Living Wave (cos)
y = cos(1/x + t)
Open in the app The dials and keys named below are the app's.
What it draws
The formula is one cosine of the phase 1/x + t. Every point of the line rises and falls with period 2π ≈ 6.28 seconds at Speed 1. Where the crests sit, though, is set by 1/x, which is small out at the edges and unbounded at the middle.
At t = 0 the crests stand at x = 1/(2πn): 0.159, 0.0796, 0.0531, …, all inside a twentieth of the plot. Across the rest of it the phase turns through less than a full cycle, so the outer curve is one long swing. At x = 0 the phase is infinite and the cosine has no value, so a gap is left there.
Why the crests crowd
The local wavenumber is the rate the phase turns with x, here 1/x². Crests are therefore 2πx² apart: 6.28 at x = 1, 1.57 at x = 0.5, 0.25 at x = 0.2. The spacing falls with the square, so near the origin the oscillations pack in tighter than the plot can resolve and the curve becomes a solid scribble.
This is the behaviour of the topologist's sine curve: the values keep sweeping the whole range from −1 to 1 and approach no limit as x → 0. Read in a complex variable, cos(1/z) has an essential singularity there, and the Casorati–Weierstrass theorem says it comes close to every value near it.
How it moves
Follow one crest. Its phase is fixed, so 1/x = C − t and x = 1/(C − t), which gives dx/dt = x². Every feature drifts to the right at a speed equal to its own x squared, 1 unit per second at x = 1. Right of the origin crests are born in the scribble and accelerate off the plot. Left of it they slide in from the edge, slow, and are swallowed. No single speed describes the picture.
Try
- Front in the deck: straight on, with the gap at x = 0 standing between the two scribbles.
- Turn Span of x down to 0.5: what was a smudge fills the plot, and the crests at 0.159 and 0.0796 stand well apart.
- Type cos(1/x² + t): even in x, so the two halves match, with crests at 1/√(2πn) = 0.399, 0.282, … that drift outward on both sides.
- Raise Ribbon depth to 40: the trailing copies reach further back and the crests trail out into fans.