Hit a guitar string and it vibrates in neat sine-wave patterns — halves, thirds, quarters of its length. That is why a string sounds musical: its overtones are whole-number multiples of the fundamental. Now hit a drum. A circular drumhead also has vibration modes, but they are not sines, their frequencies are not tidy multiples, and that is exactly why a drum sounds like a thud rather than a note. The functions that describe those circular modes are the Bessel functions.

They are the "sines and cosines of the cylinder". Wherever a physics problem has cylindrical symmetry — a vibrating drumhead, heat spreading through a round pipe, an electromagnetic wave trapped in a circular waveguide or optical fibre, the bright rings of light diffracting through a round aperture — the radial part of the solution is a Bessel function. This page is about that one family: the equation they solve, why they oscillate while slowly fading, and what their special zeros mean.

Where they come from: Bessel's equation

Separate the wave or heat equation in cylindrical coordinates and the radial factor must satisfy Bessel's differential equation of order n:

This looks a little like the equation for a spring, y'' + y = 0 (whose solutions are \cos x and \sin x), but the extra x\,y' term acts like a gentle, position-dependent friction, and the -n^2/x^2 piece (after dividing through) distorts things near the origin. Attack it with a power-series solution (a Frobenius series) and you get two independent solutions for each order n:

For any region that includes the centre — a solid drum, a filled pipe — the physical solution must be finite there, so you throw away Y_n and keep only J_n. From the series you can read straight off that J_0(0)=1 while J_n(0)=0 for every n\ge 1.

Seeing them: oscillation with a slow fade

The graph shows J_0(x) and J_1(x). Watch what they do: they oscillate like cosine and sine — J_0 starts at 1, J_1 starts at 0 — but their swings shrink as x grows. They are damped waves, fading like 1/\sqrt{x}.

For large x this behaviour becomes exact. The Bessel functions settle into a decaying cosine,

which pins down both the shrinking amplitude envelope \sqrt{2/\pi x} and the fact that, far out, the wiggles line up almost evenly with spacing \pi — but never quite, which is the whole difference from a plain cosine.

The zeros are the physics

The points where J_n(x)=0 — its zeros — are where the real content lives. Clamp a circular drumhead of radius a at its rim: the boundary condition "no motion at the edge" says the radial solution J_n(k\,a) must vanish there. That is only possible if k\,a lands on one of the zeros of J_n. So the allowed vibration frequencies are quantised by the Bessel zeros.

The first zero of J_0 sits at j_{0,1} \approx 2.405; the next at j_{0,2} \approx 5.520; then 8.654, and so on. Because these are not in the ratio 1:2:3, a drum's overtones are inharmonic — the reason a timpani gives a boomy pitch rather than a clean note. The same zeros set the cut-off frequencies of a circular waveguide and the radii of the dark rings in the Airy diffraction pattern of a telescope.

Worked examples

Example 1 — values at the origin. From the series J_n(x)=\sum_m \frac{(-1)^m}{m!(m+n)!}(x/2)^{2m+n}, the lowest power of x is x^{\,n}. So at x=0 only the n=0 function survives: J_0(0)=1, and J_1(0)=J_2(0)=\dots=0.

Example 2 — a drum frequency. A drumhead of radius a with wave speed c has its lowest axially-symmetric mode when k\,a = j_{0,1} \approx 2.405, so k = 2.405/a and the angular frequency is \omega = c\,k = 2.405\,c/a. Double the radius and the pitch drops by an octave-and-a-bit — the frequency halves.

Example 3 — the fading amplitude. How much has a Bessel oscillation shrunk by x=25 compared with x=1? The envelope goes like 1/\sqrt{x}, so the ratio is \sqrt{1/25}=1/5 — the swings are about five times smaller.

Example 4 — which solution to keep. Heat flows in a solid cylinder. The temperature at the axis (x=0) must be finite, but Y_n(x)\to-\infty there. So the coefficient of Y_n must be zero, and only J_n survives. For a hollow pipe (an annulus that excludes the axis) you keep both.