# Beat Interference

Wave · y = f(x, t)

`y = sin(x − t) + sin(1.1x + 0.7t)`

[Open in the app](https://www.wavelace.com/app#p=2) · [This page](https://www.wavelace.com/presets/beat-interference)

### What it draws

Two waves are added, and neither is remarkable alone. `sin(x − t)` has wavenumber 1, crests `2π ≈ 6.28` apart, and runs right at one unit per second at `Speed` 1. `sin(1.1x + 0.7t)` has wavenumber `1.1`, crests `2π/1.1 ≈ 5.71` apart, and runs left at `0.7/1.1 ≈ 0.64` unit per second. They are close enough in wavelength that the sum is not a mess but a slow rhythm.

### Why it pulses

Adding two sines gives a product of their half sum and half difference: `sin(x − t) + sin(1.1x + 0.7t) = 2 · sin(1.05x − 0.15t) · cos(0.05x + 0.85t)`. The first factor is what the eye calls the wave, with crests `2π/1.05 ≈ 5.98` apart, creeping right at `0.15/1.05 ≈ 0.14` unit per second.

The second factor is the loudness. Its wavenumber is only `0.05`, so across the plot it hardly changes at all, but its `0.85t` runs fast. Every `π/0.85 ≈ 3.70` seconds it passes through zero and the whole curve nearly flattens, falling from a height of `2` to `0.5`, then filling out again.

Watch one point instead of the whole line and the classic beat appears. At `x = 0` the two waves are tones of angular frequency `1` and `0.7`, and their sum swings inside an envelope `2 · sin(0.15t)`. It is silent at `t = 0`, loudest near `t = 10.5`, and silent again at `t ≈ 20.9` seconds. Two piano strings a hair apart in pitch do this, and a tuner listens for the beats to slow to nothing.

### Try

- `Front` in the deck, then wait: the swell and collapse every `3.70` seconds is the whole point.
- Make the two waves match exactly, `sin(x − t) + sin(x + t)`: the beats vanish and a standing wave takes their place, with nodes every `π`.
- Bring them closer instead, `sin(x − t) + sin(1.02x + 0.7t)`: the crests are longer lived and drift faster.
- Raise `Ribbon depth`: the ribbon's recent past then holds a whole swell and collapse at once.
- `Pause`, then `Step` through a quiet moment to see the two waves cancelling rather than stopping.

### Read more

- [Beat (acoustics)](https://en.wikipedia.org/wiki/Beat_(acoustics))
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)
- [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle)
- [Group velocity](https://en.wikipedia.org/wiki/Group_velocity)