# Breathing Gauss

Wave · y = f(x, t)

`y = exp(−0.4(x − 3 · sin(t · 0.5))²) · 2`

[Open in the app](https://www.wavelace.com/app#p=7) · [This page](https://www.wavelace.com/presets/breathing-gauss)

### What it draws

`exp(−0.4·u²)` is a Gaussian, the bell curve, and the trailing `2` is its height. The `0.4` sets its width: the bump is down to half height at `u = √(ln2 / 0.4) ≈ 1.32`, so it is `2.63` wide across the half-height points. The matching standard deviation is `1/√0.8 ≈ 1.12`.

Everything else is in `u = x − 3·sin(0.5t)`, which says where the bump is centred. The centre is at `3·sin(0.5t)`, sliding between `−3` and `3` and back once every `2π/0.5 = 4π ≈ 12.6` seconds at `Speed` 1. The shape itself never changes.

### The swing

The centre is a sine of the clock, so the bump moves the way a pendulum bob does. Its speed is `1.5·cos(0.5t)`, fastest through the middle at `1.5` units per second and momentarily still at each end, where it seems to hang before turning back. That is simple harmonic motion, and the ridge of peaks running back through the ribbon's recent past is a piece of that same sine.

Nothing here is a wave. A travelling wave carries a shape at a fixed speed, and a wave packet made of many frequencies spreads out as it goes. This bump is one rigid shape whose position is dictated by hand, so it neither disperses nor decays. It is a good picture of what a wave packet would look like if only it behaved.

### Try

- `Top` in the deck: looking straight down, the ribbon is the graph of the centre against time, one arch of a sine.
- Change `0.4` to `0.1`: the same swing under a bump twice as wide, half height at `√(ln2 / 0.1) ≈ 2.63` from the centre.
- Make it breathe for real. `exp(−0.4·x²)·(1 + cos(t))` pins the bump at the origin and pumps its height between 0 and 2 every `2π ≈ 6.28` seconds.
- Change `3·sin(t·0.5)` to `t` for a bump that runs off to the right at one unit per second and never comes back.
- Raise `Ribbon depth` to its top: the ribbon then holds most of a full swing, and the ridge bends back on itself.

### Read more

- [Gaussian function](https://en.wikipedia.org/wiki/Gaussian_function)
- [Full width at half maximum](https://en.wikipedia.org/wiki/Full_width_at_half_maximum)
- [Simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion)
- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)