Ripple tank.

Note

The equation is ∂²u/∂t² = c² ∇²u, where u is the height of the water above rest. On a grid it becomes one line of arithmetic per cell: the next height is twice the current height, minus the previous height, plus c² times the discrete Laplacian (the sum of the four neighbours minus four times the cell). Here c² is 0.45. The scheme is only stable while c² stays under 0.5 in two dimensions, so there is not much room to go faster per step.

A real tank has edges, and edges reflect. To fake an open sea, the outer 28 cells on every side are a sponge: each step multiplies the height by a factor that drops smoothly from 1 to 0.94 toward the border. A wave that enters the sponge fades before it can come back. The outermost row also uses Mur's boundary condition, which predicts where an outgoing wave should be on the next step and puts it there. Walls are the opposite: cells pinned to zero, which reflect everything.

In the double slit, the dashed rays leave the midpoint between the slits at the angles where d sin θ = m λ for whole numbers m. Along those directions the two path lengths differ by a whole number of wavelengths and the waves arrive in step. Thomas Young showed this with light in 1801, and the argument is the same for water. Shorten the wavelength and the rays fan in; the bright bands in the intensity view follow them. Near the slits the match is loose, because the formula assumes a screen far away. The brightness is set by the right half of the tank, which is why the side in front of the barrier burns out to white.

Waves on a square grid do not travel at quite the same speed in every direction. Short wavelengths along the diagonals move a few percent faster than along the axes. Drop a single source with the wavelength at 8 and the rings come out faintly squarish. That is numerical dispersion, an artefact of the grid, not of water.