# Spinning Drum Mode

Surface · z = f(x, y, t)

`z = 5 · besselJ(1, 1.403r) · cos(θ − t) +(r < 5 ? 0 : 0/0)`

[Open in the app](https://www.wavelace.com/app#p=176) · [This page](https://www.wavelace.com/presets/spinning-drum-mode)

### What it draws

A circular drum of radius 5, its skin vibrating in one of its natural patterns. The height is `5 · J₁(1.403 · r) · cos(θ − t)`, where `J₁` is the Bessel function of order one, `r` the distance from the centre and `θ` the angle round it.

The radial factor sets the rings. `J₁` is zero at 0, at 3.832 and at 7.016, and `1.403 ≈ 7.016/5` puts that last zero on the rim, so the edge stays still. The zero at 3.832 lands at `r ≈ 2.73`: a still circle inside the drum. Between them the skin rises to 2.91 near `r ≈ 1.31` and dips to −1.73 near `r ≈ 3.80`.

The last term, `(r < 5 ? 0 : 0/0)`, adds nothing inside the rim and is not a number outside it, so the sheet ends at the frame.

### Why it turns

The angular factor `cos(θ − t)` gives the drum a hump on one side and a hollow on the other, split by a still diameter. Because `θ − t` grows with the clock, that whole pattern turns about the centre. It goes round once every `2π ≈ 6.28` seconds at `Speed` 1.

The identity `cos(θ − t) = cos θ · cos t + sin θ · sin t` says what is happening. A drum has two such modes with the same pitch, one with its still diameter along the y axis and one along the x axis. Played together a quarter period apart, their sum is a pattern that travels round the drum instead of rocking in place. The still circle and the rim never move; the still diameter sweeps.

### Try

- Replace `cos(θ − t)` with `cos(θ) · cos(t)`: one mode alone, its halves rocking against each other across a still line along the y axis.
- Write `cos(θ + t)` instead: the same pattern turning the other way.
- Change 1.403 to 0.766, which puts the first zero of `J₁` on the rim: the simplest lopsided mode, with no still circle inside and its peak at `r ≈ 2.40`.
- `Top`: the still circle at `r ≈ 2.73` keeps the ground's colour while the dividing line turns through it.

### Read more

- [Vibrations of a circular membrane](https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane)
- [Bessel function](https://en.wikipedia.org/wiki/Bessel_function)
- [Normal mode](https://en.wikipedia.org/wiki/Normal_mode)
- [Standing wave](https://en.wikipedia.org/wiki/Standing_wave)