# Living Wave (cos)

Wave · y = f(x, t)

`y = cos(1/x + t)`

[Open in the app](https://www.wavelace.com/app#p=14) · [This page](https://www.wavelace.com/presets/living-wave-cos)

### 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.

### Read more

- [Topologist's sine curve](https://en.wikipedia.org/wiki/Topologist%27s_sine_curve)
- [Essential singularity](https://en.wikipedia.org/wiki/Essential_singularity)
- [Casorati–Weierstrass theorem](https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem)
- [Oscillation (mathematics)](https://en.wikipedia.org/wiki/Oscillation_(mathematics))