the Bessel functions
/ BESS-el /
What Legendre polynomials are to spheres, Bessel functions are to cylinders and drums. Strike a circular drumhead and it vibrates in patterns that fall off and oscillate as you move out from the centre -- not as clean sines, but as Bessel functions, which look like a sine wave whose amplitude slowly decays and whose wavelength drifts. They are the natural radial shapes wherever there is cylindrical symmetry.
The Bessel functions of order nu solve Bessel's equation x^2 y'' + x y' + (x^2 - nu^2) y = 0. The regular solution J_nu(x) (Bessel function of the first kind) is finite at x = 0; the second independent solution Y_nu(x) (the Neumann function) blows up there. J_nu oscillates and decays for large x roughly like sqrt(2/(pi x)) cos(x - nu pi/2 - pi/4). Like Legendre polynomials they are orthogonal, but with weight x and over a finite radius: the integral from 0 to a of J_nu(k_m r/a) J_nu(k_n r/a) r dr vanishes for different zeros k_m, k_n of J_nu -- this Fourier-Bessel orthogonality is what lets you expand a radial profile.
They appear whenever Laplace's or the wave and heat equation is separated in cylindrical coordinates: the modes of a circular drum and an optical fibre, the diffraction pattern of a circular aperture (the Airy disk, set by the first zero of J_1), the field around a wire, and frequency-modulation sidebands in signal processing. The zeros of J_nu set the allowed frequencies, just as n pi/L sets them for a string.
A circular drum of radius a vibrates in modes shaped like J_0(k r), and the allowed k are fixed by requiring the rim to be still: J_0(k a) = 0. The lowest zero, about 2.405, sets the fundamental drum tone.
The zeros of a Bessel function fix a drum's allowed frequencies.
Only J_nu is finite on the axis; the second solution Y_nu diverges at r = 0, so it is dropped for any solid region containing the axis but kept for an annulus or the exterior of a cylinder. Which solutions survive is dictated by the geometry.