How it works
Why ripples make bright points in a pool
On a sunny day the floor of a pool is covered in moving lines of light. They are called caustics, from the Greek for “burnt”, because a lens that gathers sunlight this way can set things alight. In this game you look into a pool from the side, as if through a glass wall, and make that light yourself, one ripple at a time. In a side view, each of those lines shows up as a bright point.
Worked out from physics
- The light. Sunlight comes down as rays side by side. Where each ray enters the water, it bends by Snell’s law, by an amount set by how steep the surface is there. Where neighbouring rays crowd together, the water and the floor are bright; where they spread apart, they are dim.
- The bright points. A crest (the top of a ripple) is curved like a magnifying glass, so it bends the rays under it toward each other. Light travels n = 1.333 times slower in water than in air, and the rays meet about n/(n−1) ≈ 4 times the crest’s radius of curvature below it (the radius of the circle that fits the top of the crest; 1.333 ÷ 0.333 ≈ 4). Below that point the rays cross and spread apart again.
- The ripples. They spread at the real speeds of water waves, set by gravity and surface tension: ω² = gk + (σ/ρ)k³ (ω: how fast the water bobs up and down; k: how close together the crests are; g: gravity; σ: surface tension; ρ: the water’s density). A drop makes a ring of ripples, and a ring gets lower as it spreads, so its crests get flatter and their bright points form deeper.
- The sun. It is a disc about half a degree across, which softens the bright points a little more the deeper they are. A low sun comes in at a slant, and its light lands off to one side.
Rules we set for the game
- We show a thin slice of the pool, straight through each drop: the light is worked out in that slice.
- The three sizes of ripple (small 3 cm, medium 6 cm and big 12 cm from crest to crest) and how tall they start. Real ripples last longer: we fade them faster so the pool calms down between drops.
- Only light more than 4 times as bright as sunlight through calm water counts. A pearl lights once enough of it has added up, like a magnifying glass that only burns when it is focused. The gold ring around a pearl shows how much has added up.
- The pools, their depths and the pearls. A drop can’t land right above a pearl: its ripple has to travel to it.
- The look: the water’s surface is drawn 10 times taller than it is (the light comes from the true ripples); the brightest spot under each crest gets a sparkle; brightness and colours are set for the screen. In the daily pool the sun sinks far faster than in a real day.
From the videos
This game grew out of our pool videos, where viewers asked for the math behind the light. Same physics, now in your hands.