Bessel Functions: the Drumhead Lab

Why a circle gives you Jₙ instead of sin/cos

On a rectangle, separating variables in the wave or heat equation hands you sines and cosines, because the boundary conditions are at fixed x and y. On a disk, the natural coordinates are radius r and angle θ — and when you separate variables in polar form, the angular part is still sin/cos, but the radial part solves x²y″+xy′+(x²−n²)y=0: Bessel's equation. Its solution that stays finite at the center (x=0) is Jₙ(x). The other independent solution, Yₙ(x), blows up at the origin — so it's only used when the domain excludes the center, like an annulus or a pipe with a hole.

Zeros are eigenvalues

A drumhead clamped at its rim needs Jₙ(x)=0 right at the boundary — so the zeros of Jₙ aren't just a curiosity, they're literally the allowed vibration frequencies (eigenvalues) of that drumhead mode. Compare this lab's zero list to the Root Finding lab — finding them is exactly the same bisection/Newton problem, just on a transcendental function instead of a polynomial.

A short history

Daniel Bernoulli used something like these functions as early as 1732, studying oscillations of a hanging chain. Euler ran into them again solving the vibrating circular membrane. But it was Friedrich Bessel, in 1824, who systematically tabulated and named them — not for vibrations at all, but while computing planetary perturbations from Kepler's equation. The same function turned out to describe a drumhead, a cylindrical heat sink, and a planet's orbit, because all three reduce to the same separated-variables ODE.