Bessel Function Gallery

Differential Equations · Axiom Academy

A visual encyclopedia of the oscillatory, damped waves that solve x^2y''+xy'+(x^2-n^2)y=0 — the equation behind drum vibrations, radio waves, and heat flow in cylinders. Damped waves born from one differential equation Bessel's equation x^2y''+xy'+(x^2-n^2)y=0 shows up whenever a physical problem has circular symmetry — a vibrating drumhead, a wave in a circular waveguide, heat spreading through a cylinder. Its solutions, the Bessel functions J_n(x) , look almost like sine and cosine — except their amplitude keeps shrinking as x grows. Watch J_0(x) and J_1(x) sweep out together. Each crossing of the axis is a zero — and the dots that settle on them are the exact zeros, not estimates. The zeros of J_0 (≈2.4048, 5.5201, 8.6537, …) and J_1 (≈3.8317, 7.0156, 10.1735, …) never repeat, and their spacing keeps closing in on as x grows. Click a card to plot that function. The first kind ( J_n ) is finite everywhere, including at x=0 — these are the physically realistic solutions on a disk. The second kind ( Y_n ) blows up at the origin, but together J_n and Y_n span every solution of Bessel's equation. J_0(0)=1 and J_1(0)=0 — finite. Y_0 and Y_1 both plunge to as , which is exactly why only J_n can describe a physically bounded disk (a drumhead can't have infinite displacement at its center). Zooming out: the asymptotic envelope

This is the written version of the interactive lesson above. See the full Differential Equations course.