Bessel's equation
/ Bessel: BESS-ul /
Anything that vibrates or diffuses on a circular or cylindrical domain — a drumhead, the cross-section of an optical fibre, heat in a round pipe, ripples spreading on a pond — leads, after separating variables, to one equation for the radial part. Bessel's equation is that equation: the natural radial companion to Legendre's angular equation. It is to cylinders what Legendre is to spheres.
It is x^2 y'' + x y' + (x^2 - n^2) y = 0, where n (the order) is a parameter and x is essentially a scaled radius. The origin x = 0 is a regular singular point, so you solve it by the method of Frobenius; the indicial equation r^2 - n^2 = 0 gives exponents r = +n and r = -n. The root +n produces the Bessel function of the first kind J_n(x), which is finite and oscillatory and stays bounded at the centre. A second, independent solution Y_n(x), the Bessel function of the second kind, blows up logarithmically at the origin (a consequence of the integer-order indicial roots differing by an integer), so it is discarded for any region that includes the axis.
These functions behave like 'decaying cosines': for large x, J_n(x) looks like a cosine wave whose amplitude fades as 1/sqrt(x), capturing how a circular ripple spreads and weakens. Their zeros set the resonant frequencies of a drum and the cutoff modes of a waveguide; J_0 governs the temperature in a cooling cylinder. Bessel's equation is one of the most-used equations in all of applied physics precisely because cylindrical symmetry is everywhere.
A circular drumhead clamped at radius a vibrates in modes whose radial profile is J_0(k r). The boundary condition J_0(k a) = 0 forces k a to equal a zero of J_0 (the first is about 2.405), and these zeros set the drum's overtone frequencies — which is why a drum, unlike a string, has an inharmonic, non-integer overtone series.
The zeros of J_0 are the drum's resonances — and they are not evenly spaced, unlike a string's harmonics.
J_n is not a polynomial: unlike Legendre or Hermite, Bessel's series does not truncate, so J_n is a genuine infinite series (a transcendental function). The 'order' n need not be an integer — half-integer orders give the spherical Bessel functions used in scattering.