cylindrical symmetry. The expansion in these functions is the round-geometry cousin of the Fourier series.

Here is how it arises. Separating Laplace's equation or the wave equation in polar coordinates (r, theta) splits it into an angular part — which gives ordinary sines and cosines in theta, since angle wraps around periodically — and a radial part, which becomes Bessel's equation. Its bounded solutions are the Bessel functions J_m(k r). The boundary condition at the rim of the disk (say u = 0 at r = a) forces k a to be a zero of J_m, quantizing the allowed radial wavenumbers just as the ends of a string quantized n pi / L. You then expand the radial profile of your data as a Fourier-Bessel series, a sum of J_m(k_j r) with coefficients found by orthogonality — but now orthogonality holds with respect to a WEIGHT function r dr, the area element of the disk.

This is the same separation-of-variables machinery, just with the right special functions for the geometry. It is how you find the vibration modes of a circular drum (whose overtones, unlike a string's, are NOT simple integer multiples — which is why a drum sounds less 'musical' than a string), the heat flow in a cylinder, and fields with cylindrical symmetry throughout physics and engineering. Spheres lead instead to Legendre and spherical-harmonic expansions; the philosophy is identical, only the special functions change.