As we did before, using n = 1.332 (red) and n = 1.344 (violet), we easily determine that red appears at a cone half-angle of 50.6° and violet at 53.7°. Clearly, in the secondary rainbow the colors are reversed from the primary rainbow. Violet is now on the outside of the bow and red is on the inside; all the other colors—orange, yellow, green, and blue—are in between.
Some Other Features of Rainbows
The sketch displayed in figure 18.7 gives a qualitative summary of what we have covered so far. Now for a few additional remarks.
Everyone knows that the secondary rainbow is not frequently seen and even when it is visible, it is much less intense than the primary rainbow. There are two main reasons for this: (a) the light rays creating the secondary bow lose more intensity because there are two, instead of one, reflections within the raindrops and (b) as we see in a comparison of figures 18.4 and 18.6, the emergent rays of the secondary bow are much less concentrated in the vicinity of the minimum point than those of the primary bow; in other words, there is much more scattering of light in the case of the secondary bow.
FIG. 18.7
Main features of primary and secondary rainbows.
Alexander's Dark Band
As we have seen, light rays of the sun are concentrated along a circular arc of about 42° radius to create the primary bow and along another circular arc of around 50° radius to produce the secondary bow. This leaves a region of about 8° in which there is much less light than elsewhere. Anyone who has observed vivid displays of rainbows has noticed this relatively dark zone between the primary and secondary bows. This zone is called Alexander's dark band, named after the Greek philosopher Alexander of Aphrodisias (c. 200 A.D.), who was evidently the first to correctly explain this feature of rainbows.
Tertiary and Quarternary Rainbows
In the preceding analysis we note that for both the primary rainbow and the secondary rainbow, there are two refractions of a light ray—one when the ray enters a raindrop and another when it leaves. In addition, for the primary bow there is one internal reflection and, for the secondary bow, there are two.
So a logical question is, can there be rainbows with three, four, or more reflections of the light rays within the raindrops? The answer: certainly, but it is highly unlikely you will be able to see them.
As we did before, we construct an equation that gives the total angle change, Ø, between the incident and emergent rays. The result is
where k is the number of internal reflections. The first bracketed term on the right-hand side of this equation describes the angle changes due to the entering and departing refractions. If k = 1, we get equation (18.3) and if k = 2, we get equation (18.6).
Again utilizing (18.2) in (18.8) and then differentiating with respect to i the following result is obtained:
From this relationship, using k = 1 and k = 2, we regain the relations (18.5) and (18.6) we obtained earlier. Then, proceeding as before, we let k = 3 and k = 4 to provide values of im corresponding to the tertiary and quarternary rainbows, respectively. With this information, we compute the magnitudes of the minimum angle change Øm, the corresponding cone half-angle, and so on.
An interesting result of this analysis is that both the (k = 3) tertiary and (k = 4) quarternary rainbows appear as circles around the sun! In other words, they are in the opposite direction from the primary (k = 1) and secondary (k = 2) rainbows. However, to put our minds at ease, the k = 5 bow coincides closely with the secondary bow and the one for k = 6 falls inside the primary.
Remember that these k-numbers specify the number of reflections of a light ray within a raindrop. The geometrical optics have been worked out for rainbows up to k = 20. With the help of a computer, this could be extended easily to values of k as large as we please. This might be worthwhile; interesting geometrical or numerical sequences might be generated. In any event, this topic of higher-order rainbows is well covered by Boyer (1987).
Supernumerary Bows
The contributions of many investigators, especially those of Descartes and Newton, provide a consistent and complete mechanism for the description and analysis of rainbows. This is true as far as the main features of rainbows are concerned. All we need are principles of geometrical optics including information about the variation of the refractive index with color.
Nevertheless, this relatively simple methodology cannot explain certain other features of the rainbow, including the appearance of so-called supernumerary bows. These rarely seen faint arcs appear near the inner edge of the primary bow and at the outer edge of the secondary.
To account for the appearance of these admittedly minor features of rainbows requires more than simple geometrical optics. It is necessary to utilize the wavelike properties of light to explain the phenomenon of supernumerary bows. Suggested references to pursue this topic are Boyer (1987) and Nussenzveig (1977). A word to the wise: fairly quickly, one can get rather deeply involved in mathematical physics on this and closely related topics.
Concluding Remarks
It is not surprising that a great deal of myth, legend, and folklore about rainbows has been generated over the centuries. An extensive coverage of such matters is given by Boyer (1987).
Along these lines, we now look at and hopefully resolve the following two topics:
The singer asserts, in the song at the start of this chapter, that bluebirds can fly over rainbows. The singer also expresses wonderment as to why he or she cannot do the same thing. Well, we simply state that it is highly unlikely, from the viewpoint of both aerodynamics and optics, that either the singer or bluebirds can accomplish this remarkable feat.
For a very long time, it has been believed that there are pots of gold at the ends of rainbows. This legend is easily dismissed. Were it true, the Internal Revenue Service, long ago, would have discovered and taxed this enormous source of income.
19
Making Mathematical Mountains
In this chapter and the one to follow, we are going to fabricate and analyze various kinds of mathematical mountains. To do this, we shall need some algebra and trigonometry and some analytic geometry and calculus. In addition, for problems with large numbers and diverse shapes of mountains, it will be necessary to utilize various topics of statistics and probability theory.
These are the basic mathematical tools used in the science of geomorphology: the study of the characteristics, origins, and changes of land forms.
Cones and Paraboloids
We begin our studies of the subject with a very simple mountain: a circular cone. What we want to do in this case, and in those to follow, is to determine the hypsometric (or hypsographic) curve of a particular land form, for example, a cone. Such a curve describes the mathematical relationship between the horizontal cross-sectional area of the land form and the elevation above a specified datum plane.
FIG. 19.1
Plots of the (a) elevation profile and (b) hypsometric curve of a circular cone.
Our cone is shown in figure 19.1(a). The radius of the circular base of the cone is r1 and the height is z1. At any elevation z, the radius is r. Some analytic geometry gives the following equation for the edge of the cone:
Since the cross-sectional area is A = r2, the equation of the hypsometric curve is
This relationship is plotted in figure 19.1(b).
What is the average elevation of the cone, ? You may remember that the volume of a circular cone is equal to one-third the volume of the circumscribing circular cylinder, that is, V = (1/3)A1z1 So, by definition, = V/A1 = (1/3)z1
Had we not remembered the one-third relationship concerning the volume of a cone, we could have computed its volume by using integral calculus. We shall do that in our next example.
However, before we leave the subject of “cones,” here is an interesting little problem.
PROBLEM If you are a pizza lover, or even if you're not, please answer the following. You cut a slice of angle α from your circular pizza and consume it without delay at the parlor. T
hen you take the remaining pizza, form it into a nice cone, tape the bottom seams together, and load it up with your favorite topping at the nearby self-service counter. Fill it to a nice flat even top; no heaping on.
FIG. 19.2
Plots of the (a) elevation profile and (b) hypsometric curve of a paraboloid.
Question. At what angle α should you slice the pizza to provide the maximum volume of topping in your conical pizza to take home with you? Note that this problem involves some geometry and some differential calculus.
Answer. a = 66°.
Back to hypsometry and our next mountain. A good question: What is the mathematical shape of a land form whose hypsometric curve is a straight line? Well, from analytic geometry, we easily determine that the required equation of the hypsometric line is
This equation is shown in figure 19.2(b). Also, since (A/A1) = (r/r1)2, it is easy to establish that the equation of the corresponding land form is
This equation describes an axially symmetric shape called a parabola of revolution or paraboloid. It is shown in figure 19.2(a). Beautiful Sugar Loaf peak, about 400 meters high, overlooking the harbor of Rio de Janeiro, seems to resemble a paraboloid.
Now, what is the mean elevation of our parabolic mountain? This question is easily answered by using integral calculus. We stack up an array of thin circular disks of thickness Δz and varying area Aj, to match the parabola shown in figure 19.2(a). Accordingly, the volume of the paraboloid is
where, in the limit, the number of disks becomes infinitely large and the thickness of each shrinks to zero.
Solving equation (19.3) for A, substituting into (19.5), and carrying out the integration gives V = (1/2)A1z1. In addition, we have the equation V = A1, where, by definition, is the average height. So we easily obtain = (l/2)z1. As in the case of our conical mountain, specifies the height of the melted volume.
The Great Pyramids of Egypt
On the west bank of the Nile River near Giza, not far from Cairo, are three incredible structures we shall refer to as the Great Pyramids of Egypt. They were built during the Fourth Dynasty (c. 2650 to 2500 B.C.) and are identified by the name of the kings who built them.
The main dimensions of the three pyramids are listed in table 19.1. These are the estimated original dimensions; not surprisingly, 4500 years has caused considerable erosion. The pyramids are listed in the table in chronological order of construction.
Now what we are going to do, perhaps for the first time in history, is determine the hypsometric curve of the Great Pyramids. This is an interesting problem and one that forms the basis for more complicated ones we consider later.
TABLE 19.1
Source: Edwards (1993).
It is assumed that the three pyramids have the same datum plane, z = 0. For pyramid 1, Cheops, we write the profile equation, z = z1(1 – r/r1), where z1 is the height and r1 is the base length. Since the horizontal sections of the pyramids are squares, we have A/A1 = (r/r1)2 and so A = A1 (1 – z/z2, where A1 is the base area. Therefore, for the three pyramids the hypsometric equation is
For the single-cone case, given by equation (19.2), we expressed our hypsometric equation in the explicit form, z = ƒ(A), instead of the implicit form, A = ƒ(z). The explicit form is usually preferable if we want to compute slopes or calculate volumes. If we have two cones or two pyramids, it is still fairly easy to obtain the explicit form although we get some messy quadratic equations. For three or more peaks, the tedious algebra is not worth the trouble.
Accordingly, we regard equation (19.6) as the final result. Putting the numerical values listed in table 19.1 into this equation and carrying out the simple calculations produces the hypsometric curve shown in figure 19.3(b). The half-profile of the corresponding equivalent single structure is displayed in figure 19.3(a).
Utilizing the fact that the volume of a pyramid is one-third the volume of the containing rectangular prism, the average height of the three pyramids is = 45.8 m. The total volume VT = 5.23 million m3. This is about twice the volume of Hoover Dam on the Colorado River in Arizona-Nevada.
FIG. 19.3
The Great Pyramids of Egypt. (a) Equivalent single structure (half base only); total base length re = 338 m, total base area AT = 114,233 m2, height z = 147 m, average height = 45.8 m. The structure is slightly concave upward. (b) Hypsometric curve.
The ancient Egyptians had an incredible ability in civil engineering. Cheops is nearly as high as the United Nations building (thirty-nine stories) in New York and its base covers the area of fifteen football fields—a fantastic engineering accomplishment, even by present-day standards. And it was built when power meant humans and animals!
The Egyptians also had a surprisingly large knowledge of mathematics and astronomy. They had computed the circumference of the earth to nearly modern-day accuracy and they knew that the value of was around 3.1. In their structural and irrigation engineering projects, it is possible that they were aware of the right-triangle relationship: 32 + 42 = 52.
Here is another example. In various other chapters we have met the interesting quantity called the divine proportion or so-called golden number, . The ancient Egyptians may have known about this number also. It has been suggested by Ghyka (1978) that the geometries of the Great Pyramids feature the ratios
Computing the height and hypotenuse of each of the pyramids using these relationships and the base lengths shown in table 19.1, yields results surprisingly close to the actual values.
Over the centuries, just about everything involving engineering, science, and technology—from Egyptian pyramids and medieval cathedrals to supersonic aircraft and maglev trains—has utilized mathematics to one extent or another. The history of mathematics is fascinating and there are many excellent books devoted to the subject. Two that are especially readable are those of Boyer (1991) and Resnikoff and Wells (1984). In addition, there is an interesting four-volume set of books edited by Newman (1956) entitled The World of Mathematics and an equally interesting three-volume set by Campbell and Higgins (1984) called Mathematics: People, Problems, Results. All of these references are recommended to those who would like to learn more of the history of mathematics.
Looking Ahead: Mountain Ranges and Molehills
This is an appropriate place to pause in our study of geomorphology and hypsometric curves. We continue with these topics in the next chapter, in which we build a mathematical mountain range. For this, as the title of the chapter indicates, there will be need for a certain number of molehills. Once we obtain the mathematical description of such a mountain range, we can calculate its hypsometric curve.
20
How to Make Mountains out of Molehills
In the preceding chapter, in an introduction to the subject of geomorphology, the so-called hypsometric curve was defined and several examples were computed. So we ask the question: Why do geomorphologists study things like hypsometric curves and related topics? The following paragraphs may provide some answers.
Our Always-Changing Planet
Geomorphology deals with the characteristics, origins, and changes of land forms. The incredibly powerful forces of nature—aided and abetted by mankind these days—continually alter the topography and geography of our world. Rainfalls and snowfalls, floods, droughts, winds, tornadoes, hurricanes, typhoons, landslides, avalanches, earthquakes, volcano eruptions, forest fires, ocean waves, and tides—not to mention glacier movements and periodic ice ages—never stop changing the physical features of our earth.
These forces determine the extents of soil erosion, surface runoff and drainage, mud slides and debris flows, river meandering and delta formation, reservoir siltation and waterway clogging.
All these things are better understood and dealt with when we study land mass shapes and their never-ending changes. Hypsometric curves—the quantitative relationships between land area and elevations above a datum—provide valuable information for projects dealing with agriculture and irrigation, city and regional planning, highway and airport l
ocation, river and coastal engineering, and so on.
Mathematical modeling is utilized by geomorphologists to examine these numerous land-changing phenomena. An interesting model for the study of the hydrology of drainage basins was devised by Chorley and Morley (1959). They fabricated a mathematical basin by selecting, as their mountain, a cylinder of lemniscate cross-section whose equation is
which looks something like a narrow leaf or a pointed circle. They intersected this solid with a circular cone which has the shape equation
In these expressions, (x, y, z) are the rectangular coordinates and a and c are constants. It turns out that the region contained by the two spatial surfaces, equations (20.1) and (20.2), looks much like an actual drainage basin. These researchers needed to calculate the hypsometric curve in order to study the hydrology and drainage characteristics of the basin.
Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks) Page 14