Where is the sun when you see a rainbow?
This is a good question to start thinking about the physical process that gives rise to a rainbow. Most people have never noticed that the sun is always behind you when you face a rainbow, and that the center of the circular arc of the rainbow is in the direction opposite to that of the sun. The rain, of course, is in the direction of the rainbow.
Why is the sky brighter inside a rainbow?
Notice the contrast between the sky inside the arc and outside it. When one studies the refraction of sunlight on a raindrop one finds that there are many rays emerging at angles smaller than the rainbow ray, but essentially no light from single internal reflections at angles greater than this ray. Thus there is a lot of light within the bow, and very little beyond it. Because this light is a mix of all the rainbow colors, it is white. In the case of the secondary rainbow, the rainbow ray is the smallest angle and there are many rays emerging at angles greater than this one. Therefore the two bows combine to define a dark region between them - called Alexander's Dark Band, in honor of Alexander of Aphrodisias who discussed it some 1800 years ago!
What are Supernumerary Arcs?
In some rainbows, faint arcs just inside and near the top of the primary bow can be seen. These are called supernumerary arcs and were explained by Thomas Young in 1804 as arising from the interference of light along certain rays within the drop. Young's work had a profound influence on theories of the physical nature of light and his studies of the rainbow were a fundamental element of this. Young interpreted light in terms of it being a wave of some sort and that when two rays are scattered in the same direction within a raindrop, they may interfere with each other. Depending on how the rays mesh together, the interference can be constructive, in which case the rays produce a brightening, or destructive, in which case there is a reduction in brightness. This phenomenon is clearly described in Nussenzveig's article "The Theory of the Rainbow" in which he writes: "At angles very close to the rainbow angle the two paths through the droplet differ only slightly, and so the two rays interfere constructively. As the angle increases, the two rays follow paths of substantially different lengths. When the difference equals half of the wavelength, the interference is completely destructive; at still greater angles the beams reinforce again. The result is a periodic variation in the intensity of the scattered light, a series of alternately bright and dark bands."
Mikolaj and Pawel Sawicki have posted several beautiful photographs of rainbows showing these arcs.
The "purity" of the colors of the rainbow depends on the size of the raindrops. Large drops (diameters of a few millimeters) give bright rainbows with well defined colors; small droplets (diameters of about 0.01 mm) produce rainbows of overlapping colors that appear nearly white. And remember that the models that predict a rainbow arc all assume spherical shapes for raindrops.
There is never a single size for water drops in rain but a mixture of many sizes and shapes. This results in a composite rainbow. Raindrops generally don't "grow" to radii larger than about 0.5 cm without breaking up because of collisions with other raindrops, although occasionally drops a few millimeters larger in radius have been observed when there are very few drops (and so few collisions between the drops) in a rainstorm. Bill Livingston suggests: " If you are brave enough, look up during a thunder shower at the falling drops. Some may hit your eye (or glasses), but this is not fatal. You will actually see that the drops are distorted and are oscillating."
It is the surface tension of water that moulds raindrops into spherical shapes, if no other forces are acting on them. But as a drop falls in the air, the 'drag' causes a distortion in its shape, making it somewhat flattened. Deviations from a spherical shape have been measured by suspending drops in the air stream of a vertical wind tunnel (Pruppacher and Beard, 1970, and Pruppacher and Pitter, 1971). Small drops of radius less than 140 microns (0.014 cm) remain spherical, but as the size of the drop increases, the flattening becomes noticeable. For drops with a radius near 0.14 cm, the height/width ratio is 0.85. This flattening increases for larger drops.
Spherical drops produce symmetrical rainbows, but rainbows seen when the sun is near the horizon are often observed to be brighter at their sides, the vertical part, than at their top. Alistair Fraser has explained this phenomenon as resulting from the complex mixture of size and shape of the raindrops. The reflection and refraction of light from a flattened water droplet is not symmetrical. For a flattened drop, some of the rainbow ray is lost at top and bottom of the drop. Therefore, we see the rays from these flattened drops only as we view them horizontally; thus the rainbow produced by the large drops is is bright at its base. Near the top of the arc only small spherical drops produce the fainter rainbow.
What does a rainbow look like through dark glasses?
This is a "trick" question because the answer depends on whether or not your glasses are Polaroid. When light is reflected at certain angles it becomes polarized (discussed again quite well in Nussenzveig's article), and it has been found that the rainbow angle is close to that angle of reflection at which incident, unpolarized light (sunlight) is almost completely polarized. So if you look at a rainbow with Polaroid sunglasses and rotate the lenses around the line of sight, part of the rainbow will disappear!
Other Questions about the Rainbow
Humphreys (Physics of the Air, p. 478) discusses several "popular" questions about the rainbow:
"What is the rainbow's distance?" It is nearby or far away, according to where the raindrops are, extending from the closest to the farthest illuminated drops along the elements of the rainbow cone.
Why is the rainbow so frequently seen during summer and so seldom during winter?" To see a rainbow, one has to have rain and sunshine. In the winter, water droplets freeze into ice particles that do not produce a rainbow but scatter light in other very interesting patterns.
"Why are rainbows so rarely seen at noon?" Remember that the center of the rainbow's circle is opposite the sun so that it is as far below the level of the observer as the sun is above it.
"Do two people ever see the same rainbow?" Humphreys points out that "since the rainbow is a special distribution of colors (produced in a particular way) with reference to a definite point - the eye of the observer - and as no single distribution can be the same for two separate points, it follows that two observers do not, and cannot, see the same rainbow." In fact, each eye sees its own rainbow!!
Of course, a camera lens will record an image of a rainbow which can then be seen my many people! (thanks to Tom and Rachel Ludovise for pointing this out!)
"Can the same rainbow be seen by reflection as seen directly?" On the basis of the arguments given in the preceding question, bows appropriate for two different points are produced by different drops; hence, a bow seen by reflection is not the same as the one seen directly".
