# Cross Ripple

Surface · z = f(x, y, t)

`z = sin(x + y + t) + cos(x − y + t)`

[Open in the app](https://www.wavelace.com/app#p=29) · [This page](https://www.wavelace.com/presets/cross-ripple)

### What it draws

Each term is a plane wave running along a diagonal. `sin(x + y + t)` has its crests on the lines `x + y = constant`, and `cos(x − y + t)` has its crests on the lines `x − y = constant`. The two families cross at right angles, and the sheet is the sum of the two.

Each wave has a wavevector of length `√2`, so its crests are `2π/√2 ≈ 4.44` apart measured across them. Each carries `t` with coefficient 1, so each runs at `1/√2 ≈ 0.71` units per second at `Speed` 1. Both share that speed, which makes the sum an exact solution of the two-dimensional wave equation. The whole picture repeats every `2π ≈ 6.28` seconds.

### Why the pattern

The crossing is an illusion of the sum. Written as a product, `sin(x + y + t) + cos(x − y + t) = 2 · sin(x + t + π/4) · cos(y − π/4)`. That is one corrugation of wavelength `2π ≈ 6.28` sliding along `−x` at 1 unit per second. The profile in `y` that multiplies it never moves, so it holds the corrugation between fixed ridges. The zeros of that profile, `y = 3π/4 + nπ`, are nodal lines, and the sheet stays flat along them while everything between them heaves. Where the two waves agree the height reaches 2, twice either one alone.

### Try

- `Top` in the deck: the nodal lines are the straight creases the moving ridges never cross.
- Flip the sign on the second clock, `sin(x + y + t) + cos(x − y − t)`: the roles of `x` and `y` swap, so the ridges lie the other way and the corrugation marches along `−y`.
- Speed one wave up, `sin(x + y + t) + cos(x − y + 2t)`: the nodal lines come unstuck and drift along `+y` at half a unit per second while the ridges run at 1.5.
- Drop `Span of x, y` to 4 for a single cell of the pattern, one ridge between two nodal lines.

### Read more

- [Plane wave](https://en.wikipedia.org/wiki/Plane_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)