Symmetry of the Relativistic Doppler Effect

Appendix A

Holmes explains relativity theory to Watson

Holmes explains the dual nature of space and time

The day was a dreary one. A thin rain was falling, and a dense wet fog lay low over London. Holmes looked out at the mud-colored clouds and muddy streets. He exclaimed, “Watson, I cannot live without brain work. What else is there for me? Stand by the window here. See how the yellow fog swirls down the street and drifts across the dun-colored houses. What is the use of having an energetic mind, Doctor, if one is unwilling to use it?” “My dear Holmes, the tremendous exertions in trying to understand Einstein which I have gone through during the last few days have quite worn me out — not bodily exertion, but the strain on the mind. You will appreciate that.” “Nonsense, old chap!” Holmes said, slapping me on the shoulder. “Come, Watson, we have a good day’s work before us.” “Perhaps you can bring light into something as dark and forbidding as relativity theory.” “Watson, I will bring light. The velocity of light in a vacuum is, as a round number, 300,000 kilometers per second. A light-second is defined as that distance which light travels in one second of time, and so it is equal to 300,000 kilometers. A more practical unit in astronomy is the light- year. A light-year is the distance that light travels in a year. A light-year is equal to 300,000 times 365 times 24 times 60 times 60 kilometers, or 9.5 trillion kilometers. The units light-second and light-year represent time of length. In this ‘time-of-length’ unit, the star Vega is 27 light-years away.” “Excellent! Admirable! But what use will you make of it?” “Originally, the foot and a day were considered natural units because one could measure a distance with his foot and could count the days. However, for beings who live elsewhere in the universe, those units

217

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 217 8/15/2014 11:56:21 AM 218 Remote Sensing in Action

would be quite arbitrary, so instead, natural units are now defined in terms of fundamental physical constants. One such fundamental physical quantity is the velocity of light in a vacuum, which is believed to be the same constant at all places, in all directions, and at all times. In all our discussions, unless otherwise stated, we will use natural units based on the velocity of light in a vacuum.” “They might be useful to scientists,” I said, “but beyond that, I can hardly see what use it is to me in understanding relativity theory.” “In answer, let me show you the relationship between a quantity measured in natural units and the same quantity measured in conventional units. Therefore, for the current discussion, let capital letters denote quantities in conventional units and the corresponding small letters denote the same quantities in natural units.” Holmes continued, “The velocity of light in a vacuum is 300 million meters per second; in natural units, it is one unit. In conventional units, distance is given in meters, which we may denote as capital X. In conventional units, time is given in seconds, say capital T, and velocity, capital V, is given in meters per second. In the time-of-length method, we convert all lengths to times. It may be easier for you to follow if I just write down the equations used to convert our conventional units to natural units. Here they are:

”

“Yes, I quite follow you,” I said. “I will take this opportunity to remark that the unit of x is light-second, the unit of t is second, and the unit of v is light-second per second. Even though we say light-second, the unit is actually a second, which means that both x and t have the same unit, the second, and thus, velocity v is a pure number. It is important to recognize that in natural units, velocity is dimensionless.” “Well done!” I said in an encouraging tone. “Let me mention another fact. The use of the velocity of light in defining the unit of distance has the virtue of simplifying the connection between space and time. Any message sent by light or any other electromagnetic signal from Vega to earth will take 27 years. We see that the use of light-years implies that the communication time and the distance are the same number. For example, from earth to Vega, the time of communication by an electromagnetic signal is 27 years, and the distance is also 27 light-years. For a light signal, time and distance are

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 218 8/15/2014 11:56:22 AM Appendix A: Relativity theory 219

numerically equal; for example, 27 years and 27 light-years are numerically the same. “However,” Holmes went on, “depending on the convention used for the directions of the t- and x-axes, the value of t or x might be either positive or negative. Therefore, for a light signal, we write the equation this way:

t2 – x2 = 0.

The square root of the quantity on the left is called the time-space interval. We denote the time-space interval by the letter p; that is,

time-space interval ==p tx22 − .

We therefore conclude that for any electromagnetic signal, such as light, the time-space interval is equal to zero. This equality is a vital concept in the special theory of relativity.” I said, “I am glad in my heart that you have explained the concept of the time-space interval so well, for indeed, the mystery of relativity theory was becoming too much for my nerves.” “Watson, let us return to Sir Isaac Newton. He introduced the concepts of absolute time and absolute space. Absolute space and absolute time do not depend on physical events but act as a backdrop, or stage setting, within which all physical phenomena occur.” Sherlock Holmes took a blank sheet of paper and scrawled a drawing (Figure A-1). He continued, “As before, we will express both time in seconds and distance in light-seconds. In this way, the velocity of light is unity. We plot the slow train and the fast train on the same set of axes. Remember, Watson, that I am using natural units. Velocity is distance divided by time.

In this diagram, the elapsed time t1 is the same for both trains. However,

in the case of the slow train, distance x1 is less than t1, so the slow train has

a velocity less than one. In the case of the fast train, distance x2 is greater

than t1, so the fast train has a velocity greater than one. Light goes at the speed of one.” “Surely that is evident,” I said. “Newton puts time on one leg and distance on the other leg of a right triangle. Depending on the length of each leg, velocity can be any value.” “However,” Holmes replied, “the theory of relativity postulates that the speed of light represents the maximal velocity that can be achieved in

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 219 8/15/2014 11:56:22 AM 220 Remote Sensing in Action

x-axis (for distance) (in light-seconds)

Velocity of fast train x2 = x2/t1 = 1.5 (unacceptable to Einstein).

Velocity of light t1 = t1/t1 = 1.0.

Light Fast train Velocity of slow train x1 = x1/t1 = 0.5.

Slow train t-axis (for time)

t1 = constant for three events Figure A-1. The slow train and the fast train on the same set of axes.

nature. The magnitude of the velocity of any physical object must be less than one. As my drawing shows, Newton allows the fast train to have a velocity greater than one. Such a situation is impossible. As a result, Ein- stein demands that Newtonian absolute space and absolute time must be discarded. Yes, thrown out like an old shoe!” “Most outrageous!” I exclaimed. “This is beyond anything which I could have imagined. What can be done, Holmes?” Sherlock Holmes laughed heartily at my perplexity. “My dear Watson, the answer is simple. If one universal coordinate system is impossible, it follows that each and every inertial frame must have its own coordinate system. Einstein gave the solution.” “Please tell me how Einstein accomplished that feat,” I said. “How- ever, limit your answer to those things that are absolutely essential.” “The important concept is that there is the dual nature between space and time. Newton took time as absolute and space as absolute. Einstein rejects this categorization and instead takes the speed of light as an absolute constant. Einstein supersedes the ideas of absolute time and absolute space by the notion of space-time.” I was attempting to come to mental grips with this fantastic notion when Holmes, as was his practice, suddenly seemed to change the subject.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 220 8/15/2014 11:56:22 AM Appendix A: Relativity theory 221

“It is a dark and gloomy day, Watson,” he said. “Do you observe the circle of light thrown by the lamp? The answer is found in the circle.” “The answer to what?” “The answer to what you were just thinking about, Watson, the notion of the union of time and space,” Holmes retorted. He took another sheet of paper and drew a second diagram (Figure A-2). “Einstein resolved the problem by an appropriate use of the circle to reveal the circular nature of space-time,” Holmes said. “Tell me if there is any point that I don’t make clear.” “So far, so good.” Holmes placed his finger in the center of the circle and said, “The smallest point may be the most essential, so take your time to ponder. The circle expresses the inclusivity of the universe. No individual point on the circumference is favored; they are all at the same radial distance from the center. However, each observer orients the circle according to his own time and space. The observer does the orientation in such a way that he is always at rest within his system of coordinates.” “It all seems clear enough,” I responded, “but I am not sure that I understand every detail.” “According to the observer at rest, the coordinates of the slow train

are t1 and x1, and the coordinates of the fast train are t2 and x2. Einstein, in

x p 2 p = radius of circle p = constant Fast train x = space-time interval 1 = proper time. Slow train

Velocity of slow train = x1/t1 = 0.5.

Velocity of fast train = x2/t2 = 0.9.

Figure A-2. The circular nature of space-time.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 221 8/15/2014 11:56:22 AM 222 Remote Sensing in Action

appealing to the wisdom of Pythagoras, requires that the t coordinate be the hypotenuse of a right triangle and the x coordinate the leg of the same right triangle.” “But why in the world would anyone want to do that?” “A leg of a right triangle is always less than the hypotenuse,” Holmes explained. “This means that by using this construction, the x coordinate is always less than the t coordinate. Therefore, the velocity, which is the ratio of distance over time, is always less than one. Recall that we are using natural units, in which the velocity of light is one. Einstein achieves his goal. The velocity of the train is necessarily less than one. Einstein spares himself the burden of Newton, in which the fast train has a velocity greater than the speed of light.” “It sounds incredible, but from what you assert, there can be no other conclusion. Indeed, the circle depicts the relativistic view of nature.” “You are dead on, Watson,” said Holmes. “It seems far-fetched, but relativity theory is nothing more than the symmetry of space-time. The circle represents the foundation of relativity theory, and it governs movement through time and space.” “It will take time for me to entirely grasp everything you have told me,” said I. “My dear Watson, the more unconventional a thing is, the more care- fully it deserves to be examined. The very point which appears to com- plicate a case is, when duly considered and scientifically handled, the one which is most likely to elucidate it. In space-time, the separation between two events is measured by the time-space interval between the two events, which takes into account not only the spatial separation between the events but also their temporal separation.” “In the case of the circle, what is the time-space interval?” “The space-time interval is radius p of the circle. It is also called the proper time.” “Proper?” I said. “To the contrary, this whole discussion seems most improper — most outrageous. I must insist on some further explanation. I admit, my dear Holmes, that this infernal problem about understanding Einstein is consuming me.” Sherlock Holmes, sitting comfortably in an armchair, puffed sedate- ly at his cigarette. “By a supreme effort, you can overcome the difficulties,” he said. “I have been turning it over in my mind. No matter how difficult things seem to be, you must never give up. And now, my dear Watson, without referring to my notes, I cannot give you a more detailed account of the curious involvement of the circle. However, I do affirm

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 222 8/15/2014 11:56:23 AM Appendix A: Relativity theory 223

that if you bear with me, nothing essential will be left unexplained in our story.” Holmes then paused, leaned back in his chair, and assumed a pose which I had seen so often, one which he regularly adopted when thinking hard and which I knew not to interrupt. Finally, he leaned forward and said, “It is this dual nature of Einstein’s system in regard to space and time that we must reconcile.” Dual nature and reconcile! Those words had resonated within me for years as I tried unsuccessfully to reconcile the dual nature of my companion in his roles as artist and detective. I had always been conscious of his dual nature, and then he had the impertinence to mention the dual nature of space and time! Sherlock Holmes is a connoisseur of the arts and a good judge of literature. He is also an enthusiastic musician — a capable performer and a composer as well. Holmes would frequent the major concert halls and would be engulfed in perfect happiness. At those times, the artist Holmes, with his languid, dreamy eyes, was as unlike the other Holmes, the relentless, keen-witted detective, as it was possible to conceive. The extreme exactness and astuteness that the detective Holmes used in solving a case appeared to be the reaction against the poetic and con- templative artist Holmes. In an instant, his nature could transform from extreme languor to devouring energy. Although Sherlock Holmes had retired from detective work, the lust of the chase came upon him once again on reading Einstein’s paper, and my long association with him left me no option but to encourage him to continue. “My dear Holmes, I hope you will explain this remarkable subject of space-time in more detail. If you would only elaborate, I could gain a better understanding of the concept.” “The symmetry of special relativity involves a totally new geometric concept of distance between any two events,” said Holmes. “The space- time interval must be invariant. This invariant interval involves the separation in time between any two events as well as their separation in space.” “I am afraid I still do not quite follow you, Holmes. It is all very strange and a mystery to me.” “It is a mistake to confound strangeness with mystery. The most commonplace assumption is often the most mysterious because it presents no features from which deductions may be drawn. Your assumption is a belief in the Newtonian concept of absolute space and time. However, this most commonplace assumption cannot accommodate that fact that

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 223 8/15/2014 11:56:23 AM 224 Remote Sensing in Action

the ultimate velocity is the fixed and finite velocity of light. As we have seen, the constant velocity of light ties together space and time into a new entity, space-time. The concept of space-time certainly is strange to your sensibilities, Watson, but it is not mysterious because it explains observable physical phenomena, such as electromagnetism, which otherwise cannot be explained. It is in this way that Einstein has changed the world.” “Holmes, you talk about space and time. The concept of space pre sents to the mind an image of such vastness that imagination can hardly conceive it. The concept of time, as treated in art and poetry, is always puzzling. When Isaac Newton founded classical mechanics, he viewed space as something distinct from anything external, and he viewed time as something that passes uniformly with no regard to whatever happens in the world. He defined absolute space and absolute time in that way, so as to distinguish those entities from the various ways by which they are measured.” “The definitions proposed by Newton have proved adequate for the velocities at which we conduct our daily business. But in other circumstances, they are not sufficient. The special theory of relativity uses the intriguing fact that the velocity of light is constant to unite the concepts of space and time. The idea of the time-space interval stimulates the imaginations of layperson and scientist alike. It is a difficult concept, but it can be understood by appealing to the power of the imagination.” “I do not understand why the velocity of light must be constant under all conditions,” I said. “You are not alone, Watson. That feature of relativity theory is the most difficult to grasp. A body subject to no external forces moves by inertia at a constant speed in a straight line. This body, as well as all other objects moving along with it at the same speed in the same direction, constitutes an inertial frame in which the body and these other objects are at rest. Let us look at the example of a train and the passengers seated on it. They regard the train and themselves as being at rest. They cannot tell whether the train is stopped or is moving uniformly with respect to the embankment unless they look out the windows and see for themselves.” “All well and good, Holmes! But why can’t the absolute space and time coordinates of Newton serve all trains as well as all other moving objects in the universe?” “It would be nice, arguably, if that were the case. But it is not, and we must do what nature and Einstein require, which is to assign each inertial frame its own unique time and space coordinates. What do you make of that?” “It is very bewildering.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 224 8/15/2014 11:56:23 AM Appendix A: Relativity theory 225

“But, Watson, it should not be. Nature and Einstein require that the velocity of light must be the same in all frames, most definitely in conflict with classical theory. Because the velocity of light is a ratio of space over time, the time coordinates in the various frames must adjust to keep the critical ratio, the velocity of light, constant. Therefore, time is no longer an absolute quantity that is the same everywhere. Instead, time is per- ceived differently in different inertial frames. This is the most surprising thing about relativity theory. Tell me, Watson, what have you learned?” “Both Newton and Einstein start with the right triangle. Newton puts time on one leg and distance on the other leg. The velocity is the ratio of distance to time. Depending on the length of each leg, velocity can be either less than one or greater than or equal to one. According to the principle of the constancy of the velocity of light, a physical velocity greater than or equal to one is impossible. Thus, Newton’s theory collapses. “Einstein corrected the mistake by placing time on the hypotenuse,” I continued. “Because the hypotenuse is necessarily greater than the leg, it follows that velocity must always be less than one, thus satisfying the principle. In conclusion, the open linear nature of Newton’s concept of absolute space and absolute time allows for the possibility of velocities of any magnitude. The closed circular nature of Einstein’s concept of space-time eliminates the possibility of velocities greater than the speed of light.” My companion smiled approvingly. “You sum up the difficulties of the situation succinctly and well,” he said. He looked out the window. “The weather has somewhat cleared. What do you say to a ramble through London?” At this point, I was weary of our erudite discussions and gladly acqui- esced. We strolled along Fleet Street, watching the ever-changing kalei- doscope of life as it ebbs and flows. It was two hours before we reached Baker Street again.

Holmes explains the dilation of time “Relativity is certainly very surprising,” I said after we had returned to Baker Street. “There are several most instructive points about it. In any case, Holmes, I am glad that I am well past the point of being examined as if I were a schoolboy.” “Some scientific arguments hinge on unfathomable mathematics,” said Holmes. “A person might understand the basic principles involved but still be impeded by some mathematical difficulty. But why? Often the reason is

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 225 8/15/2014 11:56:23 AM 226 Remote Sensing in Action

that the instruction was not clear or was not presented competently. Now I shall present the essential mathematics of the special theory of relativity in a manner that you will understand, and therefore, you will no longer have difficulty in understanding the basic underpinnings of this subject.” “What?” I cried. “You expect me to be a schoolboy again! I would never want to experience that part of my life again. You will have to convince me that there is a very good reason for doing so, and that will be difficult. It is possible that some armchair lounger might enjoy your neat mathematical exercises in the seclusion of his own study. But it is not practical for most people.” Holmes ignored this outburst, took one of my favorite books from the bookshelf, and said, “Watson, let us get down to what is practical. I know this is a valuable book. It is a first edition, and it has a handsome leather binding in good condition. It has been passed down in your family for generations, so this particular copy has some personal value to you. I want this book. I will pay you four shillings for it.” “I would only take at least 16 shillings for it,” I responded, without knowing what Holmes was driving at. “So, Watson, the bid is four, and the offer is 16. How can we arrive at a sale?” “Quite simply, Holmes! As I remember from the days when I really was a schoolboy, Aristotle taught that the golden mean was the desirable middle between two extremes, one of excess and the other of deficiency. For example, courage, a virtue, if taken to excess, would manifest as reck- lessness and if deficient as cowardice. In our case, your bid of four shillings is deficient, and my offer of 16 shillings is excessive. The answer, Holmes, is simple. We would settle at the golden mean.” “Brilliant,” said Sherlock Holmes, “perhaps more brilliant than you realize. I contend that if we can determine the numerical value of the golden mean, we will have discovered the meaning of Einstein’s theory of relativity.” “Do you mean that the essence of relativity theory is the determination of the mean value of two extremes?” “Exactly, Watson,” Holmes said. Then he lapsed into silence with his head thrown back and his eyes closed, an attitude which might seem list- less to a stranger but which I knew was a signal of the most intense inner reflection. “Your question must be answered,” he said at last. “The correct answer is of the utmost importance. And to arrive at it, we must start at the beginning, which means that I must now teach you about Pythagoras.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 226 8/15/2014 11:56:23 AM Appendix A: Relativity theory 227

I was at a loss as to what to say and could only mutter, “What did Pythagoras do?” “None of the writings of Pythagoras survives. All that we know about him comes from the works of later writers, and from them, we know that Pythagoras introduced, among many other things, the concept of the arithmetic mean and the concept of the geometric mean. Now let us do the mathematics. In our case, the two extremes are four and 16. To find the arithmetic mean, we add the two extremes and divide the result by two. Do you agree, Watson?” “Yes. That is quite a simple computation.” “The sum of four and 16 is 20. I divide 20 by two, and the result, 10, is the arithmetic mean.” For an instant, I thought the key to the riddle was in my hands. I said, “The arithmetic mean is halfway between the two extremes. In other words, the distance between four and 10 is six. The distance between 10 and 16 is also six. If we choose the arithmetic mean as the settlement price, we would have a situation that is equally distant between what you want and what I want. The arithmetic mean splits the difference. Such a solution appears to be fair, and hence, it would be difficult to refuse. As the seller, I would accept the price of 10 shillings.” I felt that my reasoning was flawless, but my companion’s reaction was an enigmatic smile. “Very well, Watson, as the seller, you would accept the price of 10 shillings. However, as the buyer, I would not accept that price. There would be no sale.” “A good many people, perhaps the majority, would accept that price. It is mystery why you would not.” “A mystery, is it?” Holmes cried, rubbing his hands. “This is very pi- quant, Watson. In splitting the difference, you are using what is called arithmetic reasoning. The selling price is six shillings above the bid. The offer- ing price is six shillings above the selling price. In other words, four plus the distance six is equal to 10, and 10 plus the distance six is equal to 16.” “Exactly, Holmes. Clearly, the answer is that the selling price would be the arithmetic mean 10.” “Possibly, but we must take every option into account. Are you familiar with geometric reasoning?” “No. What is geometric reasoning?” “The mathematics is only slightly more difficult, and again, it involves the two extremes, which are four and 16 in this case. To find the geometric mean, we multiply the two extremes and then take the square root of their product. Do you follow, Watson?”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 227 8/15/2014 11:56:23 AM 228 Remote Sensing in Action

I nodded and said, “I agree that this does not seem overly taxing.” “The product of four and 16 is 64, and the square root of 64, or eight, is the geometric mean.” “Of what use is the geometric mean?” “The ratio between the geometric mean eight and my bid of four shillings is eight divided by four, which is two. The ratio between your offer of 16 and the geometric mean eight is 16 divided by eight, which is also two. The geometric mean gives the proper proportion. If we choose the geometric mean as the settlement price, we would have a number that is proportional to what you want and what I want. I believe that such a solution appears fair and hence would be difficult to refuse. As the buyer, I would accept the price of eight shillings.” “Holmes, I am afraid that I do not follow your train of thought.” Holmes responded with another quick sketch (Figure A-3). “This diagram will help,” he said. “It shows that the arithmetic mean splits the difference, whereas the geometric mean proportions the difference.” Holmes continued, “Watson, I am endeavoring to tell you everything which may have any bearing on your understanding of relativity theory. I beg that you will question me on any point which I do not make clear.” “On the contrary, Holmes, your statement is singularly lucid.”

Figure A-3. The arithmetic mean splits the difference, whereas the geometric mean proportions the difference.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 228 8/15/2014 11:56:23 AM Appendix A: Relativity theory 229

“You have begun to appreciate the meaning of my elucidations. I hope that now you are as stimulated as I am. We have two numbers, Watson — my bid of four shillings and your offer of 16. If we split the difference, we find that the selling price is the arithmetic mean 10. In other words, I would pay six shillings more than my bid, and you would take six shillings less than your offer. That is the solution that you, as the seller, would want.” “My dear fellow, it is only fair.” “Not so fast, Watson! Let me give the buyer’s point of view. We have two numbers, my bid of four and your offer of 16. If we proportion the difference, we find that the selling price is the geometric mean eight. In other words, I would pay twice my bid, and you would take half of your offer.” “Please go over that again.” “My bid is four, and I double it to reach the geometric mean eight. Your offer is 16, and you reduce it by half to reach the geometric mean eight. In other words, I multiply my bid by two, and you multiply your offer by one-half. What could be fairer than that?” “The arithmetic mean 10 is better for me as seller, and the geometric mean eight is better for you as buyer. Obviously, I would choose the arithmetic mean, and you would choose the geometric mean. We disagree.” “Yes, Watson, let us agree to disagree.” “Such trivia, Holmes, really seems to have little to do with this mysterious creature that you call relativity theory.” “Watson, it most certainly does pertain to what you so colorfully call a ‘creature.’ I have no doubt that you will see the connection after you become familiar with the terminology used in relativity theory. Now, as a man of letters, how do you define the word period ?” “I am not averse to comfortable chairs and the latest periodicals, but I do not relish a long period of mathematical work.” Holmes took little notice and responded, “In physics, the word period denotes the duration of one complete cycle of a wave or oscillation. The period is the reciprocal of the frequency. In other words, a period represents an interval of time marked by the recurrence of some phenomenon. More particularly, the period of a wave is the time difference between two successive crests. The signals used in relativity theory are electromagnetic waves, so this definition is important to those who attempt to understand the theory.” “The work of Einstein must strongly appeal to you as a man of a precisely scientific mind. My dear Holmes, I am a practical man. I still do not

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 229 8/15/2014 11:56:23 AM 230 Remote Sensing in Action

understand why I must turn my attention to something that seems of little everyday use.” “You and all other practical men should do so because we are getting to the essence of remote detection, which is the source of most worthwhile knowledge. I am sure practical men would agree that knowledge is useful.” “Obviously, Holmes. So please continue.” “Let us assume that I am on a train, and you are at the station. You want to determine the velocity of the train, something that often is of practical value. You send out an electromagnetic wave of a given period, called the sending period. The wave travels to the moving train, where it is reflected. The reflected wave returns to the station, where it is received. At the receiving point, the wave has a given period, called the receiving period. In summary, at the station, you measure the sending period and the receiving period of the wave. The sending period and the receiving period are two positive numbers. They provide you with the necessary information for the determination of the velocity of the train” (Figure A-4). “Holmes, this is the kind of puzzle that makes you happy. Pray tell me how I, at the station, would determine the velocity of the moving train.” “The mathematics is the same as the way by which you determined the selling price of your book. Let the sending period be four and the receiving period be 16. According to you, the reflection period at the moving train would be the arithmetic mean 10. The distance is the difference of 10 minus four, which is six. The velocity is distance divided by time, which is six-tenths, or 0.6. The train is moving at six-tenths the speed of light, a huge velocity, well beyond anything that could be attained in the foreseeable future.” “So be it! Yet the mathematics is not clear to me.”

Sent signal going

Sending period Reflection period Receiving period

Reflected signal returning

Figure A-4. At the station, you measure the sending period and the receiving period.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 230 8/15/2014 11:56:24 AM Appendix A: Relativity theory 231

Holmes’ face showed his disappointment and annoyance. “Not clear? Well, if that isn’t clear, what could be clear?” I meekly answered, “It is not clear because the units are jumbled. You say that the distance is the difference of 10 minus four. But 10 and four are in units of time, whereas distance must be in units of space.” Holmes lowered his voice and said, “You have already forgotten the key point, Watson. We are measuring distance as the bees do, in terms of time. It is quite important to remember that we are using natural units. Time is measured in the unit of seconds. Distance is measured in the unit of light-seconds, and the velocity of light is a pure number equal to one.” “It is an explanation, but in no way is it as clear as crystal. What is the object of all of this mathematics, Holmes?” “Although the mathematics we have used so far is simple, it is about to reveal something very much deeper than appears at first glance. The reflection period as recorded by me on the moving train is entirely different from the reflection period as computed by you from the sending and receiving periods recorded on the platform. The problem is entirely analogous to the problem that we have already solved as to the sale of your book.” “We did not solve that problem at all. All we did was to agree to disagree. We must define the situation a little more clearly. ” “I am afraid, Watson, that there is no logical explanation except the one given by Einstein. Recall that you as seller accept the arithmetic mean 10 as the correct selling price. Yet I as buyer accept the geometric mean eight as the correct selling price.” “Quite true, Holmes.” “The example still remains the only solid thing with which we have to deal, and we must not permit our attention to wander from it. I will now make the logical conclusion, the conclusion that Einstein made in his paper. You, at the station, accept the arithmetic mean 10 as the reflection period at the moving train. This vehicle is remote from you. Yet I, on the moving train, accept the geometric mean eight as the reflection period. There are two answers for the same thing. Both are correct.” “How can that be?” I asked in disbelief. “It is true, Watson, because a key conclusion of Einstein is that there is not one absolute reference frame, as put forth by Newton. Einstein’s reasoning displaces that of Newton.” “But what application could this arcane reasoning possibly have?” “The Wright brothers have just developed a fixed-wing aircraft, something that could be considered a moving vehicle. One of our battleships,

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 231 8/15/2014 11:56:24 AM 232 Remote Sensing in Action

by comparison, could be considered a fixed platform. If an observer on the ship could determine the velocity of the airplane by means of reflected radio signals, I think you would agree that would be very practical information — particularly if it were an enemy aircraft.” “As an old soldier, I have to agree.” “In common terminology,” Holmes said, “the arithmetic mean is called the midpoint time, and the geometric mean is called the proper time. The arithmetic mean is always greater than the geometric mean. Consequently, the midpoint time is always greater than the proper time. For that reason, we say that the midpoint time is dilated with respect to the proper time.” “Are you saying that when the clock at the station reads the midpoint time 10, the clock on the moving train reads the proper time eight? Are you saying that these two times are different even though they both refer to the same incident?” “Precisely! This is where Einstein’s theory takes scientific thought in a new direction. The possibility of two values of time for the same incident has never been proposed before. Think about that a little, about how truly revolutionary that concept is.” “It is deeply interesting, Holmes, but a good many people would not accept this idea and in fact would consider it an outrage. I am one, at least for the moment.” “I quite understand your thinking,” Holmes said. “You have been quite abruptly brought into contact with something which seems strange and bizarre but which I contend is perfectly logical. There is no way that you, for the moment, can reconcile your own experience with your respect for my abilities. But cheer up, Watson. I am sure there are ways to make the light of reason eventually dawn on you — at least some small glimmer of that light.” “It is only the respect I have for you that gives me any hope to gain at least some understanding of relativity theory,” I said. “Let us go back to the example we have been discussing. The sending period is four, and the receiving period is 16. As a result, the midpoint period is 10, and the proper period is eight. At the station, the clock reads 10, while at the same instant on the moving train, the clock reads only eight. Thus, the moving clock is running more slowly than the stationary clock. Both clocks are identical, but the moving clock is ticking at a slower rate than the stationary clock. This enigma, the dilation of time, is often interpreted as time slowing down on the moving clock.” “A mystery it is, as great a mystery as any I have faced with you.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 232 8/15/2014 11:56:24 AM Appendix A: Relativity theory 233

“It is natural to not timidly accept this enigmatic conclusion,” Holmes replied. “In this respect, you are no different from most of mankind. To help clarify things, I have prepared a table that shows six cases, each with the midpoint time of 10. The table shows that as the distance decreases, the time dilation, which is the excess of midpoint time over proper time, also decreases. It is of the highest importance in remote detection to recognize, out of a set of several facts, which are incidental and which are vital. Otherwise, your energy and attention would be dissipated instead of being concentrated” (Table A-1). “All I see is a collection of numbers. Please be more particular.” “Case 0 refers to one who can ride on a light beam. Such a person who travels at the speed of light itself would be seen to age the proper time zero in midpoint time 10 and would therefore live for eternity. At the other extreme is Case 5, in which the person stays on the fixed platform. Such a person would be seen to age the proper time 10 in midpoint time 10 and would therefore age as we do now. In between these two extremes is Case 2, in which the person travels at 0.6 times the velocity of light. Such a person would be seen to age the proper time eight in midpoint time 10 and therefore appear to age more slowly than we do.” “You seem to want to find the entire cosmos lurking in this rather un- remarkable arithmetic,” I said thoughtfully. “Why not just accept the midpoint time for everything and forget the idea of proper time altogether? In other words, why not just split the difference and forget all the rest? That is what Sir Isaac Newton did. But I can see by your eyes that you are not satisfied with the absolute time of Newton but instead want to venture into this newfangled area of Einstein’s. Why?”

Table A-1. Six cases, each with the midpoint time of 10.

Sending Receiving Midpoint Proper Case time time time time Distance Velocity 0020 10 0.0 10 1.0 1218 10 6.0 8 0.8 2416 10 8.0 6 0.6 3614 10 9.2 4 0.4 4812 10 9.8 2 0.2 5101010010.0 0.0

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 233 8/15/2014 11:56:24 AM 234 Remote Sensing in Action

“I think the answer to that question will become apparent as we pro- ceed. However, with your permission, let us adjourn for a while. I will take a small stroll on the crowded London streets and experience the hum of human industry. Will you come with me?” “No, thank you, Holmes,” I replied. “I really think I will be more com fortable inside than out.”

Holmes explains the stereographic projection I was puzzled by Holmes’ desire to take another stroll. Outside, the wind howled down Baker Street. There was still a drizzle of rain. How- ever, an hour later, when Holmes returned from his solitary excursion, he did indeed seem mentally reinvigorated. He pulled his chair up to the fire and pushed his wet feet toward the blaze. “By Jove!” he said. “It is good that young Einstein has come up with something that at last breaks our dead monotony.” I was perplexed and disappointed in myself that I was failing to grasp much about relativity theory, so I finally asked, at what I felt was an appropriate moment, “What is the starting point in the chain of reasoning that will lead me to understand Einstein’s theory? Will the answer be found somewhere at the end of a tangled line?” Holmes looked steadily at me and replied, “I am sure that your geometry teacher, back during the first time you were a student, stressed that the circle and triangle deserve attention. Geometric shapes correspond to comparable forms in the nervous process. The circle and the triangle are basic symbols. The circle indicates things that are cyclic, and the triangle indicates things that are eternal.” “Of course, Holmes! I recall that in one of your cases, we encountered this very unusual mark, a triangle inside a circle” (Figure A-5). “Watson, what do you remember about the Pythagorean theorem?” “If a triangle has a right angle, then the side opposite the right angle is called the hypotenuse, and the sides containing the right angle are called the legs. Pythagoras is credited with discovering the most amazing fact about the right triangle. If you make a square on each of the three sides, then the largest square has the same area as the two smaller squares added together. My geometry teacher pounded that fact into my head. I still get headaches thinking about it” (Figure A-6). “Tradition says that Pythagoras learned about the right triangle during his long stay in Egypt,” Holmes said. “The right triangle was the symbol of universal nature among the ancient Egyptians. It represented the trinity

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 234 8/15/2014 11:56:24 AM Appendix A: Relativity theory 235

Figure A-5. A triangle inside a circle.

Figure A-6. The largest square has the same area as the two smaller squares added together.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 235 8/15/2014 11:56:25 AM 236 Remote Sensing in Action

Figure A-7. The symbol of universal nature among the ancient Egyptians.

Figure A-8. Horus is depicted in the form of an ancient symbol for the all-seeing eye.

in their mythology. One leg represented the god Osiris, the other leg the goddess Isis, and the hypotenuse their son Horus. The right triangle can be divided into two similar right triangles, one for Isis and the other for Osiris, as shown in this drawing” (Figure A-7). “I see that, Holmes. But when you divided the original triangle into two triangles, one for Isis and one for Osiris, what became of Horus?” With a fervor which made it evident that the subject interested him very deeply, Holmes said, “Look what happens, Watson, when I rotate the Isis triangle by 180 degrees. I get the figure that represents Horus in the form of a primitive symbol for the all-seeing eye” (Figure A-8). “So the concept of the all-seeing eye can be traced to the beginnings of civilization.” “As you can see, Watson, the primitive symbol was refined over time and ultimately became the Egyptian hieroglyph for the Eye of Horus. It was a powerful image used as protection from evil. It also was considered to confer wisdom, health, and prosperity.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 236 8/15/2014 11:56:26 AM Appendix A: Relativity theory 237

“But, Holmes, I must ask a simple question. This is very interesting, but what is its scientific meaning? What does the primitive symbol for the eye have to do with Einstein’s theory of relativity?” Holmes replied, “There are curious and suggestive details about the circumstances. I am convinced that the all-seeing eye is a factor which has to be taken into consideration, but I have not yet got quite to the heart of the matter. It is a very cold scent on which I have to run. I have taken diverse factors into consideration, but none of them alone quite gets to the heart of the matter.” Holmes sat in silence for some time. He then remarked, “In this hand, I hold the journal which contains Einstein’s original paper on relativity theory. Einstein tackles one of the most difficult puzzles that has perplexed a man’s brain, and he has come up with ideas that seem strange. I have made some progress toward grasping what he means, but I lack the one or two perceptions which I need to complete my understanding. But I’ll have them, Watson, I’ll have them! The reason is that I firmly believe they will be revealed by the Eye of Horus.” For a split instant, his eyes kindled, and a slight flush sprang into his thin cheeks. When I glanced again, his face had resumed that composure which had made so many people regard Sherlock Holmes as a machine rather than as a living, breathing human being. Holmes finally sat forward in his chair, lit his pipe, spread out some documents on his knees, and sat for some time smoking and turning them over. “I’ll tell you what,” he said. “I should like to have a quiet little glance into the details of the Eye of Horus. There is something in it which fascinates me extremely. If you will permit me, Watson, I will draw some pictures to test the truth of one or two little fancies of mine. And then we will return to the inquiry about this primitive symbol.” I had hoped that in some way, I could coax my companion to drop what I thought to be a fruitless line of inquiry, but one glance at his intense face told me how vain my expectation was. He again sat for some time in silence. He then went to the desk. He looked for paper and finally found my best-quality stationery, bonded and watermarked. It had cost sixpence a packet. He pulled out several dozen sheets. I cringed when Holmes became absorbed in a strange drama. He made absurd little diagrams on my expensive paper, giving each diagram a long scrutiny, and then he crumbled up the sheet and tossed it into the fire. At last a joyous cry showed that Holmes had found what he wanted. “I have come upon something,” he said. “On the face of it, it would appear to be something both exceptional and unexceptional. Look here, Watson —

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 237 8/15/2014 11:56:26 AM 238 Remote Sensing in Action

on the Eye of Horus, I drew a circle. I think this might be a remarkable discovery” (Figure A-9). “Your drawing is a very usual geometric construction. It does not seem remarkable at all! Any schoolchild could draw it.” “Watson, do you not recognize that my figure is that of the stereographic projection, which we previously talked about at some length? Yes, yes, the ancient Eye of Horus is a manifestation of the stereographic projection. Look at the slanting line at the top of the diagram. The line starts at the North Pole, crosses the circle at point P, and intercepts the horizontal line at point R. Clearly, point R is the stereographic projection of point P.” “This simple diagram seems to have done wonders for your composure, but I fail to see how such a commonplace association advances our inquiry.” My readers know that I would do anything for Holmes to assist him in solving a crime. However, this excursion into geometry seemed to be asking too much of my patience. I was tempted to call off this exercise, but I hesitated to do so because I could sense that Holmes was hot on the scent. His face flushed and darkened. His brows were drawn into two hard black lines, and his eyes shone.

North Pole N

O S R

South Pole M

Figure A-9. Circle and the Eye of Horus.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 238 8/15/2014 11:56:26 AM Appendix A: Relativity theory 239

Holmes extended the upward slanting line in his drawing to the point where it intersected the circle. Holmes said, “Look, Watson, the downward slanting line and the upward slanting line meet at point P on the circle. Also notice that lines MP and NR meet at a right angle at point P” (Figure A-10). To cover my ignorance, I tendered a favorite expression of his, “So much is observation. The rest is deduction.” “Now is the time for deduction, Watson. As we see, the slanting line from the South Pole pointing upward projects point S onto point P. Simi- larly, the slanting line from the North Pole pointing downward projects point P onto point R. In brief, point S projects into point P, and point P projects into point R.” “For some reason, geometry makes me hungry. Some cold beef and a glass of beer would be nice,” I said, ringing the bell. “I have been too busy to think of food, Watson, and I am likely to be busier still. By the way, I know you still harbor misgivings about geometry. However, you will probably think better of it when I stipulate that geometry can open up an avenue to understanding relativity theory that does not require equations.”

North Pole N

O S R

South Pole M

Figure A-10. Lines MP and NR meet at a right angle at point P.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 239 8/15/2014 11:56:27 AM 240 Remote Sensing in Action

“We have had some dramatic adventures together, Holmes, and I have always been willing to go wherever you directed, no matter what the dan- ger. But tackling geometry is different, and I fear that you ask too much when you expect me to understand it.” Holmes, not unexpectedly, ignored my comment and continued to concentrate on the sketch. He drew the radius from the origin to P and said, “We recognize radius OP as the partially raised obelisk, so now you can clearly see that line OR is the stereographic projection of line OP ” (Figure A-11). Holmes paused again and then resumed, “In The Hound of the Basker- villes, you wrote, ‘We looked back on it now, the slanting rays of a low sun turning the streams to threads of gold and glowing on the red earth new turned by the plough and the broad tangle of the woodlands.’ ” “Yes, Holmes, the golden and wonderful slanting rays of the setting sun.” “Let me depict the rays of the sun. We will need two sets of rays — one for the rays streaming upward and the other for those streaming downward. The upward rays strike the horizontal opaque line OS, which blocks the sunlight. The downward rays strike the opaque radial line OP, which

North Pole N

O S R

South Pole M

Figure A-11. Line OR is the stereographic projection of line OP.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 240 8/15/2014 11:56:27 AM Appendix A: Relativity theory 241

blocks the sunlight. We see that the shadow of OS is OP, and the shadow of OP is OR” (Figure A-12). “Holmes, you have an extraordinary genius for minutiae,” I remarked. “I fear, Watson, that you have not quite grasped the significance of what has been accomplished. Let me state the results. Line OP is the stereographic projection of line OS. In turn, line OR is the stereographic projection of line OP. ” “Mere details, Holmes.” “When I was quite young, I learned to appreciate the importance of details. Never trust general impressions, but concentrate on details.” “Well, then, explain the importance of these details. You have made so much out of the stereographic projection, but frankly, Holmes, it is just a circle with some added lines. It seems little more than a rather commonplace decoration or something to temporarily amuse a schoolboy.” “Commonplace? You see nothing remarkable?” Holmes almost shout- ed, and then he paused to reflect on my lack of understanding. “What use will you make of it?” I exclaimed. “I should like a few more particulars before I will understand what all this geometry means. It

North Pole N

O S R

South Pole M

Figure A-12. The shadow of OS is OP, and the shadow of OP is OR.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 241 8/15/2014 11:56:28 AM 242 Remote Sensing in Action

would be noteworthy if you would show what the stereographic projection had to do with relativity theory.” Holmes sat intently alert. “My dear fellow,” he replied at last, “shadows are associated with sunlight. What else is associated with sunlight?” “I can also quote from The Hound of the Baskervilles. I believe I wrote: ‘Outside the sun was sinking low and the west was blazing with scarlet and gold. Its reflection was shot back in ruddy patches by the distant pools which lay amid the great Grimpen Mire.’ ” “Excellent, Watson. The reflections indicated the presence of the distant pools on Grimpen Mire. Do not forget the word reflection! It is para- mount! Reflections reveal the presence of distant objects. Reflections can be used for remote detection.” “Have we gone astray? What does remote detection have to do with the matter which we are investigating, relativity theory?” “Do not forget that the title of Einstein’s paper may be more accurately translated as ‘On the Remote Detection of Moving Bodies by Elec- tromagnetic Signals.’ ” “I cannot say that I am in a state of horrible suspense, Holmes, but I am willing to do my best to understand what you say.” “How is active remote detection done, Watson? It is done by reflection, by reflection! Reflection is the basic component of active remote detection. It is just an application of the mathematics we discussed earlier with the example of the moving train. The observer transmits a wave of a given period, called the sending period. The moving target reflects the signal. The reflected wave returns, and it is received by the observer. Its period is called the receiving period.” Holmes took a piece of my expensive foolscap and made a sketch. He said, “This diagram depicts the process of normal-incidence reflection spread out in time. Light emerges from the segment OS of the observer’s timeline, is reflected by the target, and returns the segment OR of the observer’s timeline” (Figure A-13). “You know this is entirely new to me, Holmes. I must beg you to be more explicit.” Holmes responded with another quick sketch. “Look at this diagram, Watson. Point T is the midpoint between point S and point R. As you see, I have added a line that goes from P to T. As it turns out, line PT is tangent to the circle at point P” (Figure A-14). He continued, “Now I am ready to show how the stereographic projection describes relativity theory.” “Please do.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 242 8/15/2014 11:56:28 AM Appendix A: Relativity theory 243

North Pole N

P ward light rays wn Do

O S R Upward light rays

South Pole M

Figure A-13. Emergent light from OS is reflected at OP to received light OR.

North Pole N

Tangent Radius

O S T R

South Pole M

Figure A-14. Line PT is tangent to the circle at point P.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 243 8/15/2014 11:56:28 AM 244 Remote Sensing in Action

“The observer has its own time axis and distance axis. The target has its own time axis and distance axis, which are different. The two sets of axes are:

1) Observer: Time axis is horizontal line. Distance axis is tangent line. 2) Target: Time axis is radius. Distance axis is vertical line.

“Watson, what path does the signal take?” “In your eagerness, Holmes, you have wandered far into the chasms of the unknown. It is no easy matter to pick out such a path.” “I will tell you. The signal goes from S to P. The reflection occurs at P. The return signal goes from P to R.” “Holmes, pray sum up the whole argument.” “Point T is the midpoint between S and R. According to the observer, the reflection is a remote event. It occurs at time OT and distance TP. According to the target, the reflection is a local event. It occurs at time OP and distance zero. For the observer, the period of the reflection is the midpoint OT. For the target, the period of the reflection is the radius OP.” Holmes paused to let me consider what he had said, and then he continued. “Because OT is the hypotenuse and OR is the leg of a right triangle, it follows that OT is greater than OP. In other words, the period OT as measured by the observer is greater than the period OP as measured by the target. For this reason, period OT is called the dilated period, and period OP is called the proper period.” “Pray put it in terms that I can understand.” “Very well, Watson. You remember that in the sale of your book, the bid was four shillings, and the offer was 16 shillings.” “Of course, Holmes, in your attempt at buying my book, I demanded 16 shillings, and you offered four.” Holmes took another piece of my foolscap and sketched a diagram, to which he added numbers. “Look, Watson. The matter is not inexplicable. The sending period OS is my bid of four shillings. The receiving period OR is your offer of 16 shillings. From these two numbers, we can find all the rest. The arithmetic mean of four and 16 is 10, which is the length of the line segment OT. The geometric mean of four and 16 is eight, which is the length of the radius OP. Line OT is the midpoint period, and line OP is the proper period” (Figure A-15). “Most rewarding,” I said with some asperity.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 244 8/15/2014 11:56:28 AM Appendix A: Relativity theory 245

“I shall lay it all before you, Watson. Length six of line segment PT is the distance. Line segments ST and RT each have the same length, six. The velocity of the moving target is the distance divided by the midpoint period. In other words, the velocity is PT/OT, which is six divided by 10, or 0.6 times the velocity of light.” “My dear Holmes, these hard, dry statements need some little editing to soften them into the terms of real life.” “There is no difficulty at all,” said Holmes. “Your desired settlement price of 10 and my desired settlement price of eight mean there can be no sale unless one of us gives in to the other.” “Precisely.” “In the physical world, as described by the laws of Sir Isaac Newton, a sale is reached. More particularly, the buyer surrenders his position and accepts the terms of the seller. In other words, I would abandon my offer of eight and would pay you 10 shillings to meet your demand. In the world of Newton, the sale would go through at the price given by the arithmetic mean.”

North Pole N

PT = 6 OP = 8

O S T R OS = 4

OT = 10

OR = 16

South Pole M

Figure A-15. Line OT is the midpoint period, and line OP is the proper period.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 245 8/15/2014 11:56:29 AM 246 Remote Sensing in Action

“Holmes, since I accept Newton’s laws, you have convinced me that as the seller, I should hold my ground and make the buyer meet my desired settlement price.” “Yes, Watson. That is the position of Newton. Newton says the reflection period on the fixed platform is the same as the reflection period on the moving vehicle. Both are equal to the arithmetic mean 10. In Newtonian physics, there is no room for disagreement.” “I have known you long enough to appreciate that you are leading up so something, and I expect that it is Einstein. So what does the young Einstein say?” “Unfortunately, I am unable to describe the solution given by Einstein in simple language. I will have to act it out. The best way of successfully acting a part is to be it,” said Holmes. “I only ask a little patience, Watson, and all will be clear. However, you probably will not like what I say because it will show that nature satisfies both the buyer and seller simultaneously. According to Einstein, I, the buyer, would hand you eight shillings. I would count the shillings as I passed them to you — one, two, three, four, five, six, seven, eight — but at the same time, you would count the shillings as you received them from me — one, two, three, four, five, six, seven, eight, nine, 10.” “But that is incredible! You count out eight shillings to me, and simultaneously I count 10 shillings. How can you act out a play like this? It does not conform to real life.” “I must edit a previous remark,” Holmes replied. “In relativity theory, we do not agree to disagree. Instead, we just disagree. We each demand our due, and moreover, we each get our due. I pay you eight shillings, and you get 10 shillings from me. In the same way, the reflection period has value eight from the vantage point of the moving vehicle, and the exact same reflection period is seen to have value 10 from the vantage of the stationary platform.” I was unable to respond to this, so Holmes continued. “Grasping that point is difficult. So before turning to those aspects of the matter which present the greatest difficulties, we should begin by mastering the fundamental things.” Holmes grabbed another sheet of my dear foolscap and scrawled some lines. “The top sketch shows a single line. The bottom sketch shows the most basic part of the stereographic projection” (Figure A-16). “Where is this going, Holmes?” “On the diagram, the top gives Newton’s view of space and time and the bottom Einstein’s. Newton allows only one time axis. According to

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 246 8/15/2014 11:56:29 AM Appendix A: Relativity theory 247

O S T R P

O S T R

Figure A-16. The upper diagram gives Newton’s view of space and time, and the bottom gives Einstein’s.

Newton, one horizontal line serves as the time axis for both the observer and the target. As a result, the reflection period is the arithmetic mean OT. In our example, OT is 10.” “And what does Einstein do?” “Einstein requires two time axes, one for the observer and another for the target. According to Einstein, the reflection period for the observer is the arithmetic mean OT. The reflection period for the target is the geometric mean OP. In our example, OT is 10, and OP is eight.” “This is curious and still seems unreasonable,” I said. “I think you know me well enough, Watson, to realize that I am by no means an unreasonable fellow. I require facts and logic before I draw a conclusion. At the same time, it is folly rather than courage to refuse to recognize the facts as presented by Einstein, even though they seem illogi- cal at times. I must again ask for some quiet. I must think over all that we have discussed and what it means before we continue.”

Holmes explains the relativistic Doppler factor I picked up a book, volume three of Isaac Disraeli’s Curiosities of Literature, and started to read the chapter “Of a History of Events Which Have Not Happened.” Einstein’s theory certainly is bizarre and outside the conventions of the routine of everyday life! Could Holmes be ex- pounding the story of an event which has not happened? If so, I could avoid this excruciating torture of mathematics with no end in sight!

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 247 8/15/2014 11:56:29 AM 248 Remote Sensing in Action

My thoughts then wandered back to the ill-assorted and difficult path- way that we were taking to understand Einstein. Could there be, I wondered, some radical flaw in my companion’s reasoning? Might he be suffering from some huge self-deception? Was it not possible that his nimble mind had used faulty premises to adopt this wild theory that Newton was mistaken? Holmes was rarely wrong, and yet the keenest logician may occasion- ally be deceived. He was likely, I thought, to fall into error through the overrefinement of his logic, his preference for the bizarre explanation of Einstein’s when a plainer and more commonplace one lay at hand. Yet I had heard the reasons for his deductions, and I had seen the evidence myself. When I looked back on the chain of facts, many of them trivial in themselves but all tending in the same direction, I convinced myself that even if Holmes’ explanation was convoluted, the theory of Einstein must be studied no matter how unconventional and startling it appeared when first encountered. Holmes suddenly leaned forward, obviously excited, and said almost exactly what I had been thinking: “The key to understanding Einstein’s paper is his section on the Doppler effect.” “As I recall from earlier discussions, this is complicated.” “And about to get more complicated, Watson, because the Doppler factor as given by Einstein differs from the Doppler factor as derived by Doppler about half a century ago. Einstein’s is called the relativistic Dop- pler factor, whereas Doppler’s is called the classical Doppler factor.” Holmes then pointed to some equations in Einstein’s article. “Just seeing those equations sends cold shivers up and down my spine,” I said. “You said the article contains just a handful of not overly daunting equations. However, the equations are not only overwhelming but also fright- ening to me.” Holmes, as usual, continued, with no regard for my mental agony. “As I was about to say, Watson, the classical Doppler factor results from the use of Newtonian, or absolute, time. The relativistic Doppler factor results from the use of Einsteinian, or relative, time. The Doppler factor is defined as the receiving period at one location divided by the corresponding sending period at another location. Watson, do you have any more paper? It seems that all your nice stationery is used up.” Reluctantly I took out a few sheets of foolscap, all that I had left, and handed him one sheet. “Here, use this to draw your picture.” He drew this sketch (Figure A-17). “This drawing represents Newton’s point of view. According to New- ton, the signal goes from S to T to R. The Doppler factor from S to T,

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 248 8/15/2014 11:56:29 AM Appendix A: Relativity theory 249

T/S = 10/4 = 2.5 R/T = 16/10 = 1.6

O = 0 S = 4 T = 10 R = 16

Figure A-17. There must be two classical Doppler factors.

P/S = 8/4 = 2 R/P = 16/8 = 2

O = 0 S = 4 P = 8 R = 16

Figure A-18. There must be just one relativistic Doppler factor.

designated by the symbol k1, is the quotient of the reflection period 10 divided by the sending period four, which is 2.5. The Doppler factor from

T to R is designated by the symbol k2. It is the quotient of the receiving period 16 divided by the reflection period 10, which is 1.6. Because the

factors k1 and k2 are different, we see there must be two classical Doppler factors.” The inconsistency suddenly dawned on me. I exclaimed, “There is an incongruity. The outgoing classical Doppler factor 2.5 differs from the returning classical Doppler factor 1.6. That would seem to be a problem.” Holmes lowered his voice and practically whispered, “At last, Watson, you see a glimmer of what was in the mind of Einstein!” Although there still was plenty of room on the first sheet of foolscap, Holmes grabbed another sheet. “Excellent foolscap,” said Holmes, scribbling another sketch. “This shows that according to Einstein, there must be just one relativistic Dop- pler factor, namely, k equal to two” (Figure A-18).

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 249 8/15/2014 11:56:30 AM 250 Remote Sensing in Action

Holmes explains the symmetry in Einstein’s theory “Holmes, what are you driving at?” “Watson, remember what I told you. Whereas Newton allows only one time axis, Einstein requires two time axes, one for each party.” He snatched another sheet of my foolscap and drew a sketch (Figure A-19). “Look at this drawing. There are two time axes. According to Ein- stein, the signal goes from the sending time four to the reflection time eight. The reflection time is measured on the other time axis. It then was sent to the receiving time 16. The relativistic Doppler factor from four to eight is designated by the symbol k. It is the quotient of the reflection period eight divided by the sending period four. In other words, k is eight divided by four, which is two. “The relativistic Doppler factor from eight to 16 is also designated by the symbol k,” Holmes went on. “It is the quotient of the receiving period 16 divided by the reflection period eight, which is two. In other words, the two relativistic Doppler factors are the same, as required.” “This is all very well,” I said, “but the thing becomes more unintel- ligible than ever.” “Good old Watson! You are determined to remain the one fixed point in a changing age. But both of us must change, Watson. I have based my career on logic and scientific principles. But I am going to have to change. There’s a scientific revolution coming….”

Relativistic Doppler factor = k = 16/8 = 2.

Relativistic Doppler factor = k = 8/4 = 2.

Figure A-19. The two relativistic Doppler factors are the same, as required.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 250 8/15/2014 11:56:30 AM Appendix A: Relativity theory 251

Holmes suddenly stopped in midsentence, and a rare expression of surprise appeared on his face. “I think, Watson,” he said, “that you are in the presence of one of the most absolute fools in Europe. I deserve to be kicked from here to Victoria Station.” He sank back into a state of intense and silent thought, and it seemed to me, accustomed as I was to his every mood, that some new possibility had suddenly dawned on him. Sherlock Holmes sat silent for a few minutes with his fingertips pressed together. Then he said, “We have to remember that what Einstein accomplished is certainly very extraordinary. Everywhere you look in nature, you see symmetry. If you look at plants and animals, you will find that they have symmetrical body shapes and patterns. If you divide a leaf in half, you will often find that one half has the same shape as the other half. There is symmetry everywhere, so why not here? But how slow- witted I have been, for I did not understand this until I read Einstein’s article.” “To me, relativity theory is a strange conundrum! I do not have the least clue as to what you could be thinking,” I said. Holmes chuckled and wriggled in his chair, as was his habit when in high spirits. He said, “The clue is symmetry. Symmetry! Reading Einstein led me to focus on that feature, and everything I have described since then was the logical result of that factor. Hence the things that now perplex you and make relativity theory difficult for you to understand have served to enlighten me and to strengthen my conclusions.” Holmes took another sheet of my costly foolscap and drew a picture (Figure A-20). “But how about this?” he asked. “Can you make anything out of the tangle? What is true for Observer A is also true for Observer B. I trust, Watson, that you see the symmetry.” “I know what is good when I see it, and I see the symmetry now. But there is one more thing, Holmes. Since Einstein’s case for relativity seems so overwhelmingly strong, it may not be necessary to bring forward all the details. However, there is still something that bothers me. I can’t quite put my hand on it, but I am confused as to why the proper time is not the same as the midpoint time. I see no symmetry in that situation at all. Can you supply those missing links which would make this remarkable chain whole?” “I shall try to be clear and concise,” said Holmes. He took yet another sheet of my cherished foolscap and drew a sketch (Figure A-21). “See, Watson, there is symmetry here. Each observer has his own coordinate system specified by a time axis and distance axis. Follow the

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 251 8/15/2014 11:56:30 AM 252 Remote Sensing in Action

Figure A-20. What is true for Observer A is also true for Observer B. Distance for Distance

Observer B = 6

Figure A-21. “See, Watson, there is symmetry here.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 252 8/15/2014 11:56:31 AM Appendix A: Relativity theory 253

arrows. You will see that in his own coordinate system, each observer, by remote detection, finds that the other observer is at midpoint time 10 and distance six. However, the other observer is at proper time eight and distance zero in his own coordinate system. In other words, the midpoint time 10 is on the time axis of the one doing the remote detection, and the corresponding proper time eight is on the time axis of the other. The reason why midpoint time 10 is not the same as the corresponding proper time eight is that two different coordinate systems are used.” “I am always impressed by your extraordinary powers of observation in following the clue and clearing up those mysteries which had been abandoned as hopeless,” I said. “Relativity theory seems the very mad- dest, queerest thing that I have ever come across. I will never understand it.” “Understanding is not important at this stage. What is important is learning the mechanics. To understand the steam engine, one must understand thermodynamics. To use the steam engine, one must know the mechanics. Before Einstein, one would measure the dimensions of the table in front of us in the familiar three-dimensional coordinates of length, breadth, and height. One would measure the passage of time with the entirely separate and independent coordinate of Newtonian time. But now Einstein has changed everything. Instead of space and time, we have space-time. No longer is the time coordinate independent of the spatial coordinates. They are all interlinked, and we have seen how the relativistic Doppler factor is used to find the relationship. So, Watson, what have you absorbed to this point?” “Holmes, I understand that you think out little puzzles, and what you have done today wants more thinking out than I am used to. I am still trying to appreciate how the concept of symmetry….” “The appreciation of symmetry is most important,” said Holmes. “It fills a gap in your understanding which I had previously been unable to bridge in this most complex affair. You have become aware that the geometric mean prevails in the definition of the relativistic Doppler factor. It demonstrates why relativity theory must be used to explain the cosmos. Our understanding of the cosmos is ruled by projective precepts.” “I believe that I now understand, at least a little bit.” “My explanation may be difficult, but the rewards are great if you stick with it until you understand. William Blake wrote, ‘To see a world in a grain of sand.’ Do you remember that I wrote, ‘From a drop of water, a logician could infer the possibility of an Atlantic or a Niagara.’ As an example, from the geometric mean, which he conceived more than 2,000

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 253 8/15/2014 11:56:31 AM 254 Remote Sensing in Action

years ago, Pythagoras might have been able to infer the possibility of a relativistic structure of space and time.” “Of course I remember, Holmes. But when I wrote A Study in Scar- let, I purposely left out the sentence containing the term geometric mean because if a mathematical expression had been used, no one at all would read the book.” “It is a pity because the geometric mean, originally devised by Py- thagoras two-and-a-half millennia ago, is the key that opens the door. All the mathematics of special relativity can be derived from the relativistic Doppler factor, which essentially is the geometric mean. More specifically, the relativistic Doppler factor is the geometric mean divided by the sending period.” “Holmes, I fear all this is too deep for me. But let me see if I can sum up what I have learned to this point. In Newtonian theory, you split the difference between sending and receiving time to obtain the arithmetic mean, which is called the midpoint time. In relativity theory, you proportion the difference to get the geometric mean, which is proper time. Because the arithmetic mean of two positive numbers is always greater than the geometric mean, it follows that the midpoint time is stretched, or dilated, with respect to the proper time. This phenomenon is called the dilation of time.” “Your summary is a very remarkable account,” said Holmes. “It is a fitting finale to an extremely interesting session.” He gave a sigh of relief, but I couldn’t tell whether it was in reaction to his teaching or to my slight understanding.

Classical Doppler effect

Historical background

When direct measurement of something, particularly a physical entity, is not possible (and this is often the case), knowledge of the object can be acquired only by methods that belong to the field of remote sensing (or remote detection). Important examples of knowledge gained by remote sensing, which rank with some of the most impressive intellectual achievements in history, are the distances to stars, the structure of the interior of the earth, and the locations of petroleum reserves and other natural resources. Not surprisingly, this is a field that uses advanced technology and sophisticated computer algorithms. Again not surprisingly, a fundamental tool used in remote sensing is the Doppler effect. However, and perhaps surprisingly, many of the key principles used in calculating the Doppler effect are based on mathematical concepts developed more than 2000 years ago. The story of how the mathematics of the ancients became the un- derpinning of modern remote sensing is long and complicated and will involve some somewhat lengthy detours, but the journey is well worth the effort because it will link Pythagoras with Einstein. The journey will begin with the three Pythagorean means: the arithmetic mean, the harmonic mean, and the geometric mean. The reason is that the classical Doppler effect (which uses mechanical waves, such as sound waves) requires two different Doppler factors. (1) The classical Doppler factor from a fixed source to a moving receiver is essentially the arithmetic mean. (2) The classical Doppler factor from a moving source to a fixed receiver is essentially the harmonic mean. The fact that both the arithmetic mean and the harmonic mean are required represents a fundamental asymmetry in the classical Doppler effect and ultimately undermines Newtonian physics. The geometric mean is the key to the symmetrical relativistic Doppler factor used by Einstein, and it is the subject of Appendix C.

255

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 255 8/15/2014 11:57:24 AM 256 Remote Sensing in Action

The mathematics of the ancients extends from the Ionic Greek mathematician Pythagoras (c. 570–c. 495 b.c.) to the Alexandrian Greek mathematician Hypatia (c. a.d. 360–415). Hipparchus of Nicaea (c. 190–c. 120 b.c.), a Greek astronomer, geog- rapher, and mathematician of the Hellenistic period, discovered the precession of the equinoxes and was the prime figure in the development of trigonometry. He is usually credited with the invention or adaptation of several astronomical instruments, including the astrolabe. He formalized mathematical projection as a method for solving astronomical problems. In particular, he understood the stereographic projection. As shown earlier in this book, the astrolabe is based on the mathematics of the stereographic projection. Alexandria was the center of mathematics for about 700 years, rough- ly from Euclid (300 b.c.) to Hypatia (a.d. 400). Hypatia is regarded as the most important woman mathematician in history. Although none of her writings are extant (as with many ancient scientists), historical accounts indicate that she was fascinated by conic sections and developed scientific instruments, including an apparatus for water distillation, an instrument for measuring water level, and a hydrometer for determining the specific gravity of a liquid. Most important, Hypatia’s pupil Synesius described her designs for the astrolabe. Hypatia apparently was devoted to understanding the astrolabe’s relationship to Pythagorean musical intervals. Her problem was that Py- thagorean musical intervals pertain to classical physics, whereas the astronomical intervals provided by the astrolabe refer to relativistic physics. Hypatia was unable to make the transition from classical to relativistic physics.With the benefit of hindsight, we can now make that connection.

Principles of remote sensing

Remote sensing (the acquisition of data about an object by the use of sensing devices that are not in physical contact with the object) is charac- terized by intelligent use of signals that penetrate the unknown. A critical key to accurate interpretation of these signals is correct understanding of the Doppler effect. The Doppler effect may be described simply in the following way: If a wave is reflected by something moving away from the position of the receiver, the reflected wave has lower frequency than the transmitted wave. If a wave is reflected by something moving toward the receiver, the reflected wave has higher frequency than the transmitted wave.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 256 8/15/2014 11:57:24 AM Appendix B: Classical Doppler Effect 257

Any substance that continually flows under an applied shear stress is called a fluid. Sound waves propagate in a fluid medium such as air or water. For such waves, the classical Doppler effect takes into account the velocity of the source, the velocity of the observer, and the velocity of the medium. Each of these effects is analyzed separately. In this book, for the sake of simplicity, we regard the medium as stationary, and we will deal only with source and receiver. At the start of any discussion of remote sensing, it is necessary to have a good grasp of the relationship between a period of time and the corresponding frequency. A day, which includes both daylight and dark, represents one cycle of a periodic event. That cycle repeats itself day after day. There are approximately 365 days in the year, so the frequency is 365 days per year, or 365 cycles per year. The period is 1/365 years per cycle, which is one day per cycle. The main point is that the frequency is the reciprocal of the period, and conversely, the period is the reciprocal of the frequency. A clock face is circular for good reason. The second hand, the longest hand, points to the 60 divisions on the circumference of the circle. The next-longest hand, the minute hand, also points to the 60 divisions on the circumference. The shortest hand, the hour hand, points to the numbers. In this example, the period of the second hand is the time it takes for that hand to make one complete revolution. Each mark on the circumference represents one second. In one cycle (or revolution), the second hand passes 60 marks. As a result, the period of the second hand is 60 seconds per cycle. Similarly, the period of the minute hand (the time it takes for that hand to make one complete revolution) is 60 minutes per cycle. Finally, the period of the hour hand is 12 hours per cycle. In summary, the period of the second hand is 60 seconds per cycle, the period of the minute hand is 60 minutes per cycle, and the period of the hour hand is 12 hours per cycle. The frequency (i.e., the reciprocal of the period) of the second hand is one cycle per 60 seconds. Because 1/60 is 0.0167, the frequency of the second hand is 0.0167 cycles per second (cps). The frequency of the minute hand is one cycle per 60 minutes, so because 1/60 is 0.0167, the frequency of the minute hand is 0.0167 cycles per minute. The frequency of the hour hand is one cycle per 12 hours. Because 1/12 is 0.0833, the frequency of the hour hand is 0.0833 cycles per hour. The measurements of period and frequency are usually described in terms of seconds. By definition, one cycle per second is 1 Hertz, where the word Hertz is abbreviated as Hz. The period of the second hand is 60

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 257 8/15/2014 11:57:25 AM 258 Remote Sensing in Action

seconds per cycle. The frequency of the second hand is (1/60) cycles per second, or 0.0167 Hz. The period of the minute hand is 60 minutes per cycle. The quantity 60 minutes is 3600 seconds. The frequency of the minute hand is (1/3600) cycles per second, or 0.000278 Hz. The period of the hour hand is 12 hours per cycle. The quantity 12 hours is 43,200 seconds. The frequency of the hour hand is (1/43,200) cycles per second, or 0.0000231 Hz.

Wave motion Waves can be classified as either mechanical or electromagnetic. A mechanical wave requires a medium composed of physical matter. An electromagnetic wave can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. For that reason, an electromagnetic wave does not require a medium through which to travel. In fact, any medium other than a vacuum will take energy out of the wave. In the case of a mechanical wave, the substance of the medium is de- formed. When the medium is elastic, there is no permanent displacement of the particles of the medium. As an example, sound waves propagate when air molecules collide with their neighbors, transferring energy from neighbor to neighbor down the line, causing a cascade of collisions in a given direction. Because the collisions are elastic, the air molecules os- cillate about their equilibrium positions. This keeps the molecules from continuing to travel in the direction of the wave. As a result, each mol- ecule stays in the same place, whereas energy travels in the direction of the wave. Waves are either “standing” or “traveling.” A standing, or stationary, wave remains in a constant position. The waves on a string of a musical instrument are examples. Such waves are the result of interference of traveling waves propagating in opposite directions. The main property of a traveling wave is that it transfers energy from one point to another. Traveling waves are found everywhere in nature. They even occur in chemical reactions and epidemic outbreaks. The basic parameters used to describe a traveling wave are amplitude, period, frequency, wavelength, and velocity (Figure B-1). In the case of a sinusoidal wave (a very common occurrence and one that is readily analyzed mathematically), the amplitude is the magnitude of the maximum disturbance from the central value, the period is the length of time taken by one cycle, and the frequency is the reciprocal of period. The wavelength is how far the wave moves during one cycle. In

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 258 8/15/2014 11:57:25 AM Appendix B: Classical Doppler Effect 259

Time axis

Wavelength = velocity × period. Frequency = 1/period.

Figure B-1. The basic parameters used to describe a traveling wave.

other words, the wavelength is the distance from one crest to the next crest. The wave velocity is how fast the wave is moving. Because velocity is the ratio of distance over time, the velocity of the wave is the quotient of wavelength over period. Alternatively, the velocity is equal to wavelength multiplied by frequency. The symbol f is used for frequency, T is used for period, λ is used for wavelength, and v is used for velocity. The following equations hold:

Period = 1/frequency, or T = 1/f. Frequency = 1/period, or f = 1/T. Velocity = wavelength/period = wavelength × frequency, or

λ vf==λ . T Standing waves

Pythagoras, best known for the invaluable Pythagorean theorem, differentiated the humming of the strings and the music of the spheres. The humming of strings represents the earthly music of mechanical waves of sound. The music of the spheres represents the celestial interplay of electromagnetic waves of light. Pythagoras discovered the connection of number and harmony in sound waves, and he was a seminal figure in the development of ideas about “music” that were strongly metaphysical. He

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 259 8/15/2014 11:57:25 AM 260 Remote Sensing in Action

spoke of the association among music, mathematics, and the universe. His goal was to reduce the formulation of the laws of nature to a consideration of pure numbers. In this appendix, we deal with mechanical waves; in Ap- pendix C, we will deal with electromagnetic waves. Sounds can be described in terms of frequency. A sound wave creates peaks of higher air pressure and troughs of lower air pressure. Each peak followed by a trough makes up one cycle. The sounds that a person hears result from these pressure changes. With music, the frequency at which these peaks strike your ear controls the pitch that you hear. Frequency is measured in hertz (where 1 Hz is equal to one vibra- tion per second). A sinusoidal sound wave consists of successive traveling cycles of one wavelength each. Each cycle contains a positive bump representing high pressure and a negative bump representing low pressure. The ear detects these pressure changes and interprets them as the sounds we hear. The four most important Pythagorean musical notes are shown in Table B-1. The note C has a frequency of about 262 Hz, meaning that when C is played, 262 peaks of higher air pressure strike your ear each second. As seen in the table, F has a frequency of about 349 Hz, G has a frequency of about 392 Hz, and C1 has a frequency of about 523 Hz. The notes C and C1 complement rather than clash with each other. Every two peaks of C1 line up perfectly with every peak of C. A ratio compares values; it says how much of one thing there is compared with another thing. Ratios can be written in different ways. One way is to use the word to, as in “2 peaks of C1 to 1 peak of C.” A second way is to use a colon to separate the values, as in 2:1. A third way is to use a short oblique stroke to separate the values, as in 2/1. We prefer this third way. A ratio of two numbers can be expressed as a fraction when appropriate.

Table B-1. The four most important Pythagorean musical notes.

Frequency × Frequency × Note Frequency 1/261.626 6/261.626 C 261.626 1 6 F 349.228 4/3 8 G 391.995 3/2 9 C1 523.251 2/1 12

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 260 8/15/2014 11:57:25 AM Appendix B: Classical Doppler Effect 261

An octave is a musical interval encompassing eight staff positions. The ascending interval from C to C1 is an octave, as there are eight staff positions from C to C1, namely C, D, E, F, G, A, B, C1. C and G also complement each other because the frequency of G is 3/2 times the frequency of C. Every three peaks of G line up perfectly with every two peaks of C. A perfect fifth is a musical interval encompassing five notes on the musical staff. The ascending interval from C to G is a perfect fifth because there are five staff positions from C to G, namely, C, D, E, F, G. The frequency of F is 4/3 times as large as that of C. Every four peaks of F line up perfectly with every three peaks of C. As a result, the notes C and F fit together well. A perfect fourth is a musical interval encompassing four staff positions. The ascending interval from C to the F is a perfect fourth because there are four staff positions from C to F, namely, C, D, E, F. In mathematics, a fourth is the fraction 1/4, and a fifth is the fraction 1/5. However, in music, the words fourth and fifth do not have the same meanings as in mathematics. For that reason, we will always use the term perfect fourth for the Pythagorean ratio 4/3 and the term perfect fifth for the Pythagorean ratio 3/2. For example, 12/9 is a perfect fourth. The ears do not hear frequencies as such but essentially hear the loga- rithms of frequencies. As a result, the separation between the frequencies of two sounds can be measured by dividing the frequency of one by the frequency of the other. Such a separation represents a musical interval. There are seven basic musical intervals from the second to the octave (Figure B-2). Pythagoras discovered that the most pleasing musical intervals were the octave (frequency ratio = 2/1), the perfect fifth (frequency ratio = 3/2),

Major Major Major Major Major Major Perfect second third fourth fifth sixth seventh octave

Figure B-2. Seven basic musical intervals from the second to the octave.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 261 8/15/2014 11:57:26 AM 262 Remote Sensing in Action

and the perfect fourth (frequency ratio = 4/3). Those ratios are all contained in the simple numbers 1, 2, 3, 4, which are the numbers in the tetractys:

O O O

O O O O

The four rows add up to 10, the number of fingers, and can be considered to represent the organization of space. The first row represents a point defined by one point (zero dimensions). The second row represents a line defined by two points (one dimension). The third row represents a triangle defined by three points (two dimensions). The fourth row represents a tetrahedron defined by four points (three dimensions). The rows of the tetractys (going up) can be read as the ratios of 4/3 (perfect fourth), 3/2 (perfect fifth), and 2/1 (octave), forming the basic intervals of the Pythagorean scales. Aristotle wrote that the followers of Pythagoras saw that the ratios of musical scales were expressible in numbers, that all things seemed to be modeled on numbers, and that numbers seemed to be the first things in the whole of nature. Pythagoras believed that the world was digital, not analog, centuries before the invention of the modern computer. In ancient times, the two main branches of mathematics were arithmetic and geometry. Arithmetic, which deals with numbers, is digital. Geometry, which deals with forms and shapes, is analog. In the Renaissance, these two branches were brought together, and new branches of mathematics arose.

Pythagorean means

In Pythagorean philosophy, the concept of number is highly struc- tured and highly geometric. The geometry of numbers was abstract and malleable at the same time. The Pythagorean view is exemplified in the set of the three Pythagorean means. The starting point for defining each is two extremes, two positive numbers usually denoted by s and r. Each of the three means lies between the two extremes. The primary mean is the

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 262 8/15/2014 11:57:26 AM Appendix B: Classical Doppler Effect 263

geometric mean, denoted by the letter p. On either side of the geometric mean are the arithmetic mean t and the harmonic mean q. The value of any of the three means does not depend on whether r is greater than s or less than s. As a matter of convention, we will choose s (for source or sending or start) to represent the initial signal and r (for return or receiving or finish) to represent the final signal. In other words, the flow of time goes from s to r. The arithmetic mean t is defined as one-half of the sum of the extremes, or

sr+ t = . 2

The harmonic mean q involves a combination of the reciprocals of the extremes (1/s and 1/r) and their arithmetic mean (1/s + 1/r)/2. The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals, or 1 q = . 11 1 + 2 sr This equation can be rearranged to obtain

sr sr q ==2. rs+ t This equation shows that the harmonic mean is the product of the two extremes divided by the arithmetic mean. It follows that the arithmetic mean is the product of the two extremes divided by the harmonic mean:

sr t = . q The symmetry existing between the arithmetic and harmonic means is incorporated in the equation sr = qt.

This equation says that the product of the extremes is equal to the product of the means.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 263 8/15/2014 11:57:26 AM 264 Remote Sensing in Action

The geometric mean p is defined as the square root of the product of the extremes. It follows that the geometric mean is the square root of the product of the other two means. The two expressions for the geometric mean are p== sr;. p qt In the time of Pythagoras, just these three means (arithmetic, geometric, and harmonic) were known. However, two later Pythagoreans, Myonides and Euphranor, added a fourth, the gain, defined as

x = (r – s)/2 = t – s = r – t.

If r is greater than s, then the sequence s, r is said to be ascending. For an ascending sequence, the gain is positive. On the other hand, if r is less than s, then the sequence s, r is said to be descending, and consequently, the gain is negative. The Pythagorean rate v is the quotient of the gain and the arithmetic mean: x rs− v == . t rs+

Rate of change, or simply rate, is the ratio of a change in one variable over the corresponding change in another variable. The word rate can refer to any number of things. For example, it can refer to the rate of interest. The word rate also can refer to velocity; for example, the rate of 60 miles per hour. The velocity V of an object is the rate of change of distance X with respect to time t. In the case of constant velocity, we can write V = X/t. To qualify as a Pythagorean rate, the number v must be less than one in magnitude. We can do so as follows: If V is the velocity of a moving object and c is the speed of sound, then the Mach number is v = V/c. We will always take c as a positive number so that v is positive when V is positive, and v is negative when V is negative. Because we consider only objects moving at speeds less than the speed of sound, the value of v will always be less than one in magnitude. For example, the cruising velocity of the Airbus is V = 865 kilometers per hour or V = 537 miles per hour. The Pythagorean rate v = V/c is a pure number; it has no attached units of measurement such as kilometers per hour or miles per hour. At the cruising altitude of the Airbus, the speed of sound is c = 1054 kilometers per hour, or c = 655 miles per hour. The Py-

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 264 8/15/2014 11:57:26 AM Appendix B: Classical Doppler Effect 265

thagorean rate is 865/1054 = 0.82, or 537/655 = 0.82. This demonstrates that the Pythagorean rate does not depend on the units of measurement used. In conclusion, the Mach velocity of an airplane traveling at less than the velocity of sound serves as the Pythagorean rate v. The Airbus travels at Mach velocity 0.82. One advantage of the use of the Mach velocity is that distance can be expressed in terms of time. Because X = Vt, it follows that (X/c) = (V/c)t, or x = vt. If time t is expressed in terms of hours and because v is a pure number, x will also be expressed in hours. As an example, consider an Airbus crossing the Atlantic from New York to Paris. The Mach velocity v = 0.82, and the crossing time is t = 6.75 hours. Thus, the distance from New York to Paris is

x = vt = 0.82 × 6.75 hours = 5.535 hours. In other words, it takes the crossing time of 6.75 hours to travel the distance of only 5.535 hours. The distance 5.535 divided by the time 6.75 gives the Pythagorean rate v = 0.82, which is the Mach velocity. In using Pythagoras, we do not deal with two units of measurement, such as miles and hours, but with only one, which in this case is hours. In the case when the extremes and the means represent periods, the Pythagorean rate v = x/t can be interpreted as a physical velocity (i.e., distance over time in which both distance and time are expressed in seconds, hours, and so forth). For this reason, this book uses the period of a wave instead of the frequency of the wave when referring to v as velocity.

Reconciliation

Pythagoras expressed music in the scientific terms of the means and extremes. The frequency of note C can be represented by s = 6, the frequency of note F by q = 8, the frequency of note G by t = 9, and the frequency of note C1 by r = 12. These frequency values comprise the ascending sequence

C, F, G, C1 = s, q, t, r = 6, 8, 9, 12.

If s and r are the extremes, then q is the harmonic mean and t is the

arithmetic mean. The musical interval of a perfect fifth is k1 = t/s = 9/6 = 3/2 (which will be shown to be the first Doppler factor). The musical

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 265 8/15/2014 11:57:26 AM 266 Remote Sensing in Action

interval of a perfect fourth is k2 = q/s = 8/6 = 4/3 (which will be shown to be the second Doppler factor). Pythagoras laid the foundations of music by studying the sounds of stretched strings, which produce consonance or dissonance when certain ones are struck simultaneously. Pythagoras used only four strings, identical except for their lengths. The period of the sound produced by a string is proportional to its length. Because the frequency is the reciprocal of the period, the fundamental frequency of each string is inversely proportional to its length. In other words, long strings have low frequencies. For example, the keys on the left side of a piano keyboard strike long strings and produce low frequencies. The four basic notes in the musical octave are C, F, G, and C1. Table B-2 gives their frequency and period. There is no need to remember these long numbers because by replac- ing the frequency 261.626 of note C by 1, Table B-2 simplifies to Table B-3. Note that the officially designated frequencies given in Table B-2 are not quite in keeping with the precise mathematical frequencies dictated by Pythagoras given in Table B-3.

Table B-2. Frequency and period of the four major notes in an octave.

Note Frequency (Hz) Period (s) C (the first note) Start = 261.626 Start = 0.00382 F (the fourth note) Harmonic mean = 349.228 Arithmetic mean = 0.00286 G (the fifth note) Arithmetic mean = 391.995 Harmonic mean = 0.00255 C1 (the eighth note) Finish = 523.251 Finish = 0.00191

Table B-3. Frequency and period of the four major notes in an octave when using 1 as the frequency for C.

Note Frequency (Hz) Period (s) C (the first note) Start = 1 Start = 1 F (the fourth note) Harmonic mean = 4/3 Arithmetic mean = 3/4 G (the fifth note) Arithmetic mean = 3/2 Harmonic mean = 2/3 C1 (the eighth note) Finish = 2 Finish = 1/2

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 266 8/15/2014 11:57:26 AM Appendix B: Classical Doppler Effect 267

Table B-3 reveals some interesting relationships:

1) Each frequency is the reciprocal of the corresponding period. 2) Each period is the reciprocal of the corresponding frequency. 3) The frequencies from start C to finish C1 form an ascending sequence from 1 to 2. 4) The periods from start C to finish C1 form a descending sequence from 1 to 1/2. 5) In frequency, C is the start, F is the harmonic mean, G is the arithmetic mean, and C1 is the finish. 6) In period, C is the start, F is the arithmetic mean, G is the harmonic mean, and C1 is the finish. 7) In frequency, the rate is:

(finish− start) rs−−21 1 v = ===. (finish+ start) rs + 2+ 1 3

8) In period, the rate is: 1 − (finish− start) rs− 1 1 v = = =2 =− . + + 1 (finish start) rs +1 3 2

9) In frequency, the musical interval from C to G is the perfect fifth 3/2, and the musical interval from G to C1 is the perfect fourth 2/(3/2) = 4/3. 10) In period, the musical interval from C to G is the inverse perfect fifth 2/3, and the musical interval from G to C1 is the inverse perfect fourth (1/2)/(2/3) = 3/4. 11) In frequency, the musical interval from C to F is the perfect fourth 4/3, and the musical interval from F to C1 is the perfect fifth 2/(4/3) = 3/2. 12) In period, the musical interval from C to F is the inverse perfect fourth 3/4, and the musical interval from F to C1 is the inverse perfect fifth (1/2)/(3/4) = 2/3.

An important thing to remember is that in frequency, note F is the harmonic mean and G is the arithmetic mean, whereas in period, they are the opposite.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 267 8/15/2014 11:57:26 AM 268 Remote Sensing in Action

The one-way Doppler effect

Imagine a commutation channel with a source (otherwise known as sender or transmitter) at one end and an observer (otherwise known as receiver or listener) at the other end. The source transmits a sinusoidal wave of period s. This wave travels at wave velocity c. We will require the wave velocity to be a positive number. For example, if the signal is sound, then the wave velocity (under given conditions) would be c = 331 m/s.

Either the source and/or the observer may be moving. We will let Vs

denote the velocity of the source and Vo denote the velocity of the observ-

er. The source velocity Vs can be positive or negative or zero. Likewise, the observer velocity V can be positive or negative or zero. However, the o magnitude of each must be less than the wave velocity. Various conventions are used. Our convention is that the velocity of an object is positive when the object is moving away from the other object, and the velocity is negative when the object is moving toward the other object. In other words, our convention is:

• Vs is positive if the source is moving away from the observer.

• Vo is positive if the observer is moving away from the source.

• Vs is negative if the source is moving toward the observer.

• Vo is negative if the observer is moving toward the source.

The observer receives the sinusoidal wave that was transmitted by the source. If both the source and the observer are stationary or if both are traveling at the same velocity, then the observed wave has the same period as that of the transmitted wave. Otherwise, the observed wave has a different period from that of the transmitted wave. Two cases will be analyzed using the basic equation

Wavelength = wave velocity × period.

Case 1: A stationary source (Vs = 0) and a moving observer (Vo ≠ 0) The stationary source transmits a sinusoidal wave of period s, and the moving observer receives a sinusoidal wave of period t. Because the observer is moving, we must account for the distance that the observer

moves in time period t. In essence, the situation is that given s, c, Vo, we want to find t. We can do so by using the following:

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 268 8/15/2014 11:57:26 AM Appendix B: Classical Doppler Effect 269

A) The distance that the wave travels during time period s is the wave-

length given by λs = cs. B) The distance that the wave travels during time period t is the wavelength given by λt = ct. C) The distance that the observer travels during time period t is the velocity V of the observer times the period t, or V t. o o For Case 1, the fundamental equation (B) = (A) + (C) becomes

Case 1: λt = λs + Vot.

Case 2: A moving source (Vs ≠ 0) and a stationary observer (Vo = 0) Again, the moving source transmits a sinusoidal wave of period s, and the stationary observer receives a sinusoidal wave of period q. Because the source is moving, we must account for the distance that the source moves in time period s. Given s, c, V, we want to find q. In this case, the pertinent information is:

A) The distance that the wave travels during time period s is the wavelength λs = cs. B) The distance that the wave travels during time period q is the wave-

length λq = cq. C) The distance that the source travels during time period s is the velocity V of the source period s, that is, V s. s × s So for Case 2, the fundamental equation (B) = (A) + (C) becomes

Case 2: λq = λs + Vs s.

Note that in Case 1 (stationary source and moving observer), we add-

ed distance Vot that the observer travels in time period t, whereas in Case

2 (moving source and stationary observer), we added distance Vss that the source travels in time period s. The equations for the two cases are

Case 1: ct = cs + Vot.

Case 2: cq = cs +Vss. The two Mach velocities (or Pythagorean rates) are

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 269 8/15/2014 11:57:27 AM 270 Remote Sensing in Action

vo = Vo/c and vs = Vs/c.

Dividing the equations by c produces

Case 1: t = s +vot.

Case 2: q = s + vss. Solving for t and q, respectively, produces

Case 1: t – vot = s, or t = s/(1 – vo).

Case 2: q = s + vss, or q = s(1 + vs). The two Doppler factors are therefore

tq1 s+ vss kk12= =; = = =+1. vs sv1− o ss The above equations are expressed more compactly as

Case 1: t = k1s.

Case 2: q = k2s. The first Doppler factor refers to Case 1 (stationary source and moving observer) and the second to Case 2 (moving source and stationary observer).

Frequencies are defined as fs = 1/s, ft = 1/t, fq = 1/q, so in terms of frequencies, these equations are

11 1 Case1: ft== = ffss =()1 − v o . t ks11 k

11 1 1 Case 2 : fq== =ffss = . q ks22 k1+ vs

This provides what is needed to address the more complicated Case 3.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 270 8/15/2014 11:57:27 AM Appendix B: Classical Doppler Effect 271

Case 3: A moving source (Vs ≠ 0) and a moving observer (Vo ≠ 0) The answer is the combination of Case 1 and Case 2, given by

1−−voo1/− ()Vco cV  Case 3: ffos== fs =fs  1++vss 1/() V c cV + s 

If V , V are positive, the observer is moving away from the source, and s o the source is moving away from the observer. From the above equation, we find that the observed frequency f is less than the source frequency f . o s If the objects approach rather than recede, then negative numbers are used for V , V , and the above equation reveals that the observed frequency f is s o o greater than the source frequency fs. Why have we made the seeming detour into the mathematics of music and the Pythagorean means? We have done so because the Pythago- rean means incorporate the entire theory of the Doppler effect. This seems amazing at first, but in fact, it is easily demonstrated. The Pythagorean extremes are s and r, the harmonic mean is q, and the arithmetic mean is t. As shown earlier, sr = qt. The factors k and k 1 2 can be defined as rt k ==; 1 qs rq k ==. 2 ts Because the distance is x = t – s = r – t and the rate is v = x/t, the two factors can be expressed in terms involving the rate v:

tt 11 k == = = ; 1 s t−− x1/1() xt − v r tx+ k2 = = =+1()xt / =+ 1. v tt

The two factors just defined are indeed the two Doppler factors.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 271 8/15/2014 11:57:27 AM 272 Remote Sensing in Action

The two-way Doppler effect

Reflection is the key feature of active remote sensing. In Shake- speare’s words, “And since you know you cannot see yourself so well as by reflection, I, your glass, will modestly discover to yourself that of yourself which you yet know not of.” In active remote sensing, the observer both transmits and receives. There are two platforms. The observer is on one, and the mirror is on the other. The observer transmits the signal to the mirror, and the reflection returns to the observer, necessitating calculation of the two-way Doppler effect. Assume one platform is stationary and the other is moving with velocity V. V is positive if the moving platform is going away from the stationary platform and negative if the moving platform is coming toward the stationary platform. As before, the rate is v = V/c, the quotient of platform velocity divided by signal speed. There are two possible situations:

Case 1: The observer is on the fixed platform, and the target is the moving platform. Case 2: The observer is on the moving platform, and the target is the fixed platform.

In Case 1, the signal is transmitted (with period s) from a fixed platform and is incident (with period t) on the moving platform. The Doppler

factor k1 (which is for “fixed to moving”) applies, meaning t = k1/s. The signal is then reflected (with period t) from the moving platform and is re-

ceived (with period r) at the fixed platform. The Doppler factor k2 (which

is for “moving to fixed”) applies, meaning r = k2t. The sequence s, t, r is a Pythagorean sequence made up of the arithmetic mean flanked by the two extremes. In Case 2, the signal is transmitted (with period s) from the moving platform and is received (with period q) at the fixed platform. The Dop-

pler factor k2 (which is for “moving to fixed”) applies, meaning q = k2s. The signal is then reflected (with period q) from the fixed platform and

is received (with period r) at the moving platform. The Doppler factor k1

(which is for “fixed to moving”) applies, meaning r = k1q. The sequence s, q, r is a Pythagorean sequence made up of the harmonic mean flanked by the two extremes. The first Doppler factor is the arithmetic mean t divided by the source period s. The second Doppler factor is the harmonic mean q divided by the

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 272 8/15/2014 11:57:27 AM Appendix B: Classical Doppler Effect 273

source period. In effect, the first Doppler factor is the normalized arithmetic mean, and the second Doppler factor is the normalized harmonic mean. In other words, the Doppler factors are none other than normalized Pythagorean means. This connection between Doppler and Pythagoras has not been made before.

The need for two classical Doppler factors

Why are there two Doppler factors? Why is one not enough? Back to the ancients, in this case Zeno of Elea (c. 490–c. 430 b.c.), a pre-Socratic Greek philosopher best known for his paradoxes. Zeno’s most famous paradox states that a quicker runner can never overtake a slower one. Achilles is faster, and we take his speed as unity, say one mile per minute. Tortoise is slower; he runs at one-half the speed of Achilles, or v = 0.5. Mount Olympus represents the fixed platform. Tortoise represents the moving object. Achilles represents the signal. At time zero, both Achilles and Tortoise are at Mount Olympus. Tor- toise immediately starts to run. Let s = 1 be the head-start time, in which Tortoise runs one-half mile while Achilles waits. At the end of the head- start time, Achilles starts to run to catch Tortoise, who has a head start of one-half mile. While Achilles runs this half mile, Tortoise runs one-fourth mile and is still ahead. While Achilles runs one-fourth mile, Tortoise runs another one-eighth mile. While Achilles runs one-eighth mile, Tortoise runs one-sixteenth mile. This continues ad infinitum. By this reasoning, Achilles can never catch Tortoise. One of the most important types of infinite series is the geometric series. A geometric series is one in which there is a constant ratio v between each element and the one preceding it. Fortunately, a geometric series is the easiest type of series to analyze. We not only can determine its conver- gence and divergence easily, but we also can find a closed expression for it. The behavior of the terms depends on the common ratio v. If v is greater than negative one and less than one, then the individual terms of the series become smaller and smaller, and the series converges to a finite number. If v is greater than or equal to one, or is less than or equal to negative one, then the series diverges. The rule is: The geometric series a + av + av2 + av3 + av4 + ... converges to a/(1 – v) if –1 < v < 1 and diverges otherwise. Consider Table B-4. We recall that v = 0.5. If infinitely more steps are added, the partial sums in Table B-4 become the geometric series

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 273 8/15/2014 11:57:27 AM 274 Remote Sensing in Action

distance = x = v + v2 + v3 + v4 + … = v(1 + v + v2 + v3 + …) = v/(1 –v); time = t = 1 + v + v2 + v3 + v4 + … = 1/(1 – v),

which are

distance = x = 1/2 + 1/4 + 1/8 + 1/16 + … = 0.5/(1 – 0.5) = 1;

time = t = 1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 1/(1 – 0.5) = 2.

This result states that Achilles will catch Tortoise at a distance of one mile and at a time of two minutes. If we divide distance by time, we obtain the velocity v = x/t = 1/2, as we would expect. We now want to define the first classical Doppler factor. It refers to a signal sent from a fixed platform to a moving object. The signal sent from the fixed platform has period s, and the signal received at the moving object has period t. By definition, the velocity of the signal is unity. The velocity of the moving object is v. For the signal to catch the moving object, the magnitude of v must be less than unity. Because Achilles travels at the speed of unity, the distance he covers is the same as the time he runs. In other words, distance x is equal to the difference of received period t and head-start period s, that is, x = t – s. The velocity is the quotient of distance over time, that is, x = v/t. The first classical Doppler factor is defined as the quotient of the finishing time t, when Achilles catches up with Tortoise, divided by Achilles’ starting time s. In other words, the first classical Doppler factor is

time period received at moving object tt 1 1 k = == = = . 1 −−x time periodsent byfixed platform s tx1− 1 v t

Table B-4. Partial sums of distances and times.

Step Partial sum of distances Partial sum of times 0 0 1 = 1 1 1/2 = 0.5 1 + 1/2 = 1.5 2 1/2 + 1/4 = 0.75 1 + 1/2 + 1/4 = 1.75 3 1/2 + 1/4 + 1/8 = 0.875 1 + 1/2 + 1/4 + 1/8 = 1.875

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 274 8/15/2014 11:57:27 AM Appendix B: Classical Doppler Effect 275

Thus, k1 = t/s = 2/1 = 2.

We now add another run. As soon as Achilles catches up with Tor- toise, Achilles turns around and runs back to Olympus. Now Achilles runs from a moving object (Tortoise) to a fixed object (Olympus). The second classical Doppler factor is defined as the quotient of the starting time t of Achilles divided by the finishing time r when Achilles reaches Olympus. In other words, the second classical Doppler factor is

time period received at fixed platform r tx+ x kv= = = =+1 =+ 1. 2 time period sent by moving object tt t

Thus, the second Doppler factor is k2 = r/t = 3/2 = 1.5.

In summary, the first Doppler factor is used for a signal sent from a fixed source to a moving receiver (that is, for catching a flying duck), and the second Doppler factor is used for a signal sent from a moving source to a fixed receiver (that is, for catching a sitting duck). The reason for two Doppler factors is that Zeno knew it is easier to catch a sitting duck than a flying one.

The ball game

The following example of how this mathematics can be used is called the ball game. It does not involve any wave motion at all. All that is needed are two ballplayers and a person with a stopwatch. One ballplayer will stand still, and the other will run. The ball is the signal, the runner is the moving object, and the fixed player is the stationary object. The speed of the ball is one, and the velocity of the runner is v. Because the Pythagorean rate is the quotient of velocity of runner divided by speed of signal, v is actually the Pythagorean rate. Our convention is that the rate is positive if the runner is going away from the fixed player, and the rate is negative if the runner is moving toward the fixed player. Both players are at the same location at time zero. There are two cases.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 275 8/15/2014 11:57:27 AM 276 Remote Sensing in Action

Case 1 (when rate v is positive): The ball goes from the fixed player to the runner and back.

Outbound passage

At time zero, the runner starts to move away from the fixed player. The fixed player waits until time s and then throws the ball to the runner, who catches it at time t. Question: How far does the ball travel? Answer: The ball is in the air for the elapsed time of t – s. The distance is speed multiplied by elapsed time. Because the speed of the ball is one, it follows that the distance is t – s. Question: How far is the runner from the fixed player when he catches the ball? Answer: The runner has been in motion since time zero and catches the ball at time t. The elapsed time is t. The distance is velocity multiplied by elapsed time. Because the velocity of the runner is v, it follows that the distance is vt. Conclusion: The two distances must be equal: t – s = vt. Thus,

s t −=vt sor t = . 1− v Return passage At time t, the runner throws the ball back to the fixed player, who catches it at time r. Question: How far has the ball traveled? Answer: The ball is in the air for the elapsed time of r – t. The distance is speed multiplied by elapsed time. Because the speed of the ball is one, it follows that the distance is r – t. Question: How far is the runner from the fixed player when he throws the ball? Answer: The runner has been in motion since time zero and throws the ball at time t. The elapsed time is t. The distance is velocity multiplied by elapsed time. Because the velocity of the runner is v, it follows that the distance is vt. Conclusion: The two distances must be equal: r – t = vt. Thus, r = vt + t, or r = t(1 + v).

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 276 8/15/2014 11:57:27 AM Appendix B: Classical Doppler Effect 277

Case 2 (when rate v is positive): The ball goes from the runner to the fixed player and back.

Outbound passage

Both the fixed player and the runner are at the same location at time zero. At time zero, the runner starts to move away from the fixed player. The runner waits until time s and then throws the ball to the fixed player, who catches it at time q. Question: How far has the ball traveled? Answer: The ball is in the air for the elapsed time of q – s. The distance is speed multiplied by elapsed time. Because the speed of the ball is one, it follows that distance is q – s. Question: How far is the runner from the fixed player when he throws the ball? Answer: The runner has been moving since time zero and throws the ball at time s. The elapsed time is s. The distance is velocity multiplied by elapsed time. Because the velocity of the runner is v, it follows that distance is vs. Conclusion: The two distances must be equal: q – s = vs. Thus, q = s + vs, or q = s(1 + v).

Return passage

At time q, the fixed player throws the ball to the runner, who catches it at time r. Question: How far has the ball traveled? Answer: The ball is in the air for the elapsed time of r – q. The distance is speed multiplied by elapsed time. Because the speed of the ball is one, it follows that distance is r – q. Question: How far is the runner from the fixed player when he catches the ball? Answer: The runner has been running since time zero and catches the ball at time r. The elapsed time is r. The distance is velocity multiplied by elapsed time. Because the velocity of the runner is v, it follows that distance is vr. Conclusion: The two distances must be equal: r – q = vr. Thus,

q r−= vr q, or r = . 1− v

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 277 8/15/2014 11:57:27 AM 278 Remote Sensing in Action

Let us now do Case 1 and Case 2 when rate v is negative. We will use countdown when rate v is negative: … , 5, 4, 3, 2, 1, players meet. We have just derived the equations for Case 1 and Case 2 when rate v is positive. If we use countdown, the same equations apply when rate v is negative. The fixed player and the runner meet at the same location at countdown time zero, which can be demonstrated by a numerical example. Case 1 (when rate v is positive): Let s = 2, v = 1/2. We will count up to r. Outbound passage

s 22 t = = ==4. 1−−v 1() 1/2 1/2 Return passage r = t(1 + v) = 4[1 + (1/2)] = 4(3/2) = 6.

Case 1 (when rate v is negative): Let s = 6, v = –1/2. We will count down to r.

Outbound passage

s 66 t == ==4. 1−v 1 −−() 1/2 3/2 Return passage

r = t(1 + v) = 4[1 + (–1/2)] = 4(1/2) = 2.

Case 2 (when rate v is positive): Let s = 2, v = 1/2. We will count up to r. Outbound passage

q = s(1 + v) = 2[1 + (1/2)] = 2(3/2) = 3. Return passage

q 33 r = = ==6. 1−−v 1() 1/2 1/2

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 278 8/15/2014 11:57:27 AM Appendix B: Classical Doppler Effect 279

Case 2 (when rate v is negative): Let s = 6, v = –1/2. We will count down to r. Outbound passage q = s(1 + v) = 6[1 + (–1/2)] = 6(1/2) = 3. Return passage

q 33 r == = =2. 1−v 1 −−() 1/2() 3/2

The following section makes use of the periods of waves. The equations in that section are the same as the equations given in this section for a ball game.

Two-way Doppler effect in terms of periods

Because we want to deal with physical velocities, we will use periods and not frequencies. Thus, we will have a time axis and a distance axis. The signal will be a sound wave. In Mach units, the signal has a velocity equal to one, meaning a sound wave travels a distance of one unit in one unit of time (usually taken to be one second). The signal that we will use is the period of the sound wave. A two-way signal travels from one object to another object, where it is reflected and returns to the original object. The period from the original object is denoted by s, and the period to return to the original object is r. These two numbers, the extremes, can be measured directly on the original object, so they represent the known quantities. Figure B-3 graphically depicts the case of positive velocity (Figure B-3a) and the case of negative velocity (Figure B-3b). One object is moving, and the other is stationary (i.e., fixed in place). The velocity of the moving object is positive when the distance between the two objects increases with time and is negative when the distance decreases with time. For an ascending sequence (r is greater than s), the velocity is positive. On the other hand, for a descending sequence (r is less than s), the velocity is negative. Figure B-4 graphically depicts the two-way Doppler effect from stationary to moving and back to stationary for positive velocity (Figure B-4a) and negative velocity (Figure B-4b). The signal is sent from and

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 279 8/15/2014 11:57:27 AM 280 Remote Sensing in Action

returned to a fixed object. Figure B-4a depicts the ascending case. On the horizontal axis, the two extremes, s = 2 and r = 6, are plotted in ascending order. A vertical line is plotted at the arithmetic mean t = 4. The height of this vertical line is equal to the gain, which is x = 4 – 2 = 6 – 4 = 2. There is a slanting line from the origin to the top of the vertical line, whose slope is the velocity v = x/t, which is 2/4 = 1/2. On the horizontal axis in Figure B-4b, the two extremes, s = 6 and r = 2, are plotted in descending order, and a vertical line is again plotted at the arithmetic mean t = 4. The height of this vertical line is the gain, which is x = 4 – 6 = 2 – 4 = –2. There is a slanting line from the origin to the bottom of the vertical line, whose velocity is v = x/t, which is –2/4 = –1/2. On each platform, the signal goes from s to t and then back again (by reflection) to r. Both s and r are measured on the fixed object, and t is measured on the moving object. In Figure B-4a, a wave of period 2 was sent, and a wave of period 6 was returned. Because the period increases, the moving object is going away from the stationary object. This condition is expressed by the positive value of the velocity, that is, v = 1/2.

a) 3 distance axis

1 Distance axis Distance Movingmoving object object with with velocity velocity = = v v = = 1/2. 1/2

Stationarystationary object object with with velocity velocity 0 0 Timetime axis 0 1 2 3 4 5 6

Stationarystationary object object with with velocity velocity 0 0 time Time axis axis b) 0 1 2 3 4 5 6 Moving object with velocity = v = –1/2. moving object with velocity = v = -1/2

–1

–2 Distance axis Distance

–3 distance axis

Figure B-3. The case of (a) positive velocity and (b) negative velocity.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 280 8/15/2014 11:57:28 AM Appendix B: Classical Doppler Effect 281

In Figure B-4b, a wave of period 6 was sent, and a wave of period 2 was returned. Because the period decreases, the moving object is going toward the stationary object. This condition is expressed by the negative value of the velocity, that is, v = –1/2. As established earlier, the two Doppler factors (as shown in Figure B-4) are:

For fixed source s to moving object t: k1 = t/s.

For moving source t to fixed object r: k2 = r/t.

If both objects were at rest or if both objects were moving with the same velocity, the period would never change, i.e., r = t = s. In the case of a fixed source s and moving object t, in time period t, the moving object shifts distance vt. Thus the period t is equal to the sent period s plus time vt because of the movement of the receiver; that is, t = s + vt. Thus,

a) 3

1 k1 = t/s = 4/2 = 2/1 k2 = r/t = 6/4 = 3/2 Moving object with velocity = v = x/5 = 1/2.

0 s = 2 t = 4 r = 6

r = 2 t = 4 s = 6 b) 0

Moving object with velocity = v = x/t = –1/2.

k2 = r/t = 2/4 = 1/2 –1 k1 = t/s = 4/6 = 2/3

–2

–3

Figure B-4. Two-way Doppler effect from stationary to moving and back to stationary for (a) positive velocity and (b) negative velocity.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 281 8/15/2014 11:57:28 AM 282 Remote Sensing in Action

t 1 t()1− vs = , so k1 == . sv1− With the case of a moving object t to fixed object r, in time period t, the moving object shifts distance r. Thus period r is equal to period t plus time vt because of the movement of the sender; that is, r = t + vt. Thus,

r rt=()1 + v , so k2 = =+ 1 v . t For the ascending sequence (s, t, r) = (2, 4, 6), the two Doppler factors are

tr4 63 kk===2; ===. 12st2 42 For the ascending sequence, the velocity is v = (r – s)/(r + s) = (6 – 2)/(6 + 2) = 1/2. In terms of velocity, these two Doppler factors are

1 1 13 k= = =2;kv =+ 1 =+ 1 = . 121−−v 1 0.5 2 2 In the descending sequence (s, t, r) = (6, 4, 2), the two Doppler factors are

tr42 21 kk===;. === 12st63 42 For the descending sequence, velocity v = (r – s)/(r + s) = (2 – 6)/(2 + 6) = –1/2. In terms of velocity, the two Doppler factors are

1 12 1 k12= = =;kv =+ 1 =+− 1() 0.5 = . 1−v 1 −−() 0.5 3 2

Next is the case in which the sending signal and the receiving signal are both measured on the moving object. Figure B-5 graphically depicts the two-way Doppler effect from moving to stationary and back to moving for positive velocity (Figure B-5a) and negative velocity (Figure B-5b). On each diagram, the signal goes from s to q and then back again (by reflection) to r. Both s and r are mea-

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 282 8/15/2014 11:57:29 AM Appendix B: Classical Doppler Effect 283

sured on the moving object, whereas q is measured on the fixed object. The two Doppler factors are:

For moving source s to fixed object q: k2 = q/s.

For fixed object q to moving source r: k1 = r/q.

Now it is time to use the fundamental result of Pythagoras, namely, the product of the means is equal to the product of the extremes: qt = sr. Thus,

k2 = q/s = r/t.

k1 = r/q = t/s.

a) 3

k1 = r/q = 6/3 = 2/1 1

Moving object with kvelocity2 = q/s = = 3/2 v = x/t = 1/2.

0 s = 2 q = 3 r = 6

r = 2 q = 3 s = 6 b) 0

Moving object with kvelocity1 = r/q = = 2/3 v = x/t = –1/2.

–1 k2 = q/s = 3/6 = 1/2

–2

–3 Figure B-5. Two-way Doppler effect from moving to stationary and back to moving for (a) positive velocity and (b) negative velocity.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 283 8/15/2014 11:57:29 AM 284 Remote Sensing in Action

For the ascending sequence (s, q, r) = (2, 3, 6), the two Doppler factors are

For the ascending sequence, velocity v = (6 – 2)/(6 + 2) = 1/2. In terms of velocity, these two Doppler factors are

For the descending sequence (s, q, r) = (6, 3, 2), the two Doppler factors are

For the descending sequence, velocity v = (2 – 6)/(2 + 6) = –1/2. In terms of velocity, the two Doppler factors are

Examples of one-way Doppler effect

We now have all the material we need to explain the classical Doppler effect in the one-way case. We have two observations — the signal sent and the signal returned.

1) If the source is fixed and the receiver is moving, denote the signal sent

by s and the signal received by t. The Doppler factor is then k1 = t/s. Alternatively, denote the signal sent by q and the signal received by r.

The Doppler factor is then k1 = r/q. 2) If the source is moving and the receiver is fixed, denote the signal sent

by t and the signal received by r. The Doppler factor is then k2 = r/t. Alternatively, denote the signal sent by s and the signal received by q.

The Doppler factor is then k2 = q/s.

What does this have to do with Pythagoras? The Doppler factor k1 is the arithmetic mean t divided by the sending period s. The Doppler factor

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 284 8/15/2014 11:57:30 AM Appendix B: Classical Doppler Effect 285

k2 is the harmonic mean q divided by the sending period s. The divisor s acts as a normalization factor. We can conclude that the first Doppler factor is the normalized arithmetic mean of Pythagoras, and the second Doppler factor is the normalized harmonic mean of Pythagoras. In effect, the classical Doppler effect is fully incorporated within the mathematics of Pythagoras.

Let Tsource be the period of the source and Treceiver be the period of the receiver. Then the two Doppler factors can be written as:

From fixed source to moving receiver:

from moving source to fixed receiver:

where vreceiver is the velocity of the moving receiver, and vsource is the velocity of the moving source. These two Doppler factors can be used in two ways. (1) If we know both time periods, we can compute the velocity. (2) If we know one time period and the velocity, we can compute the other time period. Following are four examples of the use of Method 1.

1) Question: Suppose an observer on the fixed platform sends a signal of period 6 to the moving vehicle. The observer on the moving vehicle receives a signal of period 9. What is the Doppler factor, and what is the velocity? Answer: Because the signal goes from fixed source to moving re-

ceiver, we use k1. Tsource = 6, and Treceiver = 9. The Doppler factor k1 =

9/6 = 3/2 = 1/(1 – vreceiver). Solving this equation gives 1/3 as the velocity of the receiver.

2) Question: Suppose an observer on the moving vehicle sends a signal of period 9 to the fixed platform. The observer on the fixed platform receives a signal of period 12. Answer: Because the signal goes from moving source to fixed re-

ceiver, we use k2. Tsource = 9, and Treceiver = 12. The Doppler factor k2 =

12/9 = 4/3 = 1 + vsource. Solving this equation gives 1/3 as the velocity of the source.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 285 8/15/2014 11:57:30 AM 286 Remote Sensing in Action

3) Question: Suppose an observer on the fixed platform sends a signal of period 12 to the moving vehicle. The observer on the moving vehicle receives a signal of period 9. What is the Doppler factor, and what is the velocity? Answer: Because the signal goes from fixed source to moving

receiver, we use k1. Tsource = 12, and Treceiver = 9. The Doppler factor

k1 = 9/12 = 3/4 = 1/(1– vreceiver). Solving this equation gives –1/3 as the velocity of the receiver.

4) Question: Suppose an observer on the moving vehicle sends a signal of period 9 to the fixed platform. The observer on the fixed platform receives a signal of period 6. Answer: Because the signal goes from moving source to fixed

receiver, we use k2. Tsource = 9, and Treceiver = 6. The Doppler factor

k2 = 6/9 = 2/3 = 1 + vsource. Solving this equation gives –1/3 as the velocity of the source.

Doppler effect in terms of frequencies

Because electrical engineers deal with frequencies, the above results need to be transformed from periods to frequencies. This is straightfor- ward because frequency is the reciprocal of the period:

fs = 1/s, ft = 1/t, fq = 1/q.

The two basic equations are:

s Fixed object s to moving object t: t== sk . 1 1− v

Moving object s to fixed object q: q = sk2 = s(1 + v).

The reciprocals of these equations are: 1 11 1 Fixed object fs to moving object ft: = =−()1v , or t sk1 s 1 ffts= =− f s()1. v k1

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 286 8/15/2014 11:57:30 AM Appendix B: Classical Doppler Effect 287

1 11 1 1 == , or q sk s1+ v Moving object f to fixed object f : 2 s q 11 ffqs== f s . kv2 1+

Recall that the velocity is positive if the objects are moving apart and is negative if the objects are moving together.

1) A moving ambulance with its siren emitting at a frequency of 300 Hz moves with a speed of 20 m/s toward a stationary pedestrian. This case is that of moving object to fixed object. Take the speed of sound in air to be 343 m/s. What is the frequency of the sound heard by the pedestrian? First, the actual speed of the ambulance must be transformed to the speed normalized by the signal speed. Because the objects are moving together, the velocity is negative. Thus, the known velocity is v = –20/343 = –0.0583. This is a case of passive remote sensing, in which the goal is to find the received signal. The signal is emitted by a moving body (ambulance) and is received by a fixed object (pedestrian). Denote the

known emitted signal by fs = 300 and the unknown received signal by

fq. The known velocity is negative, so the received signal fq is greater

than the emitted signal fs. The received signal is computed as

2) Calculate the frequency of the sound heard by the pedestrian after the ambulance has passed and is receding in the distance. The known emitted signal is s = 300, and the unknown received signal is denoted by q. The known velocity v = 0.0583 is positive, so the received signal

fq is less than the emitted signal fs. The received signal is computed as

3) The ambulance is stationary and emits a sound of frequency 300 Hz. A driver steers his car toward the ambulance at a speed of 20 m/s. What is the frequency of the sound heard by the driver? This is again a case of passive remote sensing, in which the goal is to find the re-

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 287 8/15/2014 11:57:31 AM 288 Remote Sensing in Action

ceived signal. The signal is emitted by a fixed body (ambulance) and

is received by a moving object (driver). The known emitted signal fs

= 300, and the unknown received signal is denoted by ft. The known

velocity v = –0.0583 is negative, so the received signal ft is greater

than the emitted signal fs. The received signal is computed as

ft = fs (1 – v) = 300(1 + 0.0583) = 317.49 Hz.

4) What is the frequency of the siren as heard by the driver after the driver passes the ambulance and recedes from it? The known emitted

signal fs = 300, and the unknown received signal is denoted by ft. The known velocity v = 0.0583 is positive, so the received signal f is less t than the emitted signal fs. The received signal is computed as

ft = fs(1 – v) = 300(1 – 0.0583) = 282.51 Hz.

5) In a concluding train example, suppose that observer A is on the platform of a railroad station and that observer B is on a train moving at one-third the velocity of sound. Suppose that a horn on the train is sounding the musical note C, and a horn on the platform is also sounding the same note. Each observer hears both horns. As the train approaches the station, observer A hears the musical interval of a perfect fifth, whereas observer B hears a perfect fourth. The train passes the platform, all the while maintaining its speed. As the train recedes from the station, observer A hears a perfect fourth, and observer B hears a perfect fifth.

Graphical depiction of the classical Doppler effect

The Doppler effect is the most powerful tool that the modern scientist has at his disposal in remote sensing. Pythagoras discovered this effect for mechanical waves in the form of the musical interval and called it the humming of the strings. Pythagoras knew that the shortening of a string changed its pitch, or frequency, and he knew how to play octaves, perfect fifths, perfect fourths, and other intervals. The octave is the frequency interval with ratio of 2/1; the perfect fifth, 3/2; and the perfect fourth, 4/3. This section will show the graphical depiction of the classical Doppler effect. A source emits a mechanical signal of period s. A receiver would

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 288 8/15/2014 11:57:31 AM Appendix B: Classical Doppler Effect 289

detect the same signal whether both source and receiver are fixed in place or both are moving at the same velocity. We again consider two cases:

Case 1: The source is fixed, and the receiver is moving. Case 2: The source is moving, and the receiver is fixed.

The Mach velocity of sound is one. We require that the moving object has a constant Mach velocity v which is less than one. Our convention is that the Mach velocity is negative when the two objects are moving closer together and is positive when they are moving farther apart. There are two cases, each of which has two subcases. Figure B-6 graphically depicts Case 1, fixed source (left side of diagram) and moving receiver (right side of diagram). Figure B-5a shows the wavefronts as emitted by the source. The two subcases are: Subcase 1a: The Mach velocity is negative, so the receiver is moving closer to the fixed source. The left side of the right diagram shows that on the negative-velocity side, the wavefronts arriving at the moving receiver are crowded together. In other words, the moving receiver perceives a decrease in period, or an increase in frequency, of the signal emitted by the fixed source. Subcase 1b: The Mach velocity is positive, so the receiver is moving away from the fixed source. The right side of the right diagram shows that on the positive-velocity side, the wavefronts arriving at the moving receiver are spread out. In other words, the moving receiver perceives an increase in period, or a decrease in frequency, of the signal emitted by the fixed source. Figure B-7 graphically depicts the mathematical explanation of Case 1, fixed source and moving receiver. Because periods are intrinsically positive, the numbers on the negative-velocity side are positive, as are those on the positive-velocity side. The dashed circle represents the wavefront emitted by the fixed source. The emitted wavefront has the same value, s = 1, on both sides. The solid circle represents the wavefront detected by the moving receiver. This wavefront has value t = 0.6 on the negative-velocity side and t = 3 on the positive-velocity side. The reason is that value s on the negative-velocity side is reduced by quantity vt = –0.4 because of the crowding, whereas value s on the positive-velocity side is augmented by quantity vt = 2 because of the spreading. The equations are:

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 289 8/15/2014 11:57:31 AM 290 Remote Sensing in Action

Figure B-6. Case 1, fixed source (left diagram) and moving receiver (right diagram).

v = 2/3 = 0.67 v = –2/3 = –0.67

–1 –0.5 0 0.5

s = 1 s = 1 vt = 2

vt = –0.4 t = s + vt t = s + vt = 3

Figure B-7. Mathematical explanation of Case 1, fixed source and moving receiver.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 290 8/15/2014 11:57:32 AM Appendix B: Classical Doppler Effect 291

Figure B-8. Case 2, moving source (left diagram) and fixed receiver (right diagram).

Figure B-8 graphically depicts Case 2, moving source (left) and fixed receiver (right). The left diagram shows the wavefronts as emitted by the source. The two subcases are: Subcase 2a: The Mach velocity is negative, so the source is moving closer to the fixed receiver. The left side of the right diagram shows that on the negative-velocity side, the wavefronts arriving at the fixed receiver are crowded together. In other words, the fixed receiver perceives a decrease in period, or an increase in frequency, of the signal emitted by the moving source. Subcase 2b: The Mach velocity is positive, so the source is moving away from the fixed receiver. The right side of the right diagram shows that on the positive-velocity side, the wavefronts arriving at the fixed receiver are spread out. In other words, the fixed receiver perceives an increase in period, or a decrease in frequency, of the signal emitted by the fixed source. Figure B-9 graphically depicts the mathematical explanation of Case 2, moving source and fixed receiver. Because periods are intrinsically positive, the numbers on the negative-velocity side are positive, as are those on the positive-velocity side. The dashed circle represents the wave-

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 291 8/15/2014 11:57:32 AM 292 Remote Sensing in Action

v = –2/3 = –0.67 v = 2/3 = 0.67

–1 –0.5

s = 1 s = 1 vs = 0.67 VS = 0.67

vs = –0.67 q = s + vs = 1 + 0.67 = 1.67 q = s + vs

= 0.33 Figure B-9. Mathematical explanation of Case 2, moving source and fixed receiver.

front emitted by the moving source. The emitted wavefront has the same value, s = 1, on both sides. The solid circle represents the wavefront detected by the fixed receiver. This wavefront has the value q = 0.33 on the negative-velocity side and q = 1.67 on the positive-velocity side. The reason is that value s on the negative-velocity side is reduced by quantity vs = 0.67 because of the crowding, whereas value s on the positive-velocity side is augmented by quantity vs = 0.67 because of the spreading. The equations are:

Given s = 1, v = ± 2 / 3. 0.33 ifv =− 2 / 3 q = s + vs, so q = s(1 + v) = . 1.67 ifv = 2 / 3

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

0510_01_AppenB_v4--SCS.indd 292 8/15/2014 11:57:33 AM Appendix C

Relativistic Doppler effect

Mathematics

The relativistic Doppler effect uses electromagnetic waves such as light waves. The relativistic Doppler factor from a fixed source to a moving receiver is essentially the geometric mean, whereas the relativistic Doppler factor from a moving source to a fixed receiver is also the geometric mean. The fact that only the geometric mean is required represents a fundamental symmetry in the relativistic Doppler effect. In particular, Einstein based his theory of special relativity on this symmetry. The mathematics of special relativity can be expressed in terms of the geometric mean of Pythagoras and the stereographic projection of Hipparchus, Hypatia, and François d’Aguilon. “Beauty is truth, truth beauty, that is all ye know on earth, and all ye need to know.” This line by John Keats reveals the essence of mathematics. At the foot of the Spanish steps in Rome, a fountain, Fontana della Barcaccia (“Fountain of the Old Boat”), depicts a sinking boat half full of water. In 1598, the Tiber River overflowed and inundated the lower section of Rome. When the water subsided, a boat was stranded on the spot where the fountain now stands. This inspired Bernini to conceive of the fountain as a commemoration of the great flood. John Keats, born in England in 1795, poor and soon orphaned, strug- gled against poverty, disease, and the cruel persecution of the English crit- ics who were oblivious to the beauty and imagination shown in his poems. He took care of his sick brother Tom, who died of tuberculosis in 1818. By 1820, John Keats showed increasingly serious symptoms of tuberculosis. At the suggestion of his doctors, he agreed to move to Italy. Keats took a room close to the Barcaccia so he could see and hear the fountain and enjoy the scent of the flowers. However, Keats’ health grew steadily worse. He wrote, “You speak of Lord Byron and me; there is this great difference between us. He describes what he sees. I describe what I imagine. Mine is the harder task.” John Keats died on 23 February 1821, aged 25. He is buried in the Protestant Cemetery in Rome, where he lies in the

293

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 293 8/15/2014 11:58:46 AM 294 Remote Sensing in Action

shadow of the pyramid of Cestius and the old Roman wall. Within a few years, Matthew Arnold would write of Keats, “He is with Shakespeare.” Why start a chapter on mathematics with a preamble about imagination? The reason is that there could be no advances in mathematics without imagination. Many people think mathematics is just a sterile exercise of logic, but that is wrong. The great mathematicians achieved that distinction because of imagination. Their ideas are present in the conve- niences of our everyday lives, and we continually benefit from the fruits of their imagination. This book has speculated on the imagination of one of the greatest of all mathematician/scientists, Christiaan Huygens, and it asks the readers to use their imaginations in a similar manner … and to use only the basic elements of arithmetic, algebra, and geometry, as taught in high school.

Galileo and motion Galileo Galilei (1564–1642), born in Pisa, Italy, ranks with the most imaginative scientists. His most important idea was that theories can be tested by conducting experiments. That seems obvious today, but it was revolutionary at the time because it was in direct confrontation with Aris- totle, whose theories had not been challenged for about 2,000 years. As a result, science had stagnated. Perhaps Galileo’s greatest contribution to physics is his formulation of the concept of inertia, i.e., that an object in a state of motion possesses an inertia that causes it to remain in that state of motion unless an external force acts on it. This is counterintuitive because in our daily experience, most objects in a state of motion do not remain in that state. For example, a block of wood pushed at constant speed across a table quickly comes to rest when we stop pushing. Aristotle held that objects at rest remained at rest unless a force acted on them, but objects in motion did not remain in motion unless a force acted constantly on them. However, Aristotle failed to account for a “hid- den” force, the frictional force between the surface and the object. Thus, there are two opposing forces, the force associated with the push and the friction force that acts in the opposite direction. Galileo realized that as the frictional forces were decreased (for example, by placing oil on the table), the object would move farther and farther before stopping. From this, he abstracted a basic form of the law of inertia: If the frictional forces could be reduced to exactly zero (not possible

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 294 8/15/2014 11:58:47 AM Appendix C: Relativistic Doppler effect 295

in a realistic experiment, but it can be approximated to high precision), an object pushed to reach a given speed across a frictionless surface of infinite extent will continue at that speed forever after the pushing is stopped. Galileo’s most famous experiment involved falling bodies of different weight. According to Aristotle, heavier objects fall faster than lighter ones, and an object that weighs twice as much as another would fall twice as fast. However, Galileo tested this hypothesis and found that a cannon- ball weighing 10 pounds will not reach the ground ahead of a musket ball weighing less than a pound if both are dropped from the same height. Galileo also showed conclusively, by a short and clear argument, that a heavier body does not move more rapidly than a lighter one. Galileo’s simple argument assumed two stones in free fall, one light and one heavy, which are connected by a very thin and weak string. When the stones are dropped, the string does not break. The light stone could not pull up on the heavy stone, resulting in a slower fall. The two could not work together as a heavier object, resulting in a faster fall. Therefore, two objects of different weight fall at the same speed and hit the ground at the same time (and simultaneously knock down a theory that had been accepted for millennia). Galileo’s study of motion is one of the greatest scientific break- throughs, and we shall discuss it in some detail because it is a primary building block for remote detection. Let Δt be any increment of time t, and let Δx be the corresponding increment of distance x. Galileo’s definition for uniform motion requires that velocity, defined as the ratio v = Δx/Δt, is a constant. In other words, to have uniform motion, it was required that the distances traversed by the moving particle during any equal intervals of time are themselves equal. The concept of velocity defined in this way is one of the core ideas of differential calculus, which would begin to evolve into its modern form about 50 years after Galileo’s initial work and would serve as the springboard for what we now call “classical physics.” The next problem considered by Galileo, accelerated motion, requires some mathematical imagination. Galileo sought an answer by studying bodies falling with acceleration, as occurs in nature. In conformity with his definition of uniform as equal times and equal distances, he defined uniform accelerated motion as when, during any equal interval of time, an equal increment of velocity occurs. In other words, Galileo applied the same idea to acceleration as he did to velocity. Let Δt be any increment of time t, and let Δv be the corresponding increment of velocity v. Galileo’s definition for uniform acceleration is that acceleration, defined as the ratio a = Δv/Δt, is a constant.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 295 8/15/2014 11:58:47 AM 296 Remote Sensing in Action

In 1638, Galileo’s book Two New Sciences described his theories of motion via experimentation with an inclined plane, a board with a groove, down which he rolled a small metal ball (Figure C-1). Galileo’s experiment with an inclined plane concentrated on acceleration, a stage of motion ignored by Aristotle. In Principia Mathematica (1687), Newton wrote, “By the first two Laws [of motion], Galileo discovered … that the motion of projectiles was in the curve of a parabola…. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time.” Galileo’s remarkable experiment with the inclined plane is one of the most important in history. The key point is that the plane has the fixed height CB no matter what its angle of inclination relative to a horizontal surface. If a ball rolls down the plane, it will achieve a “terminal” velocity when it reaches the bottom. The question is: Does the terminal velocity depend on the inclination of the plane? Galileo demonstrated that for planes with the same fixed height, the terminal velocity is the same no matter what the inclination. In other words, the terminal velocity obtained by a ball rolled down plane CA is the same as that obtained by a ball rolled down plane CD. Today, it is universally agreed that this experiment marked the beginning of modern science. Galileo explained this result by an experiment in which the pivot point of a pendulum is changed in mid-swing (Figure C-2). A ball at the end of a string is permanently tethered at nail A. The ball falls along arc

Figure C-1. Diagram of experiments with an inclined plane.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 296 8/15/2014 11:58:47 AM Appendix C: Relativistic Doppler effect 297

Figure C-2. Galileo’s 3D drawing with three nails sticking out at points A, E, and F.

CB (with pivot point A). Its momentum upon reaching B is just enough to carry it up a similar arc BD (again with pivot point A) to the same height. Momentum is the product of velocity and mass. Mass is an intrinsic property of the body itself. A second nail is driven into the wall at E. The ball will fall along the arc CB (with pivot point A), but the ball will rise along the arc BG (with pivot point E). The same momentum makes the ball rise to the same height. If a third nail is driven into the wall at F, the same effect occurs. Because arcs CB and DB are equal and similarly placed, the momentum gained by the fall through arc CB is the same as that gained by a fall though arc DB. In addition, the momentum acquired at B, because of the fall through CB, lifts the same body through arc BD. Therefore, the momentum acquired in the fall from D to B is equal to that which lifts the same body through the same arc from B to D. Therefore, in general, momentum acquired by a fall through an arc is equal to that which can lift the body through the same arc. The experiment shows that these momenta which cause a rise through arcs BD and BG are equal because they are produced by the same momentum gained by a fall through CB. The conclusion is that both momenta (and hence velocities) gained by a fall through arcs DB and GB are equal. In other words, the terminal velocities are equal. Two basic forms of energy are potential energy (stored energy) and kinetic energy (energy of motion). Potential energy is equal to the weight of the ball multiplied by the elevation of the ball. Because elevation is the

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 297 8/15/2014 11:58:48 AM 298 Remote Sensing in Action

same at both at D and G, the potential energy of the ball is the same at both points. During the fall, potential energy is converted into kinetic energy. Hence, the kinetic energies (and the velocities) must be the same for each case. Galileo correctly introduced and used the law of conservation of mechanical energy. His demonstration is still used widely to illustrate this most basic law of physics. As stated earlier, Galileo defined uniform velocity as v = Δx/Δt = constant, and he defined uniform acceleration as a = Δv/Δt = constant. Using these definitions, he used mathematics to determine how distance increased with total fall time. Suppose a ball travels 128 inches down an inclined plane in four seconds. In other words, the initial distance is zero, and the final distance is x = 128 inches. The initial time is zero, and the final time is t = 4 seconds. The average velocity for that time interval is 128/4 = 32 inches per second.

Galileo step 1: The question involves dealing with uniformly accelerated motion in which velocity changes continually. Galileo rea- soned that for any quantity that changes uniformly, the average

value is halfway between the beginning value and the final value.. In this case, the average velocity V is 32, and if the average velocity is halfway between zero and the final velocity, it follows that v = 2 × 32, or 64 inches per second. Galileo step 2: Because Δv = v – 0 = 64 and Δt = t – 0 = 4, the constant is a = v/t = 64/4 = 16. Thus, v = at. Galileo step 3: Galileo’s definition of uniform velocity is V = Δx/Δt, meaning that Δx =VΔt, which is x = v t/2. Because v = at, Galileo obtains

x = (at/2)t, or x = at2/2.

This is a remarkable result because the equation relates total distance x to total time t without involving any velocity term. When t = 4, this equation gives (as expected)

x = at2/2 = (16)(4)(4)/2 = 128.

In summary, the variable x denotes the distance traversed in time t when a body starts at rest and is accelerated uniformly. Because the body starts at rest, the initial velocity is zero. If the final velocity is denoted by v, the average velocity is V = (0 + v)/2 = v/2.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 298 8/15/2014 11:58:48 AM Appendix C: Relativistic Doppler effect 299

Suppose the same body moves at the uniform speed given by the average velocity V = v/2. It would traverse the given distance x in the same time, t. In other words, Galileo transformed the case of uniform acceleration to an equivalent case of uniform velocity. In the case of uniform velocity V, the distance is equal to velocity multiplied by time, so x = Vt, which is x = vt/2. In the case of uniform acceleration, the velocity is equal to acceleration multiplied by time, so v = at. By combining the two equations, Galileo obtained his law of motion, x = vt/2 = (at)t/2. Thus Galileo found that the distance covered by a body starting from rest with uniform acceleration is proportional to the square of the time elapsed in traversing this distance. Sir Isaac Newton used this most cel- ebrated result about half a century later to establish the law of universal gravitation. Galileo’s law of motion can be used to find the final velocity, namely, the velocity of the ball when it reaches the bottom of the inclined plane (something that was impossible to measure at the time). Let a be the actual acceleration along the plane, and let g be the downward acceleration caused by gravity. Length x of the inclined plane will vary according to its slope. A steep slope will mean that x is small and a gentle slope that x is large. Figure C-3 graphically depicts the variables involved. The first step is to solve for t in the equation for the law of motion x = at2/2. The result is

This value of t is the time it takes for the ball to roll down length x of the plane. Now the final velocity will be

Figure C-3 shows that a/g = H/x, meaning ax = gH. As a result, the final velocity is

which depends only on H and g, both of which are constants. This establishes the key and perhaps counterintuitive result that the final velocity, namely, the velocity of the ball when it reaches the bottom of the inclined plane, depends on the height of the plane but is independent of its slope.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 299 8/15/2014 11:58:49 AM 300 Remote Sensing in Action

a/g = H/x

x a

H g

Figure C-3. Diagram illustrating Galileo’s revolutionary experiment with an inclined plane.

Galileo wrote, “Philosophy is written in this grand book, the universe…. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is hu- manly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.” By the use of the triangles and circles that describe the inclined plane and pendulum, Galileo put science on a firm footing. Albert Einstein, in his book Ideas and Opinions, placed Galileo’s achievement in stark perspective: “Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality. Because Galileo saw this, and particularly because he drummed it into the scientific world, he is the father of modern physics, indeed, of modern science altogether.”

Christiaan Huygens Many discoveries of the brilliant seventeenth-century scientist Chris- tiaan Huygens (1629–1695) were discussed earlier in this book. Amaz- ingly, that was just a partial list. In addition, Huygens wrote the first book on probability theory and derived the formula for the centrifugal force

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 300 8/15/2014 11:58:49 AM Appendix C: Relativistic Doppler effect 301

exerted by an object describing circular motion. In The Celestial Worlds Discover’d: Or, Conjectures Concerning the Inhabitants, Plants and Pro- ductions of the Worlds in the Planets, Huygens gave the first systematic explanation of possible extraterrestrial life. The many remarkable contributions by Christiaan Huygens began with major improvements to the two most important navigational tools of the seventeenth century, the telescope and the clock. He and his brother Constantijn designed and constructed a greatly improved telescope, the first one accurate enough to make significant astronomical observations. The lens was finished and ready for use in February 1655, and stunning results were quick in arriving. Just a few weeks later, on 25 March 1655, Huygens discovered Saturn’s moon Titan. He was also struck by the curious extension of Saturn, first observed in 1610, which had intrigued astronomers since Galileo. Huygens gave the solution: Saturn is encircled by a ring — thin, flat, nowhere touching, and inclined to the ecliptic. At that time, the central problem of navigation was the determination of longitude. The correct longitude is required for the location of a ship at sea and for mapping the globe. Because the earth rotates around its axis every day, longitude is correlated with time. If the difference in local time at two points is known, the longitudinal distance between them is known. In seventeenth-century exploration, improving the accuracy of the clock was at the center of scientific attention. The existing mechanical clocks, accurate to within only about 15 minutes a day, were unsuitable for navigation. Huygens’ invention of the pendulum clock was a breakthrough in timekeeping. Its escapement counts the swings while a driving weight provides the push. In effect, the escapement is a feedback regulator that controls the speed of a mechanical clock. Huygens produced his first pendulum clock in December 1656; it was accurate to within 10 seconds a day. His pendulum clocks were accurate enough to allow measurement of the orbital periods of the satellites of Jupiter during the span of a year. In fact, pendulum clocks were the most accurate clocks in the world for the next 300 years. Galileo believed that a pendulum is isochronic, in other words, that the period of a pendulum does not depend on the amplitude of its swing. However, Huygens used mathematics to find that a pendulum swinging through the arc of the circle is not isochronic. It appears to be isochronic only when the length of the arc is quite short relative to the length of the pendulum. This property gives a clock with a long pendulum an

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 301 8/15/2014 11:58:49 AM 302 Remote Sensing in Action

advantage over one with a short pendulum. However, the pendulums of early clocks were short and light to minimize the amount of energy needed to keep them in motion. As a result, early pendulum clocks had very wide pendulum swings, which decreased their accuracy. Huygens went to great efforts to make a pendulumlike mechanism that would be isochronic. The original diagram (Figure C-4) of the pendulum clock shows how the pendulum could swing through a large angle. In this work, Huygens originated the branch of mathematics known as differential geometry and introduced the cycloid, a geometric construction intimately related to the circle. In fact, the cycloid is the serial locus of a point on the rim of a circle rolling along a straight line. In Horologium Oscillatorium (1673), Huygens gives a complete mathematical description of an improved pendulum clock, called a cycloidal clock. The cycloidal clock is isochronic because its pendulum is forced to swing in the arc of a cycloid. Unfortunately, in operation, the movement of the pendulum against the metal cheeks caused an excessive amount of friction. Robert Hooke, best known for Hooke’s law, which relates stress and strain, invented the anchor escapement for a pendulum clock. The anchor escapement required a smaller angle of swing than the angle required by the escapements of the early pendulum clocks. As a result, pendulum clocks became so accurate that the cycloidal clock went out of use. Al- though Huygens’ cycloidal clock did not stand the test of time, another of his inventions certainly did. In 1675, he built a chronometer that used

Figure C-4. Original diagram of Huygens.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 302 8/15/2014 11:58:50 AM Appendix C: Relativistic Doppler effect 303

a balance wheel and spiral spring instead of a pendulum. Balance wheels and spiral springs were used in almost all watches until the invention of the quartz-crystal oscillator in the twentieth century. Although it never became widely used, the cycloidal clock is histor- ically important because it is the first successful design of an intricate apparatus based on the use of higher mathematics. In this case, the key concept was another discovery of Huygens, the evolute (the locus of the centers of the osculating circles of a curve). Other people used mechanical principles to design their inventions but not much mathematics beyond Euclidean geometry. The introduction of the use of higher mathematics to accomplish mechanical design gives Huygens a strong claim as the father of modern technology.

Huygens and remote detection

The method of remote detection refers to the acquisition of information about an object or phenomenon that is distant from the observer. By use of real-time sensing devices, it is possible to collect data from dan- gerous or inaccessible areas. The instruments are sensors that are not in physical or intimate contact with the object. The telescope as an instrument of remote detection appeared in the seventeenth century. All at once, astronomy, which had been more or less at a standstill since the time of the ancient Greeks, spurted forth a multi- tude of new discoveries. A disadvantage of the early astronomical telescopes was that higher magnification was accompanied by more spherical and chromatic aberra- tion. The result was geometric distortion and false colors. Grinding and polishing techniques for lenses improved gradually during the seventeenth century. The quality of telescopes improved, but they became longer. The telescope made by Christiaan Huygens and his brother Constan- tijn in 1655 was 23 feet long. It had a large field of view and a magnification of 100. In 1659, Huygens wrote: “In the sword of Orion are three stars quite close together. In 1656, I chanced to be viewing the middle of one of these with a telescope, instead of a single star twelve showed themselves (a not uncommon occurrence). Three of these almost touched each other, and with four others shone through the nebula, so that the space around them seemed far brighter than the rest of the heavens, which was entirely clear and appeared quite black, the effect being that of an opening in the sky through which a brighter region was visible.”

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 303 8/15/2014 11:58:50 AM 304 Remote Sensing in Action

This was a challenge to the heliocentric Copernican system, which held that the sun was the center of the universe and a surrounding celestial sphere contained all of the “fixed” stars. Huygens wrote, “all these Stars are not in the same [Celestial] Sphere, as that the Sun, which is one of them, cannot be brought to this Rule. But it is more likely they are scat- tered and dispersed all over the immense spaces of the Heaven, and are as far distant perhaps from one another, as the nearest of them are from the Sun.” Huygens concluded, “I must give my vote to have the Sun of the same nature with the fixed Stars. For then why may not every one of these Stars or Suns have as great a Retinue as our Sun, of Planets, with their Moons, to wait upon them?” In other words, Huygens said the stars are spread out all over the universe, and the sun is just another star. The Copernican system was shattered; the sun is not at the center of the universe. The celestial sphere of stars becomes nothing more than a mathematical construction. Huygens was the first person to distinguish between active and passive remote detection. He described a method of active remote detection that would give the distance of the moon from the earth. The method described by Huygens came to fruition in the twentieth century when earth-generated radar signals were reflected from the moon. Huygens also described an effective method of passive remote detection, with which he made some startling discoveries. He chose the earth-Jupiter system. He did not have the instruments that we have today, but he did have the mathematics, and his mathematics is correct. Because the earth moves in its orbit so much faster than Jupiter moves in its orbit, we may consider the two planets as moving apart for half of the year and moving together for the other half of the year. In this sense, we have a round-trip journey, a great advantage for remote detection. Of course, we have our clocks on earth. One such clock is our moon. Because of the telescope, moons of Jupiter could be seen, and they acted as clocks on that planet. The presence of an observable clock on the remote object is another great advantage for the purposes of remote detection. The earth-Jupiter system was an ideal starting point for Christiaan Huygens and for the use of his best-known contribution, namely, Huygens’ principle, or Huygens’ construction. Huygens wrote: “Each little region of a luminous body, such as the Sun, a candle, or a burning coal, generates its own waves of which that region is the center” (Figure C-5). “Thus in the flame of a candle, having distinguished the points A, B, C, concentric circles described about each

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 304 8/15/2014 11:58:50 AM Appendix C: Relativistic Doppler effect 305

Figure C-5. A candle generates its own waves.

of these points represent the waves which come from them. And one must imagine the same about every point of the surface and of the part within the flame.” Huygens used the leading edges of the wavelets to describe the propagation of waves (Figure C-6). He used the trailing edges of the wavelets to describe the reflection of waves. Despite Huygens’ brilliant insight, as Sherlock Holmes described earlier in our fictional account in this book, he faced formidable obstacles in getting people to accept his wavelet theory of wave propagation. Few accepted it, and the opponents included very prominent names. Newton claimed that light is propagated not as waves but as particles, in which case spherical wavelets are impossible. Descartes claimed that the speed of light is infinite, in which case spherical wavelets with finite radius are impossible. It was not until 1801 that Huygens’ wavelet theory of 1678 was finally vindicated, by the interference experiments of Thomas Young. Young used the side edges of the wavelets to describe the diffraction of the light into regions of geometric shadow. Whereas Huygens used a geometric approach in the seventeenth century, subsequent advances in mathematics allowed such nineteenth- century scientists as Young, Fresnel, Kirchhoff, and Maxwell to establish the mathematical foundations of Huygens’ principle. Huygens’ principle

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 305 8/15/2014 11:58:50 AM 306 Remote Sensing in Action

Figure C-6. Illustration of the propagation of waves.

can be obtained directly from Maxwell’s equations. Ampere’s law and Faraday’s law predict that every point in an electromagnetic wave acts as a source of the continuing wave, which fits perfectly with Huygens’ analysis. Today, Huygens’ wavelet theory is widely accepted — and used — to explain the propagation of electromagnetic waves and of mechanical waves. In 1672, Gottfried Wilhelm Leibniz moved to Paris. With Huygens as mentor, Leibniz began to study mathematics and physics. In a letter to Henry Oldenburg of the Royal Society in London dated 16 April 1672, Leibniz wrote, “The very noble Huygens first proposed the problem [which involved calculus] to me, and I solved it, much to the surprise of Huygens himself.” Leibniz invented the version of the differential and integral calculus in use today. However, more than a decade before the publica- tion of both Newton’s Principia and the first publications of Leibniz on calculus, Huygens, in his derivation of centrifugal force, had used the second law of motion and twice had differentiated a vector-valued function. Huygens showed how singularities can arise in the propagation of waves. His investigations of singular points were important in establish- ing the correspondence between waves and rays. In those studies, Huy- gens made use of the calculus of variations, optimization, and Hamilto- nian mechanics, all of which were officially discovered by other people

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 306 8/15/2014 11:58:51 AM Appendix C: Relativistic Doppler effect 307

years later. Huygens’ insight was based on images, allowing him to solve the advanced mathematical problems by means of elementary geometric constructions. A major breakthrough in the history of science was the determining of the finite speed of light. This was a fairly recent development. Before the seventeenth century, it was generally believed that there was no such thing as the speed of light. In other words, light could travel any distance in no time at all. Later, several attempts were made to measure light speed. Galileo concluded that “if not instantaneous, light is ex- traordinarily rapid.” Descartes was so convinced of the instantaneous transmission of light that he unwisely said he would stake all his system of philosophy on its truth. The assumption that light travels with a finite speed is fundamental to Huygens’ work. If the speed of light were infinite, his spherical wavelet would have an infinite radius, and his wavelet theory would be mean- ingless. Huygens used astronomical observations of the nearest moon of Jupiter to determine the speed of light. Since then, more sophisticated techniques have improved the precision of the value obtained by Huy- gens. Today, the speed of light is known very accurately. Relative to the earth’s frame of reference, sunlight originates from a moving source, whereas candlelight does not. For Huygens’ wavelet theory to work, wavelets emanating from a given wavefront of light have to be the same spherical shape, whatever the source of the light. The only way this condition can be met is that the speed of light is constant. In other words, the velocity of light (in a vacuum) is the same value c in each and every inertial reference frame. Maxwell established that the speed of light is a fundamental constant of nature. His famous equations give the properties of the electric and magnetic fields in terms of sources, current density, and charge density. From those equations, Maxwell established the existence of electromagnetic waves that travel at speed

where ε0 denotes the electric constant, which is a universal constant, and

µ0 denotes the magnetic constant, which is also a universal constant. As a result, according to this equation, the velocity of light is a universal constant. In this equation, the most accurate value of the speed of light is given.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 307 8/15/2014 11:58:51 AM 308 Remote Sensing in Action

Maxwell’s equations explain specifically how electromagnetic waves physically propagate through space. As soon as Maxwell calculated the numerical value of the above expression for c, he saw immediately that it was essentially the same as the speed of light in a vacuum. Maxwell concluded that light is an electromagnetic wave, even though no other type of electromagnetic wave had been identified at the time. (In 1886, Heinrich Rudolf Hertz established the existence of radio waves. These are electromagnetic waves, which were predicted by Maxwell.) Maxwell’s equations not only unify the theories of electricity and of magnetism but of optics as well. In other words, electricity, magnetism, and light can all be understood as aspects of a single entity — the electromagnetic field. Light (in a vacuum) travels with the same constant speed in all directions in each and every reference frame. In other words, Huy- gens’ wavelets are spherical. In brief, Maxwell gave physical confirmation to Huygens’ theory of the wave propagation of light. In Science, 24 May 1940, Einstein wrote, with obvious admiration, “The precise formulation of the time-space laws was the work of Max- well. Imagine his feelings when the differential equations he had formu- lated proved to him that electromagnetic fields spread in the form of po- larized waves, and at the speed of light! To few men in the world has such an experience been vouchsafed. It took physicists some decades to grasp the full significance of Maxwell’s discovery, so bold was the leap that his genius forced upon the conceptions of his fellow workers.” Maxwell had computed the speed of light successfully from the electromagnetic constants, but as Einstein said, he had to convince others. Nineteenth-century physicists expected that if they took the earth as the frame of reference and measured the speed of light, they would obtain varying results. The reason for their conclusion was that the earth in its motion around the sun constantly changes its velocity with respect to a reference frame called the aether. Luminiferous aether (or ether), meaning light-bearing aether, was the term used to describe a medium for the propagation of light. Maxwell wrote in the Encyclopedia Britannica in 1878, “Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, until all space had been filled three or four times over with aethers…. The only aether which has survived is that which was invented by Huygens to explain the propagation of light.” In 1879, Maxwell decided to measure the velocity of light in various reference frames. He looked back into the work of Huygens and used

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 308 8/15/2014 11:58:51 AM Appendix C: Relativistic Doppler effect 309

the same method of passive remote detection which Huygens had used two centuries earlier. This method, using one-way signals provided by the nearest moon of Jupiter, held promise because the signals travel over astronomical distances with relatively long traveltimes. Like Huygens, Maxwell was plagued with astronomical data that were still not sufficient- ly accurate to validate the theory. As a result, Maxwell again turned to Huygens and found the method of active remote detection written down by Huygens. Instead of using two- way signals to the moon of Jupiter as Huygens had proposed, Maxwell proposed using two-way signals over small distances on the earth. Such signals, which have short traveltimes, could be used because of great advances in instrumentation. Maxwell’s proposed two-way experiment used a beam of light that is returned to its starting point. In other words, the source and receiver are at the same point, and the signal is returned by reflection. How- ever, Maxwell died that same year, before he could try the experiment. In 1887, A. A. Michelson and E. W. Morley famously performed Maxwell’s experiment and, as is very well known, the results were revolutionary. Their experiment was designed to determine the absolute motion of the earth through the then privileged inertial frame, called the aether, which almost all scientists assumed to exist. However, no absolute motion could be detected. This null result of the experiment could not be explained in terms of classical physics. In that way, the aether was deposed from the position of a privileged frame. The aether frame, if it exists, is unobservable. In other words, the experiment showed that no privileged frame exists at all. Thus, for light, all inertial frames are of equal status. Light, now without any privileged frame, finds refuge in no frame at all, namely, a vacuum. Because it be- longs to no frame, light has the same velocity in all inertial frames … and this is the cornerstone of relativity. There is no reason for the unobservable material medium known as the aether at all because it confuses the essential difference between electromagnetic waves and mechanical waves. The propagation of electromagnetic waves (light waves, radio waves, and so forth) is essentially different from the propagation of mechanical waves (sound waves, seismic waves, water waves, and so forth). Matter (air, rocks, water, and so forth) is required for the propagation of mechanical waves. Matter is not required for the propagation of electromagnetic waves. Electromagnetic waves can propagate in a vacuum. The most basic form of Huygens’ construction consists of a circle and a sloping line tangent to it (Figure C-7). The center of the circle is point T,

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 309 8/15/2014 11:58:51 AM 310 Remote Sensing in Action

O S T R

P Figure C-7. Basic form of Huygens’ construction.

the lower extreme is point S, the upper extreme is point R, and the tangent point is point P. The distances between points are

t = OT = midpoint period x = TS = TR = TP = radius of the circle = distance v = x/t = velocity p = OP = proper period s = OS = t – x = lower extreme = sending period r = OR = t + x = upper extreme = receiving period

The midpoint time t is the arithmetic mean of the lower extreme s and the upper extreme r. In other words, t = (s + r)/2. The Pythagorean theorem gives p2 = t2 – x2.

After factoring the right side, the equation becomes

p2 = (t – x)(t + x) = sr,

which gives p= sr. This equation says the proper period p is the geometric mean of the lower extreme s and the upper extreme r. The use of the velocity of light in defining the unit of distance has the virtue of simplifying the connection between space and time. Any message sent by light or any other electromagnetic signal from Vega to earth will take 27 years. We see that the use of light-years implies that the communication time and the communication distance are both the same

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 310 8/15/2014 11:58:51 AM Appendix C: Relativistic Doppler effect 311

number. For example, from earth to Vega, the time of communication by an electromagnetic signal is t = 27 years, and the distance is x = 27 light- years. Because x = ct where c = 1, traveltime t and travel distance x are numerically equal for a light signal. We could write the equation

t – x = 0 (for a light signal).

However, depending on the convention used for the directions of the t- and x-axes, the values of t and/or x might be either positive or negative. Therefore, we write instead

t2 – x2 = 0 (for a light signal).

The square root of the quantity on the left is called the time-space interval; i.e., time-space interval = tx22−=0.

We therefore conclude that for any light signal, the time-space interval is equal to zero. For material bodies, the magnitude of velocity v must be less than c, so (in natural units) we have . For such bodies, travel distance x = vt is less than traveltime t. As a result, the time-space interval is positive, that is,

time-space interval = tx22−= p(for a material body),

where p is the proper time.

Stereographic projection

A sphere is defined as the set of all points in three-dimensional space at a fixed distance (called the radius) from a given point (called the center). The term sphere refers to the surface only, so the sphere is a two- dimensional surface. The interior of a solid sphere is properly called a ball. Any planar cross section through a sphere is a circle. In the limiting case in which the slicing plane is tangent to the sphere, the cross section reduces to a point. Where the plane defining the cross section passes

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 311 8/15/2014 11:58:52 AM 312 Remote Sensing in Action

through a diameter, the size of the circle is a maximum, and so the circle is called a great circle. A great circle of a sphere is a circle that runs along the surface so as to cut the sphere into two equal halves, as distinct from a small circle. The great circle therefore has both the same circumference and the same center as the sphere. A point on a sphere is specified by its longitude and latitude (Figure C-8). A meridian (or line of constant longitude) is an imaginary arc on the earth’s surface from the North Pole to the South Pole that connects all locations running along it with the given longitude. Each meridian is one-half of a great circle. The position of a point on the meridian is given by the latitude. On earth, the equator is a great circle, but the parallels of latitude are not great circles. They are small circles instead. Their centers are not at the center of the earth but are on the line connecting the poles. Figure C-9 graphically depicts the stereographic projection of point P on the sphere to point R on the equatorial plane. Stereography is the art or technique of depicting solid bodies on a plane surface.The polar stereographic projection is very important in cartography. Take a sphere of radius p. Either the North Pole N or the South Pole M may be chosen as the projection center. Let us chose the North Pole. The projection plane (i.e., the map plane) is the equatorial plane, i.e., the plane that goes through center O of the

Figure C-8. Illustration of longitude and latitude.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 312 8/15/2014 11:58:52 AM Appendix C: Relativistic Doppler effect 313

φ R O Equatorial plane

Radius = OP = p.

Projection = OR = r.

Projection factor = k = OR/OP = r/p. N

O φ

Figure C-9. Stereographic projection of point P.

sphere and that is perpendicular to polar axis ON. Let P be any point on the sphere. A projection ray (or light line) is a straight line originating at the North Pole. Draw the projection ray so that it passes through point P on the sphere. If point P is in the Northern Hemisphere, the projection ray must be extended to the equatorial plane. If point P is in the Southern Hemisphere, the projection ray must already cut to the equatorial plane. Let R be the point where the projection ray cuts the equatorial plane. Point R is called the stereographic projection of point P. Alternatively, we may say that line OR is the stereographic projection of radius OP. The projection ratio k is the ratio of OR to OP; i.e., the projection ratio is k = OR/OP. In summary:

• The North Pole is mapped into infinity. • All points on the sphere and in the Northern Hemisphere are mapped into the exterior of the equatorial circle. • The equator is mapped into the equatorial circle. (In other words, the equator is constant.)

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 313 8/15/2014 11:58:53 AM 314 Remote Sensing in Action

• All points on the sphere and in the Southern Hemisphere are mapped into the interior of the equatorial circle. • The South Pole is mapped into the origin of the equatorial circle.

NASA created the most detailed global color map of the planet Jupiter ever produced. It is a polar stereographic projection with the South Pole in the center and the equator at the edge (Figure C-10). The stereographic projection is conformal, which means that it preserves angles. In other words, the angle between two lines on the sphere is the same as the angle between their images in the plane. The price paid for conformality is that of real distortion. Small regions on the sphere project almost accurately on the plane, making the stereographic projection a good map projection for small areas. However, radial distortion increases outward from the center. Figure C-11 graphically depicts a North Pole stereographic projection where point R is the stereographic projection of point P. (In describing mathematics, the ideal approach is to take the simplest possible situation that preserves the generality of the problem. For ease of presentation, from now on, we shall deal only with two-dimensional figures. To this end, we will use the great circle NPM to represent the sphere.)

Figure C-10. An example of polar stereographic projection. Courtesy of the Na- tional Aeronautics and Space Administration Jet Propulsion Laboratory.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 314 8/15/2014 11:58:53 AM Appendix C: Relativistic Doppler effect 315

O R

M Figure C-11. North Pole stereographic projection.

The light source is the North Pole N. The light line (projection ray) goes from N through point P to point R on the equatorial line. The projection factor is OR/OP. Figure C-12 graphically depicts a South Pole stereographic projection where point P is the stereographic projection of point S. (Alternatively, we may say that point S is the inverse stereographic projection of point P.) The light source is now the South Pole M. The light line (projection ray) goes from M through point S on the equatorial line to point P. The projection factor is OP/OS. Figure C-13 graphically depicts a bipolar stereographic projection, which shows the projections from both poles. The figure shows the tangent line PT. Point P is the point of tangency to the circle. Point T is the point of intersection of the tangent line with the equatorial line. The two light lines to point P are NP and MP. Because NM is a diameter, it follows that the inscribed angle NPM is a right angle. In other words, the two light lines are at right angles to each other. From the North Pole, line OR is the stereographic projection of radius OP. The projection factor is OR/OP. From the South Pole, radius OP is the stereographic projection of line OS. The projection factor is OP/OS. By the logic of plane geometry, it can be shown that triangles SOP and POR are similar. Thus, OR is to OP as OP is to OS. This relationship establishes the fact that both projection factors are the same value k; that is,

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 315 8/15/2014 11:58:54 AM 316 Remote Sensing in Action

O S

M Figure C-12. South Pole stereographic projection.

O S T R

Figure C-13. Bipolar stereographic projection.

With reference to Figure C-13, the radius of the circle is OP. The above equation shows that radius OP is the geometric mean of OS and OR; that is,

Let us assign symbols to the various line segments. We define

sending period = s = OS proper period = p = OP receiving period = r = OR

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 316 8/15/2014 11:58:55 AM Appendix C: Relativistic Doppler effect 317

midpoint period = t = OT offset distance = x = TS = TR = TP pr projection factor = k = = sp

What do these symbols mean? Consider a case of active remote detection. Line segment OS is the sending period s; that is, s is the period of the outgoing signal emitted by the source. Line segment OP is the proper period p; that is, p is the period of the reflected signal as measured at the target. Line segment OR is the receiving period r; that is, r is the period of the incoming signal received back at the source location. In the outward journey, sending period s is projected to proper period p; that is, p = ks. On the return journey, proper period p is projected to receiving period r; that is, r = kp. If we combine the two equations, we obtain r = kp = k(ks) = k2s.

If we solve this equation, we obtain the fundamental expression for the projection factor given by

This equation expresses the projection factor in terms of the sending period and the receiving period. The stereographic projection factor k is the same thing as relativistic Doppler factor k. The two factors are identical. The equations that express midpoint period t and offset distance x in terms of sending period and receiving period are t = (r + s)/2 and x = (r – s)/2. The inverse equations that express the sending period and the receiving period in terms of midpoint period and offset distance are s = t – x, r = t + x. The equation OR/OP = OP/OS becomes r/p = p/s, which gives p2 = sr. In other words, the product of the sending period and the receiving period is the square of the proper period. Thus the proper period is

The midpoint period t is the arithmetic mean of the sending period and the receiving period. The proper period p is the geometric mean of the sending period and the receiving period.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 317 8/15/2014 11:58:56 AM 318 Remote Sensing in Action

In summary, and perhaps unexpectedly, the mathematics of the relativistic Doppler factor and, for that matter, all the mathematics of special relativity are incorporated in the geometry of the stereographic projection.

Symmetry of the relativistic Doppler effect In explaining the classical Doppler effect for sound waves, we must take into account the velocities of the source and receiver relative to the medium. Sound waves are mechanical waves that require a medium. In the case of electromagnetic waves traveling through a vacuum, there is no medium. Symmetry is a physical or mathematical property of a system that is preserved under some change. The type of symmetry known as relativistic symmetry requires that the behavior of two objects remains the same regardless of their absolute motion. Their behavior would depend only on their relative motion. Relativistic symmetry is amenable to mathematical formulation, and thus it can be used to simplify many problems in physics. In the case of relativistic problems, the signal speed is the speed of light in a vacuum. We recall our convention, namely, that s (for source or sending or start) represents the initial signal, and r (for return or receiving or finish) represents the final signal. In other words, the flow of time is from s to r. If r > s, then we have an ascending sequence with v > 0. If r < s, then we have a descending sequence with v < 0. In a classical (or acoustic) case of active remote detection, a mechanical signal (such as sound) is sent outward from a fixed platform. Let s be its sending period. The signal strikes a moving target, where it is reflected. The signal returns to the fixed platform. Let r be the receiving period. The reflection period is not in doubt because it must be the midpoint period t.

The classical Doppler factor from fixed object to moving object is k1= t/s.

Another expression for k1 is r/q, where q is the harmonic mean of s and r.

The classical Doppler factor from moving object to fixed object is k2 = r/t.

Another expression for k2 is q/s, where q is the harmonic mean of s and r. For the classical Doppler factors, it makes a difference which object is moving and which is stationary. In a relativistic (or optical) case of active remote detection, an electromagnetic signal (such as light) is sent outward from a fixed platform. Let s be its sending period. The signal strikes a moving target, where it is reflected. The signal returns to the fixed platform. Let r be the receiving period. The reflection period is in doubt. On the moving target, it is the

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 318 8/15/2014 11:58:56 AM Appendix C: Relativistic Doppler effect 319

proper period p. On the fixed platform, it is the midpoint period t. The relativistic Doppler factor from fixed object to moving object is k = p/s. The relativistic Doppler factor from moving object to fixed object is k = r/p. These two relativistic Doppler factors are the same. For the relativistic Doppler factor, it makes no difference which object is moving and which is stationary. Only the relative velocity between the two objects matters. It so happens that the relativistic Doppler factor k is the geometric

mean of k1 and k2. However, this result has no physical significance be-

cause k1 and k2 refer to mechanical waves, whereas k refers to electromagnetic waves.

The classical Doppler factor k1 is the arithmetic mean t divided by

sending period s. The classical Doppler factor k2 is the harmonic mean q divided by sending period s. The relativistic Doppler factor k is the geometric mean p divided by sending period s. The entire mathematics of the Doppler effect is encapsulated in the three means made legendary by Pythagoras two-and-one-half millennia ago. There is a fundamental lack of symmetry in having two Doppler factors for mechanical waves. In the depiction of the heavens by the stereographic projection, the ancients came upon the relativistic Doppler effect, which is made up of one entity, namely, the relativistic Doppler factor. The relativistic Doppler factor depends only on the relative velocity between two moving objects. The stereographic projection uses light as the signal. It follows that for the relativistic Doppler factor to apply, light or some other electromagnetic wave must be used as the signal. In conclusion, the relativistic Doppler effect shows perfect symmetry.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenC_v4--scs.indd 319 8/15/2014 11:58:56 AM This page has been intentionally left blank

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 Appendix D

Special relativity theory

Special relativity and the Lorentz transformation

Three concepts play a central role in relativity theory: invariance, time-distance, and causality. The concept of invariance is basic to the science of physics. We say that the laws of physics are invariant. This simple concept is extremely powerful. If one stands at one place or at another place, things may look different, but the underlying laws of physics are still the same. This is called translational invariance. Similarly, rotational invariance means the laws of physics do not change if we spin around and see the world while facing different directions. Invariance principles are connected to the conservation of physical quantities. The law that states that momentum is conserved corresponds to translational invariance. The law that states that angular momentum is conserved corresponds to rotational invariance. Geometry was the most advanced form of mathematics for millennia. However, in the seventeenth century, Descartes introduced algebra into geometry, and the whole new world of analytic geometry opened up. Algebraic equations can express exact quantitative relationships between different physical quantities. Not least among those relationships is the equation for speed. By instinct, people understand the difference between fast and slow, but that notion became formalized in an algebraic equation, namely, “speed equals distance divided by time,” or v = x/t, where v is speed, x is distance, and t is time. For example, if you drive 120 miles in two hours, your (average) speed is 120 divided by two. The answer is that the (average) speed is 60 miles per hour. The simple equation for speed is extremely useful, but does it invari- ably furnish a description of nature with which everyone would agree? In other words, does the equation deal with invariant quantities? As it turns out, neither distance nor time is invariant. Both depend on the perspective of the viewer. The theory of relativity devastates the commonplace understanding of time. Consequently, space and time must be merged into one entity called

321

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 321 8/15/2014 12:00:12 PM 322 Remote Sensing in Action

space-time. After making this intellectual leap, the next step is to consider causality. The order of cause and effect cannot be reversed. The effect cannot precede the cause. If this were not true, we would face irresolvable contradictions. For example, consider one period of a wave — the time difference between two successive crests. We encounter one crest first and the other crest later, so we can say that first precedes second. There- fore, even after accepting that time is not invariant, causality must still hold, and this insistence on a causal universe constrains the possibilities of melding space and time. The fact that both distance and time can be expressed in terms of the unit known as a second creates the opportunity to find an invariant quantity in the realm of space-time that can be visualized via Huygens’ construction. As previously described, the most basic form of Huygens’ construction consists of a circle and a right triangle. The center of the coordinate system is O. The center of the circle is T, and the radius of the circle is x. One leg of the triangle is a sloping line OP, which is tangent to the circle. The length of OP is the invariant period p. The hypotenuse of the triangle is the horizontal line OT. The length of OT is the midpoint period t. The other leg of the triangle is a radius TP of the circle. Figure D-1 shows two cases of Huygens’ construction. The only thing that the two have in common is that period p has the same numerical value for each; in other words, the time period p is invariant. In Case 1 (Figure D-1a), the various symbols have a subscript 1, whereas in Case 2 (Figure D-1b), they have a subscript 2. The symbol p has no subscript because it

is the same in each case. More specifically, the invariant time p = OP1 is

represented by the coordinates time t1 = OT1 and distance x1 = T1P1 of the

first person. Similarly, the invariant time p = OP2 is represented by the

coordinates time t2 = OT2 and distance x2 = T2P2 of the second person. The Pythagorean theorem means

2 2 2 (OP1) = (OT1) – (T1P1) 2 2 2 (OP2) = (OT2) – (T2P2) ,

which is the same as

In this scheme, space x and time t are no longer absolutes. They have been sacrificed in favor of an absolute space-time quantity p. The first

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 322 8/15/2014 12:00:12 PM Appendix D: Special relativity theory 323

Lowerlower extreme extreme = s=1 =s1 OS= OS1. 1 Upperupper extreme extreme = r=1 =r1 OR= OR1. 1 OO Midpointmidpoint period period = tt11 == OTOT11. T RR T11 11

SS 1 0 11 . P 1 P 1 Proper period = p = OP 1 = T proper period p = OP = T 1 1

1 . 1 Distancedistance x x

PP11

Lower extreme = s = OS . upperUpper ex extremetreme = rr == OR . lower extreme = s2 2 = OS2 2 22 2 Midpointmidpoint period period = t == OTOT. O 2 22 TT R 22 R22 SS Proper period = p = OP22 proper period p = OP 2 . P P 2 2 2 = T = T2 2x

2 . 2 Distancedistance x P P22

Figure D-1. Two cases of the Huygens’ construction. (a) Case 1 and (b) Case 2.

person measures t1 and x1. The second person measures t2 and x2. How- ever, they both obtain the same space-time invariant p. We would expect the observations of the two people to be different because the two cases have different space-time configurations. Figure D-2 depicts Case 1 and Case 2 on the same diagram. Is there a relationship between circle 1 and circle 2? The lower extremes of the

respective circles are s1 = OS1 and s2 = OS2. The upper extremes of the

respective circles are r1 = OR1 and r2 = OR2. We want to find circle 0 that transforms circle 1 into circle 2. The lower and upper extremes of circle 0

are s0 and r0, respectively. Huygens’ first rule of composition says that the lower extreme of circle 2 is equal to the product of the lower extremes of circle 0 and circle 1. The second rule of composition says that the upper extreme of circle 2 is

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 323 8/15/2014 12:00:13 PM 324 Remote Sensing in Action

Circlecircle 2 2

Circlecircle 1 1

s = OS r = OR s1 1 = OS1 1 t = OT r11 = OR1 1 t11 = OT1 1 s = OS r = OR s2 = OS2 t = OT r22 = OR2 2 2 2 t22 = OT2 2 S S T T O S1 S2 T1 T2 RR1 R 1 2 1 2 1 R2 00 x = = T TP P 11 1 11 1 x = T P x2 = T2 2P P 2 2 2 p = OP P1 p = OP1 1 1 p = OP p = OP2 2

P 2P2

Figure D-2. Case 1 and Case 2 plotted together.

equal to the product of the upper extremes of circle 0 and circle 1. The two rules are encompassed in the two equations

s2 = s0 s1;

r2 = r0 r1.

Recall that the “proper time p” is the geometric mean of s and r. Multiply- ing the above two equations produces

s2r2 = s0 r0 s1 r1,

which gives

p2 = p0 p1.

Because circles 1 and 2 have the same proper time p, the above equation becomes

p = p0 p.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 324 8/15/2014 12:00:14 PM Appendix D: Special relativity theory 325

Because p can be canceled from both sides, p0 = 1. In other words, circle 0 has proper time unity. This property ensures that circles 1 and 2 have the same proper time p. As we have seen, the two rules are

s2 = s0 s1 or t2 – x2 = (t0 – x0)(t1 – x1);

r2 = r0 r1 or t2 + x2 = (t0 + x0)(t1 + x1).

Each equation can be expanded to obtain

t2 + x2 = t0 t1 + t0 x1 + x0t1 + x0 x1;

t2 – x2 = t0 t1 – t0 x1 – x0t1 + x0 x1.

These two equations can be added and subtracted, producing two new equations

t2 = t0t1 + x0 x1;

x2 = t0 x1 + x0t1.

These equations are called the Lorentz equations, or the Lorentz transfor-

mation. They are equivalent to the Huygens’ equations s2 = s0 s1 and r2 =

r0 r1 for the composition of wavelets. Some basic algebraic manipulation of these equations produces

The velocity v and the dilation factor γ are defined as

Using these definitions, the Lorentz equations may then be written as

t2 = γ (t1 + vx1);

x2 = γ (x1 + vt1).

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 325 8/15/2014 12:00:14 PM 326 Remote Sensing in Action

The Lorentz equations, when written in this form, give time t2 and dis-

tance x2 in terms of time t1 and distance x1. Let us also define the velocities

Dividing the lower Lorentz equation by the upper one results in the cel- ebrated Einstein addition formula vv+ 1 . v2 = 1+ vv1

The above form of the Lorentz equations can be inverted to give time

t1 and distance x1 in terms of time t2 and distance x2. The Lorentz equations are “reflexive,” meaning the subscripts can be interchanged, and v can be changed to –v. The result is the form of the Lorentz equations given by

t1 = γ (t2 –vx2);

x1 = γ (x2 – vt2).

Huygens was concerned with remote detection, and his tool was the wavelet. In an isotropic medium, the wavelet must be spherical, and Huy- gens worked out the necessary mathematics. With the success of Maxwell’s electromagnetic theory, Einstein, in 1905, realized that the finite speed of light (which is an intrinsic part of electromagnetic theory) must be applied to mechanics as well. Maxwell’s theory required no change because it was developed on the basis of finite signal speed. However, Newton’s laws of motion had to be rewritten. Einstein did so by proposing two postulates on which relativity theory is based. The first postulate is that all the laws of physics are the same in every inertial reference frame. The second postulate is that light is always propagated in empty space with a definite velocity c, which is independent of the state of motion of the emitting body. The second postulate is equivalent to the condition that the ultimate signal speed is the velocity c of light in a vacuum. One result of the rewriting of Newton’s laws of motion was Einstein’s best-known equation, E = mc2. It says that energy is equal to mass times the square of the velocity of light. This equation transformed physics and, as everyone knows, many other areas also.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 326 8/15/2014 12:00:15 PM Appendix D: Special relativity theory 327

Relationship to Huygens

The Huygens’ construction incorporates the mathematics of special relativity (Figure D-3). To understand the spherical wavelets of Huygens, we must go back to geometry. Again, for ease of exposition, we will use circles instead of spheres. A theorem from geometry states: If a tangent and a secant intersect outside a circle, then the tangent is the geometric mean of the secant and its external segment. This theorem is basic to the understanding of Huygens, as will be shown in the following example. The horizontal line in Figure D-3 contains the origin O, the sending point S, and the receiving point R. The circle with diameter SR is called the Huygens’ circle. The center of circle T is the midpoint of diameter SR. The radius of the circle is x = TP = TS = TR. The tangent from origin O to the circle touches the circle at P. The length of line OP is designated by p. The secant is r = OR (the receiving time), and the external portion is s = OS (the sending time). The tangent is p = OP (the proper time). The Huygens’ circle (on right in Figure D-4) can be added to the bipolar stereographic projection. The result is the bicircular bipolar stereo-

Tangent = OP = p.

O S T R External segment Secant = OR = r. = OS = s.

Figure D-3. Geometric illustration of the mathematics of special relativity.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 327 8/15/2014 12:00:15 PM 328 Remote Sensing in Action

O S T R

M Figure D-4. Huygens’ circle is the one on the right.

graphic projection. We recall that the Huygens form of the Lorentz transformation is

s2 = s0s1, r2 = r0r1, p0 = 1, and p1 = p2 = p.

From these equations, we form

which is

This result is the stereographic form (or equivalently the relativistic Doppler form) of the Lorentz transformation.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 328 8/15/2014 12:00:16 PM Appendix D: Special relativity theory 329

World lines

In his 1908 address “Space and Time” delivered at Cologne, Germa- ny, Hermann Minkowski took time into account as a fourth dimension. Time had to be treated mathematically in a different way from the three spatial dimensions. Characteristic features of the Minkowski concept can be illustrated by means of the so-called Minkowski plane. Only space- time geometry is described by the Minkowski construction. One of the coordinate axes necessarily must represent the time axis because in the special theory of relativity, purely spatial geometry (the geometry of x, y, z) remains Euclidean. Thus, in the choice of reference frames, it is conve- nient to consider the x-, t-planes and to suppose that the relative velocity between two frames is in the direction of the x-axis. A particle at a given location at a given time represents an event. Ev- ery event in the real physical world occurs at a definite world point of the Minkowski diagram. No matter how the particle moves or not, the continuous sequence of events represented by this particle yields a certain curve in space-time called the world line of the particle. We shall deal in natural units, so the velocity of light is one. A light particle (a photon) has no mass. The velocity of a photon is equal to one. The velocity v of any particle that has mass must be less than one (v < 1). Figure D-5 graphically depicts this concept. Case 1 (Figure D-5a) represents the world line of a particle at rest. Case 2 (Figure D-5b) represents the world line of a particle moving at velocity v. The x-, t-axes are at right

angles to each other. In Case 1, a particle at rest is at the location x = x1. Its world line is a straight line parallel to the t-axis. In Case 2, a particle moves uniformly along the x-axis at velocity v. Its world line is a straight line inclined at angle q to the t-axis. Because v < 1, angle q cannot exceed 45°. For a light particle (a photon), v = 1, so angle q is 45°. Figure D-6 graphically depicts rectangular axes (Figure D-6a) and oblique axes (Figure D-6b). A light line is defined as a world line of light. The first step in constructing a Minkowski diagram is to pick a point as origin. Then draw the two light lines passing through the origin. One represents the northgoing light line and the other the southgoing light line. The two light lines are drawn at right angles to each other. The second step is to draw one of the two axes, either t or x. This axis will make a certain angle with one of the light lines. The other axis is drawn to make the same angle on the other side of the light line in question. As a result, the light line will bisect the angle between the two co-

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 329 8/15/2014 12:00:16 PM 330 Remote Sensing in Action

a) b) x x1

World line of World line of particle particle at rest moving at velocity v x v 1 θ x1

O t O t1

Figure D-5. Illustration of the world lines of particle: (a) Case 1, at rest, and (b) Case 2, in motion.

a) b)

O O

Figure D-6. Light lines as they appear on (a) rectangular axes and (b) oblique axes.

ordinate axes. The simplest case is when the angle between the two axes is a right angle. In such a case, the coordinate system is rectangular, and the light lines make angles of 45o with each axis. Figure D-6a shows the rectangular case, in which the axes are at right angles. Figure D-6b shows the oblique case, in which the axes are not at right angles. In either case, the light line bisects the angle between the axes. Now consider two inertial frames. The first will involve a certain set of variables. The second will involve the same set of variables but with a prime attached to each. The primed frame is moving at velocity v relative

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 330 8/15/2014 12:00:17 PM Appendix D: Special relativity theory 331

θ θ

Figure D-7. Rectangular and oblique axes plotted together.

to the unprimed frame. As we know, the xʹ-, tʹ-axes are related to the x-, t-axes by the Lorentz transformation,

xʹ = γ(x – vt); tʹ = γ(t – vx),

where the gamma factor is defined as γ = 1/ 1.− v2 Figure D-7 graphically depicts two sets of axes (rectangular and ob lique) plotted on the same diagram. The rectangular x-, t-axes are drawn at 45o angles to the light line. The oblique xʹ-, tʹ-axes are each at angle q to the light line. The Lorentz transformation shears the x- and t-axes into the

xʹ- and tʹ-axes. The straight line x = x1= constant is parallel to the t-axis, and

the straight line t = t1 = constant is parallel to the x-axis. The straight line xʹ

= x1ʹ = constant is parallel to the tʹ-axis, and the straight line tʹ= t1ʹ = constant is parallel to the xʹ-axis.

Relativity of simultaneity

The relativity of simultaneity can be stated in the form of a theorem: Two events O and F simultaneous in a given inertial frame are not simultaneous in any other inertial frame. Figure D-8 graphically depicts two oblique coordinate systems, one primed and the other unprimed. The two events O and F can be used to define the coordinate axes of the given inertial frame as follows: Con- nect O and F by a straight line, call this line the x-axis, and let O be the

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 331 8/15/2014 12:00:17 PM 332 Remote Sensing in Action

x'-axis x-axis

F Northgoing light line

t-axis

t'-axis

F' O Figure D-8. Two oblique coordinate systems plotted together.

origin. Draw the t-axis so that the light line bisects the angle between the x-axis and t-axis. The x-axis and t-axis form an oblique coordinate system. Any other coordinate system will have a different set of axes, say, xʹ and tʹ. The light line will bisect the angle between these axes, as shown in Figure D-8. In the xʹ-, tʹ-frame, simultaneous events lie along lines parallel to the xʹ-axis. Thus, we draw a line through F parallel to the xʹ-axis, and we let Fʹ be the intersection of this line with the tʹ-axis. By construction, events F and Fʹ are simultaneous in the xʹ-, tʹ-frame. Because Fʹ lies to the right of O on the tʹ-axis, event Fʹ occurs after event O. Therefore, event F occurs after O in the xʹ-, tʹ-frame. We conclude that events O and F, which are simultaneous in the given (unprimed) frame, are not simultaneous in the other (primed) frame. Events which are simultaneous with reference to one inertial frame are not simultaneous with respect to another inertial frame. Every inertial frame has its own particular time. Unless we are told the inertial frame to which the statement of time refers, there is no meaning in the statement of the time of an event. Before the advent of the special theory of relativity, it had always been assumed tacitly in physics that the statement of time had an absolute significance, i.e., that it is independent of the inertial frame in question, and therefore, time was the same in all inertial frames. However, this assumption of absolute time was incompatible with the tenets of relativity theory and thus had to be abandoned.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 332 8/15/2014 12:00:18 PM Appendix D: Special relativity theory 333

Einstein’s train

In the textbook treatments of relativity theory, the relativity of simultaneity is not presented as a mathematical theorem as we have done, but instead, it is given in the form of examples. The premium example of simultaneity, from which all the others are modeled, was given by Einstein. It is called Einstein’s train. A train travels north along a straight railroad track with constant velocity v. The passengers use the train as an inertial frame in which they are at rest, and they regard all events in reference to the train. The railroad track also represents an inertial frame, and workers on the track are at rest in this frame. Lightning has struck the track (depicted by the x-axis) at two places — M (in the south) and N (in the north). Make the additional assumption that these two lightning flashes occurred simultaneously. How do we determine whether there is sense in the statement of this assumption? We offer the following method with which to test the simultaneity. By measuring along the track, the length of the connecting line MN should be determined. Take the origin at the midpoint of distance MN. If a worker at the midpoint of the track perceives the two flashes of lightning at the same time, then they are simultaneous. There are two world lines. One is the world line of the center O of the track, which may be considered as a particle at rest. The other is the world line of the center Oʹ of the southgoing train, which may be considered as a particle moving south at constant velocity v. Figure D-9 shows two events M and N, which are simultaneous in the track frame but are not simultaneous in the train frame. Figure D-10 is an alternative diagram for the two events M and N, which are simultaneous in the track frame but not in the train frame. We will show that this alternative diagram is in the form of a stereographic projection. In the following discussion, either diagram may be used. The flash from M travels along the northgoing light line, and the flash from N travels along the southgoing light line. The worker at O perceives the two flashes reaching him at the same time if — and only if — the two light lines intersect the world line of O at the same event P, as depicted in Figures D-9 and D-10. Thus, either of the two diagrams shows the simultaneity of events M and N with respect to the track. In such a case, the world line of O can serve as the t-axis of the inertial frame of the railroad track, and line MN can serve as the x-axis of the inertial frame of the track.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 333 8/15/2014 12:00:18 PM 334 Remote Sensing in Action

N (north point) World line of O Southgoing

R World line of O' O' and O S (center of southbound train)

Southgoing train travels in direction from N to M.

M (south point)

Figure D-9. Two events M and N that are simultaneous in the track frame but not in the train frame.

World line of O N (north)

S R O' and O World line of O’

M (south) Figure D-10. Alternative diagram for M and N that is in the form of a stereographic projection.

The two light lines do not intersect on the world line of Oʹ of the southgoing train. As a result, the two flashes would not be simultaneous with respect to the world line of Oʹ. As seen in Figures D-9 and D-10, event R occurs after event S. Thus, the diagrams show the nonsimultaneity of

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 334 8/15/2014 12:00:19 PM Appendix D: Special relativity theory 335

events M and N with respect to the train. Whereas the world line of Oʹ can serve as the tʹ-axis of the inertial frame of the railroad train, line MN cannot serve as its xʹ-axis. Another line must be used as the xʹ-axis, namely, the line chosen so that the light line through the origin bisects the angle between the xʹ-axis and tʹ-axis. The question is: Are two events (e.g., the two strokes of lightning M and N) which are simultaneous with reference to the track frame also simultaneous with reference to the train frame? Let us turn to the analysis given by Einstein. He shows in the following way that the answer must be in the negative. When we say that lightning strokes M and N are simultaneous with respect to the track, we mean that the rays of light emitted at places M and N, where the lightning strikes occur, meet each other at midpoint O of length MN of the track. However, events M and N also correspond to positions M and N on the train. Let Oʹ be the midpoint of distance MN on the southbound train. Just when the flashes of lightning occur (as judged from the track), point Oʹ naturally coincides with point O, but the southbound train moves southward with velocity v of the train. A worker sitting in position O on the track does not possess this velocity, so he remains permanently at O. The light flashes emitted by lightning strikes M and N would reach him simultaneously, i.e., they would meet just where he is situated (event P). A passenger at midpoint Oʹ of the southbound train (considered with reference to the railroad track) is hastening toward the flash of light coming from southern point M while he rides ahead of the flash coming from northern point N. As seen in Figures D-9 and D-10, the northgoing flash from M hits the world line of Oʹ at S, and the southgoing flash from N hits the same world line at R. Because S is below R on this world line, the observer at Oʹ sees the beam of light emitted from M earlier than he will see that emitted from N. Thus, a passenger in the train frame, knowing that Oʹ is the midpoint of the train and seeing the flash from M arrive before the flash from N, must conclude that the lightning strike occurred at M before it occurred at N. Figure D-11 establishes our claim that the alternative diagram shown in Figure D-10 is in the form of a stereographic projection. In conclusion, a passenger in the train frame would claim that the two lightning strikes are not simultaneous, despite the fact that observers in the track frame would say they are.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 335 8/15/2014 12:00:19 PM 336 Remote Sensing in Action

N (North Pole) World line of O' P (center of track)

S T R O’ and O World line of O (center of southbound train)

M (South Pole)

Figure D-11. Graphic proof that the diagram in Figure D-10 is a stereographic projection.

Dilation of time

A small railroad station is located on a straight railroad track with vis- ibility two miles in each direction. High-speed bullet trains do not stop at the station. As a safety warning, a sign at the two-mile mark in each direction instructs the train engineer to sound the whistle to alert the people at the station to get off the track. The whistle would then sound continuously until the train is through the station. The sign in the station reads: “When you first hear the whistle, the train is still two miles away. Immediately get off the track.” The first day, a bullet train traveling at 0.18 miles per second hits and destroys a wheelbarrow on the track at the station. The person who owned the wheelbarrow sues the railway company. The grounds for the suit are that when the person heard the whistle, the train was much too close for him to get the wheelbarrow off the track. The railway company argues that when the person heard the whistle, the train was two miles away, and thus the owner had plenty of time to get the wheelbarrow off the track. The speed of sound is approximately one mile in five seconds (= 0.20 miles per second). We use capital C for the speed of sound. The attorney for the owner argues that when the owner heard the whistle, the train was only 0.20 miles from the station and not two miles away, as the sign in the station claims. In view of this fact, the owner did not have enough time to get the wheelbarrow off the track.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 336 8/15/2014 12:00:19 PM Appendix D: Special relativity theory 337

C = velocity of sound = 0.20 miles per second. t = time elapsed = 10 seconds. Distance = Ct = (0.20) (10) = 2.0 miles.

V = velocity of train = 0.18 miles per second. t = time elapsed = 10 seconds. X = Distance = Vt = (0.18) (10) = 1.8 miles.

Actual distance = 2.0 – 1.8 = 0.2 miles.

Figure D-12. Calculation of the actual distance in miles.

1 sonic-second = 0.20 miles. c = C/C = 1 sonic-second per second = 1. t = time elapsed = 10 sonic-seconds. Distance = ct = (1) (10) = 10 sonic-seconds.

v = V/C = velocity of train = 0.18/0.20 = 0.9. t = time elapsed = 10 seconds. x = distance = vt = (0.9) (10) = 9 seconds.

Newton proper time = t – x = 10 – 9 = 1 second.

Figure D-13. Calculation of Newton proper time.

When asked by the judge to explain his reasoning, the attorney responded, “In the time interval of 10 seconds after the whistle was first sounded, the sound traveled 0.20 times 10 equals 2.0 miles, namely, the distance from the two-mile mark to the station. In the same time interval, the train traveled 0.18 times 10 equals 1.8 miles. Thus, when my client first heard the whistle, the train was only 2.0 minus 1.8 miles away; that is, the train was at the actual distance of 0.2 miles from the station and not two miles away at all” (Figure D-12). The attorney won the case for the owner of the wheelbarrow. Figure D-13 graphically depicts the Newton proper time. (Here, the term proper time is modified by the word Newton. Without any modifica- tion, proper time always refers to Einstein proper time.) To convert the relevant distances to time, because sound is used as the signal, the conversion factor is the speed of sound, namely C = 0.20 miles per second. The

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 337 8/15/2014 12:00:20 PM 338 Remote Sensing in Action

distance from the warning sign to the station is two miles, which corresponds to the time of 2/0.2 = 10 seconds. The time interval t = 10 seconds is called the dilated time. The distance that the train travels in the time interval t = 10 seconds is 0.18 × 10 = 1.8 miles. This distance in terms of time is x = 1.8/0.2 = nine seconds. The actual distance of 0.2 miles in terms of time is p = 0.2/0.2 = one second. The time interval p is called the Newton proper time. The judge determined that the Newton proper time p = 1 should have been used, and not the dilated time t = 10. The critical Newtonian equation is p = t – x. In summary, the time interval of 10 seconds is called the dilated time, and the time interval equal to one second is called the Newton proper time. The lesson is that the Newton proper time is appropriate, not the dilated time, when dealing with sound waves. An example of the use of relativistic (or Einstein) time involves the muon, the first of the short-lifetime particles found by physicists (discovered in 1938). The muon is similar to the electron but with about 200 times the mass of an electron. Cosmic rays hitting the earth from outer space create muons by collisions in the atmosphere. These muons, which are very energetic particles, can move with 199/200 of the velocity of light. In other words, they travel at speed v = 199/200 = 0.995. They cover a distance of x = 19.9 light-microseconds on the trip to the earth before they decay. A point in the atmosphere where a muon is created corresponds to the marked warning point on the railroad track in the previous example. The earth represents the railroad station. A photon (i.e., a particle of light) corresponds to the sound pulse. The photon goes to the earth at a speed of c = 1. The muon corresponds to the train. The muon goes to the earth at a speed of v = 199/200, and the distance traveled is x = 19.9 light-microseconds = 5970 meters. Velocity v and distance x are physical observations that we can measure. Time t is then computed. The traveltime is t = x/v = 19.9/(199/200) = 20 microseconds. Figure D-14 graphically depicts the Einstein proper time. What is the Einstein proper time p? In the train example, the Newton proper time is given by p = t – x. However, the muon example must use relativistic (or Einstein) time, and the answer is given by the Pythagorean theorem. The dilated time t is the hypotenuse of a right triangle. The distance x is one leg of the right triangle, and the Einstein proper time p is the other leg. Thus the Einstein proper time for a stationary muon is found by solving the famous Pythagorean formula

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 338 8/15/2014 12:00:20 PM Appendix D: Special relativity theory 339

c = velocity of light = 1 light-microsecond per microsecond = 1. v = velocity of muon = 199/200 = 0.995.

t = dialated time = 20 seconds. p = proper time = 2 seconds. x = distance = vt = (0.995) (20) = 19.9 light microseconds.

Figure D-14. Calculation of Einstein proper time.

Stationary muons (i.e., muons fixed at one place) can be produced artificially in the laboratory. These muons have the Einstein proper lifetime of about p = two microseconds when they are measured at rest in a laboratory on earth. It seems puzzling that a muon lasts for only p = two microseconds in the laboratory, but muons can travel a distance of x = 19.9 light-microseconds (which is 300 × 19.9 = 5970 meters) in the atmosphere. Let us examine this situation. The period of the internal lifetime clock of the muon is p = two microseconds. For a muon moving at speed v = 0.995, the period is dilated to t = 20 microseconds. This computation is based on the fact that the muon hurls through the atmosphere a distance that we measure on earth as x = v t = 19.9 light-microseconds. This situation is not different from the train example except that now we are dealing in relativistic time, which requires Pythagoras. The train took 10 seconds to go a distance of nine sonic-seconds. The Newton proper time is one, which is the difference given by 10 minus nine. The muon took 20 microseconds to go a distance of 19.9 light-microseconds. The Einstein proper time is two, which is the other leg of the right triangle with hypotenuse 20 and leg 19.9. Another explanation is as follows. A stationary muon is not moving through space, so it travels through time as fast as possible. It therefore ages rapidly and dies in the proper time of p = two microseconds. A moving muon travels through time at a reduced speed, so it dies in the dilated time of t = 20 microseconds. (A trip at a reduced speed takes longer; this is a trip at reduced speed through time.)

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 339 8/15/2014 12:00:20 PM 340 Remote Sensing in Action

Light clocks and some interesting conclusions

Imagine two sticks with the same length. One stick is stationary, and the other is placed on a moving spaceship so that it is perpendicular to the direction of motion (like the mast of a ship moving horizontally on the sea). A person on the ground and a person on the spaceship can each make a mark on the other’s stick as they pass each other. By symmetry, one mark cannot be above or below the other, and thus the two marks are at the same coordinate. Therefore, the length of each stick does not change. In other words, the stick neither stretches nor shrinks. To an observer on the ground, the moving stick perpendicular to the direction of motion on the spaceship has the same length as the stationary stick on the ground. The conclusion is:

Result 1: A stick moving with a constant velocity in a direction perpendicular to itself has the same length as the stick when it is at rest.

Now imagine a special clock constructed to find how fast moving clocks run compared with stationary clocks. The clock is made from a stick with a mirror at each end, and a beam of light is made to flash back and forth between the two mirrors. Amazingly, this simple device will show why an observer sees time on a moving clock as going slower than time on an identical stationary clock. Two light clocks with the same length are constructed and synchro- nized. One is placed on a spaceship and mounted perpendicularly to the direction of motion. Figure D-15 graphically depicts the light clock at rest (Figure D-15a) and a moving light clock (Figure D-15b). The spaceship, as shown in Figure D-15b, travels uniformly in a horizontal direction from left to right so that the stick is in a vertical position. The other clock is on a fixed platform on the ground. If the length of the stick is g, then it takes time g/c for the light pulse to go from the bottom mirror to the top mirror on the fixed clock and another g/c for the reflected pulse to go from the top mirror to the bottom one. Because we are using natural units, c = 1, each of these time durations is simply g, the length of the stick. A round trip takes time 2 g. This is the time interval between ticks of the clock when the clock is at rest. This time interval is called the proper time p; that is, p = 2 g. However, when the observer on the ground looks at the clock going by, he sees that the light, in going from mirror to mirror, takes a zigzag

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 340 8/15/2014 12:00:20 PM Appendix D: Special relativity theory 341

a) b) Top

g t/2 t/2 g Light line down Light Light line up Light

Bottom A vt/2 B vt/2 C

Light clock at rest Light clock moving to right

Traveltime = 2g = p = proper time. Traveltime = 2(t/2) = t = dilated time.

Figure D-15. Light clock (a) at rest and (b) moving.

path. Let t be the time interval between ticks (i.e., one round trip) when the stick is moving sideways. In time t/2, the bottom of the stick moves distance vt/2 from A to B and vt/2 again from B to C. In going from the bottom mirror to the top mirror, light travels distance ct/2 from A to D and ct/2 again from D to C. Because c = 1, the quantity ct/2 becomes simply t/2. Line AD is the hypotenuse of a right triangle, one side of which is g and the other vt/2 . The Pythagorean theorem gives

2 Because 1− v is less than one, t is greater than p. This result shows that an observer on the stationary ground perceives that one tick of the moving clock takes a longer time than one tick of the stationary clock. In other words, the dilated time t for one click of the moving clock is greater than the proper time p for one click of the stationary clock. The greater the velocity, the more slowly the moving clock appears to run. Therefore, we have:

Result 2: If a light clock takes time p between ticks when it is at rest, then when its stick moves perpendicularly to the motion at velocity v, it appears to an observer at rest that the clock takes the longer (or dilated)

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 341 8/15/2014 12:00:21 PM 342 Remote Sensing in Action

time tp=−/1 v2 between ticks.

Because t > p, we see that the clock runs more slowly when it is moving than when it stands still. The rate of a clock is inversely proportional to the time between ticks. Because 1/t is less than 1/p, we can say that an observer at rest finds that the rate of a clock moving past him is less than the rate of a clock at rest with him. The conclusion is:

Result 3: If a light clock takes time p between ticks when it is at rest, then when the clock moves with velocity v, it takes the longer (or dilated) time tp=−/1 v2 between ticks. This result does not depend on whether the clock moves perpendicularly or parallel to its length. Fur- thermore, this result is true for any clock whatsoever.

Michelson-Morley experiment and length contraction

As discussed earlier, the premier experiment of special relativity was conceived by Maxwell and was carried out by Albert A. Michelson and Edward W. Morley in 1887. The Michelson-Morley instrument was in es sence two perpendicular light clocks. The experiment verified that both clocks kept the same time regardless of their orientation in respect to the earth’s motion around the sun or the sun’s motion around the center of the galaxy and so forth. This was the first empirical verification of Result 3 above. Some further analysis will lead to another, astonishing, result. When the instrument is at rest, the two clocks tick at the same rate. This is also true when they are in motion. Now imagine that one stick moves parallel to the motion, and the other stick moves perpendicularly to the motion. Both clocks tick at the same rate. When the moving instruments pass a stationary observer, the perpendicular stick (by Result 1) has rest length p. The length of the stick moving parallel to the direction of motion is as- signed the unknown value h. Can the value of h be deduced from the knowledge that the parallel- stick clock keeps the correct time? In the stereographic projection shown in Figure D-16, proper time p on the moving frame is stretched to dilated time t on the fixed frame. Length g on the moving frame is compressed to contracted length h on

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 342 8/15/2014 12:00:21 PM Appendix D: Special relativity theory 343

With extremes OS and OR, OQ = harmonic mean, OU = geometric mean, J G OT = arithmetic mean.

E P OF = h = contacted length OF = g length

OP = p = proper time of signalS O U T R O OT = t = dilated time of signal

Figure D-16. The stereographic projection shows how the proper time on the moving frame p is “stretched” to the dilated time t on the fixed frame.

the fixed frame. The time and distance axes (t, x) of the fixed observer are OT and OE, respectively. The time and distance axes (tʹ, xʹ) of the moving clock are OP and OF, respectively. In the moving frame, the parallel-stick clock with length g is shown at two positions, OF and PG. The pulse of light leaves the lower mirror at point O at time zero and distance zero in either coordinate system. The pulse arrives at the upper mirror at point A where the pulse is reflected. Finally, the pulse arrives back at the lower mirror at point P. The proper time is p = OP, and the dilated time is t = OT. In other words, the observer in the fixed frame sees proper time p as dilated to value t. Similarly, the observer in the fixed frame sees length g contracted to value h. This shrinking of moving things as seen by an observer at rest is called the FitzGerald contraction and sometimes also the Lorentz contraction. The conclusion is:

Result 4. A stick of length g (at rest) moving with a constant velocity v in a direction parallel to itself has the contracted length h when seen by an observer at rest.

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenD_v4--SCS.indd 343 8/15/2014 12:00:22 PM This page has been intentionally left blank

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 Index

Aberdeen, 183 midpoint time, and Newtonian theory, absolute space, 219, 220, 223, 224 254 absolute space and time, Newtonian concept normalized, 273 of, 223 symmetry with harmonic mean, 263 absolute time, 101, 219, 220, 223, 224, 242 Arnold, Matthew, 97, 294 and relativity theory, 242 ‘A Scandal in Bohemia,’ 41 absolute velocity, 21 ascending sequence, 264 absolute zero, 25 Asteroid, The Dynamics of an, 1, 42, 52, Académie Royale Des Sciences, 146 115, 183, 194 accelerated motion, 295 asteroid belt, 115 Achilles, and tortoise, 273–275 asteroids, 8 acoustic Doppler equations, 51 astrolabe, 29, 47, 70, 72, 73, 74, 75, 109, active remote detection, 94, 304 110, 116, 140, 141, 147, 160, 161, 165, and Christiaan Huygens, 304 166, 201, 256 Adams, John Couch, 189, 194, 195 and description, time and space, 141 Adler, Irene, 41 and stereographic projection, 141, Admiralty, The British, 44, 45, 46 147, 161, 166 Adonaïs, 64 invention of, 256 Adventure of the Empty House, 2, 4, 207 Astrolabe, A Treatise on the, 73 Aegean Sea, 97 astronomy, 9 aether, and aether theory, 181, 182, 309 A Study in Scarlet, 28, 173, 254 the privileged inertial frame, 309 Atlantic Ocean, 120, 253 Aguilon (see d’Aguilon) A Treatise on the Astrolabe, 73 Airbus, cruising velocity, 264, 265 A Treatise on the Binomial Theorem, 4 Alexandria, 256 Aubrey, John, 66 alpha particles, 8 Aubrey holes, 66–67 altitude and azimuth, 69–70 Avebury, stone circle, 65 Ampere’s law, 306 azimuth and altitude, 69–70 amplitude of a wave, 258 invention of, 70 analog computer, 161 anchor escapement for pendulum clock, 302 angular momentum 20 Babbage, Charles, 190, 192 Annalen der Physik, 7 Babbage’s difference engine, 192 Anne Wardwell (see Wardwell, Lady Anne) Babylonians, 66 Archimedes, 147, 149 bacteria, 122 Aristarchus, and radius, earth’s orbit, 155 Baker Street, 31, 37, 38, 40, 41, 42, 56, 57, Aristotle, 20, 28, 64, 122, 226, 262, 294, 64, 144, 180, 193, 225, 234 295 ball, defined, 311 and the golden mean, 226 baritsu, 3, 207 arithmetic mean, 86, 101, 227–229, 232, Banquo, 117, 205 245, 254, 255, 263, 265, 266, 267, Barrow’s translation of Euclid, 194 272–273 Basel, 32, 61 and geometric mean, 227–229 Baskerville, Sir Henry, 21 and midpoint time, 232, 254 Baskerville Hall, 21 defined, 263 Baskervilles, The Hound of the, 21, 240, 242 345

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 345 8/15/2014 12:01:24 PM 346 Remote Sensing in Action

Battle of Hastings, 81, 82, 83, 84–88, 89, Christiaan Huygens (see Huygens, Chris- 91, 94 tiaan) and remote detection, 83 clock, and periodicity of hands, 257–258 Battle of New Orleans, 212 coal bed, seismic reflection from, 52–53 bee dance, 11 coal seams, 52–54 Belgium, 187 Collier’s, 65 Bern, 213, 214, 215 Colloden, Daniel, 185, 186, 201 Bernini, 293 and speed of sound in water, 185, 186 Besso, Michele, 215 Colossus, 140 Bibliothéque Royale, 146 commutation channel, 268 binomial theorem, 4, 115 Conrad, Joseph, 96–97 Binomial Theorem, A Treatise on the, 4 conservation laws, 20 bipolar stereographic projection, 315–316 conserved quantities, fundamental, 20 Black Forest, 166 control theory, 195–196 Blake, William, 138, 253 conventional units compared with natural ‘Bohemia, A Scandal in,’ 41 units, measurement of a quantity, 218 Boltzmann, Ludwig, and entropy, 197 Copernican system, 304 Bombay, 88 Copernicus, 74, 140 Boole, George, logician, 192 Cordelia, daughter of King Lear, 105, 106 Boyle, 131 Cosmotheoros, 122 Britain, 65, 106, 113 Coudenberg Castle, Brussels, 32, 34, 50, British Empire, 92 106, 107, 109, 113, 117, 118–127, 128, British Museum, 69 136, 137, 144, 161 British Navy, 46 Sherlock Holmes, handwritten ac- Brussells, 32, 43, 105, 106, 107, 108–117, count of experiences in, 121–127 118, 119, 120, 136, 137, 138, 141, 144, Cremona, 117 147, 160, 161, 170, 171, 172, 185, 201 Curiosities of Literature, 247 train to, 25 April, 1891, 108–117 cycloid, 302 Buckingham Palace, 42 cycloidal clock, 302 Burgundy, 105 Burgundy, duke of, 106, 107, 116, 119, 120, 121, 165 d’Aguilon, François, 29, 51, 109, 110, 111, Burgundy, House of, 123, 124, 125 112, 113, 115, 123, 130, 135, 136, 141, Byron, Augusta Ada, Countess of Lovelace, 144, 145, 147, 149, 159, 160, 161, 162, 189–194, 211 163, 165, 166, 169, 172, 201, 293 and Six Books of Optics, Long Valued by Philosophers and Mathemati- calculus, differential, 295 cians, 145, 147, 159 calculus of variations, and Huygens, 306 and stereographic projection, 293 Calisto, 150 Dalton, John, 8 Cambridge, 147 ‘Darkness,’ 188, 189 Cambridge, University of, 51, 113, 189, 194 da Vinci, Leonardo, 101 Canterbury, 32, 59, 61, 62, 64, 65, 75, 94 deductive reasoning, 109 Canterbury Cathedral, 65 Democritus, 8 Canterbury Tales, The, 73 De Morgan, Augustus, 192 Cassini, Jean-Dominique, 155, 156, 175, Dendera, Egypt, 70 176, 177 Descartes, and velocity of light, 305, 307 celestial circle, 172 descending sequence, 264 celestial sphere, 130 detection, 9, 10 Ceres, 115, 194 described, 9 Cestius, pyramid of, 294 remote, 9, 10 Charles the Bold, 120 Detection, The Whole Art of, 5, 10 Chaucer, 73 Dieppe, 32, 61, 96, 105 Childe Harold’s Pilgrimage, 190 differential geometry, 302

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 346 8/15/2014 12:01:25 PM Index 347

digital computer, 215 moving source, fixed receiver, digital computer algorithm, first, 192 289–292 Dijon, 120 Doppler effect, one way, 268–269, 271 dilated period, 244 moving source and moving observer, dilation of time, 254 271 Diogenes Club, 40, 42, 43, 123 moving source and stationary ob- Disraeli, Isaac, Curiosities of Literature, server, 269 247 stationary source and moving ob- distance, unit of, definition, and velocity of server, 268–269 light, 310 Doppler effect, optical, 103, 159, 172, 173, ‘Does the Inertia of a Body Depend on its 174, 176, 177, 179, 182 (see also Dop- Energy Content?’, 23 pler effect, relativistic) Doppler, Christian Johann, 62 Doppler effect, relativistic, 75, 102, 103, Doppler effect, 51, 55, 61, 62–63, 75, 89, 104, 293–319 (see also Doppler effect, 98–99, 100, 101, 102, 103, 104, 156, optical, and Doppler effect, relativistic, 159, 172, 173, 174, 176, 177, 179, 182, symmetry of) 248, 255–292, 293–319 (see also Dop- Christiaan Huygens, 300–303 pler effect, types, below) Doppler effect, relativistic, symmetry acoustical, 51, 100, 101, 103 of, 318–319 and electromagnetic waves, 103 Galileo and motion, 294–300 fixed source and moving receiver, 61 Huygens and remote detection, moving source, fixed receiver, 62–63 303–311 primary use, 156 mathematics, relativistic Doppler ef- Doppler effect, classical, 51, 61, 98–99, fect, 293–294 100, 101, 102, 103, 255–292, 318 (see stereographic projection, 311–318 also Doppler effect, classical, graphical Doppler effect, relativistic, symmetry of, depiction) 318–319 and arithmetic mean, 255 and classical Doppler effect, relative and harmonic mean, 255 to sound waves, 318 and mechanical waves, 255 and relativistic Doppler factor, 319 ball game, 275–279 and relativistic symmetry, 318 described, 256 classical Doppler factor, 318 Doppler factors, classical, two, need fixed object to moving object, 319 for, 273–275 symmetry, preservation of, 318, 319 fundamental asymmetry of, 255 Doppler effect, theory of, and Pythagorean graphical depiction, 288–292 means, 271 historical background, 255–256 Doppler effect, two-way, 272–273, 280, in terms of frequencies, 286–288 281–284 one-way, 268–271, 284–286 moving to stationary and back, posi- Pythagorean means, 262–265 tive and negative velocity, 283 reconciliation, 265–267 positive and negative velocity, illus- remote sensing, principles, 256–258 trated, 280 sound waves, 318 signal transmitted from fixed standing waves, 259–262 platform, received at moving stationary observer, 98–99, 101, 102, platform, 272 103 signal transmitted from moving plat- two-way, 272–273 form, received at fixed platform, two-way, and periods, 279–284 272 wave motion, 258–259 stationary to moving and back, posi- Doppler effect, classical, graphical depic- tive and negative velocity, 281 tion, 288–292 fixed source, moving receiver, 289–292

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 347 8/15/2014 12:01:25 PM 348 Remote Sensing in Action

Doppler factor, 29, 51, 54, 60, 61, 89, 90, Doppler factor, symmetrical relativistic, 255 98–100, 101, 102, 103, 137, 159, 172, Doppler factors, 265, 266, 270, 271, 273, 174, 179, 182, 183, 247–249, 250, 251, 274, 275, 282, 284, 285 253, 254, 255, 293, 317–319 (see also ascending sequence, 282, 284 Doppler factors) classical, two, need for, 273–275 acoustical, 100, 101 defined, 271 classical, 98–99, 248, 249, 318 (see descending sequence, 282, 284 also Doppler factor, classical) first, and arithmetic mean of Pythago- classical, and Newtonian (absolute) ras, 285 time, 248, 249 first classical, 265 mathematics of, and geometry of first classical, defined, 274 stereographic projection, 318 fixed source to moving receiver, use optical, 51, 54, 100, 101, 103, 109, of, 285 137, 153, 159, 172, 174, 177, 179, moving source to fixed receiver, use 182, 183 of, 285 relativistic, 100 relative to normalized Pythagorean relativistic, and mathematics of spe- means, 273 cial relativity, 254 second, 266, 285 relativistic, and stereographic projec- second, and normalized harmonic tion factor, 317 mean of Pythagoras, 285 Doppler factor, classical, 98–99, 248, 249, second classical, defined, 275 318 double-refraction problem, Iceland spar, 176 and distinction of stationary object Dover, 32, 59, 60, 61, 64, 82, 83 and moving object, 318 Doyle, Conan, 108 and Newtonian (absolute) time, 248, Duke of Marlborough, 118 249 Duke of Wellington, 212 fixed object to moving object, 318 Duke Philip, 32, 34, 43, 50, 51, 107, 109, moving object to fixed object, 318 114, 119, 120, 121, 122, 123, 124, 125,

Doppler factor k1, 60 126, 128, 129, 136, 137, 144, 147, 172, Doppler factor, optical, 51, 54, 100, 101, 201 103, 137, 159, 172, 174, 179, 182, 183 Dunsinane Hill, 76 and relativistic Doppler factor, 54 Dupin, 173 of Huygens, 51, 54 Durham, University of, 194 Doppler factor, relativistic, 54, 100, 101, Dynamics of an Asteroid, The, 1, 42, 52, 102, 103, 247–249, 250, 251, 253, 254, 115, 183, 194 255, 293, 319 and electromagnetic waves, 103 and mathematics of special relativity, earth, 150, 152, 156, 157, 158 254 and rendering of proper period on and optical Doppler factor, 54 Jupiter, 158 and sending period, 254 diameter, 157 definition of, and geometric mean, earthquakes, 52 253, 254 earthquake seismology, 83 fixed object and moving object, no and passive remote detection, 83 distinction, and necessity of elec- earth’s moon, as a clock, 304 tromagnetic wave as signal, 319 Eastbourne, 32, 65, 81, 82, 92, 94 fixed source to moving receiver, and Easter, 140 geometric mean, 293 East Indies, 188 geometric mean, fundamental sym- echolocation, 46 metry, and Einstein’s theory of ecliptic, 75 special relativity, 293 Edward the Confessor, 82 moving source to fixed receiver, and E = mc2, 23 geometric mean, 293 Egypt, 17–19, 234, 236 symmetrical relativistic, 255 obelisks, 17–19

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 348 8/15/2014 12:01:25 PM Index 349

Einstein, Albert, 1–35, 40, 54, 55, 64, 74, English Channel, 32, 96, 97, 105, 190 80, 98, 101, 180, 181, 182, 187, 189, entropy, 193, 197, 200, 202–203 208, 215, 217, 220, 221, 222, 223, 224, and order, 197 225, 226, 229, 231, 232, 233, 234, 237, concept, 193, 197 246, 247, 248, 250–254, 255, 293, 300, described, 203 308 equinox, precession of, 256 and evaluation of Maxwell’s formula- Euclid, 9, 27, 56, 70, 71, 194, 256 tion of time-space laws, 308 axioms of, 9 and special relativity, 182 Barrow’s translation of, 194 and time axes, 250 Euclidian geometry, 303 Ideas and Opinions, 300 Eudoxus, 145 relativistic time, 101 Euphranor, 264 relativity theory, 20, 101, 226, 237, Europa, 150 250–254 Europe, 204, 209, 250 Einstein’s system, space and time, dual evolute, 303 nature, 223 Eye of Horus, 236, 237, 238 Einstein’s theory of relativity, 226, 237, and stereographic projection, 238 250–254 essence of, and mean values, 226 symmetry in, 250–254 Faraday’s law, 306 Einstein’s theory of special relativity, 293 Fates, the three, 140 Einstein’s view of space and time, compared Faustus, Dr., 165 with Newton’s view, illustrated, 247 Fibonacci, Leonardo, 72 electric charge, 20 Fibonacci number, 72 electric constant, Maxwell’s, 307 Fibonacci sequence, 72 Electrodynamics of Moving Bodies, On the, ‘Final Problem, The,’ 4, 30, 33, 39, 59, 60, 7 207 electromagnetic field, and aspects of elec- Fleance, 117, 205 tricity, magnetism, and light, 308 Fleet Street, London, 225 electromagnetic signals, 9, 219 fluid, defined, 257 and special theory of relativity, 219 Fontana della Barcaccia, Rome, 293 electromagnetic wave, color of, as related to “Fountain of the Old Boat,” Rome, 293 frequency, 132 France, 106, 123, 146 electromagnetic wave, intensity of, as Frankenstein, Victor, 76, 193, 199 related to amplitude, 132 Frankenstein; or, The Modern Prometheus, electromagnetic wave equation, 23 189, 193, 202 electromagnetic waves, 8, 23–24, 46, 103, Frederick, Emperor, 120 309 French Academy, 156, 175 and relativistic Doppler effect, 103 French Guiana, 156 speed relative to speed of light, 23–24 French Royal Academy, 174 propagation, and vacuum, 309 French Royal Observatory, 174, 175 propagation, as distinguished from frequencies, separation between, measure- propagation, mechanical waves, ment of, 261 309 frequency, compared with period, 62, 91 electromagnetism, 224 frequency, measurement in hertz, 260 electron, date of discovery, 8 frequency and time, 257 elementary particles, 8 frequency of waves, 258 elm, shadow of, 15–17, 153 Fresnel, Augustin, 181, 305 Emperor Frederick, 120 and mathematics of Huygens’ prin- Empty House, Adventure of The, 2, 4, 207 ciple, 305 Encyclopaedia Britannica, 181, 308 and wave theory of light, 181 energy, 20, 22–26 From the Earth to the Moon, 23 and mass, explained, 22–26 England, 146, 174, 187, 188, 195, 214

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 349 8/15/2014 12:01:25 PM 350 Remote Sensing in Action

gain, defined, 264 giant rat of Sumatra, 58 Galilei, Galileo (see Galileo) Gibbs, Willard, 103 Galileo, 146, 150, 294–300, 307 Gilbert, W. S., 82 and accelerated motion, 295 Giza, great pyramid of, 138 and beginning of modern science, 296 Glasgow, 13 and book, Two New Sciences, 296 Godwin, Mary, 76, 188 and concept of inertia, 294 Godwinson, Harold, 82 and discovery of satellites of Jupiter, Goethe, 138 150 golden mean, 226 and Einstein’s statement of opinion, golden ratio, 70, 72 300 golden rectangle, 70, 71, 72 and experiment with inclined plane, ‘Governors, On,’ 196 296, 300 Grande Hotel, Strasbourg, 108, 138, 143 and experiment with pendulum, gravitation, universal, law of, 299 296–297 gravity, as reference, 13 and law of conservation of mechani- Great Birnam Wood, 76 cal energy, 298 great circle of a sphere, defined, illustrated, and law of inertia, 294 312 and law of motion, 299–300 Greece, 166 and law of universal gravitation, 299 Griffin, Charles, Glasgow, 5 and pendulum, 146 Grimpen Mire, 242 and velocity of light, 307 gunpowder, 123 definition of uniform accelerated motion, 295, 298, 299 definition of uniform motion, 295 Hamiltonian mechanics, 306 definition of uniform velocity, 295, harmonic mean, 255, 263, 265, 266, 267, 298, 299 272, 273, 318 experiment, falling bodies of different defined, 263 weight, 295 normalized, 273 parabola, and motion of projectiles, symmetry with arithmetic mean, 263 296 Hastings, 32, 65, 81, 82, 83, 84, 88, 89, 91, uniform acceleration transformed to 94 uniform velocity, 299 Battle of, 81, 82, 83, 84–88, 89, 91, Ganymede, 150 94 Gauss, Carl Friedrich, 194 Hathor, goddess, 70 Gemmi Pass, 201 heelstone, Stonehenge, 67, 68–69 Geneva, 32, 61, 105, 136, 185–187, 188, Hertz, Heinrich Rudolf, 178, 181, 308 189, 193, 206 and existence of radio waves, 308 Geneva Public Library, 201 electromagnetic waves, and radio Geneve, Le Journal de, 1 waves, 178 geometric mean, 100, 101, 103, 154, 222, Hipparchus of Nicaea, 29, 73, 256, 293 227–229, 232, 253, 254, 255, 262–263, and astrolabe, invention of, 256 264, 293, 316 and precession of equinoxes, 256 and proper time, 222, 232, 254 and stereographic projection, 256, 293 and Pythagoras, 103, 293 and trigonometry, development of, and relativistic Doppler factor, 253, 256 254 projection of, 73 and symmetrical relativistic Doppler Hippocrates, 111 factor, 255 Holland, 113, 146, 175, 176 defined, 264 geometric mean, and arithmetic mean, 227–229 proportionality, 228–229 geometric series, rule of, 273

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 350 8/15/2014 12:01:25 PM Index 351

Holmes, Mycroft, 6, 7, 23, 25, 28, 29, 30, and determination of velocity of light, 31, 34, 38, 39, 40, 42, 43, 44, 45, 47, 176 51, 52, 55, 92, 93, 94, 95, 96, 105, 106, and differential geometry, 302 107, 108, 110, 111, 114, 115, 120, 123, and differentiation, vector-valued 124, 129, 135, 142, 145, 147, 148, 149, function, 306 150, 159, 160, 162, 163, 164, 165, 169, and dilation of time, 177 182, 186, 187, 189, 192, 199, 201, 205, and discovery of optical Doppler fac- 206, 207, 208, 209, 210, 211, 212, 213, tor, 172–177 214, 215 and discovery of Saturn’s rings, 183 Holmes, Oliver Wendell, 199, 200 and discovery of Titan, 301 honeybee, 11 and extraterrestrial life, 301 (see The and sun compass, 11 Celestial Worlds ...) Hooke, Robert, 131, 302 and first pendulum clock, 146 and anchor escapement for pendulum and French Academy, 175 clock, 302 and Hamiltonian mechanics, 306 and balance-spring method of time- and laws of motion, 175 keeping, 131 and Liebniz, 148 Hooke’s law, 302 and mentoring of Leibniz, 306 Horologium Oscillatorium, 302 and optical Doppler effect, 176, 179 Horus, 236 and passive remote detection, 304, Hotel Le Méridien, 116 309 Hound of the Baskervilles,The, 21, 240, 242 and pendulum clock, 174, 301, 302 Hudson, Mrs., 5, 37, 41, 55, 56, 116, 185 and probability theory, 300 Hugo, Victor, 138 and propagation, light waves, in Humpty Dumpty, 84, 200 vacuum, 46 Huygens, Christiaan, 23, 28, 31, 46, 47, and radius, earth’s orbit, 155–156 48–53, 54, 55, 56, 75, 104, 109, 110, and relative view of space and time, 111, 112, 113, 114, 115, 116, 117, 119, 147, 149, 162, 163 121, 122, 123, 124, 125, 127, 128, 129, and relativistic Doppler factor, 138 130, 131, 132, 135, 137, 138, 139, 140, and remote detection, 303–311 141, 142, 143, 145, 146, 147, 148, 149, and ring of Saturn, 301 150, 151, 153, 154, 155–156, 157, 159, and Royal Society of London, 146 160, 161, 162, 163, 164, 165, 166, 167, and second law of motion, 306 168, 171, 172–177, 179, 180, 181, 182, and singularities in propagation of 183, 189, 195, 201, 294, 300–311 waves, 306 anagram, 122 and speed of light, 161 and Académie Royale Des Sciences, and spherical propagation of light, 146 183 and active remote detection, 304 and spherical-wave construction, 137 and Bibliothéque Royale, 146 and telescope for astronomical obser- and calculus, and inverse problem, vations, 301, 303 176 and telescopes, 174 and calculus of variations, 306 and the evolute, 303 and center-of-mass law, 175 and Titan, 301 and centrifugal force, 300–301 and Treatise on Light, 147, 148, 149, and concept of wavelet, early ex- 150, 151, 182 (see also Treatise ample, 151 on Light) and connection between time and and union of space and time, 142 space, 180 and use of images, 307 and conservation of kinetic energy in and wavelet theory, Young’s experi- collisions, 175 ments on interference, 305 and conservation of momentum, 175 as “father of modern technology,” 303 and cycloid, 302 and cycloidal clock, 302

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 351 8/15/2014 12:01:25 PM 352 Remote Sensing in Action

distinction between and methods of inclined plane, height and slope of plane, active and passive remote detec- relative to velocity of ball, 299 tion, 304 inertial frames, 220, 224 Huygens’ circle, 53, 54 and coordinate systems, 220, 224 Huygens’ construction, 28, 49, 50, infinite series, 273 113, 310 Inspector Lestrade, 41 Huygens’ drawings, 135 Io, 150, 175 Huygens’ principle (see Huygens’ Ireland, 4, 65 principle) Isis, 236 spherical wavelet, and stereographic triangle of, 236 projection, 179 Italy, 185 spherical-wavelet principle, 131 stereographic projection, wavelet, and wave propagation, 130 Jackson, Andrew, 212 wavelet theory of propagation of Jacquard, Joseph, 192 electromagnetic and mechanical Jacquard loom, 192 waves, 305–306, 307 Journal des Sçavans, 175, 176 wave theory, reception of, 131 Jovian moon, 29 wave theory and diffraction of light, Jupiter, 115, 150, 151, 152, 153, 154, 155, 132 156, 158, 174, 175, 176, 301, 304, 309, wave theory and interference of light, 314 132 and sending period, 153 wave theory and polarization of light, measurement of orbital periods of 132 satellites, 301 Huygens, Constantijn, 110, 111, 121, 123, moons, as clocks, 304 135, 145, 301, 303 polar stereographic projection, 314 Huygens, Treatise on Light, 147, 148, 149, proper period, 151, 152, 158 150, 151, 182 Jupiter, satellites of, 174, 175 and early example of concept of and determination of longitude, 175 wavelet, 151 as natural chronometer, 175 Huygens’ anagram, 122 eclipses of, 174 Huygens’ circle, 53, 54 Jupiter, the god of Roman mythology, 202 Huygens’ construction, 28, 49, 50, 113, 310 reflection of waves, 49, 50 Huygens’ drawings, 135 Keats, John, 293, 294 Huygens’ principle, 29, 35, 48, 53, 109, Keats, Tom, 293 113, 123, 127–136, 137, 139, 140, 172 kinetic energy, relative to potential energy, and optical Doppler factor, 109 297–298 and Strasbourg Cathedral, 139, 140, King Harold II, 81, 82, 91 172 King Henry V, 126 origin, 127–136 King James, 205 Hypatia, mathematician, 256 King Lear, 95–96, 105–108 and astrolabe, designs for, 256 folly of, 105–108 and Pythagorean musical intervals, Kirchhoff, and mathematics of Huygens’ 256 principle, 305 hydrometer, invention of, 256

Lady Anne Wardwell (see Wardwell, Lady Iceland spar, 176 Anne) Ideas and Opinions, 300 Lake Geneva, 185–186, 187, 201 imagination, value of, 294 speed of sound in water, 185–186 inclined plane, Galileo, and beginning of latitude and longitude, illustrated, 312 modern science, 296 leagues, converted to toises, 157 Leiden, University of, 123

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 352 8/15/2014 12:01:25 PM Index 353

Leith Hill Music Festival, 75 London, 11, 13, 21, 30, 32, 46, 60, 123, Le Journal de Geneve, 1 136, 141, 145, 147, 193, 198, 201, 205, L’embouchure d’un fleuve, 118 217, 225, 234 Lestrade, Inspector, 41 London Times, 2, 3 Leuk, 201 longitude, determination of, 301 Liebniz, Gottfried Wilhelm, 131, 148, 174, Lord Byron, 32, 76, 187, 188, 190, 192, 176, 306 193, 201, 211, 293 and calculus, invention of, 148, 174, Lovelace, Ada Byron, 211, 215 306 Low Countries, 106 light, 8, 25, 28, 49, 78, 80, 115, 131, 132, Lowther Arcade, 57 157, 175–176, 178, 179, 181, 217, 218, luminiferous aether, 308 219, 220, 222, 224, 225, 305, 307, 309, Luxembourg, 32, 61 310 amplitude, and energy, 132 and massless particles, 132 Macbeth, 76, 117, 118, 135, 142, 171, 205 finite speed of, and ultimate signal Mach number, defined, 264 speed, 25 Mach units, 279 light, as electromagnetic wave, 178 Mach velocity, 265, 269–270, 289 light, speed of, converted to toises, advantage of, and Pythagorean rate, 157 265, 269–270 light, spherical wavefront, 179 sound, negative and positive, 289 light, velocity of, forward problem magnetic constant, Maxwell’s, 307 and reverse problem, 175–176 Marlborough, duke of, 118 light, wave theory, 181 Mars, 91, 115, 122 speed, in a vacuum, 25, 49 Marston, Mary, 37 speed of, as absolute constant, 220, mass, 22–26, 297 224, 225 and energy, explained, 22–26 speed of, as maximal velocity, 219, dependence on velocity, 25 220, 224, 225 one gram, energy equivalent, 23 timelessness of, 80 relative to momentum, 297 velocity, and natural units, 222 massless particles, and light, 132 velocity in all inertial frames, and material wave, 51 relativity, 309 mathematics, relativistic, 293–294 velocity in vacuum, 217, 218 Maxwell, James Clerk, 20, 23, 24, 34, velocity of, 78, 115, 131 51,116, 132, 133, 134, 149, 177, 178, velocity of, and definition of unit of 179, 181, 182, 183, 196, 197, 305–306, distance, 310 307, 308, 309 velocity of, as universal constant, and and active remote detection, 309 Maxwell, 307 and aether, 181 wave theory of, 28 and confirmation of Huygens’ theory Light, Treatise On, 50, 123, 124, 147, 148, of wave propagation of light, 308 149, 150, 151, 154, 155, 156, 157, 159, and electromagnetic field, 179 164, 165, 172, 173, 174, 177, 182 and electromagnetic waves, 132, 178 and relativity, 149 and mathematics of Huygens’ prin- light beam, 233 ciple, 305–306 light-hour, 152 and Maxwell’s equations, 20, light-minutes, and distance, 161 177–179, 307 light-second, 13, 22, 217, 218 and Michelson-Morley experiment, as unit of distance, 22 182, 309 light signal, time and distance, numerical and Planck, Max, 134 equality, 218 and speed of light as fundamental light-year, 13, 217 constant, 307 Literature, Curiosities of, 247 and totality of properties of light, 134

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 353 8/15/2014 12:01:25 PM 354 Remote Sensing in Action

electromagnetic theory, and energy Holmes’ meeting with Moriarty, ac- and intensity of a wave, 23, 24, count of, second part. 150–165 132, 134 Holmes’ meeting with Moriarty, ac- Maxwell’s equations and propagation count of, third part, 165–169 of electromagnetic waves through Lady Anne’s monologue about, space, 307 194–199 Maxwell’s demon, 197–198, 203 Maxwell’s equations, 177–179 Maxwell’s electromagnetic theory, and optical Doppler factor, Huygens’ energy as related to intensity of a wave, discovery of, 172–177 132 time and space, essential meaning, Maxwell’s equations, 20, 177–179, 307 179–183 measurement, and remote sensing, 255 train to Strasbourg, 137–142 mechanical energy, law of conservation of, Morspergen, Jakob, 214 298 motion, accelerated, uniform, defined, 295, mechanical wave, 51 298 Meiringen, Switzerland, 1, 33, 201, 206, Mount Olympus, 273, 275 212 moving bodies, remote detection, and elec- Mercator projection, 18 tromagnetic signals, 9 meridian, defined, illustrated, 312 moving target, in relativity theory, 22 Michelson, A. A., 182, 309 ‘Musgrave Ritual, The,’ 15–17, 153 and Michelson-Morley (Maxwell’s) musical interval, 261 experiment, 182, 309 musical notes, 260–261 Michelson interferometer, 182 complementary, 260–261 Michelson-Morley experiment, 182, 309 Pythagorean, 260 Middle Ages, 166 music of the spheres, 259 midpoint period, 152, 158, 317 Myonides, 264 defined, 152, 158 midpoint time, and arithmetic mean, 232, 254 Napoleon, 212 and Newtonian theory, 254 Napoleonic wars, 212 Milton, John, 39 Narbonne, 37 mirrors, and interchange of front and back, NASA, National Aeronautics and Space 55 Administration Jet Propulsion Labora- molecular velocities, Maxwell distribution tory, 314 of, 197 natural units of measurement, 218, 222, 231 momentum, 20 and velocity of light, 222 moon, 8, 13 compared with conventional units, distance from earth, 13 measurement, 218 motion, limits, 80 defined in terms of fundamental Morely, E. W., 182, 309 physical constants, 218 and Michelson-Morley (Maxwell’s) Neptune, 194 experiment, 182, 309 Newhaven, 32, 61, 65, 75, 81, 94, 96 Moriarty, Colonel, 2 New Orleans, 212 Moriarty, Professor James, 1, 2, 3, 4, 5, 6, Newton, Sir Isaac, 20, 21, 24, 29, 51, 77, 31, 33, 34, 38, 39, 42, 43, 44, 45, 51, 55, 78, 101, 115, 131, 132, 138, 146, 147, 56, 57, 58, 59, 60, 61, 62, 82, 92, 93, 94, 149, 162, 174, 176, 177, 180, 181, 219, 107, 108, 113, 115, 135, 137–183, 187, 220, 222, 224, 225, 231, 233, 245, 246, 189, 194–199, 200, 202, 203, 204, 205, 247, 248, 250, 296, 299, 305 206, 207, 208, 209, 210, 211, 212, 213, and absolute space and time, postu- 214, 215 late on, 146, 147 discovery of Huygens’ principle, and absolute time, 101, 138 169–172 and calculus, 174 Holmes’ meeting with Moriarty, ac- and classical mechanics, 224 count of, first part, 143–150 and law of universal gravitation, 299

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 354 8/15/2014 12:01:25 PM Index 355

and mathematical physics, 176 optical Doppler factor, 51, 54, 100, 101, and particle theory of light, 132, 180, 103, 109, 137, 153, 159, 172, 174, 177, 181 179, 182, 183 and time axis, 250 and projection ratio, 153 physics, and Maxwell, 24 optimization, and Huygens, 306 Newtonian absolute space, 220, 223 oriented circle, 78–79 Newtonian absolute time, 220, 223, 248, Orion, sword of, 303 252 Osiris, 236 Newtonian mechanics, 24, 26 triangle of, 236 and motion, slow relative to speed of Oxford University, 113 light, 24 and relativity, approximation for slow-moving bodies, 26 Pall Mall, 31, 34, 123 Newtonian physics, 246 Pandora, 202 Newtonian theory, midpoint time, and arith- Pandora’s box, 202 metic mean, 254 parabola, and motion of projectiles, 296 Newtonian time, 60 Paradise Lost, 39 Newton’s absolute coordinate system, 103 paramecia, 122 Newton’s laws, and disagreement with Paris, 61, 96, 105, 107, 146, 156, 174, 175, theory of relativity, 24 187, 265 Newton’s view of space and time, compared Paris Academy, 160 with Einstein’s, illustrated, 247 passive remote detection, and Christiaan New York, 265 Huygens, 304, 309 Niagara Falls, 253 Pasteur, Louis, 122 Nîmes, 37 patterns, as signals, 195 nonoriented circle, 78–79 pendulum clock, 301, 302 normalization factor, 285 perfect fifth, 261, 262, 265 normalized velocity, 156 defined, 261 Northern Hemisphere, 313 perfect fourth, 261, 262, 266 North Pole, 17, 69, 70, 312, 313, 315 defined, 261 North Pole stereographic projection, 314, period, 62, 91, 229 315 and cycles of waves, 229 compared with frequency, 62, 91 relative to frequency, 229 obelisks, Egypt, 17–19 Philip, Duke (see Duke Philip) octave, 261, 262, 266–267 Philosophiæ Naturalis Principia Math- defined, 261 ematica, 51 four basic notes, frequency and peri- Philosophical Transactions of the Royal ods, relationships, 266–267 Society, 176 “Of a History of Events Which Have Not photometer, 112 Happened,” 247 photons, 79, 80, 133 offset distance, 317 and inertial frame, 80 On the Electrodynamics of Moving Bodies, and one time instant, 80 7 physics, classical, 295 ‘On the Remote Detection of Moving Bod- Physik, Annalen der, 7 ies by Electromagnetic Signals,’ 242 Pisa, Italy, 72, 294 ‘On the Stability of Saturn’s Rings,’ 183 Planck, Max, and theory of electromagne- opaque radius, in stereographic projection, tism, 133–134 19 and energy of photons, 133–134 optical Doppler effect, 103, 159, 172, 173, and Maxwell’s wave theory, 134 174, 176, 177, 179, 182 and quantum mechanics, 134 and determination of velocity of light, Planck’s constant, h, 134 176 Planck’s constant, h, 134 planets, 8

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 355 8/15/2014 12:01:26 PM 356 Remote Sensing in Action

Plato, 89, 145, 178 and independence from units of mea- Poe, Edgar Allen, 173 surement, 265 polar stereographic projection, 312, 313, and speed of sound, 264 314, 315 Pythagorean sequence, 272 potential energy, relative to kinetic energy, Pythagorean theorem, 70, 98, 99, 101, 103, 297–298 154, 234, 310 primary mean, 262–263 Pythagorean triples, 70 Principia Mathematica, 146, 296 projection factor, 17, 18, 19, 153, 154, 317 in stereographic projection, 17, 18, 19 quantum mechanics, 134 projection ratio, 18, 153, 154 quantum physics, fundamental equation, and optical Doppler factor, 153 134 in stereographic projection, 18 Queen Victoria, 40, 41, 104 Prometheus, 202, 203, 215 proper period, 153, 158, 244, 316 and sending and receiving periods, Ra, sun god, 17 153, 158 radar, 46 proper time, 311 radio waves, 46 proper time, and geometric mean, 222, 232, railroads, 196 254 railway-telegraph network, 196, 197, 211 and relativity theory, 254 rate of change, one variable relative to a and space-time interval, 222 second, defined, 264 Pythagoras, 20, 70, 97–98, 99–100, 103, receiving period, 60, 85, 86, 87, 88, 152, 138, 154, 155, 158, 159, 164, 204, 222, 153, 158, 230, 242, 317 226, 227, 253, 254, 255, 256, 259–265, reference, frames of, 76 266, 271, 273, 283, 284, 285, 288, 293, reflection of waves, 47–48 310, 319 reflection period, 246 and arithmetic mean, 285 reflections, and remote detection, 242 and geometric mean, 103, 254, 293 reflection seismology, 53 and humming of the strings, 288 refraction of waves, 47 and mathematics of Doppler effect, Reichenbach, 43 319 Reichenbach Falls, 1, 2, 3, 4, 30, 31, 33, 34, and musical notes, 260 118, 127, 136, 201, 212–215 and normalized harmonic mean, 285 event at, third version, 212–215 and number and harmony in sound relative velocity, 21, 22 waves, 259 between two material bodies, 22 and Pythagorean theorem, 259 relativistic Doppler factor, and Huygens, and relativistic structure of space and 138 time, 253 relativistic Doppler factor of Einstein, 54 fundamental result of, 283 relativistic symmetry, 318 Pythagorean means, 262–265, 271, relativistic time, 101 273 relativity, 20, 22, 25, 35, 75, 80, 101, 138, Pythagorean philosophy, 262 157, 309 Pythagorean ratio, 261 and Einstein, 25 Pythagorean doctrine, 169 and light, 309 Pythagorean means, 262–265, 271, 273 Einstein’s special theory of, 20 and theory of Doppler effect, 271 principle of, 20, 25 arithmetic, 263 space-time, 157 geometric, 262–263 relativity, special theory of, and electromag- harmonic, 263 netic signals, 219, 224 normalized, relative to Doppler fac- relativity theory, 22, 35, 75, 80, 101, 222, tors, 273 229, 239, 242, 254 primary, 262–263 and absolute time, 242 Pythagorean rate, 264, 265, 269–270, 275

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 356 8/15/2014 12:01:26 PM Index 357

and electromagnetic waves as signals, Scotland Yard, 41 229 Second Afghan War, 213 and geometry, 239 sending period, 60, 85, 86, 87, 88, 153, 158, and symmetry of space-time, 222 230, 242, 254, 316 geometric mean, and proper time, 254 and relativistic Doppler factor, 254 reference relative to position of geometric mean, and relativistic Dop- observer, 22 pler factor, 254 relativity theory, explained, 217–254 sexagesimal system, 66 Doppler factor, relativistic, 247–249 shadows, use of, light source and light rays, Einstein’s theory, symmetry in, assumptions, 14 250–254 Shakespeare, 42, 76, 77–78, 126, 205, 272, space and time, dual nature, 217–225 294 stereographic projection, explained, Shelley, Mary Godwin, 188, 189, 193, 234–247 199–203 time, dilation of, explained, 225–234 Shelley, Mary Godwin, and Lovelace, Ada Rembrandt, 126, 127, 128, 129, 137, 172 Byron, 185–215 remote detection, 44, 83–85, 186, 211, 242, Shelley Percy Bysshe, 64, 76, 77, 140, 188, 303–311, 318 190 active, 83–85, 186 Sherlock Holmes, The Return of, 65 and Huygens, 303–311 signals, nature of, 195 and reflections, 242 ‘Silver Blaze,’ 83 classical (acoustic), 318 singularities in propagation of waves, 306 passive, 83–84 sinusoidal waves, 258–259 remote detection, moving bodies, by amplitude, 258 electromagnetic signals, 9, 10, 28, 304, frequency, 258, 259 318–319 period, 258, 259 active, described, 10 velocity, 259 active, relativistic (optical) case, wavelength, 258, 259 318–319 Sir Isaac Newton, 219 active and passive, distinction, 304 Six Books of Optics, Long Valued by passive, described, 10 Philosophers and Mathematicians, 145, remote sensing, 256–258, 272 147, 159 active, and reflection, 272 small circle of a sphere, defined, illustrated, principles of, 256–258 312 Renaissance, 73, 262 Socrates, 178 Reuters, p. 1 Somme, 120 Rhone River, 201 sonar, 46 Rome, 166, 293 sound, 185, 266 Rømer, Ole, 155, 175, 176 period of, relative to length of string, Rømer report, 175 266 Röntgen, Wilhelm Conrad, and discovery of speed of, in water, 185 X rays, 178 sounds and frequency, 260 Royal Academy of Science, 123 sound waves, number and harmony in, 259 Royal Society of London, 146 South Downs, 92, 97 Rubens, Peter Paul, 112, 135 South Downs, Sussex, 3, 11 Southern Hemisphere, 313, 314 South Pole, 70, 312, 314 Sahara Desert, 86 South Pole stereographic projection, Saturn, 122, 146, 301 315–316 rings of, 146, 301 space and time, dual nature, 217–225 Scarlet, A Study in, 28, 173, 254 space and time, Einstein’s system, dual Science, 308 nature, 223 scientific method, 9, 26–30, 34 and communication as a weakness, 27

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 357 8/15/2014 12:01:26 PM 358 Remote Sensing in Action

space-time, 19–22, 78, 79, 220–221, 222, projection factor and identity with 224, 252 relativistic Doppler factor, 317 circular nature, 221 South Pole stereographic projection, continuum, 78 315–316 explained, 19–22 stereographic projection, and shadows, symmetry, and relativity theory, 222 14–19 union of, 220–221 opaque radius, 19 unity, 79 projection factor, 17, 18 space-time interval, and proper time, 222 stereographic projection, explained, space-time relativity, 157 234–247, 312–314 special relativity, 223, 254, 318 and Eye of Horus, 238 and mathematics of, 254 illustrated, 312–314 and relativistic Doppler factor, 254 of space and time, 246 and symmetry of, 223 stereography, defined, and stereographic mathematics of, and stereographic projection, illustrated, 312–314 projection, 318 stone circles, 65, 147 special theory of relativity, and electromag- Stonehenge, 56–70, 139 netic signals, 219 Straker, John, 83 spectroscope, 181 Strand, The, 2, 5, 6, 59, 120, 121 sphere, defined, 311 Strasbourg, 32, 105, 108, 136, 138, 140, spherical propagation of light, and Chris- 141, 142, 143, 160, 161, 169, 170, 172, tiaan Huygens, 183 185, 201, 204, 206 spherical wavelet, 137, 140, 141, 161 Strasbourg Cathedral, 34, 138, 139, 141, and stereographic projection, 161 142, 144, 145, 161, 165, 166, 167, 168, and Strasbourg Cathedral, 140 170, 171, 204 and structure, time and space, 141 and Huygens’ principle, 139 construction, and stereographic pro- and Roman temple, 139 jection, 137 Strasbourg clock, 140 spontaneous generation, 122 submarines, 46, 186 standing waves, 259–262 summer, year without, 188 stars, 8 sun, 150, 304 Steele, Frederic Dorr, 65 sun, rate of variation in position, 14 stereographic projection, 52, 67, 69, 71, 73, sun compass, 11, 12 110, 112, 114, 131, 141, 147, 161–162, Surrey, 75 166, 168, 172, 179, 234–247, 256, Sussex, 3, 97 311–318 Swiss Alps, 108, 208 and astrolabe, 141, 147, 161, 166 Switzerland, 7, 170, 209 and geometric mean, 316 symmetry, as physical or mathematical and Huygens’ spherical wavelet, 179 property of a system, preserved, 318 and independence from equations, and relativistic symmetry, 318 179 Synesius, 256 bipolar stereographic projection, 315–316 conformal, 112 tail-wagging dance, honey bees, 12–13 geometry of, and incorporation of conversion factor, time to distance, 12 mathematics of special relativity, technological advances, and scientific 318 theory, 63 illustrated, 312–314, 315 telescope for astronomical observations, inverse, 131 construction of, 301, 303 North Pole stereographic projection, Christiaan and Constantine Huygens, 314, 315 303 projection factor, 315, 317 tetractys, 262 projection factor, fundamental expres- ‘The Adventure of the Empty House,’ 2, 4, sion for, 317 207

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 358 8/15/2014 12:01:26 PM Index 359

The Canterbury Tales, 73 ultimate velocity, 79 The Celestial Worlds Discover’d: Or, uniform velocity, defined, 298 Conjectures Concerning the Inhabitants, Urania, 64, 140 Plants and Productions of the Worlds in Uranus, 194 the Planets, 301 The Dynamics of an Asteroid, 1, 42, 52, 115, 183, 194 vacuum, and electromagnetic waves, propa- ‘The Final Problem,’ 4, 30, 33, 39, 59, 60, gation of, 309 207 van Goyen, Jan, 118 The Hound of the Baskervilles, 21, 240, 242 van Leeuwenhoek, Anthonie, 111 ‘The Musgrave Ritual,’ 15–17, 153 van Rijn, Rembrandt, 111 The Return of Sherlock Holmes, 65 Vaughn, Henry, 49 thermodynamics, 197, 200, 203, 252 Vega, 217, 218, 311 first law of, 200, 203 time of communication by electro- second law of, 197, 200, 203 magnetic signal, 311 The Strand, 2, 5, 6, 59, 120, 121 velocity, 79, 80, 156, 218, 264, 298 The Whole Art of Detection, 5, 10 defined, 264 Tiber River, Rome, 293 dimensionless, and natural units, 218 time, 78, 138, 141, 158–159, 163, 164, 177, normalized, 156 219, 225–234, 248 ultimate, 79, 80 absolute, 138 uniform, defined, 298 and speed of light, 164 Venus, 122, 156 dilation of 158–159, 163, 177, Verne, Jules, 23 225–234 Victoria Station, London, 32, 38, 57, 58, 60, perception of, relative to inertial 61, 105, 145, 250 frames, 225 Villa Diodati, Lake Geneva, 76, 187–189, relative, and Einstein, 248 190, 193, 210 time and space, meaning of, 78 volcanic ash, 188 time and space, structure, and spheri- vorticella convallaria, 122 cal wavelet, 141 Vosges Mountains, 166 time-space interval, 219 Vulcan, 202 time and frequency, 257 time axes, Newton and Einstein, 250 timeline compared with space line, 76 wagtail dance, bees, 12 Times, London, 2, 3 Wallis, John, 122 time-space interval, 311 Wardwell, Lady Anne, 37–104, 107, 108, described, 311 109, 111, 112, 116, 129, 137, 138, 148, for a material body, 311 170, 172, 173, 177, 179, 183, 189, 190, for any light signal, 311 192, 193, 194–199, 201, 206, 210–212, Titan, 122, 301 215 TNT, 23 Canterbury, 72–75 traveling waves, basic parameters for de- Canterbury, train to, 56–59 scription, 258–259 Dieppe, water passage to, 96–104 Treatise on the Binomial Theorem, A, 4 Doppler effect, classical, 60–65 Treatise on Light, 50, 123, 124, 147, 148, Doppler factor, optical, and relativis- 149, 150, 151, 154, 155, 156, 157, 159, tic, 53–56 164, 165, 172, 173, 174, 177, 182 Eastbourne, train to, 81–91 and relativity, 149 Hastings, train to, 75–81 trigonometry, development of, 256 Lady Anne, first document, 47–53 Two New Sciences, Galileo, 296 Lady Anne, second document, 65–72 221B Baker Street, 5 Moriarty, Professor, Lady Anne’s monologue about, 194–199 Mycroft’s plan of campaign, 43–47

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 359 8/15/2014 12:01:26 PM 360 Remote Sensing in Action

Newhaven, train to, 92–96 sinusoidal, 258 Sherlock Holmes meets Lady Anne, standing, 258 39–43 traveling, 258 War of 1812, 212 wave theory of light, 149 water distillation, 256 wave velocity, 259 Waterloo, 212 Wellington, Duke of, 212 wave, amplitude of, 132 Wells, H. G., 23 wave, period of, relative to frequency and Whitehall, 123 velocity, 265 Williams, Ralph Vaughan, 75 wavefront, geometric construction of, 124 William the Conqueror, 81, 82, 84, 85, 88, wavefront and secondary wavelets, 47–49 91, 94 wavelength, 258, 268 Wright brothers, and fixed-wing aircraft, velocity and period, 268 231 wavelet, concept of, 137, 172 wavelets, envelope of, 48–49 wave motion, 258–259 X-rays, 8, 178 wave-partical duality of light, 134 discovery of, 178 wave period, 258, 259 wave propagation, theory of, 109, 112, 113, 114 York, 82 waves, 97, 258, 305, 306 Yorkshire, 82 and sound, 97 Young, Thomas, 181, 305 electromagnetic, 258 and wave theory of light, 181 electromagnetic, propagation, 306 interference experiments, and wavelet mechanical, 258 theory of light, 305 mechanical, propagation, 306 oceanic, and the understanding of waves, 97 Zeno of Elea, 64, 76, 273, 275 propagation of, and wavelets, leading paradox of, 273, 275 edges, 305 Zeno’s paradox of the arrow, 64 reflection of, and wavelets, trailing Zuiderzee, 120 edges, 305

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021

Index.indd 360 8/15/2014 12:01:26 PM