# Wave Packet

Wave · y = f(x, t)

`y = exp(−0.2x²) · sin(15x + 5t)`

[Open in the app](https://www.wavelace.com/app#p=4) · [This page](https://www.wavelace.com/presets/wave-packet)

### What it draws

`sin(15x + 5t)` is the carrier. Wavenumber 15 puts its crests `2π/15 ≈ 0.42` apart, the finest ripple any of the wave presets draws. The `+5t` sends them left at `5/15 ≈ 0.33` unit per second at `Speed` 1, while a fixed point cycles every `2π/5 ≈ 1.26` seconds.

`exp(−0.2x²)` gathers them into a hump. It is at half height at `x = ±1.86` and `1/e` at `±2.24`, and under a thousandth of full height by `±6`. Nine crests live inside the half-height width.

### The physics behind the shape

This is the picture a textbook draws for a particle that is somewhere rather than everywhere. A narrow band of wavenumbers around `k = 15` is added together, cancelling away from the middle and reinforcing inside the hump. The narrower the hump in `x`, the broader the band of `k` it takes to build it, which is the whole content of the uncertainty relation.

What a real packet then does, this formula does not. Left to itself under the Schrödinger equation each wavenumber travels at its own speed, so the hump moves at the group velocity and slowly spreads while the crests inside it slip through at a different rate. Here the hump is a fixed function of `x` and only the crests move. Schrödinger looked for packets that keep their shape in 1926 and found them for the harmonic oscillator alone; a free packet always spreads.

### Try

- `Front` in the deck, then `Pause` and `Step`: the crests slide left, the hump does not.
- Send the hump with them, `exp(−0.2 · (x + 0.33t)²) · sin(15x + 5t)`: now the packet itself travels.
- Reverse the carrier with `exp(−0.2x²) · sin(15x − 5t)` to run the crests right instead.
- Raise `Ribbon depth`: the leaning stripes in the ribbon's recent past are the crests' path through the standing hump.
- Widen the hump with `exp(−0.05x²) · sin(15x + 5t)`: more crests, and a shape closer to one pure wavenumber.

### Read more

- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)
- [Group velocity](https://en.wikipedia.org/wiki/Group_velocity)
- [Dispersion relation](https://en.wikipedia.org/wiki/Dispersion_relation)
- [Uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle)
- [Matter wave](https://en.wikipedia.org/wiki/Matter_wave)