# Depth Sheet

Wave · y = f(x, t)

`y = sin(x − t) · cos(z · 0.8 + t · 0.5)`

[Open in the app](https://www.wavelace.com/app#p=20) · [This page](https://www.wavelace.com/presets/depth-sheet)

### What it draws

The formula is a product of two factors, one in `x` and one in `z`. `sin(x − t)` is the wave itself: crests `2π ≈ 6.28` apart, travelling right at one unit per second at `Speed` 1.

Each trailing slice is fed its own depth as `z`, and `cos(z·0.8 + t·0.5)` multiplies the whole of a slice by a single number. It sets how tall that slice is and nothing else, so the trailing copies stop being a ribbon and become a sheet. Its wavelength in depth is `2π/0.8 ≈ 7.85`, about one cycle over the ribbon's reach, and `t·0.5` rolls the pattern forward.

### The sheet in depth

The two feeds interact, because a slice further back in depth is also older. The clock inside the second factor therefore runs backwards along the sheet, against the `0.8z`, and partly cancels it. What is left is a slower turn from one slice to the next than the depth term alone would give.

Where that factor passes zero, a whole slice flattens onto the plane. Those flat slices march steadily toward the front, and the depth pattern comes back to itself every `2π/0.5 ≈ 12.6` seconds at `Speed` 1.

The lag shows in the other direction too. An older slice draws the same sine a little later in its travel, so its crests sit behind those of the slice in front of it. The ridges of the sheet lean off the depth axis by a fixed angle, and that lean is the wave's speed drawn as a slope.

### Try

- `Top` in the deck: straight down on the sheet, where the leaning ridges and the flat slices are plainest.
- Take the depth away, `sin(x − t)`: every slice the same height, and the sheet is an ordinary ribbon again.
- Stop the roll, `sin(x − t) · cos(z·0.8)`: with no `t` in the second factor the depth pattern holds still and the flat slices stay where they are.
- Double the depth frequency, `sin(x − t) · cos(z·1.6 + t·0.5)`: the factor turns twice as fast with depth, halving the wavelength to `3.93` and fitting more flat slices into the sheet.

### Read more

- [Plane wave](https://en.wikipedia.org/wiki/Plane_wave)
- [Separation of variables](https://en.wikipedia.org/wiki/Separation_of_variables)
- [Node (physics)](https://en.wikipedia.org/wiki/Node_(physics))
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)