# Travelling Ripple

Wave · y = f(x, t)

`y = sin(x − t) /(1 + 0.15x²) · 3`

[Open in the app](https://www.wavelace.com/app#p=1) · [This page](https://www.wavelace.com/presets/travelling-ripple)

### What it draws

`sin(x − t)` is the wave. Its wavenumber is 1, so crests are `2π ≈ 6.28` apart. The phase depends only on `x − t`, so the whole shape slides right at exactly one unit per second at `Speed` 1. A fixed point of the line rises and falls every `6.28` seconds.

Everything else is the window the wave is seen through. `3/(1 + 0.15x²)` is a bell that peaks at `3` in the middle and falls to half of that at `x = ±√(1/0.15) ≈ ±2.58`. By `x = ±6` it is down to `0.47`.

### Why the pattern

The two pieces answer different questions. The sine says where the crests are, the bell says how tall a crest is allowed to be where it stands. Because the bell is a function of `x` alone it does not move, so nothing here is a travelling pulse.

A crest is born small at the left edge, swells as it climbs the bell, reaches `3` as it crosses the middle and dies away to the right. The ribbon's recent past draws that whole life at once, each slice showing the crest one step back along the same road. A curve `1/(1 + x²)` falling off as `1/x²` is the Lorentzian, the same profile as a resonance line and the Cauchy distribution.

### Try

- `Front` in the deck: straight on, the bell is the outline the crests never cross.
- Change `sin(x − t)` to `sin(x + t)`: the same picture running left instead.
- Widen the window with `sin(x − t) / (1 + 0.02x²) · 3`: half height is now at `±7.07`, past the edge, so the crests barely fade at all.
- Raise `Span of x` to 12: nearly four crests fit, and the bell shrinks them hard at both ends.

### Read more

- [Sine wave](https://en.wikipedia.org/wiki/Sine_wave)
- [Phase velocity](https://en.wikipedia.org/wiki/Phase_velocity)
- [Envelope (waves)](https://en.wikipedia.org/wiki/Envelope_(waves))
- [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution)