Analysis of rainbows (Part II)
- David Chandler
This demo continues from Analysis of rainbows (Part I). Use the check boxes to display how light scatters from an individual raindrop or the resulting pattern from a large number of raindrops, which we call a rainbow. You can also select whether you want to see the primary or secondary bows or both. If you select to see the raindrop, you can move the drop around and see how light from the scatter pattern would enter the eye of the person. You can also alter the direction of sunlight with the golden arrow.
Using the check boxes, display the raindrop and the primary bow. The raindrop backscatters incoming light back toward the source forming a cone. The light is white except at the surface of the cone. Red forms a slightly larger cone, so the colors separate out. See Part I to understand why that is. Move the droplet around and see where it would have to be for the person to see white light, colored light, or no light from that droplet. Now display the rainbow. Notice that if you place the raindrop inside the bow it will scatter white light toward your eyes, making the sky appear bright inside the bow. If you place the droplet at the top of the bow, you can see how it would contribute to the overall color in that direction. Move the point at the center of the rainbow. The bow is not at any particular distance. When you see the bow you are seeing a cone of incoming light from droplets at different distances, with you at the vertex of the cone. Each person is seeing a rainbow created by a different collection of droplets, but always in a cone of a fixed angle centered on the anti-solar direction (i.e. around the shadow of your head). The bow appears projected to infinity, so "somewhere over the rainbow" and the "pot of gold at the end of the rainbow" are literary and imaginative references, not physical places. Notice that if the droplet is placed outside the bow the light from that droplet does not reach you at all. It sends its light elsewhere. This causes the sky to appear dark just outside the bow. Now turn off the rainbow and switch to the raindrop and secondary checkboxes. The light that reflects twice before leaving the droplet forward scatters. Again there is a cone, but it is a cone where you would not see any scattered light. Notice that the colors are reversed since red deflects the most from the forward direction, making it deflect less as measured from the direction of incoming sunlight. Move the drop around until you can find where the person sees color. Remember that the light is forward scattered, so droplets inside the cone scatter their light elsewhere and droplets outside the cone scatter white light toward you. Also note that the more times the light reflects inside the drop, the fainter it gets. The primary bow is brightest. The secondary bow is often overlooked. Now display the secondary rainbow. Note that it is larger than the primary bow, but still centered on the anti-solar direction. This time there is visible scattering from outside the bow, but no scattering from droplets inside the bow. Again, move the rainbow to different distances to remind yourself that it is coming from droplets along the surface of a cone. Now put up both primary and secondary bows. Note that the sky should appear bright inside the primary bow, dark outside the primary bow, and bright again (less bright, just not as dark) outside the secondary bow. This produces a dark band that can be seen around natural rainbows, if you look for it. The primary and secondary bows are always there, in arcs surrounding the anti-solar direction at the same angle from the incoming sunlight, provided there are droplets in that direction and sunlight can reach the droplets and the scattered light can reach your eye. Now play with the direction of the sunlight. Note that the bows are largest when the sun is near the horizon, smaller as the sun gets higher, and not visible at all if the sun is too high in the sky. The bows can appear completely circular if you are looking down into mist (at a water sprinkler or while flying). Rainbows are spectacular and things stirring imagination and awe, but they have an understandable physical cause. Understanding the causes actually gives you more phenomena to look for when you see a natual rainbow. Go out any day, if it is sunny, and create a rainbow with a spray of water. Look for rainbows in the spray of waterfalls. If you can create a beam of bright light in a dark lab you can observe the light emerging from the primary, secondary, and possibly more reflections in the drop. Challenge: Extend the Part I analysis by adding more reflections in Geogebra. The refraction as the beam enters the droplet is computed for you. The rest of the path can be constructed using the reflecting tool about lines of symmetry. Calculating the angle of refration and understanding the law of reflection are physics. The rest is geometry. For the physics of refraction, look up Snell's Law.