The same mathematical curves describe drumheads, radio waveguides and DNA's X-ray images
Strike a drum and its skin ripples in patterns described by Bessel functions. The same family of curves governs radio waves inside a cylindrical waveguide, sound radiating from speakers, friction in round pipes and the X-ray patterns that revealed the helical shapes of DNA and proteins. Anything with circular symmetry tends to summon them.
Friedrich Bessel, a German who worked in both astronomy and mathematics, gave them a thorough treatment in 1824, and they carry his name. Formally, they are solutions to one particular second-order differential equation, Bessel's equation, which contains a number called the order that fixes the shape of the solution. The order can be any complex number, though the useful cases are often simple. Whole-number orders appear in problems set in cylinders, which is why those solutions are sometimes called cylinder functions or cylindrical harmonics. Half-integer orders appear in spherical problems and are called spherical Bessel functions.
The equation turns up whenever physicists break Laplace's equation or the Helmholtz equation, the workhorses for static fields and waves, into separate pieces in cylindrical or spherical coordinates. That is why the functions are so central to wave propagation and potentials. Beyond vibrating membranes and waveguides, they appear in solutions of the Schrödinger equation for a free particle, in quantum field theory, in seismology for analysing surface waves from tiny tremors, and even in the probability distribution of the product of two normally distributed quantities. Signal engineers meet them in FM audio synthesis and in the Bessel filter.
Plotted, a Bessel function of the first kind looks like a sine or cosine wave that slowly fades as it goes. Unlike a true sine wave, though, its zero crossings are not evenly spaced, except approximately far from the origin. Because the equation is second order, it always has two independent solutions, known as the first and second kinds.
Bessel himself defined the functions through an integral, now called the Hansen-Bessel formula, and derived several of their properties from it. Their conventional scaling still reflects that origin in definite integrals rather than in differential equations.
Source: Bessel function
