Reckoning / Light / The rainbow
Light · one drop
The rainbow
Sunlight enters a raindrop, reflects once off the back of it, and comes out again. Rays that strike the drop at different heights come out turned by different amounts. Near one particular height they bunch together, and that bunching, a little different for each colour, is the bow.
The drop
Computed
The chart shows where rays from each height end up. The curves flatten at their peaks (the dots), where many neighbouring rays leave at almost the same angle. That pile-up makes the bright bow. Red peaks at a larger angle than violet, so red is on the outside.
Descartes’ calculation
A ray striking the drop at height b (as a fraction of the radius) meets the surface at angle i with sin i = b, and refracts to r with sin i = n sin r. Each refraction, on entering and again on leaving, turns it by i − r, and each reflection inside turns it by 180° − 2r. With one reflection inside, the total turn is
D is least where cos² i = (n² − 1)/3. That is at b = 0.861, i = 59.4° for yellow light. The light leaving there makes an angle 180° − D with the direction straight away from the Sun, the antisolar point, where your head’s shadow falls. Every drop at that angle sends you its bright light, so the bow is a circle around the antisolar point: 42.5° for red, 42.1° for yellow and 40.6° for violet light.
With two reflections inside, the least turn comes where cos² i = (n² − 1)/8. That gives the fainter secondary bow, from about 50.1° (red, now on the inside) to 53.5° (violet, outside): its colours are reversed. No light from one or two reflections leaves between the two bows, so the sky there is darker: Alexander’s dark band.
Theodoric of Freiberg explained the bow with water-filled glass globes about 1307: refraction, a reflection inside, refraction again. René Descartes traced rays through a drop by calculation in Les Météores (1637) (Descartes 1637) and found the 42° angle. Newton then explained the colours. Ray optics cannot explain everything. The faint pastel supernumerary bows just inside the main one come from interference, a wave effect, worked out by Thomas Young (1803) and George Airy (1838).
These are rays, the geometrical-optics picture: light is treated as lines that bend and split at surfaces. Diffraction and interference, which need the wave picture, are not shown. The bench is a flat cross-section, so a lens here is really a cylinder of glass seen end-on. That is exact for rays in the plane of the drawing. Air is taken as n = 1, and the glass and water indices are given relative to air, so the angles are right. Colours on screen are an approximation of the spectrum.