Living Wave
y = sin(1/x + t)
Open in the app The dials and keys named below are the app's.
What it draws
Everything interesting is in 1/x. Out at the edges it changes slowly and the curve is a long lazy swell. The local wavenumber is 1/x², so crests are 2π·x² apart: 25 at x = 2, 6.28 at x = 1, both wider than the plot itself.
Closer in they crowd, to 1.57 at x = 0.5 and 0.063 at x = 0.1. The zeros sit at x = 1/(nπ − t) for every whole n, an infinite crowd of them piling into the origin. At x = 0 the formula has no finite value at all, and a gap is left there rather than a line drawn through it.
Where the wiggles come from
Adding t to the phase makes the pattern move outward. A feature of fixed phase satisfies 1/x + t = constant, so it sits at x = 1/(c − t) and travels at x² units per second. That is a crawl of 0.04 at x = 0.2, one unit per second at x = 1, four at x = 2. Oscillations are manufactured at the origin without end, stretch as they run out, and leave at the edge. At any fixed x the height cycles with period 2π ≈ 6.28 seconds at Speed 1.
Frozen at one instant this is the topologist's sine curve, the standard example of a set that is connected but not path-connected. No path along it reaches the origin, since that means climbing through infinitely many crests. Near x = 0 the function takes every value between −1 and 1 infinitely often, the real shadow of the essential singularity sin(1/z) has there. The picture cannot show it: near the middle the crests are finer than the plot can resolve, and the hash drawn there is aliasing rather than the function.
Try
- Pull Span of x down to 1: the whole plot is the crowded region.
- Tame it with sin(1/x + t)·x, squeezed to nothing at the origin, which it now reaches.
- Make it worse with sin(1/(x·x) + t): even in x, crests π·x³ apart, aliasing further out.
- Turn Speed down to 0.25 and watch one crest walk out from the middle, slow at first, then faster.
- Look from Top: the tracks left through the ribbon curve away from the origin as the crests gather speed.