# Drumhead

Surface · z = f(x, y, t)

`z = ∫₀^π cos(r · sin(u)) du/π · cos(t)`

[Open in the app](https://www.wavelace.com/app#p=100) · [This page](https://www.wavelace.com/presets/drumhead)

### What it draws

The height is Bessel's integral, computed at every mesh point: `z = (1/π) ∫₀^π cos(r · sin u) du · cos t`. The integral is `J₀(r)`, the Bessel function, so the sheet is `J₀(r) · cos t`, a function of the distance `r = hypot(x, y)` from the centre alone. At the centre `J₀(0) = 1`. Out from it the function falls through zero at `r ≈ 2.405`, bottoms out at `−0.403` near `r ≈ 3.83`, and crosses zero again at `r ≈ 5.520` and at `r ≈ 8.654`. On a sheet of half-width 8 the first two circles are whole and the third clips the corners.

The factor `cos t` makes a standing wave. Every point rises and falls in step, the centre between `+1` and `−1`, while the zero circles never move. The sheet flips sign every `π ≈ 3.14` seconds at `Speed` 1.

### Why the integral

`cos(r · sin u)` is a plane wave of wavelength `2π` travelling in the direction `u`. Averaging over `u` adds up equal plane waves from every direction, and the sum is a circular ripple, `J₀(r)`: at the centre every wave has a crest, along the zero circles they cancel.

### The drum

A membrane stretched over a circular frame obeys the wave equation, and its circular standing waves are exactly `J₀(k · r) · cos(ω · t)`, with the rim pinned where `J₀` is zero. Cut this sheet at `r = 2.405` and it is the fundamental of a drum of that radius, one dome breathing up and down. Cut it at `r = 5.520` and it is the drum's second circular mode, the dome ringed by a trough with a still circle between. The zeros of `J₀` are not evenly spaced, so the frequencies of these modes are not multiples of the fundamental: the second is `5.520/2.405 ≈ 2.295` times the first. A string's overtones are exact multiples, hence a string's pitch and a drum's thud.

### History

Leonhard Euler met this function in 1764, in the vibrations of a stretched membrane. Friedrich Bessel studied the functions systematically in 1824, in the perturbations of planetary orbits, and the integral on screen is his.

### Try

- `Top`: the still circles keep the ground's colour while the rings between them swap sign with the clock.
- Add a diameter, `integral(cos(r·sin(u) − u), u, 0, π)/π · cos(θ) · cos(t)`: the integral is now `J₁(r)`, the drum's first lopsided mode, its halves rising against each other across a still line along the y axis.
- Double the wavenumber, `integral(cos(2·r·sin(u)), u, 0, π)/π · cos(t)`: that is `J₀(2r)`, every circle at half its radius, the first at `r ≈ 1.202`.
- `Side`: the profile is `J₀` itself, the trough only 0.403 of the centre dome.

### Read more

- [Bessel function](https://en.wikipedia.org/wiki/Bessel_function)
- [Vibration of a circular membrane](https://en.wikipedia.org/wiki/Vibration_of_a_circular_membrane)
- [Standing wave](https://en.wikipedia.org/wiki/Standing_wave)
- [Friedrich Bessel](https://en.wikipedia.org/wiki/Friedrich_Bessel)