# Checker Waves

Surface · z = f(x, y, t)

`z = sin(5x + t) · sin(5y + t)`

[Open in the app](https://www.wavelace.com/app#p=31) · [This page](https://www.wavelace.com/presets/checker-waves)

### What it draws

Both factors have the same shape. `sin(5x + t)` is a wave along `x` with wavenumber 5, so its crests are `2π/5 ≈ 1.26` apart, and it slides along `−x` at `1/5 = 0.2` units per second at `Speed` 1. `sin(5y + t)` is the same wave along `y`.

The product is zero wherever either factor is zero, so the sheet is pinned along two families of lines crossing at right angles. Those lines cut it into square cells `π/5 ≈ 0.63` on a side. Each cell bulges up or down, its sign flipping from neighbour to neighbour, and the whole grid of lines slides diagonally at 0.2 units per second in each direction. Everything repeats every `2π ≈ 6.28` seconds.

### Why the pattern

The product splits into two diagonal pieces: `sin(5x + t) · sin(5y + t) = (cos(5(x − y)) − cos(5(x + y) + 2t))/2`. The first half carries no `t` at all. It is a fixed corrugation whose ridges lie along the line `x = y`, with crests `2π/(5√2) ≈ 0.89` apart across them. The second half is the same corrugation turned onto the other diagonal and set moving, at `2/(5√2) ≈ 0.28` units per second along `−x, −y`. A checkerboard that slides is a standing pattern in one diagonal plus a travelling one in the other, added together.

### Try

- `Top` in the deck: the square cells and the two families of lines between them, seen as a checkerboard.
- Reverse one clock, `sin(5x + t) · sin(5y − t)`: the fixed corrugation moves to the other diagonal and the travelling one takes its place.
- Make the cells bigger, `sin(3x + t) · sin(3y + t)`: cells `π/3 ≈ 1.05` wide, and the lines slide faster, at `1/3 ≈ 0.33` units per second.
- Raise `Mesh`: the cells here are small, and a finer grid is what keeps their corners square.

### Read more

- [Standing wave](https://en.wikipedia.org/wiki/Standing_wave)
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)
- [Node (physics)](https://en.wikipedia.org/wiki/Node_(physics))
- [List of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities)