# Spiral Wave

Surface · z = f(x, y, t)

`z = sin(3θ − 2r + t)`

[Open in the app](https://www.wavelace.com/app#p=34) · [This page](https://www.wavelace.com/presets/spiral-wave)

### What it draws

The height is a single sine of `3θ − 2r + t`, written in the polar pair `r = hypot(x, y)` and `θ = atan2(y, x)`. Nothing damps it, so every crest stands as tall at the rim as at the middle.

A crest is a curve of constant phase, `3θ − 2r + t = constant`, which rearranges to `r = (3θ + t − constant)/2`. So `r` grows in proportion to the angle, which is the definition of an Archimedean spiral. The 3 on `θ` gives three of them, evenly spaced, each gaining `3π ≈ 9.42` in radius over a full turn. Along any ray from the middle the arms cross at intervals of `2π/2 = π ≈ 3.14`.

### How it turns

Hold `r` fixed and the phase stays put when `3θ + t` does. The whole figure therefore rotates toward decreasing `θ` at `1/3` of a radian per second at `Speed` 1, coming back to itself after `6π ≈ 18.85` seconds. Hold `θ` fixed instead and the same motion reads as every crest creeping outward at half a unit per second. A rotating spiral and an outgoing wave are one motion seen two ways.

`θ` jumps by `2π` across the negative `x` axis, but 3 is a whole number, so `3θ` jumps by `6π` and the sine does not notice: the sheet has no seam there. At the middle the phase has no value at all. All three arms wind into that one point, and around any small circle about it the phase runs through `6π`. That is a phase singularity, and it is what pins the spiral in place.

### Try

- `Top` in the deck: the three arms, seen as they are usually photographed.
- Add arms, `sin(5θ − 2r + t)`: five of them, turning more slowly, at `1/5` of a radian per second.
- Reverse the clock, `sin(3θ − 2r − t)`: the spiral turns the other way and the crests run inward.
- Break the whole number, `sin(2.5θ − 2r + t)`: the sine no longer matches across the negative `x` axis and a torn seam appears along it.
- Pull `Mesh` to its top stop: the arms turn solid, and the middle stays ragged whatever the grid, because there is no height to find there.

### Read more

- [Archimedean spiral](https://en.wikipedia.org/wiki/Archimedean_spiral)
- [Excitable medium](https://en.wikipedia.org/wiki/Excitable_medium)
- [Belousov–Zhabotinsky reaction](https://en.wikipedia.org/wiki/Belousov%E2%80%93Zhabotinsky_reaction)
- [Optical vortex](https://en.wikipedia.org/wiki/Optical_vortex)