Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  P-waves travel faster than S-waves. We should also note the simple, but vitally important fact that shear waves like S-waves cannot propagate in liquids. There are also waves that run along the surface of solids. They too may involve longitudinal or transverse effects, and they are named after their investigators, Rayleigh and Love. (See Bullen and Bolt. The book by Bolt gives a good, simple introduction to seismic waves.)

  The path of wave fronts may be traced out using rays. It is the spread in the times taken as waves follow particular ray paths from the earthquake that is the key information provided by the seismic-wave detectors. Because P-waves travel fastest, they arrive first, hence the notation primary (P) and secondary (S) waves.

  Like other waves, the P- and S-waves reflect at boundaries between different media (like light at a mirror or sound waves forming an echo); refract or bend as they move from one region into a different one (like light rays at the air-water interface); and follow curved paths if the elastic properties of the medium in which they are propagating vary continuously (like light rays as they give mirages in the variable refractive index lower atmosphere).

  There is one additional important property of seismic waves as they meet a surface: a mixture of waves is generated. Thus if a P-wave meets a surface there will be transmitted and reflected P-waves, but also newly generated transmitted and reflected S-waves. Because P- and S-waves have different speeds, the generated S-waves do not make the same angle with the surface as the P-waves do. This is a complication that turns out to be a key to interpreting certain seismograph results.

  One final point about P- and S-waves: as they reflect and refract, they will take different paths from the source to the observation point. There will be regions on the earth where one or another type of wave will not be observed because of their propagation characteristics. Those “shadow regions” are very important pieces of information for the theorist to explain.

  Figure 4.3 shows seismic rays within the earth. At this stage, you may prefer just to glance at it to get an idea of the complications involved as rays reflect, refract, and convert to a different form. Although many people wrestled with the problem of seismic waves and what they tell us, I have chosen three people to illustrate the gradual progress to the picture we have today. (The paper by Brush gives a more complete story.)

  Figure 4.3. Seismic rays within the earth. P- and S-waves are represented, and the various changes at interfaces can be observed. Figure created by Annabelle Boag. Drawn using data from B. A. Bolt's 2004 book (see bibliography), which should be consulted for full details.

  4.4.2 Richard Oldham and the Use of Seismic Waves

  Richard Oldham (1858–1936) left England in 1879 to join the Geological Survey of India, working there until ill health forced his return to England in 1903. He became familiar with the rapidly accumulating seismic data, and in 1900, published “On the Propagation of Earthquake Motion to Great Distances.” Here, Oldham was able to gather together data from many events and put it into a graphical form that clearly revealed groupings and trends in transmission times. His graph shows data forming three distinct curves. He was able to identify P- and S-waves, give information about their speeds, and also identify a third phase wave. Then he made the prophetic statement:

  If the curves drawn on the diagram represent the true time curves, it should be possible to deduce from them the relation between the variation of velocity of transmission and depth below the surface.6

  After much work, Oldham published his famous 1906 paper “The Constitution of the Interior of the Earth, as Revealed by Earthquakes.” He opens by summarizing the limited knowledge of the earth then available and notes that “the central substance of the earth has been supposed to be fiery, fluid, solid, gaseous in turn, till geologists have turned in despair from the subject.” He goes on to say that the days of speculation may be over as seismograph data becomes available and interpreted. Oldham then boldly makes his celebrated statement:

  Just as the spectroscope opened up a new astronomy by enabling the astronomer to determine some of the constituents of which distant stars are composed, so the seismograph, recording the unfelt motion of distant earthquakes, enables us to see into the earth and determine its nature with as great a certainty, up to a certain point, as if we could drive a tunnel through it and take samples of the matter passed through.7

  To “see into the earth” in this way requires large, detailed calculations of ray paths and disturbance arrival times for different models of the earth. Eventually Oldham became confident enough in his calculations to make statements about P- and S-wave speeds in the earth and to come to his surprising conclusion:

  From the considerations detailed in the foregoing pages, I conclude that the interior of the earth, after the outermost crust of heterogeneous rock is passed, consists in a uniform material, capable of transmitting wave-motion of two different types at different rates of propagation: that this material undergoes no material changes of physical character to a depth of about six-tenths of the radius, each change as takes place being gradual and probably accounted for sufficiently by the increase in pressure; and that the central four-tenths of the radius are occupied by matter possessing radically-different physical properties, inasmuch as the rate of propagation of the first phase is but slightly reduced, while the second-phase waves are either not transmitted at all, or, more probably, transmitted at about half the rate which prevails in the outer shell.

  A diagram showing how rays occur in a model Earth with a core was given by Oldham and is shown in figure 4.4. Richard Oldham calculated the wave speeds necessary to fit the seismograph observations and made one of science's great discoveries: the earth has a distinct inner core.

  Figure 4.4. Seismic-ray paths in an earth with a core according to (a) Oldham and (b) Lehmann. S is the source or origin of the rays. Uniform media are assumed so that the rays follow straight lines. Figure created by Annabelle Boag. Redrawn from the papers of Oldham and Lehmann (see bibliography).

  4.4.3 Inge Lehmann Refines the Core Model

  Following Oldham's 1906 paper, there was investigation and speculation by many scientists about the nature of the earth's core. Oldham himself, in a 1913 Nature paper, suggested that fluids or gases could be involved. Sir Harold Jeffreys, one of the greats of seismology, concluded in 1926 that the core is truly fluid. Remembering that S-waves cannot propagate in fluids, this explains the comments by Oldham on their absence in his 1906 paper.

  The next major step came in the work of Inge Lehmann (1888–1993). Lehmann was Danish, studied mathematics at the Universities of Copenhagen and Cambridge, and then had a career in seismology. Her personal and scientific story may be read in her 1987 reminiscences “Seismology in the Days of Old” and in the review by Kölbl-Ebert. It is unfortunate that this exceptional representative of women in science is so little known. Two quotes reported by Kölbl-Ebert sum up Lehmann's (and sadly many other women's) experiences. Of her schooling, Lehmann said that “no difference between the intellect of boys and girls was recognized, a fact that brought some disappointment later in life when I had to recognize that this was not the general attitude,” and “you should know how many incompetent men I had to compete with—in vain.”8

  Lehmann studied the seismograph records from the 1928 Mexican and 1929 New Zealand earthquakes. In particular she looked at the strength of the waves and their shadow zones over the Earth. After many calculations, she took a momentous step:

  I then placed a smaller core inside the first core and let the velocity in it be larger so that a reflection would occur when the rays through the larger core met it. After a choice of the velocities in the inner core was made, a time curve was obtained, part of which appeared in the interval where there had not been any rays before. The existence of a small solid core in the innermost part of the earth was seen to result in waves emerging at distances where it had not been possible to predict their presence.9

  Sample rays as given by Lehmann are shown in figure 4.4.
(The reader wishing to get a better understanding of this should see the very clear and detailed figure 2 in the Kölbl-Ebert paper. The paper by Rousseau is a detailed but relatively simple introduction to Lehmann's discovery of the inner core.)

  So it was that Oldham's prediction became fact, and, by analyzing the seismic records, the crust/mantle/fluid-outer-core/solid-inner-core model was discovered Although much refined and improved, it remains as today's accepted picture of the earth.

  4.4.4 Mohorovičić and His Tantalizing Discontinuity

  The Croatian Andrija Mohorovičić (1857–1936) was a pioneering seismologist who is best known for his 1910 discovery about the boundary between the earth's crust and mantle. In October 1909 there was an earthquake in Croatia with an epicenter about 40 km south-east of Zagreb. Mohorovičić had access to local seismographic data, which he plotted out and concluded:

  [it] cannot be expressed by only one curve, there are two curves: one beginning in the epicenter reaching distances up to 700 km., certainly not beyond 800 km. Second, lower curve, begins certainly at 400km, but it is possible that it has already started at 300 km.10

  Mohorovičić was convinced that the two curves related to the same type of wave. (See the papers by Herak, and by Jarchow and Thompson for the curves and further details.) Mohorovičić interpreted the data using the model shown in figure 4.5. After completing his calculations he concluded that at a depth of 54 km there was a distinct boundary between the crust and the mantle, with wave speeds changing from 5.68 km/sec to 7.75 km/sec as the boundary is crossed. Thus there is a discontinuity, and seismic rays are reflected by it to give ray paths as shown in figure 4.5. This explained the two curves and the shadow zone that Mohorovičić observed in the 1909 earthquake seismographic data.

  Figure 4.5. Mohorovičić's diagram showing seismic rays reflected at the crust-mantle discontinuity. Figure created by Annabelle Boag. Drawn from information in the paper by Herak (see bibliography).

  The Mohorovičić discontinuity, now known as the Moho, is defined by Jarchow and Thompson as “that level in the Earth where the compressional wave velocity increases rapidly or discontinuously to a value between 7.6 and 8.6 km/sec.”11 The depth of the Moho varies from 5 km to 8 km for crust below deep ocean basins and from 20 km to 70 km for continental crust. It is the comparatively small depth below the ocean that makes Mohorovičić's such a tantalizing result; can we drill down to the Moho and examine the physical changes in detail? (For a recent report on progress I refer you to “Drilling to Earth's Mantle” by Umino, Nealson, and Wood.)

  4.4.5 Discussion

  This topic is a beautiful example of the way calculation may be used to turn data into information about the physical world. In this case, the method is essential since we are exploring the details of regions of Earth that are not directly accessible. As a good candidate for the label “great,” I list calculation 10, seismic rays reveal the earth's interior.

  From a technical point of view, while the calculations of ray paths and times may be relatively simple given the properties of the material through which the waves propagate, reversing the problem to one of finding the material properties leading to a given set of ray paths can be particularly troublesome. The first problem (find the rays given the material) is called the direct problem; the second problem (find the material properties) is called the inverse problem. Inverse problems are notoriously difficult; they can be unstable, depend very sensitively on input data, and raise questions about the uniqueness of the solution.

  The work discussed here has been extended enormously and is still providing forefront research problems (see the recent reviews by Buffett and Olson). The properties of the core depend on the behavior of matter at extreme pressures and temperatures, and these are now active areas in solid-state physics and chemistry. (For a short discussion and an example about the structure of iron under extreme conditions, see the 2010 Physics Update on “Iron's Structure at Earth's Core.”) On the darker side, the use of seismic waves for detecting nuclear explosions has been a valuable tool for monitoring nuclear test ban treaties (the modern network of 337 recording facilities is described by Auer and Prior).

  Finally, we can look back to Lord Kelvin's calculation on the cooling of the earth and appreciate the complexities of the system he was attempting to model. Perry was correct when he began to identify all sorts of possibilities that invalidate Kelvin's simple model calculation.

  4.5 MOTION ON A SPINNING GLOBE

  The earth is a large sphere rotating about a north-south axis to give us our twenty-four-hour day with alternating daylight and nighttime. That rotation combined with our knowledge of the circumference of the earth leads us to conclude that we are moving at over 1,500 kilometers per hour. That enormous speed was a stumbling block for the early acceptance of the rotating Earth concept. Why are the effects of motion at such a great speed not apparent? Why aren't birds and clouds left behind? Why don't our hats fly off as they might do on even a galloping horse? These are profound questions, and they require an equally profound advance in science to answer them. If we have to pick one great early pioneer in this work, it clearly must be Galileo, and it is he who is responsible for the next calculation in my list.

  As discussed in the first chapter, mathematics plays an essential part in science. Perhaps the most famous—and certainly one of the most beautifully expressed—statements of this guiding idea came from Galileo:

  Philosophy is written in that great book which ever lies before our eyes—I mean the Universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth.12

  The work discussed below is taken from Galileo's Two New Sciences, which is readily available today. It is easy enough to read, although it does become rather tedious as it is written in terms of a long, and at times rambling, debate among the three protagonists spread over four days and an “added day.” Also, the mathematics used by Galileo is couched in a geometric form quite unfamiliar to present-day readers. Before coming to the major calculation, we need to see how Galileo introduces the types of motion involved.

  4.5.1 Basic Concepts

  The Third Day of Galileo's debate is called On Local Motion, and Galileo tells us he is introducing “a brand new science concerning a very old subject.”13 He begins by defining constant speed or uniform motion:

  Equal or uniform motion I understand to be that of which the parts run through in any equal times whatever are equal to one another.

  He can now make the important point that “motion in the horizontal plane is equable, as there is no cause of acceleration or retardation.”

  That leads him to state what we might call the idea of inertia and thus be reminded of Newton's first law:

  It may also be noted that whatever degree of speed is found in the moveable, this by its nature indelibly impressed on it when external causes of acceleration or retardation are removed, which occurs only on the horizontal plane;…From this it likewise follows that motion on the horizontal is also eternal, since if it is indeed equable it is not weakened or remitted, much less removed.14

  The old Aristotelian idea of a continuing cause for motion is removed; once in motion in the horizontal plane, an object continues that way. Thus a hat leaving your head retains its motion and is not left behind at a rate of over a thousand km/hr. Objects on the surface of the Earth all move in the same “equable” manner.

  Galileo also needs to describe motion not with constant speed, but with constant acceleration:

  I say that motion is equably or uniformly accelerated which, abandoning rest, adds on to itself equal momenta of swiftness in equal times.15

  The speed increases by the same amount in any equal time intervals.

  Galileo
deduces properties of uniformly accelerated motion by using a geometric representation of the quantities involved. (This idea goes back to Nicole Oresme (1325–1382), amongst others, but Galileo was not one for giving credit to his predecessors.) In figure 4.6 (a), the line CD represents the distance traveled; in the accompanying diagram, the time taken is measured along the line AB, and the speed at any given time is measured by the line perpendicular to AB, thus forming the line AE. The final speed is given by BE. The distance traveled is the area between the lines AB and AE (what today we refer to as the area under the speed-versus-time curve), so the total distance traveled is given by the area of the triangle AEB. Simple geometry tells us that triangle AEB and rectangle AGFB have the same area. Thus by comparing areas in his diagram, Galileo can now give his first result:

  The time in which a certain space is traversed by a moveable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same moveable carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed [EB] of the previous, uniformly accelerated motion.

  This is sometimes called the average speed rule, or the Merton rule, and was known well before Galileo's time.

  Figure 4.6. Galileo's diagrams for describing uniformly accelerated motion. (a) shows what happens in the time interval AB. (b) shows how equal time intervals AC, CI, and IO give increasingly different distances traversed. From Galileo Galilei, Discourses and Mathematical Demonstrations Relating to Two New Sciences (Discorsi e dimostrazioni matematiche, intorno à due scienze) (1638).