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 Newton’s 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 bridge’s tenuous power.
... 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 colour’d
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 form’d by spherical drops
of water.[1]
Thomas Barker, “A remarkable cloud rainbow” (1739)
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 Barker’s) is his bow’s “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).[2] 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 form’d 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have been replaced by “very small round & icy hail.”[3] However, hail is neither clear enough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.
Fig. 8-1: A cloudbow, also known as a fogbow or pilot’s bow, seen
from an airplane. The bright light immediately surrounding the airplane
shadow is not a cloudbow, but a glory.[4]
{N.B.: Not all images are reproduced in this WWW version of Chapter 8.}
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 Newton’s nor Descartes’ theories of the rainbow can account for
them.[5] 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
rainbow’s cultural symbolism. After all, they roil the simple rainbow image
considerably, and the fact that their scientific explanation was even more
abstruse than Newton’s 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.
Cloudbows: The “Circle of Ulloa”
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 Juan’s
party had seen from Mount Pambamarca in present-day Ecuador (Fig. 8-4).[6] Juan and Ulloa were far from the
first to write on cloudbows -- Theodoric of Freiberg’s 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.[7] Witelo simply accounted
for the “completely white rainbow” with a combination of thin
vapor and clear illumination.[8] No
more persuasive is Avicenna’s statement that his cloudbow’s diameter shrank
because he grew more distant from the sun as he descended the mountain toward
the bow.[9]
Fig. 8-4: Eighteenth-century illustration of a cloudbow and glories seen
by Juan’s scientific party. (Lynch and Futterman 1991, Fig. 2) The cloudbow
is the large white arc surrounding the bull’s-eye glory pattern (the relative
angular sizes of the cloudbow and glories are at odds with nature and Juan’s
accompanying text).
Leonardo da Vinci’s 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:
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.[10]
More than 200 years later, Juan and Ulloa could add nothing to the cloudbow’s
explanation, simply noting that “beyond those Rainbows [i.e.,
glories], one saw at some distance a fourth Arch of a whitish hue.”[11] Nevertheless, the widespread circulation
of Ulloa’s account did attach his name to the cloudbow as the “circle
of Ulloa.”[12] 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 rainbow’s colors were unaffected
by raindrop size.[13] In fact, Descartes
quite casually notes that raindrops’ “being larger or smaller does
not change the appearance of the arc ....”[14]
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,[15]
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).[16]
Furthermore, by the 18th century measurements showed that (1) although supposedly
fixed by Newton, the rainbow’s width varied, and (2) cloudbows had a markedly
smaller angular diameter than rainbows.[17]
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. (England’s Princess
Margaret is among those lucky enough to have seen a bright, and thus vividly
colored, lunar rainbow.)[18] But
fixing blame for the cloudbow’s 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.[19]
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.[20] 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).
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 prism’s colors.[21]
Witelo preceded Theodoric in mentioning these pastel bows in his Perspectiva
of ca. 1270-1274.[22] 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 Newton’s 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.”[23]
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 century’s more unlikely optical catalysts, Benjamin
Langwith (ca. 1684-1743), Rector of Petworth. In a letter to the
Royal Society’s Philosophical Transactions in 1723, Langwith detailed
his recent observations of supernumerary bows:
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.[24]
Bear Fig. 8-3 in mind as you read Langwith’s 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 suppos’d exact Agreement between
the Colours of the Rainbow and those of the Prism, is the reason that it
has been so little observed.”[25]
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 ....”[26]
Identifying just what this change in the drops is has proven to be a surprisingly
long-lived problem,[27] 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 Newton’s and Descartes’ theories would also prove unsuccessful
in correctly explaining the supernumeraries.[28]
For that explanation, we must move to the 19th century.
Thomas Young and the Interference Theory of the Rainbow
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.[29] 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 Langwith’s supernumeraries “admit also a very easy and complete
explanation from the same [light wave] principles” and that the “circles
sometimes seen encompassing the observer’s shadow in a mist” (glories)
were interference phenomena.[30]
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.[31]
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 wave’s crest
sits in another’s 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 analogy’s details change slightly
when we switch from water to light waves, but Fig. 8-6 nonetheless suggests
the parallels.
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 wave’s 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.[32]
Thus, far from merely being a scientific iconoclast, Young saw in his own
experiments how to overcome shortcomings in Huygens’ wave theory, and how
Newton’s wave theory of the tides had significance for optics.
Nevertheless, Young’s 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 Newton’s reputation.[33] However, Young’s 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 Young’s 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 light’s intensity, which is what polarizing sunglasses do to glare reflected
from highways.[34]
Because sunlight itself is unpolarized, Brewster’s 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 rainbow’s 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.”[35] 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.[36]
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 theory’s
polarization problem.[37] 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.”[38]
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.[39]
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
Young’s. We save that story for Chapter 9.
Interference as a Model for All of the Bows
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 land’s 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. 8-7’s 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.
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. 8-7’s 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.[40] 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 chaser’s year. Figure 8-8’s 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.
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.
A Microscopic Explanation of Rainbow Colors
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 i, 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 wave’s 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.[41]
Thus the medium’s 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 rainbow’s 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 car’s 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. 8-7’s 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 primary’s interior and the secondary’s
exterior.
Why do Cloudbows and Rainbows Look So Different?
Since Theodoric’s day one question in particular has dogged comparisons
of the rainbow and cloudbow: why do the bows look so different? Young’s
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. 8-7’s raindrop.
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 raindrop’s second supernumerary. As Young himself
noted, you can estimate the raindrop size in a shower based on the spacing
between supernumerary bows.[42]
Clearly this spacing decreases with increasing drop size.
Why should this be so?
One answer comes from translating Young’s 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. 8-9’s 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 cloudbow’s broadening
in terms of diffraction,[43]
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 color’s
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 cloudbow’s rarity and the misnomer “white rainbow”
can confuse even the most diligent author. In Walter Maurer’s 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.”[44]
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
airplane’s 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.[45]
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
Juan’s party saw from Mount Pambamarca, although they described being “enveloped
in the clouds, which [were] dissipated by the first rays of the Sun.”[46] 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 cloudbow’s “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.[47] 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 McDonald’s
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.[48]
On your next airplane flight, see what your visual impression of the cloudbow’s
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.”[49]
A Map of the Rainbow’s Colors
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 Young’s interference model.[50] 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 8-11’s 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. 8-11’s
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.
Fig. 8-11: An interference theory’s 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. 8-11’s 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 cloudbow’s breadth is huge -- it spans
more than 8°.[51] A bow 8° wide sounds impossible, but as we will see, Fig. 8-1’s 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 bow’s 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.[52]
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. 8-8’s 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. 8-11’s
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 bow’s 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.[53] 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.
Observed and Intrinsic Rainbow Colors
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 rainbow’s
background.
For now, we concentrate on the last (but far from least) of these real-world
effects, the rainbow’s 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 rainbow’s 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.[54]
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. 7-20’s 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 computer’s color monitor; conventional color televisions will have somewhat
different primaries.)
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. 7-20’s rainbow.
Now look at the tiny ellipse near the sunlight color’s chromaticity (marked
by an x). This ellipse is the entire gamut of observed colors in Fig. 7-20’s
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 television’s
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. 7-21’s 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. 7-20’s observed
rainbow colors, and 19 times greater than Fig. 7-21’s observed rainbow colors.[55]
Fig. 8-13: Figure 8-12’s analysis repeated, this time for Fig. 7-21’s
rainbow.
Even if we digitally remove the color and brightness of each rainbow’s 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. Television’s 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.
[56] 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 rainbow’s 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 8-13’s rainbow chromaticity curves to examine
their details (Figs. 8-14 and 8-15, respectively). On closer inspection,
Figure 8-12’s apparently closed ellipse of observed rainbow colors (thick
curve) opens into a G-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,[57] but which
have never been measured in nature before now.
Fig. 8-14: close-up of Fig. 8-12’s rainbow chromaticities, showing
both observed and intrinsic rainbow colors
Fig. 8-15: close-up of Fig. 8-13’s rainbow chromaticities, showing both
observed and intrinsic rainbow colors
Rainbow Brightness: Color’s Constant (and Sometimes Confusing) Companion
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 background’s 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 rainbow’s 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. 7-21’s very bright
rainbow is measurably brighter than Fig. 7-20’s 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.
Fig. 8-16: observed and intrinsic rainbow brightness for Fig. 8-14’s
chromaticity curve (See Fig. 7-20 for the corresponding rainbow.)
Fig. 8-17: observed and intrinsic rainbow brightness for Fig. 8-15’s 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.[58]
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[59] 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 rainbow’s exterior
rapidly gives way to brighter colors in its interior. Recall that one truism
of Aristotelian rainbow theory was that the rainbow’s brightest color
is red.[60] In vivid rainbows such
as Figs. 7-20 and 7-21, even naked-eye observers can readily see that red
is the rainbow’s darkest color.[61]
However, because Aristotle’s color theory made red a very bright color,
presumably his fiction about the rainbow’s 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. 8-8’s 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 rainbow’s 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. 8-8’s rainbow.
Why is the Cloudbow So Dull?
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. 8-1’s cloudbow and Fig. 8-8’s 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.
8-1’s cloudbow color gamut is only about half that of Fig. 8-8’s 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. 8-1’s cloudbow
In the sense that this cloudbow has poorer colors than Fig. 8-8’s rainbow,
this last result is not surprising. However, should not the essentially
white cloudbow have a tiny color gamut compared with Fig. 8-8’s vivid
rainbow? The answer to this seeming puzzle is that the cloudbow’s measured
color gamut is taken from a ring nearly 12° wide
in Fig. 8-1, while
Fig. 8-8’s larger color gamut spans a ring only 4° wide. Although
we cannot be certain, it seems plausible that the rainbow’s more compact
display (i.e., its smaller angular width) enhances the apparent vividness
of its measurably purer colors.
We said earlier that Fig. 8-1’s 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 rainbow’s 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.[62]
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 rainbow’s visual impact stems from its brightness,
which obviously is much greater than the cloudbow’s 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 cloudbow’s status as the visual also-ran among water-drop
bows.
Fig. 8-23: Comparison of the relative brightness and width of Fig. 8-1’s
cloudbow and Fig. 8-8’s rainbow. Brightness is scaled logarithmically here.
Turner and The Wreck Buoy: A Cloudbow in Disguise?
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.[63] However, a work by English
artist J. M. W. Turner might be taken for a cloudbow. In The Wreck Buoy
(Fig. 8-24), the bow’s 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 Turner’s
intentions very much. Turner’s biographer Walter Thornbury (1828-1876) described
the repainting as having come out “gloriously with a whitened, misty
sky and a double rainbow.”[64]
Aside from Thornbury’s 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.”[65] Even
blunter was The Spectator, which slammed Turner’s rainbow, crying
that “[nature’s] faultless arc is shaped no better than the vault of
an ill-built wine-cellar.”[66]
The Wreck Buoy was far from a critical failure, however. John Ruskin
admired it, as did others.[67] 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.
Fig. 8-24: J. M. W. Turner, The Wreck Buoy (ca. 1807;
repainted 1849; Walker Art Gallery, Liverpool, England)
Another possibility exists. Turner’s pairing of a traditional symbol of
deliverance (the rainbow) with one of disaster (shipwrecks) gives the viewer
a mixed symbolic message.[68] 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 Turner’s view of the repainted Wreck
Buoy may also reflect the pessimism of his 1843 Light and Colour
(Goethe’s Theory) -- The Morning After the Deluge -- Moses Writing the Book of
Genesis. Turner’s epigraph there bleakly notes “In prismatic guise
/ Hope’s harbinger, ephemeral as the summer fly / Which rises flits, expands
and dies.”[69] Like Light
and Colour’s 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 Turner’s keen interest in
color theory and rainbow optics[70]
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 catalog’s entry for Buttermere
Lake with an epigraph from James Thomson’s poem Spring.
[71]
Turner’s choice seems odd because Thomson explicitly describes a “grand
ethereal Bow” that “Shoots up immense; and every hue unfolds ....”
[72] That is hardly what we see in
Buttermere Lake -- or in The Wreck Buoy. In both paintings,
Turner’s agenda is less meteorological than pictorial. Turner’s 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[73] and a meteorologically impossible rainbow
(Fig 8-26; the scene’s 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 picture’s broad, wan bow
certainly is similar to a cloudbow, and perhaps Turner’s extensive travels
and eclectic reading had introduced him to the cloudbow’s ghostly form.
We may never know Turner’s 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)
The Supernumerary Road to Rainbow’s End
As vital as the cloud and supernumerary bows are and have been to the rainbow’s
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 9’s science in a single phrase -- we know where the rainbow’s
pot of gold is!
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Notes and References -- Chapter 8
[1]. Barker 1988, p. 110 [2]. Perhaps surprisingly, winter’s
freezing temperatures do not rule out water drop formation, making wintertime
rainbows or cloudbows entirely possible. (Bohren 1987, pp. 180-185) [3]. 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) [4]. See Greenler (1980, pp. 143-146)
and Anthes et al. (1981, p. 429) for discussions of the glory. [5]. Boyer 1987, p. 277 [6]. Lynch and Futterman 1991 [7]. Wallace 1959, p. 225; Theodoric
1914, pp. 179-180 [8]. Witelo 1972, p. 462 [9]. Horten and Wiedemann 1913, p.
541 [10]. Leonardo da Vinci 1956, p. 335.
See Palmer (1945, p. 204) and Greenler (1980, Plate 1-7) for 20th-century
observations of red rainbows. [11]. Lynch and Futterman 1991,
p. 3538 [12]. 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) [13]. With the benefit of hindsight,
we can generate different-colored rainbows even within the confines of geometrical
optics. [14]. Descartes 1965, p. 332 [15]. Eastwood 1966, p. 328 [16]. Churma 1995 [17]. Boyer 1987, p. 287 [18]. Botley 1970, p. 287. Smith
and Vingrys (1995) discuss in some detail how lunar rainbows’ low brightness
affects their colors. [19]. Boyer 1987, pp. 280, 307 [20]. Middleton 1965, pp. 54-57;
Boyer 1987, p. 307 [21]. Wallace 1959, p. 223 [22]. 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) [23]. Boyer 1987, p. 279 [24]. Langwith 1809, p. 623 [25]. Boyer 1987, p. 278 [26]. Langwith 1809, p. 624 [27]. Fraser 1983a [28]. Boyer 1987, pp. 279-281 [29]. Steffens 1977, pp. 92-101;
Sabra 1967, pp. 185-230 [30]. Young 1804, pp. 8, 11. In
fact, the glory’s explanation involves considerably more than Young’s 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) [31]. Young 1804, p. 11 [32]. 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. [33]. Steffens 1977, pp. 128-136 [34]. See Bohren (1991, pp. 25-48)
for a thorough qualitative explanation of polarization; Können (1985,
pp. 47-56) discusses the rainbow’s polarization. [35]. Boyer 1987, p. 289 [36]. The vibrations of a drumhead
illustrate this idea. [37]. Boyer 1987, p. 290 [38]. Brewster 1833, p. 361. Brewster’s
objection here was to some implausible optical properties of wave theory’s
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 light’s wave nature. Although
Brewster’s complaint was well-founded, his conclusion (that wave theory
was gravely flawed) was not. (Boyer 1987, pp. 290-291) [39]. Young 1804, p. 9 [40]. Recall that deviation angle
is defined in Chapter 6, n. 191. [41]. 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. [42]. 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. [43]. 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. [44]. Maurer 1967-1968, p. 374 [45]. Note that Fig. 8-1 was photographed
with a wide-angle lens. [46]. Lynch and Futterman 1991,
p. 3538 [47]. McDonald 1962, p. 243 [48]. Minnaert (1993, p. 203) clearly
analyzes this perceptual conundrum. [49]. McDonald 1962, p. 244 [50]. This is Airy theory; see Chapter
10’s “Airy’s Rainbow Theory: The Incomplete ‘Complete’ Answer.” [51]. Note that 8° = 145°-137°. [52]. See Fig. 8-23 below for estimates
of the radii of Fig. 8-1’s cloudbow and Fig. 8-8’s rainbow. [53]. Of course, bear in mind Chapter
7’s color-naming pitfalls. [54]. See Chapter 7’s “How
Many Colors Does the Rainbow Have?” for discussions of our colorimetric
and image analysis techniques. [55]. Lee 1991, p. 3403 [56]. Bohren 1987, pp. 160-163 [57]. For example, see Rösch
(1968, p. 238). [58]. Note that now we are looking
at these curves from a different vantage point than earlier. [59]. For example, see those near
u' = 0.19, v' = 0.485 in Fig. 8-15. [60]. Aristotle 1931, Meteorologica
374b. As with much of Aristotle’s rainbow explanation, ambiguity lurks here.
Aristotle’s 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) [61]. 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). [62]. Uncertainties abound in measuring
the angular size of amorphous features such as Fig. 8-1’s cloudbow. Accordingly,
this cloudbow’s 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. [63]. For a cloudbow near-miss,
see Fig. 10-28. [64]. Thornbury 1970, p. 105. A
small segment of a secondary bow is visible in Fig. 8-24. [65]. Illustrated London News
1849, p. 347 [66]. Spectator 1849, p.
592 [67]. Milner 1990, p. 76 [68]. Landow 1977, p. 369 [69]. Quoted in Schweizer (1982a,
p. 439). [70]. Kemp 1990, pp. 301-303 [71]. Wilton 1980, pp. 35-36. A portion of this poem is quoted as Chapter 3’s epigraph. [72]. Thomson 1971, p. 10 (Spring,
line 205) [73]. Wilton 1980, pp. 165-166
