Bessel function of the first kind
/ BESS-el /
Strike a drumhead, or watch ripples spread from a stone dropped in a round pond, and the pattern is not a simple sine wave: it oscillates but its amplitude fades as it spreads outward in a circle. The Bessel functions of the first kind, J_n(x), are the standard waves of circular and cylindrical geometry — the radial analogue of sine and cosine when the world has a centre rather than a flat grid.
J_n(x) is the finite, well-behaved solution that stays bounded at the centre x = 0. It can be written as a power series, J_n(x) = the sum over m from 0 of (-1)^m over (m! Gamma(m+n+1)) times (x/2)^{2m+n}, which already shows the gamma function quietly at work. For large x it settles into a decaying, phase-shifted cosine, J_n(x) is approximately square root of (2 over pi x) times cos(x minus n pi/2 minus pi/4), so it looks like an outward-travelling wave whose amplitude dies like 1 over square root of x — exactly the energy-conserving spreading of a two-dimensional ripple. Its zeros, the values where J_n vanishes, are not evenly spaced and are tabulated as fundamental constants.
Bessel functions of the first kind set the vibration frequencies of a circular drum, the modes of an optical fibre or a microwave cavity, the diffraction pattern of a circular aperture (the Airy disk), heat flow in a cylinder, and the frequency content of an FM radio signal. Here they are catalogued as functions with known series, asymptotics and zeros; the Bessel differential equation that defines them belongs to the series-solutions topic.
The lowest vibration mode of a circular drum of radius a has frequency set by the first zero of J_0, which is approximately 2.405: the fundamental is proportional to 2.405 times the wave speed over a, not a simple multiple of pi.
Unlike a string, a drum's overtones are not integer multiples of the fundamental — they are spaced by the irregular zeros of J_0.
Even when the order n is an integer, J_n is not a polynomial and its zeros are not evenly spaced; the harmonics of a drum are inharmonic, which is why a drum sounds unlike a plucked string.