# Living Wave (tan)

Wave · y = f(x, t)

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

[Open in the app](https://www.wavelace.com/app#p=15) · [This page](https://www.wavelace.com/presets/living-wave-tan)

### What it draws

The phase is `1/x + t`, the same as in the cosine version, but the tangent turns it into a run of spikes instead of a run of crests. A pole stands wherever the phase reaches `π/2` plus a whole number of `π`. At `t = 0` the poles are therefore at `x = 2/((2n + 1)π)`: `0.637, 0.212, 0.127, 0.0909, …`, mirrored on the negative side.

Between two poles the curve climbs once through every real value. Each point of the line completes that whole climb in `π ≈ 3.14` seconds at `Speed` 1, the period of the tangent, half the `2π` of a sine. `Amplitude` opens low, at 0.6, because the tangent has no ceiling of its own.

### The poles

No spike runs away: they are all cut flat at a common height, so a pole reads as a plateau rather than a line to infinity. The poles crowd toward the middle exactly as the crests do. The phase turns at the rate `1/x²`, so consecutive poles are `πx²` apart. At `x = 0` there is no value at all, and the curve is cut in two there.

### How it moves

A pole holds its phase, so `1/x = C − t` and `x = 1/(C − t)`, giving `dx/dt = x²`. Every spike slides to the right at the square of its own position: `1` unit per second out at `x = 1`, `0.04` at `x = 0.2`. On the right the spikes appear near the origin and race outward, stretching apart as they go. On the left they drift in, bunch up and disappear into the middle. Nothing repeats in place, which is the whole effect.

### Try

- `Front` in the deck: the spikes side on, each one cut at the same height.
- Set `Ribbon depth` to 1: one clean curve with no trailing copies, which is the easiest way to count the poles.
- Turn `Span of x` down to 0.5: the four poles nearest the middle spread out across the plot.
- Take the reciprocal, `1/tan(1/x + t)`: the poles and the zeros trade places, and the spikes fall between where they were.
- Raise `Amplitude` to 2: the spikes meet the ceiling sooner and become broad flat-topped walls.

### Read more

- [Trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions)
- [Asymptote](https://en.wikipedia.org/wiki/Asymptote)
- [Topologist's sine curve](https://en.wikipedia.org/wiki/Topologist%27s_sine_curve)
- [Classification of discontinuities](https://en.wikipedia.org/wiki/Classification_of_discontinuities)