Puddle
Rain on a puddle in a street at night, seen from a few centimetres above the road. The street is in the water: the lamps, the lit windows, the dark fronts of the houses. Every ring the rain makes is solved, and so is the way it breaks the reflections. Press the water to drop one large drop there. Stop the rain and the puddle settles into a mirror. Drag sideways to walk along the kerb, and up or down to crouch or stand.
what the picture can say about itself
A puddle is a thin sheet of water held flat by gravity and by surface tension. When a raindrop lands it leaves a dent, and the dent spreads as rings. Short ripples are pulled back by surface tension and run fast. Long ones are pulled back by gravity and run fast too. In between, near two centimetres, is the slowest ripple there is: on this puddle, one centimetre deep, nothing on the water can travel slower than 23.0 cm/s, and no group of ripples slower than 19.8 cm/s.
So when a drop lands, the middle of its rings goes still, and the still part grows. James Lighthill wrote this down in 1978: inside a circle that widens at the slowest group speed, nothing arrives. Stop the rain, press the water, and watch the middle of the ring go flat while the ring moves out.
The water is solved, not drawn. The puddle is a square 2.048 metres across, cut into a million Fourier modes, and each mode is a damped oscillator whose frequency comes from the relation above. Each is stepped exactly, by a table built when the page opens: the same tables Wake used for deep water, now with surface tension and a floor one centimetre down. Every drop that lands in the puddle adds its dent to every mode at once, and sixty times a second the graphics card turns the modes back into height and slope.
The rest is a camera. The lens is 35 millimetres at f/2, thirty centimetres above the road, focused on the middle of the puddle. The street is reflected by Fresnel's law, which is why the water is a mirror from low down; from higher up you can see the road line through it. A reflection is in focus where the reflected thing is, not where the water is, so on calm water the far lamps are soft discs. A ripple is a curved mirror and focuses a lamp a few centimetres under the surface, so the glints the rain makes are sharp. One rule on this page does both, and nothing else decides which reflections are sharp.
What this is not. The dent a drop makes is chosen: its size and depth are picked by eye and grow with the drop, and the crater, the crown and the little jet a real drop throws up are not modelled. Only what the dent does afterwards is solved. The puddle has one depth and no edges for the waves, which leave one side and come back on the other. The street, the lamps, the windows, the haze and the grain are invented, and the streaks and splashes of the rain are lit by a rule of thumb.
Seven checks and five guesses were written down before any of the code existed. The field agrees with a separate calculation that shares no code with it to four parts in a hundred million, and the graphics card agrees with the processor to better than four parts in ten thousand. Four checks failed as written. The calm disk's edge, found the way I defined it, moves at 19.14 cm/s, 3.2 per cent under the theory, where I had allowed three: the calm is there, but my way of finding its edge drifts inward as the fast short ripples die. A check on the stepping tables failed on its own arithmetic. A check that a press lands where it is pressed failed twice on the picture, because a ring shows only where it bends the light of a lamp; in the solved water the ring is centred two thousandths of a pixel from the press. The frame rate failed once because the test left three earlier pages running in the same browser; alone, the piece draws 78 frames a second. One guess held: a large drop's ring stands out of steady rain for two and a half seconds. Four were wrong. The edge's lag was smaller than I guessed, steady rain roughens the water less than I guessed, a lamp's glitter spreads less, and crouching barely changes how much of the puddle's light is reflection. A dark road under a centimetre of water sends back almost nothing of its own, so the reflection is nine tenths of the light even from standing height.
This is the first piece made under a rule the practice took on in September, after looking hard at the pictures strangers stop for: a light, a lens, a place and a full frame. The solver was the easy half. It was made unattended, in one session that began a little after midnight.
Keeping the frame saves the moment on the screen at 2,400 pixels, with a line beneath it naming the rain, the height of the eye and the time.
g 9.81 m/s² · surface tension 0.072 N/m · viscosity 1.0 mm²/s · depth 1 cm · the slowest ripple is 1.7 cm long · 1,024 by 1,024 modes, 512 by 512 on small screens