Checker Waves
z = sin(5x + t) · sin(5y + t)
Open in the app The dials and keys named below are the app's.
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.