Rainbows bring a smile. They may remind you of leprechauns or they may remind you of Noah. Or you may just find them pretty. They are lovely to look at and, like so many common things, have a wealth of real physics in their explanation. Rainbows are seen after rains or after sunlight passes through the fine mist of a waterfall or a fountain or an ordinary garden sprinkler.
Rainbows are seen when the sun is over the viewer's back and with the sunlight being passing through a mist of tiny spherical water droplets. This diagram illustrates what happens when a ray of light enters a spherical drop of water. There, it is refracted at the front surface, reflected inside at the back surface, and refracted again as it exits the front surface. Due to dispersion, colors are spread out at each refraction.
If we look at light rays that come into a spherical water droplet at varying distances from the centerline, we will find that some of the rays are scattered through 180° -- entirely back scattered. Other rays will be scattered less but there is a minimum scattering angle of about 138° as shown here. That means the angle between the incident light and the scattered light will be 42° (42° = 180° - 138°). In fact, most of the light will be scattered at an angle very close to this as shown
The sketch below shows what an observer would see with the Sun over her back and sunlight incident on a mist of tiny water droplets. Most of the light will be scattered through about 138°, with an angle of 42° between the incoming and scattered light. This will produce a bright arc or ring or "bow" with an angle of 42° between the center of this bow and the bow as measured by the observer as sketched in the figure. Some light will be scattered through angles greater than 138° so this light will be seen inside the bow, making this region brighter. No light will be scattered through angles smaller than 138° so no light will be scattered outside the bow, making this region dark.
Now for the color in a rainbow!
Sometimes, if the conditions are just right, a second rainbow may be seen outside the usual one. This secondary rainbow is fainter than the primary rainbow and the colors are reversed in order. The sky will be bright outside this secondary rainbow and the region between the two rainbows will be dark. Such a double rainbow is shown below. The secondary rainbow is the result of light that has been reflected twice inside the water droplet as sketched here.
(C) 2003, Doug Davis; all rights reserved