CINXE.COM

<html><head><script type="text/javascript" src="/_static/js/bundle-playback.js?v=HxkREWBo" charset="utf-8"></script> <script type="text/javascript" src="/_static/js/wombat.js?v=txqj7nKC" charset="utf-8"></script> <script>window.RufflePlayer=window.RufflePlayer||{};window.RufflePlayer.config={"autoplay":"on","unmuteOverlay":"hidden"};</script> <script type="text/javascript" src="/_static/js/ruffle/ruffle.js"></script> <script type="text/javascript"> __wm.init("https://web.archive.org/web"); __wm.wombat("http://www.usna.edu:80/Users/oceano/raylee/RainbowBridge/Chapter_8.html","20051214151844","https://web.archive.org/","web","/_static/", "1134573524"); </script> <link rel="stylesheet" type="text/css" href="/_static/css/banner-styles.css?v=S1zqJCYt" /> <link rel="stylesheet" type="text/css" href="/_static/css/iconochive.css?v=3PDvdIFv" />  <title>Raymond L. Lee, Jr.: Chapter 8--THE RAINBOW BRIDGE</title></head> <body text="#000000" bgcolor="#F0F8FF" link="#0000FF" vlink="#800080" alink="#FF0080"> <body><h1><center>©Raymond L. Lee, Jr., U. S. Naval Academy The Rainbow Bridge:  Chapter 8 What are all the colors of the rainbow?</h1> <h2><a href="https://web.archive.org/web/20051214151844/http://www.psupress.org/books/titles/0-271-01977-8.html"> Penn State Press</a> (University Park, PA), 2001 and SPIE Press (Bellingham, WA) ISBN 0-271-01977-8 Co-author: Alistair B. Fraser</h2></center> <table border><tr rowspan="1" align="left"> <td colspan="1" align="left"> Long before Newton, unusual varieties of rainbows had prompted sporadic scientific interest. After Newton, these varietal bows raised nagging doubts about the completeness of his answers, and ultimately would lead to powerful new rainbow theories in 19th-century optics. With these theories, we can see that all the colors of the rainbow are actually quite different from our preconceptions. Yet for 19th-century artists still debating the validity of Newtons rainbow colors, these new optical theories clearly were peripheral -- the divorce between the rainbows of interest to them and to scientists was nearly complete. Ironically, the 19th century produced some of the strongest claims about the unity of artistic and scientific enterprise, testimony to the rainbow bridges tenuous power. </td></tr></table> <center>... I saw in that part of the Skie, where a Rainbow would naturally be; Something, which was like one, but much broader, fainter, and though colourd yet indistinct; there was no appearance of Rain, nor do I believe there was any: And indeed the Bow was too much confused to be formd by spherical drops of water.<a name="fnB0"></a><a href="#fn0">[1]</a> Thomas Barker, A remarkable cloud rainbow (1739)</center> Although we cannot be certain what kind of bow Thomas Barker saw during the winter of 1739-1740, we can be sure that he did not see a rainbow proper. Our clue (and Barkers) is his bows indistinct colors. As we have seen earlier, two characteristics earn almost any sky feature the name rainbow -- it is an arc and colored. While not all colored arcs in the sky are rainbows, Barker is quite right to be skeptical that his broader, fainter arc is not a rainbow. What Barker calls a Frost Rainbow probably is a cloudbow (Fig. 8-1).<a name="fnB1"></a><a href="#fn1">[2]</a> At first glance, this broad, anemically colored bow looks so different from the rainbow that we may be forgiven if we side with Barker and believe that it is too much confused to be formd by spherical drops of water. Barker makes the plausible but erroneous assumption that in forming his bow, raindrops have been replaced by very small round & icy hail.<a name="fnB2"></a><a href="#fn2">[3]</a> However, hail is neither clear enough nor symmetric enough to be an adequate substitute for transparent, spherical water drops. In fact, both the rainbow and cloudbow are merely varieties of water-drop bows. In the case of cloudbows, the water drops are those of the clouds themselves. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_1.GIF" width="350" height="248" align="bottom" naturalsizeflag="3"></center> Fig. 8-1: A cloudbow, also known as a fogbow or pilots bow, seen from an airplane. The bright light immediately surrounding the airplane shadow is not a cloudbow, but a glory.<a name="fnB3"></a><a href="#fn3">[4]</a> <h3><center> {N.B.: Not all images are reproduced in this WWW version of Chapter 8.}</center> </h3> As many an inquisitive (and wet) child has discovered, making a rainbow does not require that you wait for rain -- a sunlit spray of water drops from a hose will do just fine (Fig. 8-2). So apparently, raindrops are not uniquely qualified to form rainbows (or water-drop bows, more generally). But how do water droplets in clouds yield a bow whose appearance differs so radically from a bow produced by rain droplets? Fig. 8-2: A water-drop bow seen in a spray of sunlit drops. (Photograph courtesy of Michael E. Churma) Surprisingly, explaining the essentially colorless cloudbow also tells us how some colorful, but subtle, rainbow features arise (Fig. 8-3). In Fig. 8-3, note the narrow pastel bands inside the primary rainbow -- the supernumerary bows. Although reports of the supernumeraries date from the 13th century, neither Newtons nor Descartes theories of the rainbow can account for them.<a name="fnB4"></a><a href="#fn4">[5]</a> By mid-18th century, increasingly frequent accounts of the supernumeraries and cloudbow provoked scientific interest (and consternation), ultimately leading to new theories of the rainbow. Fig. 8-3: Supernumerary rainbows inside a primary rainbow at Kootenay Lake, British Columbia, 12 July 1979. Neither the cloudbow nor the supernumeraries have contributed much to the rainbows cultural symbolism. After all, they roil the simple rainbow image considerably, and the fact that their scientific explanation was even more abstruse than Newtons made them relatively uninteresting to most lay audiences. By the 19th century, artistic interest in rainbow optics usually lagged behind the science itself (the cases of John Constable and Frederic Church are typical). Ironically, while some artists interest in rainbow science reached its zenith, scientific study of the rainbow had already moved far beyond most artists scientific expertise. Given this increasingly tenuous rainbow bridge between science and art, our story of cloudbows and supernumeraries is primarily a scientific one. Nonetheless, it is a story with great rewards for even the most casual rainbow observer. <hr><h3>Cloudbows: The Circle of Ulloa</h3> In 1748, Spanish explorers Jorge Juan y Santacila (1713-1773) and Antonio de Ulloa (1716-1795) gave an account of glories and a cloudbow that Juans party had seen from Mount Pambamarca in present-day Ecuador (Fig. 8-4).<a name="fnB5"></a><a href="#fn5">[6]</a> Juan and Ulloa were far from the first to write on cloudbows -- Theodoric of Freibergs account of cloudbows preceded the Spaniards by nearly 450 years, Witelo had mentioned cloudbows even earlier, and Avicenna clearly describes an 11th-century cloudbow. Early documentation, however, did not mean early explanation. While Theodoric correctly noted that the cloudbow (or fogbow) and the primary rainbow arose in the same way, he offered no convincing explanation of why their colors differed.<a name="fnB6"></a><a href="#fn6">[7]</a> Witelo simply accounted for the completely white rainbow with a combination of thin vapor and clear illumination.<a name="fnB7"></a><a href="#fn7">[8]</a> No more persuasive is Avicennas statement that his cloudbows diameter shrank because he grew more distant from the sun as he descended the mountain toward the bow.<a name="fnB8"></a><a href="#fn8">[9]</a> Fig. 8-4: Eighteenth-century illustration of a cloudbow and glories seen by Juans scientific party. (Lynch and Futterman 1991, Fig. 2) The cloudbow is the large white arc surrounding the bulls-eye glory pattern (the relative angular sizes of the cloudbow and glories are at odds with nature and Juans accompanying text). Leonardo da Vincis typical meteorological insightfulness is evident when he describes the cloudbow. In his Treatise on Painting, written a few years before his death in 1519, Leonardo explained why some rainbows are red when the sun is low in the sky. He then expanded on the topic, noting that: <center> That redness, together with the other colors [of the rainbow], is of much greater intensity, the more the rain is composed of large drops, and the more minute the drops, the paler are the colors. If the rain is of the nature of mist, then the rainbow will be white and so completely without hue.<a name="fnB9"></a><a href="#fn9">[10]</a> </center> More than 200 years later, Juan and Ulloa could add nothing to the cloudbows explanation, simply noting that beyond those Rainbows [i.e., glories], one saw at some distance a fourth Arch of a whitish hue.<a name="fnB10"></a><a href="#fn10">[11]</a> Nevertheless, the widespread circulation of Ulloas account did attach his name to the cloudbow as the circle of Ulloa.<a name="fnB11"></a><a href="#fn11">[12]</a> In the 18th century though, growing scientific interest in exotic water-drop bows demanded more than expanded nomenclature. Observers knew that cloud drops were much smaller than raindrops, and presumably this size difference determined the colors of the resulting bow. But how? Both Descartes and Newton believed that the rainbows colors were unaffected by raindrop size.<a name="fnB12"></a><a href="#fn12">[13]</a> In fact, Descartes quite casually notes that raindrops being larger or smaller does not change the appearance of the arc ....<a name="fnB13"></a><a href="#fn13">[14]</a> Yet rainbow observers had recognized for centuries before Descartes and Newton that not all rainbows look the same. For example, Robert Grosseteste noted the difference between the colours of one rainbow and those of another in his 13th-century De Iride,<a name="fnB70"></a><a href="#fn70">[15]</a> and Avicenna had commented on the colorlessness of his cloudbow. Not only do cloudbows and rainbows differ in color, but even a given rainbow can show color and brightness variations around its arc (Fig. 8-5).<a name="fnB14"></a><a href="#fn14">[16]</a> Furthermore, by the 18th century measurements showed that (1) although supposedly fixed by Newton, the rainbows width varied, and (2) cloudbows had a markedly smaller angular diameter than rainbows.<a name="fnB15"></a><a href="#fn15">[17]</a> <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_5.JPG" width="350" height="238" align="bottom" naturalsizeflag="3"></center> Fig. 8-5: A primary rainbow showing color variations around its circumference (State College, PA, 1973). One cloudbow explanation with a decidedly Aristotelian tinge was that clouds small drops produced too little light for rainbow colors to result, yielding only white light. As we saw in Chapter 7, reduced brightness does affect rainbow colors, making the lunar rainbow white. (Englands Princess Margaret is among those lucky enough to have seen a bright, and thus vividly colored, lunar rainbow.)<a name="fnB16"></a><a href="#fn16">[18]</a> But fixing blame for the cloudbows colorlessness on its small drops was unsatisfactory because rainbows no brighter than cloudbows were quite colorful. Some scientists thought that cloudbows and supernumeraries were formed by distorted, non-spherical drops, an idea supported into the 19th century.<a name="fnB17"></a><a href="#fn17">[19]</a> Although plausible, we now know that this idea is both wrong and inconsistent with cloud drops small size. A far less plausible explanation was that cloudbows require cloud drops in the form of hollow spheres, or vesicles. This vesicular theory of the cloudbow was already rather shopworn when revisited in 1845 by French physicist Auguste Bravais (1811-1863). In an elaborate geometrical optics analysis, Bravais claimed that the speculative (in fact, nonexistent) vesicles were required to explain cloudbows, a claim met with widespread scientific skepticism.<a name="fnB18"></a><a href="#fn18">[20]</a> However, a powerful new theory already accounted for both the cloudbow and the supernumerary bows, rendering Bravais analysis little more than a scientific rearguard action. To understand how this new theory arose, we now turn to rainbows that were not supposed to exist -- the supernumerary bows (Fig. 8-3). <hr><h3>Supernumerary Bows: Cloudbows Superfluous Cousins</h3> Theodoric had commented in his De Iride not only on cloudbows, but also on the pastel purple and green bands seen within the primary rainbow. He confidently (and incorrectly) explained them as being similar to the pale colors seen at the fringes of a sunlit prisms colors.<a name="fnB19"></a><a href="#fn19">[21]</a> Witelo preceded Theodoric in mentioning these pastel bows in his Perspectiva of ca. 1270-1274.<a name="fnB20"></a><a href="#fn20">[22]</a> Yet despite sporadic interest in these bows in the following centuries, they remained largely a troublesome curiosity for rainbow theorists. Since antiquity, philosophers and commoners alike tended to assume that the term rainbow meant a single circular arc of nonrepeating colors. In other words, the iris described by most writers was what we now call the primary rainbow. In both theory and myth, the secondary rainbow had been assigned a place of clearly secondary importance. So what were observers to make of these even more peripheral bands of pastel color? Descartes and Newtons hard-won achievements seemed threatened by these marginal additions to the rainbow proper. One common reaction to being confronted with the unexplained is to label it inexplicable, which in this case meant labeling the pastel bows spurious or supernumerary.<a name="fnB21"></a><a href="#fn21">[23]</a> The supernumerary bows thus acquired their faintly reproving name, one that has persisted long after we know that they are an integral part of the rainbow, not a vexing corruption of it. Doubtless 18th and 19th-century optics would have arrived at explanations of the supernumeraries even if these bows had not been the subject of spirited scientific discussion. But we can be equally sure that awareness of the supernumerary problem spurred on some theoreticians. With this in mind we come to one of the 18th centurys more unlikely optical catalysts, Benjamin Langwith (ca. 1684-1743), Rector of Petworth. In a letter to the Royal Societys Philosophical Transactions in 1723, Langwith detailed his recent observations of supernumerary bows: <center> The first series of colours was as usual, only the purple had a far greater mixture of red in it than I had ever seen in the prismatic purple; under this was a coloured arch, in which the green was so predominant, that I could not distinguish either the yellow or the blue: still lower was an arch of purple, like the former, highly saturated with red, under which I could not distinguish any more colours.<a name="fnB22"></a><a href="#fn22">[24]</a></center> Bear Fig. 8-3 in mind as you read Langwiths observation; he has succinctly captured many of the supernumerary bows features. Langwith even self-confidently questioned the rainbow wisdom of Royal Society president Isaac Newton: I begin now to imagine, that the Rainbow seldom appears very lively without something of this Nature, and that the supposd exact Agreement between the Colours of the Rainbow and those of the Prism, is the reason that it has been so little observed.<a name="fnB23"></a><a href="#fn23">[25]</a> In another remarkable observation, Langwith noted that supernumerary bows were absent from the bases of vivid rainbows but visible near their tops. He speculated presciently that this effect depends on some property which the drops retain while they are in the upper part of the air, but lose as they come lower ....<a name="fnB24"></a><a href="#fn24">[26]</a> Identifying just what this change in the drops is has proven to be a surprisingly long-lived problem,<a name="fnB25"></a><a href="#fn25">[27]</a> one that we will take up in Chapter 9. As the 18th century progressed, many other accounts of supernumerary rainbows surfaced in scientific journals and textbooks. Explanations ranged from the bows being merely an illusory visual artifact to their being caused by sulfur compounds dissolved in the drops. In addition, the same incorrect arguments advanced for the cloudbow were also given for the supernumeraries -- that the raindrops must either be hollow or asymmetric. Elaborate geometric variations on Newtons and Descartes theories would also prove unsuccessful in correctly explaining the supernumeraries.<a name="fnB26"></a><a href="#fn26">[28]</a> For that explanation, we must move to the 19th century. <hr><h3>Thomas Young and the Interference Theory of the Rainbow</h3> By 1800 most English scientists believed, as had Newton, that the behavior of light was best explained as a series of small particles that traveled from the light source to the eye. In the late 17th century, Robert Hooke and Christiaan Huygens had asserted that light behaved more like waves than particles.<a name="fnB27"></a><a href="#fn27">[29]</a> Throughout the 18th century the controversy had simmered, with Huygens ideas being considered, at least in England, somewhat suspect. In fact, all the explanations of the rainbow offered so far in this book just as easily could have used the words the path of the particles of light instead of the rays of light. Nothing, not even our discussion of the rainbow color, depends on light being thought of as a series of waves. However, in an 1803 lecture, English physician and scientist Thomas Young asserted that the supernumerary bows could only be explained if light were thought of as a wave phenomenon. Although Young used his wave theory to address several puzzling optical problems in addition to the supernumeraries, explaining these was clearly his signal achievement. Young noted in particular that Langwiths supernumeraries admit also a very easy and complete explanation from the same [light wave] principles and that the circles sometimes seen encompassing the observers shadow in a mist (glories) were interference phenomena.<a name="fnB28"></a><a href="#fn28">[30]</a> Young went on to say confidently that Those who are attached to the NEWTONIAN theory of light, or to the hypotheses of modern opticians, founded on views still less enlarged, would do well to endeavour to imagine any thing like an explanation of such interference patterns.<a name="fnB29"></a><a href="#fn29">[31]</a> Thus the supernumerary rainbows proved to be the midwife that delivered the wave theory of light to its place of dominance in the 19th century. The seeming disparity between the two theories of light has narrowed to the point that either one can explain a tremendous wealth of optical phenomena. Light can be thought of as either waves or particles, and only convenience and simplicity determine which approach is used to study a particular phenomenon. For the rainbow, we choose the wave model of light. Interference is the wave property that interested Young and which we use to explain supernumerary bows. One way of visualizing interference is to imagine waves on a lake. If the wakes of two boats cross, their waves will interfere with the other. Where the crests of two waves coincide, they reinforce each other to make a larger wave. However, if one waves crest sits in anothers trough, the two disturbances cancel each other and the water will be at its original level. When waves combine to make a larger wave the effect is called constructive interference; when they cancel it is called destructive interference. Our analogys details change slightly when we switch from water to light waves, but Fig. 8-6 nonetheless suggests the parallels. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_6.GIF" width="350" height="263" align="bottom" naturalsizeflag="3"></center> Fig. 8-6: Computer simulation of an interference pattern created by two expanding circular light waves. If a wave trough and crest coincide, the pattern is darker than either waves average brightness (destructive interference). If wave crests or troughs coincide, the pattern is brighter (constructive interference). In fairness, we should note that Newton himself was aware that water wave interference affected ocean tides, and that he could explain some optical phenomena by assuming that light had wave properties. Nevertheless, because Huygens was unable to explain satisfactorily how pulsating light waves could travel in straight lines, Newton rejected the wave theory of light and resolutely insisted that light consisted of streams of particles.<a name="fnB30"></a><a href="#fn30">[32]</a> Thus, far from merely being a scientific iconoclast, Young saw in his own experiments how to overcome shortcomings in Huygens wave theory, and how Newtons wave theory of the tides had significance for optics. Nevertheless, Youngs claims of scientific superiority to Newton on the rainbow did make him an iconoclast. Scathing published attacks on his theory accused him of disrespect for Newton -- not a surprising reaction given that the supernumeraries themselves were deemed an affront to Newtons reputation.<a name="fnB31"></a><a href="#fn31">[33]</a> However, Youngs theory soon faced a far more serious challenge than reactionary tirades. French engineer Étienne-Louis Malus (1775-1812) and David Brewster of England separately demonstrated in 1808 and 1815 that reflected sunlight and rainbow light both have a property not readily explained by Youngs theory. That property is polarization, which usually is invisible to us. However, we can detect polarization (and cause it) by using a polarizing device, the most familiar example of which today is polarizing sunglasses. What do polarizers such as sunglasses do? In very general terms, they block the transmission of some components of a light wave, while allowing others to pass, thus imposing undulatory order on light that may have very little (such as sunlight). This state of order is not an all-or-nothing proposition. Degrees of partial polarization (neither completely polarized nor unpolarized) are the rule, not the exception, in nature. If a light source is at least partially polarized, in certain orientations a polarizer can visibly reduce the lights intensity, which is what polarizing sunglasses do to glare reflected from highways.<a name="fnB32"></a><a href="#fn32">[34]</a> Because sunlight itself is unpolarized, Brewsters and Malus discovery that two common sources of atmospheric light were partially polarized seemed remarkable. Especially troubling for Young was the fact that the geometry of the Newtonian-Cartesian rainbow theory easily led to explanations of the rainbows high degree of polarization. By contrast, Young was at a loss to explain how his model of a light wave (which he likened to the back and forth motions in sound waves) could yield the observed polarization. As a consequence, Brewster could serenely say that observation agrees so well with the results of calculation that there remains no doubt of the truth of the Cartesian explanation. Young glumly but gamely admitted to Malus in 1811 that Your experiments show the insufficiency of a theory which I have adopted, but they do not prove it false.<a name="fnB33"></a><a href="#fn33">[35]</a> Young was correct. The insufficiency of his theory (and of previous wave theories) was its depiction of the waves themselves. Recall that Young described light waves as similar to sound waves, in which air expands and contracts (i.e., oscillates) along the same direction that the sound wave propagates.<a name="fnB34"></a><a href="#fn34">[36]</a> Such a wave is called longitudinal. Another kind of wave is generated if you tether one end of a rope and then whip the other end up and down -- the rope itself does not move forward, but a wave with up-and-down oscillations moves forward along its length. In such a transverse wave, oscillations occur perpendicular to the direction in which the wave itself moves. Nearly simultaneously, Young and a French engineer, Augustin-Jean Fresnel (1788-1827), independently conceived of the same answer to wave theorys polarization problem.<a name="fnB35"></a><a href="#fn35">[37]</a> If light waves were transverse rather than longitudinal, then they could indeed be polarized and be affected by polarizers. Consider an example. If our rope wave has up-and-down oscillations and we make it oscillate within a narrow vertical opening, the wave is unaffected (i.e., it is transmitted) beyond the opening. If, however, we turn the narrow opening so that it is horizontal, obviously the rope wave will be damped out (i.e., not transmitted) beyond the opening. The transverse rope wave thus responds to a polarizer, in this case the narrow opening. With this solution in hand, Young and Fresnel could rightly claim that the wave theory of light offered a superior explanation of the rainbow. Not only could the interference theory of the rainbow account for the primary, secondary, and supernumerary rainbows, it could also describe their polarization. Nevertheless, scientific conservatism and unsolved theoretical problems with wave theory combined to make its acceptance slow. This resistance is nicely summarized by Brewster, who would write defensively in 1833 about the wave theory of light that ... I have not yet ventured to kneel at the new shrine, and I must even acknowledge myself subject to that national weakness which urges me to venerate, and even to support, the falling temple in which Newton once worshiped.<a name="fnB36"></a><a href="#fn36">[38]</a> As for the rainbow itself, Young could use wave theory to account for the color and brightness of the supernumerary bows and even to estimate the sizes of raindrops that yielded supernumeraries.<a name="fnB37"></a><a href="#fn37">[39]</a> However, neither he nor Fresnel gave a thoroughly quantitative account of the interference theory of the rainbow. Not until the 1830s would such quantification appear, and once again a scientific temple would need repairs, this time Youngs. We save that story for Chapter 9. <hr><h3>Interference as a Model for All of the Bows</h3> As Fig. 8-6 suggests, interference by water waves was for Thomas Young an apt analogy to interference by light waves. When one portion of a light wave passes through another there may be either constructive interference, which gives a brighter light, or destructive interference, which gives darkness. Amazingly, light can cancel light to give darkness! Neither light wave is altered, for after they pass through each other, they are back to normal. For instance, interference patterns of the rainbow cannot cast shadows, and a light beam cannot block other lights. The reason is that one beam of light waves can only interfere locally with another as they cross. But can we cross two flashlight beams and get darkness at their crossing? Obviously not. Such light is said to be incoherent: flashlight beams contain a jumble of waves reminiscent of a lake surface on race day. While wave interference occurs between individual crests or troughs, no consistent pattern emerges from the chaos. Only when the light waves march across each other in orderly ranks does a coherent pattern of light and dark bands emerge. On the very fine scale of tiny raindrops even sunlight will be coherent. One way of visualizing this is that plow furrows in a hilly field may appear straight and parallel for a short distance, but over longer distances they clearly wander around the lands contours. Similarly, on a very small scale any light source will be coherent. Most raindrops are less than a few millimeters in radius, much less than the distance over which sunlight is coherent. Thus light wave interference in sunlight forms the supernumeraries -- and the primary rainbow. To see how this works, we first note that supernumerary bows are not caused by interference between two light waves. Instead, two different portions of the same light wave interfere. In Fig. 8-7, we once again show a circular slice through a raindrop, much as we did in Fig. 6-5. Now, rather than parallel light rays entering the drop, a series of wave ridges and troughs (Fig. 8-7s vertical lines) advances toward the drop as a front of parallel waves. Think of the parallel lines as representing the wave fronts of parallel rays of sunlight. Rays, which are always locally perpendicular to their corresponding waves, show the waves direction of travel. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_7.jpg" width="597" height="504" align="bottom" naturalsizeflag="3"></center> Fig. 8-7: A moiré pattern that mimics the constructive and destructive interference pattern of sunlight refracted and reflected within a raindrop. The advancing wave front is refracted into the droplet, some of it is reflected from the rear surface of the drop, and then is refracted out of the drop. As it traverses the drop, the wave front folds over on itself, as indicated by Fig. 8-7s cross-hatching. When the two portions of the wave are superimposed, they interfere to produce a pattern of bright and dark bands. This interference pattern is drawn as a moiré pattern in Fig. 8-7, and light waves refracted and reflected by drops produce much the same effect in your eye. The bright and dark bands radiating from the drop in Fig. 8-7 simulate the bright and dark bands of light that form the rainbow. In fact, the large bright region at the angle of minimum deviation is the primary rainbow.<a name="fnB38"></a><a href="#fn38">[40]</a> This region, in turn, is separated from the first supernumerary bow (the next bright band) by a band of darkness. The supernumerary bows are thus as much a part of the whole phenomenon as the primary bow itself: each represents a region of maximum brightness in the interference pattern that results when the wave front folds over on itself near the angle of minimum deviation. However, even though the supernumerary bows are an integral part of the rainbow, they are not always seen with the primary or secondary rainbow. In fact, seeing any more than a faint first supernumerary is often the highlight of a rainbow chasers year. Figure 8-8s three distinct (and ghostly fourth) supernumerary bows are as beautiful as they are unusual. We will explore the reasons for this rarity in Chapter 9. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_8.JPG" width="350" height="233" align="bottom" naturalsizeflag="3"></center> Fig. 8-8: Primary and secondary rainbow with four supernumerary rainbows inside the primary, Kootenay Lake, British Columbia (rainbow Kootenay). The fourth (innermost) supernumerary is extremely faint, even in the original slide. <hr><h3>A Microscopic Explanation of Rainbow Colors</h3> We can use Figure 8-7 to address a question whose answer eluded Descartes and which hobbled his rainbow theory: why is red light bent less than violet for a given angle of incidence <tt>i</tt>, including at minimum deviation? In other words, why is red light bent less than violet in the rainbow? We begin by reexamining refraction from a microscopic standpoint, something that Descartes tried to do, although incorrectly and in much different fashion than modern optics. Like radio waves, light may be thought of as a very rapid oscillation of an electric field. As the light traverses a transparent medium such as glass, water or even air, it sets the molecules vibrating. Strictly speaking, only the electrons in the molecules are materially influenced by the passing light waves electric field. Electrons cannot vibrate as fast as the wave, and these lagging electrons cause the light to slow down. The light vibrates just as fast as it ever did, but now travels through the transparent medium at a somewhat slower speed. Violet light, with its shorter wavelength, has a higher frequency than does red light.<a name="fnB39"></a><a href="#fn39">[41]</a> Thus the mediums electrons have even less success in keeping up with the oscillations of violet light than red light. The net result is that violet light travels more slowly than red light in a medium such as water. Merely because violet light travels more slowly through water does not by itself account for the rainbows color dispersion. How does a variation in speed result in a sideways shift? Imagine a car traveling down a narrow paved road. If its right wheels run onto the gravel shoulder, the car will swerve to the right. The wheels on the right have been slowed by the soft shoulder, and because the cars right side is now traveling slower than its left, the car will turn to the right. This is just the way that bulldozers are steered. For example, to turn to the right, the treads on the right are braked and those on the left are allowed to run at normal speed. With this image in mind, reexamine Fig. 8-7. The parallel wave fronts that approach Fig. 8-7s raindrop are each perpendicular to the corresponding beam of parallel sunlight. Where this light beam enters the raindrop obliquely, one side of the beam (or one side of the corresponding wave front) encounters the water before the other. In Fig. 8-7, the lower side of the beam enters the raindrop first. Because this part of the beam (or wave front) slows first, the beam turns more sharply into the drop (i.e., closer to its surface normal, which is a drop radius here; see Fig. 5-3). Thus the light is refracted or bent, and the amount of bending depends on how much the beam is slowed when it first enters the water drop. Because violet light is slowed more than red, violet light undergoes more refractive bending than red light. As Figs. 6-6 and 6-7 indicate, this greater deviation of violet compared to red puts violet on the primarys interior and the secondarys exterior. <hr><h3>Why do Cloudbows and Rainbows Look So Different?</h3> Since Theodorics day one question in particular has dogged comparisons of the rainbow and cloudbow: why do the bows look so different? Youngs theory holds that the sizes of raindrops generating a rainbow change its appearance. For example, Figure 8-9 shows a moiré interference pattern for a cloud drop that is 50 times smaller than Fig. 8-7s raindrop. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_9.jpg" width="590" height="421" align="bottom" naturalsizeflag="3"></center> Fig. 8-9: Constructive and destructive interference pattern of sunlight refracted and reflected within a cloud drop. Notice how Figs. 8-7 and 8-9 differ. A small cloud drop gives widely spaced bows, but for the larger drop, not only do the supernumerary bows become more tightly spaced, but each bow itself becomes narrower. In Figs. 8-7 and 8-9, the first supernumerary for the cloud droplet occurs at about the same deviation angle as the raindrops second supernumerary. As Young himself noted, you can estimate the raindrop size in a shower based on the spacing between supernumerary bows.<a name="fnB40"></a><a href="#fn40">[42]</a> Clearly this spacing decreases with increasing drop size. Why should this be so? One answer comes from translating Youngs own explanation into modern terms. Young would correctly maintain that the spacing of bright and dark bands in the folded wave front depends on the pathlength that the wave has traversed within the drop. Yet even cloud drops are many times larger than the wavelengths of light -- Fig. 8-9s cloud drop has a radius ten times larger than the wavelength of green light. So the pathlength description is a little more involved than we might first suspect. Greenler explains the cloudbows broadening in terms of diffraction,<a name="fnB71"></a><a href="#fn71">[43]</a> a phenomenon undoubtedly known to Young (and which can be subsumed within his interference explanation). How do we explain cloudbows pastel colors? First note that each colors position in any bow is determined by refraction -- red is deviated the least, violet the most. In the primary rainbow, colors thus occupy different positions, with red to the outside. But the primary bow is just the first interference maximum, and the width of that maximum for each color depends on the size of the raindrops. If the drops are very large, the width of each color band will be narrow and so the various colors will not overlap significantly, resulting in fairly pure rainbow colors. With small drops the story is quite different. Each band of color can become so broad that all colors overlap, and additive color mixing yields a pallid or even white bow. So the phrase all the colors of the rainbow is a very slippery one indeed. Water drops in the atmosphere have a tremendous range of sizes. The radius of the largest raindrops rarely exceeds 2.5 millimeters (mm), and a typical raindrop has a radius of about 0.5 mm. Drizzle drops have radii of about 0.1 mm while those of a typical cloud or fog drop are about 0.01 mm. Bows can be generated by any of these drops, but only a rain shower that has quite large drops can produce a bow with vivid colors. Indeed, only when the drop radius is larger than about 1/3 mm can we see red in the bow. For the smaller drops found in drizzle, rainbow colors become quite pastel. The bow formed by cloud drops is white, with only the faintest hint of red or yellow to the outside. As a result, when the bow is seen in clouds or fog it is sometimes called the white rainbow, but this term is really an oxymoron. Any drop big enough to be called rain is too big to give a white rainbow. Our use of the term cloudbow or fogbow is not accidental. Combining the cloudbows rarity and the misnomer white rainbow can confuse even the most diligent author. In Walter Maurers survey of the rainbow in Sanskrit literature, he notes with puzzlement that Some texts ... specify a white (sveta) rainbow, and here again one wonders whether or not some other atmospheric phenomenon is meant.<a name="fnB41"></a><a href="#fn41">[44]</a> Because both the cloudbow and the cloud against which it is seen are white, the bow can be recognized only as a curved band which is brighter than the cloud, much as Thomas Barker did. We can easily confuse the cloudbow with the glory, which looks something like a small, pastel rainbow around the airplanes shadow on the cloud (see Fig. 8-1). The glory has a radius of only a few degrees, so the complete circle is relatively easy to see. On the other hand, the cloudbow has a radius of about 40°, so seeing all of it from airplanes small passenger windows usually is difficult.<a name="fnB42"></a><a href="#fn42">[45]</a> Although we usually see cloudbows from aircraft these days, we can also see them on a bank of fog. Imagine a clear night in the fall when fog has formed over the land. In the morning, if the fog begins to thin before the sun has climbed high in the sky, you may have the eerie experience of seeing a cloudbow arching over your shadow on the fog. This in fact is what Jorge Juans party saw from Mount Pambamarca, although they described being enveloped in the clouds, which [were] dissipated by the first rays of the Sun.<a name="fnB43"></a><a href="#fn43">[46]</a> Because fog is just another name for a cloud that envelops you on the ground, no distinction need be made between the fogbow and cloudbow. The viewing geometry of cloudbows can be confusing indeed. If we look down from an airplane or mountain at a horizontal deck of clouds below us, what will the cloudbow look like? Figure 8-1 suggests that we will see a circular arc. Yet our on-scene impression may be quite different, as the following account indicates. In September 1961, atmospheric physicist James McDonald was looking down from his airplane seat at a cloudbow some 21,000 feet below. At first puzzled by what he saw, he suddenly found that the cloudbows true nature was then unambiguously revealed when I noted further that here and there along its course it assumed a most vivid rainbow banding where I was looking down through breaks in the 8,000-ft [cloud] deck at rain showers below.<a name="fnB44"></a><a href="#fn44">[47]</a> In other words, McDonald had the rare treat of seeing a cloudbow interspersed with rainbow segments as he flew along (Fig. 8-10)! Fig. 8-10: Viewing geometry for a cloudbow seen from an airplane. Point C is the airplane shadow, around which McDonald saw a glory. Some 42° from this shadow, he saw an apparently elliptical cloudbow. (from McDonald 1962, Fig. 1) As Fig. 8-10 indicates, McDonald was looking at an oblique angle below the horizon. This meant that the cloud drops producing the cloudbow, along with their companion raindrops that produced the rainbow, traced out an ellipse on the cloud deck. This fact, combined with a very human perceptual insistence on placing the rainbow or cloudbow at some position in the landscape, led McDonald to the compelling illusion that the bow was an ellipse. Of course, because every raindrop or cloud drop contributing to McDonalds arc was about 42° from the head of his shadow (the antisolar point), he was by definition seeing a circle of light. Yet had we been there, we too would have been convinced perceptually, if not intellectually, that we were looking at an ellipse.<a name="fnB45"></a><a href="#fn45">[48]</a> On your next airplane flight, see what your visual impression of the cloudbows shape is. Regrettably, you will have to contend not only with the limitations of your visual system, but with those of airplane windows as well. Not much has changed since McDonald noted with understandable frustration that his small airplane window precluded my seeing the full ellipse ..., a limitation arguing the need for glass-bottomed jet transports for meteorologically inclined passengers.<a name="fnB46"></a><a href="#fn46">[49]</a> <hr><h3>A Map of the Rainbows Colors</h3> Whatever geometric confusion the cloudbow or rainbow causes, we can be certain that the cloudbow will be essentially colorless and that rainbows will not. Rather than merely describing how rainbow colors depend on drop size, why not display the colors themselves? We have done just that in Fig. 8-11, a map of the colors predicted by a successor to Youngs interference model.<a name="fnB47"></a><a href="#fn47">[50]</a> In this map, drop radius is scaled logarithmically along the horizontal axis, and it increases from cloudbow sizes on the left (0.01-mm radius) to the size of a large raindrop on the right (1-mm radius). Figure 8-11s horizontal axis has black tick marks where too little space is available for the corresponding radius label (e.g., 0.06 mm). Otherwise, radius tick marks are red and their corresponding labels are black. Deviation angle (or angle from the sun) is Fig. 8-11s vertical axis, and it increases downward. We have, in effect, made our map look as though we were scanning the primary rainbow from outside to inside. Thus Fig. 8-11 shows us thin slices through many different rainbows, each one of which is colored as if a single drop of a particular size were responsible for the rainbow slice. <img src="https://web.archive.org/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/papers/Airy_primaries.jpg" width="690" height="490" align="bottom" naturalsizeflag="3"> Fig. 8-11: An interference theorys map of rainbow colors vs. drop size (Airy theory). The colors have been smoothed (blurred) by the 0.5° angular width of the sun. Perhaps Fig. 8-11s most striking feature is the broad expanse of nearly colorless bows on its left side. Here we are in the province of cloudbows, a region that extends up to drop radii of about 0.1 mm. Note that for the smallest drops (0.01-mm radius) the cloudbows breadth is huge -- it spans more than 8°.<a name="fnB48"></a><a href="#fn48">[51]</a> A bow 8° wide sounds impossible, but as we will see, Fig. 8-1s cloudbow is nearly that wide. Between 0.02 and 0.1 mm, the first cloudbow supernumeraries emerge, each of them as colorless as their corresponding primary. As we move rightward (to larger drop sizes) and downward (to larger deviation angles) within Fig. 8-11, the spacing and width of the supernumeraries narrows, as Young predicted. Note too that as drop size decreases, the bows angular radius (as distinct from its angular breadth or width) decreases. In principle, this decrease in radius can be as much as 2° -3° as we move from raindrops to cloud drops.<a name="fnB49"></a><a href="#fn49">[52]</a> To the right of 0.1-mm drops, rainbow colors begin to appear, beginning with an increasingly vivid red on the outside of the primary near deviation angles of about 138°. This is the familiar red exterior of the rainbow proper, although at drop sizes below about 0.2 mm it lacks equally vivid counterparts within the rainbow. At these drop sizes, the supernumeraries too have become more colorful, although as Langwith noted, their colors resemble prismatic ones very little. Instead, pastel reds and greens like the ones seen in Fig. 8-8s supernumeraries dominate. Note too that the spacing of the supernumeraries has become positively claustrophobic -- at 0.2-mm radius, patient counting reveals eight supernumeraries. At large drop sizes, the supernumeraries account for a smaller fraction of all the rainbow light than do their small-droplet cousins. Fig. 8-11s evidence for this is the gradual darkening of the supernumeraries as we look from left to right. However, because the overall brightness of the rainbow increases with drop size, the large-droplet supernumeraries will actually be brighter than those seen, say, in a cloudbow. Unfortunately, these combined color and brightness changes are too complex for us to display legibly in Fig. 8-11, so we have ignored the fact that the bows overall brightness increases with drop size. At the largest drop sizes (0.25-1 mm radius), we finally begin to see the canonical rainbow colors. If you like, you may find seven or more colors in the rainbow.<a name="fnB50"></a><a href="#fn50">[53]</a> We suspect that most readers will, like us, find at most only six distinct colors -- red, orange, yellow, green, blue (or perhaps cyan), and violet -- for the 0.3-mm radius drops. For drops larger than this, blue actually disappears from the rainbow, as suggested by Fig. 7-21. <hr><h3>Observed and Intrinsic Rainbow Colors </h3> Tempted as we may be, we should not regard Fig. 8-11 as an infallible field guide to rainbow colors. Many factors besides drop size determine rainbow colors in nature. Among these are (1) the horizontal depth of rain showers (a thin sheet of rain will produce less vivid colors than a thick one), (2) the range of raindrop sizes in a shower (many different drop sizes coexist in rainfall, not just the solitary sizes of Fig. 8-11), (3) scattering and absorption by other atmospheric particles (e.g., dust or haze can produce reddened sunlight and thus reddened rainbows), (4) flattening of falling raindrops by air drag, and (5) the illumination of the rainbows background. For now, we concentrate on the last (but far from least) of these real-world effects, the rainbows background illumination. Remember that rainbow light can interfere locally with itself in the raindrop, but interference will not affect incoherent light from the background. Thus the rainbows colors are mixed additively with those of the background. In a specially prepared scene such as Fig. 8-2, we all but eliminate background light by draping the background in black. Note that the very thin spray of droplets in Fig. 8-2 yields a bow bright enough to be seen against black velvet, but that the bow disappears against the light-colored bricks. So Fig. 8-2 suggests that if we took such extreme measures on the scale of a rain shower, we would see a rainbow with purer, more distinct colors. Obviously it is impractical to drape landscapes in black velvet on the chance that we might see a naturally occurring rainbow. However, we can achieve much the same effect if we remove the background electronically, rather than physically, when we analyze digitized images of rainbows.<a name="fnB51"></a><a href="#fn51">[54]</a> If in a digitized image we average along many different radial slices across a rainbow, we can get an accurate idea of its intrinsic colors (i.e., with background light removed), rather than merely its observed colors (i.e., with background light included). Naturally, if we include the background illumination, we will measure rainbow colors as we usually see them. We distinguish between observed and intrinsic rainbow colors because the intrinsic colors tell us how sunlight and the raindrops have contributed to the rainbow, independent of the myriad color variations in the cloud background. So let us examine these observed and intrinsic colors for a cloudbow and three separate rainbows-- Fig. 8-1 and Figs. 7-20, 7-21, and 8-8, respectively. Figure 8-12 is a colorimetric analysis of Figs. 7-20s vivid rainbow. In Fig. 8-12 we have shown the entire gamut of perceptible colors, which is bounded by the monochromatic (i.e., 100% pure) colors of the horseshoe-shaped spectrum locus. For reference, within this locus we have marked some typical red, green, and blue color limits for color television. Because television is an additive mixing system, it can generate all of the colors within the triangle bounded by the diamonds in Fig. 8-12. Clearly television can reproduce most colors, which is another way of saying that it spans much of the human color gamut. (The television primaries shown here are actually those for a computers color monitor; conventional color televisions will have somewhat different primaries.) <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_12.GIF" width="364" height="416" align="bottom" naturalsizeflag="3"></center> Fig. 8-12: The CIE 1976 UCS diagram, within which are (1) the red, green, and blue primaries of a typical color television (marked by diamonds), (2) an estimated sunlight color for Fig. 7-20 (marked by an x), (3) the observed and intrinsic colors (thick and thin curves, respectively) seen as we look radially across Fig. 7-20s rainbow. Now look at the tiny ellipse near the sunlight colors chromaticity (marked by an x). This ellipse is the entire gamut of observed colors in Fig. 7-20s splendid rainbow! We drew this elliptical chromaticity curve by connecting in sequence the individual rainbow chromaticities that we measured from the outside to the inside of the bow. The ellipse seems impossibly small, especially when we imagine a beautiful rainbow in comparison to televisions presumably pedestrian colors. There is no contest; television unquestionably can both generate more colors and more vivid colors than we see in Fig. 7-20. Surely, though, Fig. 7-21s spectacular rainbow can challenge color television. However, as we see in Fig. 8-13, once again color television has a far greater color gamut. By one measure, our color-television primaries span a color gamut some 30 times greater than that of Fig. 7-20s observed rainbow colors, and 19 times greater than Fig. 7-21s observed rainbow colors.<a name="fnB52"></a><a href="#fn52">[55]</a> Fig. 8-13: Figure 8-12s analysis repeated, this time for Fig. 7-21s rainbow. Even if we digitally remove the color and brightness of each rainbows background, the resulting intrinsic rainbow colors cannot compete with those of color television. In Figs. 8-12 and 8-13 the gamuts of the intrinsic rainbow colors are considerably larger than those of the observed colors (2.5 and 3.8 times greater, respectively). Remember, of course, that we will not see these intrinsic colors as we admire a rainbow outdoors; only the observed colors are evident. Do Figs. 8-12 and 8-13 mean that the phrase all the colors of the rainbow is a cheat? Not really. Although observed color gamuts in even the most spectacular rainbows are small, recall that the spectrum locus defines the limits of color perception. Monochromatic lights define its 100% pure colors, and such lights are all but absent from our everyday color environment. Televisions nearly pure color primaries are important exceptions, but even they are seldom seen as large areas of uniform, unmixed color. And we almost certainly do not simultaneously compare television primaries with a rainbow seen outdoors. So, while we can see light of 100% purity, we almost never do see it in nature. For example, the blue sky has a theoretical maximum purity of only about 41%, and observed sky purities will be even smaller. <a name="fnB53"></a><a href="#fn53">[56]</a> Vivid phenomena like the green, red, and blue flashes occasionally seen with a rising or setting sun have higher purities than either the rainbow or blue sky, although the flashes colors usually are quite short-lived. The colors of parhelia and some other ice-crystal optics may persist longer than a rainbow and likely are its equal in purity. Yet deserved or not, it is a vivid rainbows long arc of reasonably pure hues that sticks in the popular imagination as a paragon of color variety. Next we zoom in on Fig. 8-12 and 8-13s rainbow chromaticity curves to examine their details (Figs. 8-14 and 8-15, respectively). On closer inspection, Figure 8-12s apparently closed ellipse of observed rainbow colors (thick curve) opens into a <tt>G</tt>-shaped curve. This curve opens even more when we remove the background colors (thin curve), revealing that the outside of the bow would be quite red (i.e., have high u' values) were it not for the bluish cloud background. In fact, removing the pale blue of the clouds rotates the entire chromaticity curve. Similar shifts occur in Fig. 8-15, our close-up view of Fig. 8-13. These changes in the chromaticity curves are significant because they reveal color gamuts that rainbow theory has long predicted,<a name="fnB54"></a><a href="#fn54">[57]</a> but which have never been measured in nature before now. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_14.GIF" width="371" height="423" align="bottom" naturalsizeflag="3"></center> Fig. 8-14: close-up of Fig. 8-12s rainbow chromaticities, showing both observed and intrinsic rainbow colors Fig. 8-15: close-up of Fig. 8-13s rainbow chromaticities, showing both observed and intrinsic rainbow colors <hr><h3>Rainbow Brightness: Colors Constant (and Sometimes Confusing) Companion</h3> Figures 8-16 through 8-19 take two different tacks in showing us how brightness changes across the rainbows of Figs. 7-20 and 7-21. In Figs. 8-16 and 8-17, we have drawn curves of relative rainbow brightness vs. angle from the sun, both for the observed rainbow (sky backgrounds brightness included; thick curve) and the intrinsic rainbow (background brightness removed; thin curve). In both figures, brightness is relative to that of a perfectly reflecting white card seen under the same illumination. As we look from left to right in Figs. 8-16 and 8-17, we are looking from outside the primary rainbow to inside. The primary rainbows maximum brightness is evident as the large peaks on the figures left sides. To the right of this peak is a smaller one showing us the first and only supernumerary bow evident in Figs. 7-20 and 7-21. As we might expect, Fig. 7-21s very bright rainbow is measurably brighter than Fig. 7-20s rainbow (compare the vertical axes in Figs. 8-17 and 8-16). Figures 8-16 and 8-17 also show us that removing the fairly uniform sky brightness makes the intrinsic rainbows darker than the observed ones, but that the resulting brightness curves have essentially the same shape for a particular rainbow. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_16.GIF" width="401" height="404" align="bottom" naturalsizeflag="3"></center> Fig. 8-16: observed and intrinsic rainbow brightness for Fig. 8-14s chromaticity curve (See Fig. 7-20 for the corresponding rainbow.) Fig. 8-17: observed and intrinsic rainbow brightness for Fig. 8-15s chromaticity curve (See Fig. 7-21 for the corresponding rainbow.) So far we have looked at separate chromaticity and brightness curves for the rainbows of Figs. 7-20 and 7-21. What if we combined this information in a single, unified curve for each observed rainbow? We see such curves in Figs. 8-18 and 8-19, where we have drawn a perspective view of the u', v' chromaticity plane and added brightness as a vertical dimension. For sake of comparison, we have duplicated the chromaticity curves of Figs. 8-14 and 8-15 on the gridded u', v' chromaticity plane.<a name="fnB55"></a><a href="#fn55">[58]</a> Above these chromaticity curves proper, we have drawn the combined chromaticity and brightness curves, which loop and twist in three dimensions. Fig. 8-18: perspective view of combined rainbow chromaticity and brightness changes drawn separately in Figs. 8-14 and 8-16 (observed rainbow) Fig. 8-19: perspective view of combined rainbow chromaticity and brightness changes drawn separately in Figs. 8-15 and 8-17 (observed rainbow) Now some mysterious features of Figs. 8-14 and 8-15 are explainable. The convoluted squiggles on the inside of the observed rainbow chromaticity curves<a name="fnB56"></a><a href="#fn56">[59]</a> are the simultaneous changes in color and brightness in the supernumerary bows. When we project this three-dimensional curve onto a plane (as we have in Fig. 8-15), the line crosses over itself, yielding a very complicated two-dimensional squiggle. Also note that for both rainbows, the darker orange on the rainbows exterior rapidly gives way to brighter colors in its interior. Recall that one truism of Aristotelian rainbow theory was that the rainbows brightest color is red.<a name="fnB57"></a><a href="#fn57">[60]</a> In vivid rainbows such as Figs. 7-20 and 7-21, even naked-eye observers can readily see that red is the rainbows darkest color.<a name="fnB58"></a><a href="#fn58">[61]</a> However, because Aristotles color theory made red a very bright color, presumably his fiction about the rainbows red represents the triumph of color theory over rainbow observation. We see much the same pattern in Fig. 8-20, our close-up perspective view of rainbow color and brightness for Fig. 8-8. As Fig. 8-8s multiple supernumeraries suggest, Fig. 8-20 will have many chromaticity squiggles in it. These are fairly clearly resolved in the three-dimensional curve, but we have also labeled their broad expanse in Fig. 8-20. Note that the supernumeraries brightness decreases as we move toward the rainbows center. This trend supports the rainbow model shown in Fig. 8-11, where the supernumeraries brightness decreases with increasing angular distance from the sun. Fig. 8-20: perspective view of combined rainbow chromaticity and brightness changes for Fig. 8-8s rainbow. <hr><h3>Why is the Cloudbow So Dull?</h3> Although we know why the cloudbow and rainbow look different, just how different are their colors? One part of our answer comes in Fig. 8-21, in which we compare the chromaticity curves of Fig. 8-1s cloudbow and Fig. 8-8s rainbow. Several features are noteworthy here. First, the cloudbow is not very colorful -- it does not stray far from the color of sunlight (marked by an x). Second, the cloudbow curve in Fig. 8-21 looks as smooth as the rainbow curve (thick curve). Although we can measure some small-scale brightness fluctuations in Fig. 8-1 (probably due to photographic film grain), no regular patterns are visible beyond those of the glory, cloudbow, and supernumeraries. Third, Fig. 8-22 indicates that the reddish outside of the cloudbow is its most colorful region because chromaticity here is the farthest from white of any cloudbow colors. Fig. 8-1 corroborates this visually. Finally, Fig. 8-1s cloudbow color gamut is only about half that of Fig. 8-8s rainbow. Fig. 8-21: comparison of cloudbow and rainbow chromaticity curves (See Figs. 8-1 and 8-8 for the rainbow photographs.) Fig. 8-22: principal colorimetric features of Fig. 8-1s cloudbow In the sense that this cloudbow has poorer colors than Fig. 8-8s rainbow, this last result is not surprising. However, should not the essentially white cloudbow have a tiny color gamut compared with Fig. 8-8s vivid rainbow? The answer to this seeming puzzle is that the cloudbows measured color gamut is taken from a ring nearly 12° wide in Fig. 8-1, while Fig. 8-8s larger color gamut spans a ring only 4° wide. Although we cannot be certain, it seems plausible that the rainbows more compact display (i.e., its smaller angular width) enhances the apparent vividness of its measurably purer colors. We said earlier that Fig. 8-1s cloudbow was quite wide compared to the rainbow. Figure 8-23 indicates just how great the difference is. The cloudbow itself spans about 6°, and its first supernumerary maximum occurs more than 7° from the cloudbow maximum. Compare these figures with the rainbows 2° width and the mere 1° separation between the rainbow and first supernumerary. As Fig. 8-11 predicts and Fig. 8-23 demonstrates, the cloudbow has a smaller angular radius (as distinct from its width) than the rainbow; the cloudbow radius in Fig. 8-23 is estimated to be about 1° less than the rainbow radius.<a name="fnB59"></a><a href="#fn59">[62]</a> Thus the cloudbow is angularly broader than the rainbow, but its total angular size is somewhat smaller (although the difference in radius is much less obvious than the difference in breadth). Visually speaking, however, there is little doubt that the cloudbow is a poor second to the narrower, more vivid rainbow. Part of the rainbows visual impact stems from its brightness, which obviously is much greater than the cloudbows in Fig. 8-23. Figure 8-23 supports a claim we made about our rainbow color map (Fig. 8-11). We said then that cloudbow supernumeraries are not much dimmer than the cloudbow itself, which is what we see in Fig. 8-23. However, this figure also makes clear that the brightness of the rainbow supernumeraries falls off much more dramatically and within a much smaller angular distance. The combination of pastel colors and low, slowly changing brightness virtually guarantees the cloudbows status as the visual also-ran among water-drop bows. Fig. 8-23: Comparison of the relative brightness and width of Fig. 8-1s cloudbow and Fig. 8-8s rainbow. Brightness is scaled logarithmically here. <hr><h3>Turner and The Wreck Buoy: A Cloudbow in Disguise?</h3> We have found no unequivocal representations of fogbows or cloudbows in older paintings, and we are unaware of any modern commercial art that suggests them.<a name="fnB60"></a><a href="#fn60">[63]</a> However, a work by English artist J. M. W. Turner might be taken for a cloudbow. In The Wreck Buoy (Fig. 8-24), the bows nearly colorless appearance and great breadth hint that Turner may have had a fogbow in mind. However, the background does not much resemble fog. Contemporary reaction to The Wreck Buoy does not clarify Turners intentions very much. Turners biographer Walter Thornbury (1828-1876) described the repainting as having come out gloriously with a whitened, misty sky and a double rainbow.<a name="fnB61"></a><a href="#fn61">[64]</a> Aside from Thornburys suggestion of rainbows in the mist, however, Victorian comment on these pastel bows was often more merciless than meteorological. The Illustrated London News frostily described The Wreck Buoy as Evidently a picture painted twenty years ago, left lumbering about, and then cleaned up, or intended to be so, by the insertion of two or three new bright rainbows.<a name="fnB62"></a><a href="#fn62">[65]</a> Even blunter was The Spectator, which slammed Turners rainbow, crying that [natures] faultless arc is shaped no better than the vault of an ill-built wine-cellar.<a name="fnB63"></a><a href="#fn63">[66]</a> The Wreck Buoy was far from a critical failure, however. John Ruskin admired it, as did others.<a name="fnB64"></a><a href="#fn64">[67]</a> But why should Turner have painted such a wan (and obviously discomfiting) rainbow here? One possibility is that Turner merely wanted to give the impression of a rainbow, and so only its grossest features seemed pertinent to him. <center><img src="/web/20051214151844im_/http://www.usna.edu/Users/oceano/raylee/RainbowBridge/RB_images/Fig8_25.GIF" width="350" height="260" align="bottom" naturalsizeflag="3"></center> Fig. 8-24: J. M. W. Turner, The Wreck Buoy (ca. 1807; repainted 1849; Walker Art Gallery, Liverpool, England) Another possibility exists. Turners pairing of a traditional symbol of deliverance (the rainbow) with one of disaster (shipwrecks) gives the viewer a mixed symbolic message.<a name="fnB65"></a><a href="#fn65">[68]</a> Of course, the subject of The Wreck Buoy is itself unusual, for it is both an image of maritime disaster and of a warning device for avoiding future disasters. In this sense, the rainbow (or fogbow) serves its usual function as a harbinger of hope. Yet Turners view of the repainted Wreck Buoy may also reflect the pessimism of his 1843 Light and Colour (Goethes Theory) -- The Morning After the Deluge -- Moses Writing the Book of Genesis. Turners epigraph there bleakly notes In prismatic guise / Hopes harbinger, ephemeral as the summer fly / Which rises flits, expands and dies.<a name="fnB66"></a><a href="#fn66">[69]</a> Like Light and Colours prismatic wash of color, the wan and broad Wreck Buoy rainbow scarcely resembles the traditional rainbow of hope. Before making too much of this, however, we should note that Turners keen interest in color theory and rainbow optics<a name="fnB67"></a><a href="#fn67">[70]</a> was paired with a penchant for painting nearly colorless rainbows such as Buttermere Lake (Fig. 8-25). Fig. 8-25: J. M. W. Turner, Buttermere Lake (1798; Tate Gallery, London) Turner accompanied the Royal Academy exhibition catalogs entry for Buttermere Lake with an epigraph from James Thomsons poem Spring. <a name="fnB72"></a><a href="#fn72">[71]</a> Turners choice seems odd because Thomson explicitly describes a grand ethereal Bow that Shoots up immense; and every hue unfolds .... <a name="fnB68"></a><a href="#fn68">[72]</a> That is hardly what we see in Buttermere Lake -- or in The Wreck Buoy. In both paintings, Turners agenda is less meteorological than pictorial. Turners Romantic interest in a sublime view of nature meant that mere literal details could always be altered to achieve a desired epiphanic effect. As an example, in his watercolor Rome: The Forum with a Rainbow Turner presents us with a topographically rearranged view of the Forum<a name="fnB69"></a><a href="#fn69">[73]</a> and a meteorologically impossible rainbow (Fig 8-26; the scenes lighting is inconsistent with the rainbow). This combining of ruined antique grandeur and splendid natural beauty into a sublime whole would have been for Turner a far more important goal than simply transcribing on-scene details. Similarly, Turner may have had meteorology in mind when painting The Wreck Buoy. The pictures broad, wan bow certainly is similar to a cloudbow, and perhaps Turners extensive travels and eclectic reading had introduced him to the cloudbows ghostly form. We may never know Turners intentions here, but the idea remains a tempting one. Fig. 8-26: J. M. W. Turner, Rome: The Forum with a Rainbow (1819; British Museum, London) <hr><h3>The Supernumerary Road to Rainbows End</h3> As vital as the cloud and supernumerary bows are and have been to the rainbows story, their comparative rarity makes them a rainbow dessert course, not an entrée. But is there even more to the visually delicious supernumerary bows than we have suggested? In fact, the supernumeraries are far from uniform, and this lack of uniformity partly explains why they were considered spurious bows. We will see next what supernumerary variations within a given rainbow tell us about rainbow itself. For the practically minded, however, we can recast Chapter 9s science in a single phrase -- we know where the rainbows pot of gold is! <hr><h3>Bibliography -- Chapter 8</h3> Anthes, Richard A., John J. Cahir, Alistair B. Fraser, and Hans A. Panofsky, 1981: The Atmosphere (Charles E. Merrill Publishing Co., Columbus, OH, 3rd edition, 531 pp.). Aristotle, 1931: The Works of Aristotle, v. 3: Meteorologica, De Mundo, De Anima, Parva Naturalia, De Spiritu, W. D. Ross, ed. (Oxford University Press, London, n. p.). Barker, Thomas, 1988: The Weather Journals of a Rutland Squire: Thomas Barker of Lyndon Hall, John Kington, ed. (Rutland Record Society, Rutland County Museum, Oakham, UK, 217 pp.). Bohren, Craig F., 1991: What Light through Yonder Window Breaks? More Experiments in Atmospheric Physics. (John Wiley and Sons, New York, 190 pp.). Bohren, Craig F., 1987: Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics (John Wiley and Sons, New York, 195 pp.). Botley, Cicely M., 1970: Lunar rainbow, Weather 25, 287-288. Boyer, Carl, 1987: The Rainbow: From Myth to Mathematics (Princeton University Press, Princeton, NJ, 376 pp.; reprinting, with new photographs, of Thomas Yoseloff edition (New York, 1959)). Brewster, David, 1833: Observations on the Absorption of Specific Rays, in reference to the Undulatory Theory of Light, London and Edinburgh Philosophical Magazine and Journal of Science 2 (3rd series, May 1833), 360-363. Churma, Michael E., 1995: The Modeling and Remote Sensing of the Rainbow (M.S. thesis, Pennsylvania State University, University Park, PA, 105 pp.). Descartes, René, 1965: Discourse on Method, Optics, Geometry, and Meteorology (Bobbs-Merrill Co., Inc., Indianapolis, IN, 361 pp.; English translation by Paul J. Olscamp). Eastwood, Bruce S., 1966: Robert Grossetestes theory of the rainbow: A chapter in the history of non-experimental science, Archives Internationales dHistoire des Sciences 19, 313-332. Fraser, Alistair B., 1983a: Chasing rainbows: Numerous supernumeraries are super, Weatherwise 36, 280-289. Greenler, Robert, 1980: Rainbows, Halos, and Glories (Cambridge University Press, Cambridge, UK, 195 pp.). Horten, Max, and Eilhard Wiedemann, 1913: Avicennas Lehre vom Regenbogen nach seinem Werk al Schifâ, Metorologische Zeitschrift 30, 533-544. Illustrated London News, 1849: Exhibition of the Royal Academy of Arts, 1849 Supplement to the Illustrated London News 14 (26 May), 345 ff. Juan, Jorge, and Antonio de Ulloa, 1964: A Voyage to South America (Alfred A. Knopf, New York, 245 pp.; English translation by John Adams). Kemp, Martin, 1990: The Science of Art (Yale University Press, New Haven, CT, 375 pp.). Können, Günther P., 1985: Polarized Light in Nature. (Cambridge University Press, Cambridge, UK, 172 pp.). Landow, George P., 1977: The rainbow: A problematic image, in Nature and the Victorian Imagination, U. C. Knoepflmacher and G. B. Tennyson, eds., (University of California Press, Berkeley, CA, 519 pp.), pp. 341-369. Langwith, Benjamin, 1809: Concerning the Appearance of several Arches of Colours, contiguous to the inner Edge of the common Rainbow, observed at Petworth in Sussex, by the Rev. Dr. Langwith, Philosophical Transactions of the Royal Society of London, from their commencement, in 1665, to the year 1800; abridged, with notes and biographic illustrations, 6 (1713-1723), 623-624. Lee, Raymond L., Jr., 1991: What are all the colors of the rainbow?, Applied Optics 30, 3401-3407, 3545. Leonardo da Vinci, 1956: Treatise on Painting [Codex Urbinas Latinus 1270], v. 1 (Princeton University Press, Princeton, NJ, 443 pp.; English translation by A. Philip McMahon). Lynch, David K., and Susan N. Futterman, 1991: Ulloas observations of the glory, fogbow, and an unidentified phenomenon, Applied Optics 30, 3538-3541. Maurer, Walter H., 1967-1968: The rainbow in Sanskrit literature, Adyar Library Bulletin 31-32, 360-381. McDonald, James E., 1962: A gigantic horizontal cloudbow, Weather 17, 243-245. McDonald, James E., and B. M. Herman, 1964: On a point of terminology in the optics of the rainbow, Bulletin of the American Meteorological Society 45, 279-280. Middleton, W. E. Knowles, 1965: A History of the Theories of Rain and Other Forms of Precipitation (Oldbourne Book Co. Ltd., London, 223 pp.). Milner, Frank, 1990: J. M. W. Turner: Paintings in Merseyside Collections (National Museums and Galleries on Merseyside, Liverpool, 83 pp.). Minnaert, Marcel G. J., 1993: Light and Color in the Outdoors (Springer-Verlag, New York, 417 pp.; English translation by Len Seymour). Palmer, Frederic, 1945: Unusual rainbows, American Journal of Physics 13, 203-204. Rösch, Sigmund, 1968: Der Regenbogen in Wissenschaft und Kunst, Applied Optics 7, 233-239. Sabra, Abdelhamid I., 1967: Theories of Light from Descartes to Newton (Oldbourne Book Co. Ltd., London, 363 pp.). Schweizer, Paul D., 1982a: John Constable, rainbow science, and English color theory, Art Bulletin 64, 424-445. Smith, George, and Algis Vingrys, 1995: Seeing red, New Scientist 147 (15 July), 52-53. Spectator, 1849: The arts. Royal Academy: The landscapes, The Spectator (London) (23 June), 591-592. Steffens, Henry J., 1977: The Development of Newtonian Optics in England (Science History Publications/USA, New York, 190 pp.). Theodoric, 1914: Dietrich von Freiberg, Über den Regenbogen und die durch Strahlen erzeugten Eindrüke, Joseph Würschmidt, ed. (Aschendorffsche Verlagsbuchhandlung, Münster, 204 pp.). Thomson, James, 1971: Poetical Works, J. Logie Robertson, ed. (Oxford University Press, London, 516 pp.; reprinting of 1908 edition). Thornbury, Walter, 1970: The Life of J. M. W. Turner, R. A.: Founded on Letters and Papers Furnished by His Friends and Fellow-Academicians (Ward Lock Reprints, London, 636 pp.; reprinting of Chatto and Windus edition (London, 1877)). Wallace, William A., 1959: The Scientific Methodology of Theodoric of Freiberg (University Press, Fribourg, Switzerland, 395 pp.). Wilton, Andrew, 1980: Turner and the Sublime (British Museum Publications, London, 192 pp.). Witelo, 1972: Item Vitellonis Thuringopoloni Libri X (Perspectiva), bound with Opticae Thesaurus Alhazeni Arabis Libri Septem, Nuncprimum Editi (Johnson Reprint Corp., New York, 474 pp.; reprint of Friedrich Risner edition (Basel, 1572)). Young, Thomas, 1804: The Bakerian Lecture. Experiments and Calculations relative to physical Optics, Philosophical Transactions of the Royal Society of London 94, 1-16. <hr><h3>Notes and References -- Chapter 8</h3> <a name="fn0"></a><a href="#fnB0">[1]</a>. Barker 1988, p. 110 <a name="fn1"></a><a href="#fnB1">[2]</a>. Perhaps surprisingly, winters freezing temperatures do not rule out water drop formation, making wintertime rainbows or cloudbows entirely possible. (Bohren 1987, pp. 180-185) <a name="fn2"></a><a href="#fnB2">[3]</a>. Barker 1988, p. 111. A century earlier, Descartes had also hopefully invoked many very round and transparent grains of hail mingled among the rain to explain reports of another kind of unusual rainbow. (Descartes 1965, p. 344) <a name="fn3"></a><a href="#fnB3">[4]</a>. See Greenler (1980, pp. 143-146) and Anthes et al. (1981, p. 429) for discussions of the glory. <a name="fn4"></a><a href="#fnB4">[5]</a>. Boyer 1987, p. 277 <a name="fn5"></a><a href="#fnB5">[6]</a>. Lynch and Futterman 1991 <a name="fn6"></a><a href="#fnB6">[7]</a>. Wallace 1959, p. 225; Theodoric 1914, pp. 179-180 <a name="fn7"></a><a href="#fnB7">[8]</a>. Witelo 1972, p. 462 <a name="fn8"></a><a href="#fnB8">[9]</a>. Horten and Wiedemann 1913, p. 541 <a name="fn9"></a><a href="#fnB9">[10]</a>. Leonardo da Vinci 1956, p. 335. See Palmer (1945, p. 204) and Greenler (1980, Plate 1-7) for 20th-century observations of red rainbows. <a name="fn10"></a><a href="#fnB10">[11]</a>. Lynch and Futterman 1991, p. 3538 <a name="fn11"></a><a href="#fnB11">[12]</a>. Boyer 1987, p. 277. This name is ironic because Juan, not Ulloa, observed the cloudbow in 1737 and authored its scientific account. However, Ulloa seems to have been the abler (and thus more memorable) writer. (Juan and Ulloa 1964, pp. 16-17, 112) <a name="fn12"></a><a href="#fnB12">[13]</a>. With the benefit of hindsight, we can generate different-colored rainbows even within the confines of geometrical optics. <a name="fn13"></a><a href="#fnB13">[14]</a>. Descartes 1965, p. 332 <a name="fn70"></a><a href="#fnB70">[15]</a>. Eastwood 1966, p. 328 <a name="fn14"></a><a href="#fnB14">[16]</a>. Churma 1995 <a name="fn15"></a><a href="#fnB15">[17]</a>. Boyer 1987, p. 287 <a name="fn16"></a><a href="#fnB16">[18]</a>. Botley 1970, p. 287. Smith and Vingrys (1995) discuss in some detail how lunar rainbows low brightness affects their colors. <a name="fn17"></a><a href="#fnB17">[19]</a>. Boyer 1987, pp. 280, 307 <a name="fn18"></a><a href="#fnB18">[20]</a>. Middleton 1965, pp. 54-57; Boyer 1987, p. 307 <a name="fn19"></a><a href="#fnB19">[21]</a>. Wallace 1959, p. 223 <a name="fn20"></a><a href="#fnB20">[22]</a>. Witelo describes his rainbow under a rainbow as follows: whenever more rainbows are seen with the same placement of colors, one is under the other, as first red then green: and then purple: and again red: and again green: and finally purple .... He confidently (although incorrectly) claims that a diversity of matter in the different bands explains these bows. (Witelo 1972, pp. 464-465) <a name="fn21"></a><a href="#fnB21">[23]</a>. Boyer 1987, p. 279 <a name="fn22"></a><a href="#fnB22">[24]</a>. Langwith 1809, p. 623 <a name="fn23"></a><a href="#fnB23">[25]</a>. Boyer 1987, p. 278 <a name="fn24"></a><a href="#fnB24">[26]</a>. Langwith 1809, p. 624 <a name="fn25"></a><a href="#fnB25">[27]</a>. Fraser 1983a <a name="fn26"></a><a href="#fnB26">[28]</a>. Boyer 1987, pp. 279-281 <a name="fn27"></a><a href="#fnB27">[29]</a>. Steffens 1977, pp. 92-101; Sabra 1967, pp. 185-230 <a name="fn28"></a><a href="#fnB28">[30]</a>. Young 1804, pp. 8, 11. In fact, the glorys explanation involves considerably more than Youngs suggestion that it is perhaps more nearly related to the common colours of thin plates as seen by reflection (i.e., interference colors). (see Anthes et al. 1981, p. 429) <a name="fn29"></a><a href="#fnB29">[31]</a>. Young 1804, p. 11 <a name="fn30"></a><a href="#fnB30">[32]</a>. Steffens 1977, pp. 111-113. Huygens believed that light waves consisted of longitudinal pulses rather than periodic transverse waves. For more about these two kinds of waves, see our discussion of polarization below. <a name="fn31"></a><a href="#fnB31">[33]</a>. Steffens 1977, pp. 128-136 <a name="fn32"></a><a href="#fnB32">[34]</a>. See Bohren (1991, pp. 25-48) for a thorough qualitative explanation of polarization; Können (1985, pp. 47-56) discusses the rainbows polarization. <a name="fn33"></a><a href="#fnB33">[35]</a>. Boyer 1987, p. 289 <a name="fn34"></a><a href="#fnB34">[36]</a>. The vibrations of a drumhead illustrate this idea. <a name="fn35"></a><a href="#fnB35">[37]</a>. Boyer 1987, p. 290 <a name="fn36"></a><a href="#fnB36">[38]</a>. Brewster 1833, p. 361. Brewsters objection here was to some implausible optical properties of wave theorys speculative luminiferous ether, a supposedly omnipresent medium required for the transmission of light. Young and Fresnel separately gave this ether contradictory physical properties, and it was eventually abandoned as being both untenable and unnecessary for explaining lights wave nature. Although Brewsters complaint was well-founded, his conclusion (that wave theory was gravely flawed) was not. (Boyer 1987, pp. 290-291) <a name="fn37"></a><a href="#fnB37">[39]</a>. Young 1804, p. 9 <a name="fn38"></a><a href="#fnB38">[40]</a>. Recall that deviation angle is defined in Chapter 6, n. 191. <a name="fn39"></a><a href="#fnB39">[41]</a>. Mathematically, in any medium the speed of light V is given by: V = frequency x wavelength. Frequency is the number of wave cycles that pass a point per second. For a given V, frequency increases as wavelength decreases. <a name="fn40"></a><a href="#fnB40">[42]</a>. Young 1804, p. 9. Although Young estimated the range of drop sizes required to produce supernumeraries of a particular radius, he offered no explanation for his estimate. <a name="fn71"></a><a href="#fnB71">[43]</a>. Greenler 1980, pp. 10-12. See McDonald and Herman (1964) on the confusing and sometimes contentious use of the word diffraction in modern rainbow theory. <a name="fn41"></a><a href="#fnB41">[44]</a>. Maurer 1967-1968, p. 374 <a name="fn42"></a><a href="#fnB42">[45]</a>. Note that Fig. 8-1 was photographed with a wide-angle lens. <a name="fn43"></a><a href="#fnB43">[46]</a>. Lynch and Futterman 1991, p. 3538 <a name="fn44"></a><a href="#fnB44">[47]</a>. McDonald 1962, p. 243 <a name="fn45"></a><a href="#fnB45">[48]</a>. Minnaert (1993, p. 203) clearly analyzes this perceptual conundrum. <a name="fn46"></a><a href="#fnB46">[49]</a>. McDonald 1962, p. 244 <a name="fn47"></a><a href="#fnB47">[50]</a>. This is Airy theory; see Chapter 10s Airys Rainbow Theory: The Incomplete 锟紺omplete Answer. <a name="fn48"></a><a href="#fnB48">[51]</a>. Note that 8° = 145°-137°. <a name="fn49"></a><a href="#fnB49">[52]</a>. See Fig. 8-23 below for estimates of the radii of Fig. 8-1s cloudbow and Fig. 8-8s rainbow. <a name="fn50"></a><a href="#fnB50">[53]</a>. Of course, bear in mind Chapter 7s color-naming pitfalls. <a name="fn51"></a><a href="#fnB51">[54]</a>. See Chapter 7s How Many Colors Does the Rainbow Have? for discussions of our colorimetric and image analysis techniques. <a name="fn52"></a><a href="#fnB52">[55]</a>. Lee 1991, p. 3403 <a name="fn53"></a><a href="#fnB53">[56]</a>. Bohren 1987, pp. 160-163 <a name="fn54"></a><a href="#fnB54">[57]</a>. For example, see Rösch (1968, p. 238). <a name="fn55"></a><a href="#fnB55">[58]</a>. Note that now we are looking at these curves from a different vantage point than earlier. <a name="fn56"></a><a href="#fnB56">[59]</a>. For example, see those near u' = 0.19, v' = 0.485 in Fig. 8-15. <a name="fn57"></a><a href="#fnB57">[60]</a>. Aristotle 1931, Meteorologica 374b. As with much of Aristotles rainbow explanation, ambiguity lurks here. Aristotles geometric arguments erroneously require rainbow colors to become progressively darker as we move from red to green to purple. Apparently Aristotle regards yellow as the brightest of all colors, yet red is the brightest of his rainbow colors. In an attempt to make a flawed theory agree with observation, he confusingly places the brighter yellow between red and green. (Meteorologica 375a) <a name="fn58"></a><a href="#fnB58">[61]</a>. Modern rainbow theories realistically predict that reds on the outside of the primary rainbow are its darkest colors (see Figs. 8-11 and 10-29). <a name="fn59"></a><a href="#fnB59">[62]</a>. Uncertainties abound in measuring the angular size of amorphous features such as Fig. 8-1s cloudbow. Accordingly, this cloudbows radius might be as much as 1° -2° smaller than that estimated in Fig. 8-23. See Minnaert (1993, p. 201) and Greenler (1980, pp. 11-12) for other estimates of cloudbow radii. <a name="fn60"></a><a href="#fnB60">[63]</a>. For a cloudbow near-miss, see Fig. 10-28. <a name="fn61"></a><a href="#fnB61">[64]</a>. Thornbury 1970, p. 105. A small segment of a secondary bow is visible in Fig. 8-24. <a name="fn62"></a><a href="#fnB62">[65]</a>. Illustrated London News 1849, p. 347 <a name="fn63"></a><a href="#fnB63">[66]</a>. Spectator 1849, p. 592 <a name="fn64"></a><a href="#fnB64">[67]</a>. Milner 1990, p. 76 <a name="fn65"></a><a href="#fnB65">[68]</a>. Landow 1977, p. 369 <a name="fn66"></a><a href="#fnB66">[69]</a>. Quoted in Schweizer (1982a, p. 439). <a name="fn67"></a><a href="#fnB67">[70]</a>. Kemp 1990, pp. 301-303 <a name="fn72"></a><a href="#fnB72">[71]</a>. Wilton 1980, pp. 35-36. A portion of this poem is quoted as Chapter 3s epigraph. <a name="fn68"></a><a href="#fnB68">[72]</a>. Thomson 1971, p. 10 (Spring, line 205) <a name="fn69"></a><a href="#fnB69">[73]</a>. Wilton 1980, pp. 165-166 <hr><center> [ <a href="contents.html">Rainbow Bridge table of contents</a> | <a href="https://web.archive.org/web/20051214151844/http://www.usna.edu/Users/oceano/raylee/RLL_cv.html"> Raymond Lees WWW homepage</a> ]</center>  <script type="text/javascript" language="JavaScript">var site="sm5RainbowBridge"</script> <script type="text/javascript" language="JavaScript1.2" src="https://web.archive.org/web/20051214151844js_/http://sm5.sitemeter.com/js/counter.js?site=sm5RainbowBridge"> </script> <noscript> <a href="https://web.archive.org/web/20051214151844/http://sm5.sitemeter.com/stats.asp?site=sm5RainbowBridge" target="_top"> <img src="https://web.archive.org/web/20051214151844im_/http://sm5.sitemeter.com/meter.asp?site=sm5RainbowBridge" alt="Site Meter" border="0"></a> </noscript>   </body></html>