Spiral Wave
z = sin(3θ − 2r + t)
Open in the app The dials and keys named below are the app's.
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.