# Egg Carton

Surface · z = f(x, y, t)

`z = sin(x) · cos(y + t)`

[Open in the app](https://www.wavelace.com/app#p=30) · [This page](https://www.wavelace.com/presets/egg-carton)

### What it draws

The expression is a product of two one-variable pieces. `sin(x)` depends on `x` alone and never moves. Its zeros, `x = nπ`, are straight nodal lines, and the sheet is pinned flat along every one of them. `cos(y + t)` depends on `y` and the clock. It slides along `−y` at 1 unit per second at `Speed` 1, so the whole pattern repeats every `2π ≈ 6.28` seconds.

Extremes sit where both factors are extreme, at `x = π/2 + mπ` and `y + t = nπ`. That is a grid of hills and hollows spaced `π ≈ 3.14` apart in each direction. Their signs alternate like a chessboard, which is the egg carton, and each one reaches `±1` before `Height` scales it.

### Why the pattern

Split the product into a sum and the moving parts appear: `sin(x) · cos(y + t) = (sin(x + y + t) + sin(x − y − t))/2`. Those are two plane waves of equal strength running along the two diagonals. Each has a wavevector of length `√2`, so each travels at `1/√2 ≈ 0.71` units per second, and their sum solves the two-dimensional wave equation with that speed. Where they meet in step a hill rises, where they meet out of step a hollow drops, and along `x = nπ` they cancel exactly for all time. A product of a fixed profile and a travelling one is the same thing as two waves crossing. The picture only decides which description is easier to see.

### Try

- `Top` in the deck: the chessboard, with the still nodal lines running the length of the sheet.
- Stop the clock inside the formula, `sin(x) · cos(y)`: a fixed carton, the standing pattern the two crossing waves make.
- Halve the cells in one direction, `sin(2x) · cos(y + t)`: twice as many nodal lines, spaced `π/2 ≈ 1.57`.
- Pull `Mesh` to its top stop: the hills and hollows read as one lit solid, and the flat lines between them stand out.

### Read more

- [Standing wave](https://en.wikipedia.org/wiki/Standing_wave)
- [Node (physics)](https://en.wikipedia.org/wiki/Node_(physics))
- [Wave equation](https://en.wikipedia.org/wiki/Wave_equation)
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)