The Physics of Rainbows
As we are coming down to the final weeks of winter, the days are getting longer, and it is slowing starting to warm up. We are all looking forward to springtime, with its promises of flowers and rain. Along with this rain brings reminders of rainbows. As Donald Ahrens says in the Meteorology Today magazine, “rainbows are one of the most spectacular light shows observed on earth (About).” In fact, one of the best ways to view a rainbow at it’s utmost beauty is when half of the sky is still dark with clouds, and the observer is standing at a spot where the sky is clear (Rainbow). All of the different colors of the rainbow are very recognizable and memorable to all who observe them. Many children are taught the “Roy G. Biv” mnemonic as young children, learning the basic knowledge about rainbows. Whenever I see one, I am reminded of the beauty of the earth and how amazing creation is, with its power to boost our spirits and remind us of God’s promise. A rainbow is defined as an optical and meteorological phenomenon that causes a nearly continuous spectrum of light to appear in the sky when the sun shines onto droplets of moisture in the Earth’s atmosphere (Rainbow). A rainbow takes the shape of an arc with its various colors of red, orange, yellow, green, blue, indigo, and violet. Getting more acquainted with rainbows and how they are formed, however, can help to understand and appreciate the full effects of such an amazing feature of the Earth.
There are two main aspects that influence a rainbow, and these include the altitude of the sun and the size of the raindrops (The Rainbow). The sunlight is refracted, which means the bending of light as it passes from one medium to another. This refraction makes different wavelengths, or colors, of the white light from the sun to separate. Determining whether or not the wavelengths will go through the raindrops or reflect always depends on the angle that the light falls on the backside of the raindrop (The Rainbow). Red light is refracted by a smaller angle than blue light, and when leaving the raindrop, the red rays of light have gone through a smaller angle than that of the blue rays (Rainbow). If a certain wavelength hits the back of a raindrop at a less than 48 degree angle, the light will pass through a spherical raindrop, but if the light does not go through and strikes back, the light will be reflected. The process of light bouncing off of the raindrop through reflection is repeated for many different raindrops, thus creating a rainbow. The light from the sun will pass over the watcher, hit the rain droplets, and come back to the observer, so we can view the rainbow (The Rainbow). It is also important to note that whenever someone observes a rainbow, the sun is always at their back (Rainbow).
In addition to how rainbows are formed through sunlight and raindrops, what actually makes the shape, or “bow” of the rainbow? This idea was first discussed by Rene Descartes in 1637. Descartes had many ideas and drawings of how the shape of the rainbow can be understood and explained. He asked people to imagine how light is reflected through the raindrop, how it is reflected by curved, mirror-like insides of a raindrop, and how it refracts as it leaves the drop. Furthermore, if we think about how this process is applied to many, many raindrops, we can then start to perceive the shape of the rainbow.
Descartes also calculated the standard deviation for the red light ray to be about 138 degrees. Descartes drew a ray of light that showed the smallest angle of deviation of all of the rays on the raindrop, which was called the Descartes ray, or the rainbow ray. This concentration of the rays closest to the minimum deviation gives the rainbow its arc shape in the sky (About). It is also very surprising to know that a rainbow, in reality, is not a two-dimensional shape of an arc, but rather a three-dimensional cone figure where the highest point, or the apex, is at the observer’s eye level. There is no concept, or evidence of distance, therefore, it seems to us as observers that a rainbow is two-dimensional. To the observer of the rainbow, only the raindrops at the highest point of the cone make it possible for us to actually visualize the shape that we see, and actually see the rainbow. In reality, this implies that each and every person looks at a different rainbow, because each observer’s eyes are at an apex of different cones. In addition, the outside layers of the cone are red, and the inside is violet, which matches up with the order of the colors in the spectrum (The Rainbow). Finally, the reason that we can’t see a full circle of a rainbow is because the Earth is in our way. The lower that the sun is to the ground, the more of a circle that we will see, and if the sun is higher in the sky, we will see a smaller arch in the rainbow (About).
Colors of the Rainbow
In addition to seeing the shape of the rainbow, the array of colors is what really stands out to us. The rays of light that are dispersed as a result of reflection and refraction show a series of colors called the visible spectrum (Serway). From our mnemonic of “Roy G. Biv,” we know that these colors are red, orange, yellow, green, blue, indigo, and violet (Rainbow). The angle of deviation of these colors depends on the wavelength. Red light deviates the least, and violet light deviates the most. Therefore, the remaining colors in the spectrum fall in between these two extremes of wavelengths (Serway). The Red light will emerge from the raindrop at an angle of 42 degrees, while the violet light emerges at a 40 degree angle to the rainbow observer’s line of vision. It is important to note that only one color of the spectrum is given off from each individual raindrop, and that the combination of all the many raindrops is what will produce the wide array of colors that we can see with the unaided eye (Formation). Isaac Newton explained that each color in the array has its own angle of deviation, and that all of these colors can be combined again to form the original white sunlight (Serway). He discovered this property of sunlight in 1666. This concept is the first way to explain why we can see colors arise from a rainbow. The other reason is that light from different colors can be refracted by different amounts as it passes from one medium, such as air, into another medium, such as water. The reason that we see the blue light on the inside of the arc of the rainbow is because blue light is refracted more than red light, which is on the top of the “bow (About).” For this reason, the color bands of the rainbow have always been in the same order; therefore, “Roy G. Biv” should never change his name (Formation).
Occasionally, after the rain comes, we can see a double rainbow. A second, dimmer, and thicker rainbow, called a secondary rainbow is seen above the primary rainbow, which is brighter and more defined. What causes this to happen? The secondary rainbow is caused by the double reflection of light that is inside the raindrops (Rainbow). These drops will appear at angles of 50-53 degrees rather than 40-42 degrees in the primary rainbows. Here, the blue light emerges from the drops at a bigger angle of 53 degrees, and the red light emerges at 50 degrees. The colors in the secondary rainbow are thus seen to an observer in the opposite order of the primary one. Red is seen on the inside, and blue and violet are seen on the outside, or the top of the rainbow (Formation). Another reason for this double reflection of light is because not all of the energy of the rays of light escape from the raindrop after the first reflection. When the light is reflected for the second time, the ray will travel along the inside of the raindrop and emerge, flowing into the secondary rainbow (About). Every now and then, a third rainbow, or a triple rainbow, can be seen, but these are unusual circumstances. There have even been a few rainbow watchers that have claimed to have seen quadruple rainbows. In these instances, the bows on the outermost arcs had pulsating and rippling appearances, differing from the primary and secondary rainbows. Also, these rainbows were spotted on the same side of the Sun, making them very hard to notice (Rainbow).
Renee Descartes' sketch of how primary and secondary rainbows are formed.
Another question that comes to mind when getting the opportunity to view a double rainbow is, “Why is the sky brighter inside the rainbows?” When we study the refraction of the sunlight on the raindrops, it is noticed that there are many multiple rays that emerge at angles that are smaller than the rays of the actual rainbow. But, no light from single internal reflections are at angles that are greater than this ray of light. Therefore, there is a lot of light that is within the inside of the rainbow, and very little that is beyond, or above and behind the rainbow. Like mentioned earlier, this bright light within the rainbow is formed from the combining of all of the colors in the rainbow spectrum, which forms the bright white light (About). In addition, when two rainbows are present, the dark area of dimmer, unlit sky between the primary and secondary rainbows is called Alexander’s Band (Rainbow). This was named in honor of Alexander of Aphodisias, who discussed this dark band around 1800 years ago (About).
Alexander's Band is seen between the primary and secondary rainbows.
Sometimes, we can see and observe another beautiful phenomenon that can be part of a striking rainbow display. Some rainbows have faint, dimmer looking bows, or arcs that are just inside and close to the top of the rainbow can be seen (About). Very rarely, they can be seen outside the secondary rainbows, as well. They are to some extent detached from the main part of the rainbow and they have a pastel color to their bows that do not match up with the usual color of the rainbow. These bows are known as supernumerary arcs, and it is difficult to explain their existence through geometric optics, like other parts of a rainbow can be explained. These fainter looking rainbows are said to have been caused by the interference between the rays of light that follow somewhat different paths with varying lengths within the raindrops. Supernumerary bows are seen the clearest in color when the raindrops are smaller and more similar in size. With the different angles of refraction for rays of varying colors, the patterns of interference are different for rays of light of different colors. Therefore, each band is diverse in color, which creates a miniature rainbow (Rainbow). In 1804, Thomas Young was the first to explain how supernumerary rainbows were indications of the wave nature of light. He explained light in relation to it being a wave of a certain kind and that when two rays of light are scattered in the same direction inside a raindrop, they might interfere with each other (About). When studying physics and mathematics, the scattering theory is defined as, “a framework for studying the scattering of waves and particles.” The scattering of waves correlates to the colliding and scattering of a wave with a different material object. In this case, sunlight is scattered by the raindrops to form a rainbow (Scattering). Depending on how the rays come together, they can be brightening, which is constructive, or they could be destructive, and show a decrease in brightness (About).
Photograph of a supernumerary arc, showing the additional green and purple arcs inside the primary rainbow.
Reflection Rainbows and Reflected Light Rainbows
There are two other types of rainbows that have to do with light reflecting off of a body of water. These are called reflection rainbows and reflected light rainbows. Reflection rainbows are produced by the reflection of the source of incident light, which is usually the sun, and when sunlight reflects off water before reaching the raindrops (Rainbow, About). Reflection rainbows have the same endpoints as normal rainbows, but they embody a bigger arc when all of it is visible. In contrast, a reflected rainbow is formed when light that has first been reflected inside the raindrops then reflects off of a body of water before it reaches the observer. This is not an exact image of the primary rainbow, but it is displaced away from it a bit depending on the height of the sun (Rainbow). Furthermore, it is possible for a rainbow to be formed from indirect sunlight. It is then noticed that a rainbow produced by a reflected light source will appear to be higher in the sky than one formed by direct sunlight (Formation).
It is evident that rainbows are not only wonders of beauty, but are intricate structures involving the nature of light and the laws of geometric optics. By learning more about the how rainbows are formed and the complexity to which they give off their radiant colors, I have learned to appreciate these structures much more than before.
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