# Damped Pulse

Wave · y = f(x, t)

`y = exp(−0.1x²) · sin(6x − 4t)`

[Open in the app](https://www.wavelace.com/app#p=3) · [This page](https://www.wavelace.com/presets/damped-pulse)

### What it draws

`sin(6x − 4t)` is the wave inside the pulse. Its wavenumber is 6, so crests sit `2π/6 ≈ 1.05` apart. The `4t` makes a fixed point of the line rise and fall every `2π/4 ≈ 1.57` seconds at `Speed` 1. The phase `6x − 4t` holds still for a crest moving right at `4/6 ≈ 0.67` unit per second.

`exp(−0.1x²)` is the Gaussian bell that shapes it. It is `1` at the middle and half height at `x = ±2.63`, down to `1/e` at `±3.16` and to `0.027` by `±6`. Five crests fit inside the half-height width, and beyond it the line lies flat.

### Why the pattern

The bell holds only `x`, so it is nailed to the plot: the pulse is a window, not a parcel. Crests are made at the left edge of the bell, grow as they climb it, cross the middle at full height and shrink away on the right. All of them travel at the same steady `0.67` unit per second.

That is the difference between this and a real pulse on a string or in a fibre, where the carrier and its envelope travel together and the shape moves as one body. Writing `exp(−0.1 · (x − 0.67t)²)` for the bell puts the envelope on the same journey as the crests and turns the picture into a genuine travelling packet.

### Try

- `Front` in the deck: the bell is the outline the crests grow into and out of.
- Send the window along with the wave, `exp(−0.1 · (x − 0.67t)²) · sin(6x − 4t)`: one pulse crosses the plot and leaves.
- Tighten the bell to `exp(−0.4x²) · sin(6x − 4t)`: half height at `±1.32`, and only a couple of crests survive.
- Flip the sign of the `4t` to send the crests left through the same window.
- Raise `Ribbon depth`: the ribbon's recent past fills with the diagonal stripes the crests leave behind as they cross the fixed bell.

### Read more

- [Gaussian function](https://en.wikipedia.org/wiki/Gaussian_function)
- [Envelope (waves)](https://en.wikipedia.org/wiki/Envelope_(waves))
- [Phase velocity](https://en.wikipedia.org/wiki/Phase_velocity)
- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)