Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  in which we return to applications of classical mechanics; and see how some of the great techniques of applied mathematics were discovered.

  I have discussed the successes of classical, or Newtonian mechanics, in describing the solar system and how the modifications imposed by relativity and quantum theory have taken science to new triumphs. In this chapter, I return to some of the earliest work in classical mechanics that has led to mathematical techniques of universal importance in both science and mathematics. In the twentieth century, this early work has been combined with new computational techniques to carry out calculations that have changed our views on dynamical systems and their behavior.

  This chapter is a little more mathematically detailed than the previous ones, especially for the first two topics, but even skimming over the details should allow everyone to appreciate the results.

  12.1 HARMONICS AND NEW WAYS OF THINKING

  The pendulum was of great importance in the early development of mechanics by Galileo, Newton, and Huygens. Its motion was explained theoretically, and it provided a tool for exploring the laws of motion and the effects of gravity. The next step was to deal with continuous systems; so for example, the pendulum (a weight at the end of a light string) would be replaced by a heavy rope or chain hanging down from a suspension point. I return to the motion of such a suspended system in section 12.2. The problem which received much attention and led to some revolutionary mathematical thinking was that of a vibrating string such as we find in a guitar or other stringed instrument. It may come as a surprise to many people, but it was from studies of vibrating strings that some of the most important of all physical concepts and mathematical descriptions evolved.

  Assume that a string has length L, mass ρ per unit length, and is under a force giving it a tension T. The string is stretched along the x-axis and vibrates so that its shape at time t is given by y(x,t). See figure 12.1. The dynamical problem is to find the motion of the string given the initial shape y(x,0) = yin and assuming the string is released from rest. (This is a standard problem found in introductory physics texts and books on waves such as that by Coulson and Jeffrey.) The sound frequencies produced by vibrating strings were considered by Pythagoras and the evaluation of the string's shape was a key problem in the history of mathematical physics.

  By necessity, this topic uses some detailed mathematics; in section 12.1.2, I have summarized and interpreted the results and their implications in a general way. (Readers wishing to see more details of the history of this topic may consult Morris Kline's comprehensive history, and check the relevant papers (referred to below) and commentaries on them in the book by Cannon and Dostrovsky and in the sourcebooks edited by Magie and Struik.)

  12.1.1 Discovering the Theory of the Vibrating String

  In 1636, Marin Mersenne, taking up where Pythagoras left off, we might say, wrote about “Musical Tones Produced by Strings” and gave rules explaining how these tones depended on string length, tension, and so on. A new era began in 1713 when Brook Taylor considered the dynamics of the particles making up a string and used analogies with pendulum motions to show that a stretched string vibrated with a frequency ν1 = √(T/ρ)/2L and had the shape of a sine curve.

  Taylor knew that the vibrating string did not always have this shape, but he assumed that after a number of vibrations it would settle down into this proper shape. (Incidentally, this is the Taylor who remains famous for the mathematical Taylor series.)

  Both Joseph Sauveur (inventor of the term acoustics) in 1713 and John Wallis in 1741 wrote about other forms that the vibrating string might assume, noting in particular that there could be nodes—points on the string that remained at rest. Examples are shown in figure 12.1.

  Figure 12.1. (a) General form of vibrating string of length L. (b)–(e) Shape of vibrations corresponding to the first four frequencies for a vibrating string. Note that in (c)–(e) there are nodal points at which the string remains stationary. Figure created by Annabelle Boag.

  The full mathematical analysis of the vibrating string was largely the work of three of the premier mathematicians of the eighteenth century: Daniel Bernoulli (1700–1782), Jean le Rond d'Alembert (1717–1783), and Leonard Euler (1707–1783). In one first step, d'Alembert used a model of the string as a system of linked particles, applied a limiting process, and thus derived the famous wave equation for the displacement y(x,t):

  D'Alembert gave a solution for this equation in terms of traveling waves (see Coulson and Jeffrey for example). But the solution of greatest interest, the one championed most by Daniel Bernoulli, was the one written as a sum of the string harmonic shapes and frequencies:

  This solution corresponds to the string of length L being released from rest in some form y(x,0) = yin(x). The constants an in the sum are chosen to match that initial release condition (see below).

  For many years there was much controversy surrounding the use of equation (12.2) (see Kline for the intriguing story), and Euler and Joseph-Louis Lagrange battled with Daniel Bernoulli over its validity. On the mathematical front, a question arises when we use equation (12.2) for the initial t = 0 case:

  Euler had argued that the initial shape of the string could take a very general form, which included shapes with abrupt changes in slope (that is, continuous curves, but not curves with a continuous derivative). Euler and Lagrange did not accept that such general curves could be matched by a series of sine functions; Daniel Bernoulli claimed that there were an infinite number of constants, an, and that allowed all functions to be accommodated. At the heart of this controversy was the definition of a function. It turns out that Bernoulli was correct, but a number of years went by until Joseph Fourier and others provided the background and rigor to settle the question in his favor.

  Equations (12.2) and (12.3) give us the solution of the vibrating string problem, and it only remains to find a way to calculate the constants an for a given initial displacement of the string. Eventually it was discovered that each constant may be found by carrying out a particular integration:

  We now have a complete solution for the vibrating string problem; evaluating equations (12.2)–(12.4) tells us exactly how the string shape y(x,t) evolves from any given input shape yin(x). That is one of the great triumphs of mathematical physics.

  12.1.2 Interpretation

  A stretched string can vibrate in particular modes or patterns. The nth mode vibrates with a frequency νn depending on the string parameters T, ρ, and L: νn = nc/2L = n√(T/ρ)/2L; and has a pattern with n–1 nodes. The n=1 mode is called the fundamental and modes with larger n are the harmonics. Examples are shown in figure 12.1. Mathematically, the modal pattern is described by a sine function, and the vibration has amplitude varying periodically in time according to a cosine function. Thus each term in equation (12.2) represents one of these modes given an initial amplitude an.

  If the initial shape of the string is exactly in the form of one of those modal patterns or sine functions, then the string will vibrate so that pattern is maintained for all time with its amplitude varying with the modal frequency νn. There are an infinite number of these modes.

  It is most likely that the initial shape of the string does not exactly match one of the modal patterns. In this case, equation (12.2) gives us the remarkable result that the motion of the string will be a mixture of modes simply added together with appropriate amplitudes to match the initial displacement. Here is the way Daniel Bernoulli put it:

  My conclusion is that all sonorous bodies include an infinity of sounds with a corresponding infinity of regular vibrations…. We remark that the chord AB [the string, in this case] cannot make vibrations only conforming to the first figure [fundamental] or second [harmonic] or third and so forth to infinity, but that it can make a combination of these vibrations in all possible combinations.1

  An example will help to appreciate this result, and then I will discuss the wider implications for mathematics and science.

  12.1.3 Playing Pizzicato


  Some music requires stringed instruments like violins to be played pizzicato, that is they are plucked rather than excited using a bow. The plucked string is the most famous example of the use of the above theory. The problem is illustrated in figure 12.2. The string is initially in a triangular shape with its midpoint raised a distance h above the x axis.

  Figure 12.2. The initial shape of a plucked string, and the shape of the modes excited when it is released. (Not to scale; the harmonics have much smaller amplitudes than those shown here.) Figure created by Annabelle Boag.

  It is easy to find the modal amplitudes using equation (12.4) and

  Only the odd-numbered modes are excited because the sine term in equation (12.5) is zero whenever n is even. The odd-mode amplitudes oscillate in sign and because of the n2 term they rapidly decrease in magnitude. In the case of the plucked string we can describe its motion by writing its shape at time t after release as

  This result is shown graphically in figure 12.2. Obviously the excited modes must have the same symmetry as the starting configuration, and that is why those modes with n = 2, 4, 6,…are not present in the string's motion.

  We can see that plucking a string at its midpoint strongly excites the fundamental mode; other modes are also excited (in theory, an infinite number of them), but their importance rapidly decreases with increasing mode number. Representing the motion of the string in terms of the modal vibrations gives us a simple way to see which frequencies are generated and just how the harmonics enter to give that attractive composite plucked-string sound.

  12.1.4 Going Further

  The theory outlined above is often called Fourier theory since it was Joseph Fourier (1768–1830) who fully developed and expanded the ideas encapsulated in equations (12.2)–(12.4). Fourier showed how these same ideas could be used to understand the conduction of heat and described his work in his 1822 book The Analytical Theory of Heat. Lord Kelvin (as William Thomson then) was one of the first great enthusiasts for Fourier theory (“Fourier is a mathematical poem”2 he said), and if you look back to chapter 4, you will see that Kelvin used Fourier's approach in his studies of the age of the earth and in his evaluation of tides. Lord Kelvin recognized the power of Fourier's methods when he and Peter Guthrie Tait wrote in their famous textbook that they must consider

  Fourier's Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.3

  Kelvin would have been interested to see the growth of Fourier optics in which Fourier's methods are used to describe optical effects and to evaluate imaging instruments. The resolution of image details discussed earlier in section 9.3.2 is beautifully described in terms of spatial frequencies (see the review by Westheimer and the book by Wandell).

  The theory has been generalized so that the sine functions used above are replaced by other functions found by solving the differential equation relating to the physical problem under investigation (as sine waves were found by solving the wave equation, equation (12.1), for the motion of a stretched string). If these new functions are ψn(x), and suitable conditions are satisfied, the function to be found may be written as

  All of the ideas discussed above about calculation methods and interpretation of solutions carry over into this new area. Further examples will be given in the next section. (A comprehensive review and history of this area may be found in the book by Elena Prestini.)

  This wonderful method becomes quite general: split the calculation into two parts. The first part involves finding the suitable basis functions like sines or ψn(x) in general; the second part involves finding the coefficients an to match the conditions imposed by a particular problem.

  We sometimes speak of the method of superposition, because that is what is happening in equations (12.2) and (12.6) for example; many basis solutions are being superposed to build up any required solution. The building up of solutions using the simplest basis solutions is used for linear systems in general (Feynman may be consulted for an excellent general introduction). The wave equation, equation (12.1), Fourier's heat conduction equation, electromagnetic wave equations, and quantum theory are all linear systems, and I will say more about such things in section 12.3. It is interesting to note that Paul Dirac's superb book The Principles of Quantum Mechanics (discussed in section 11.1) opens with a chapter called “The Principle of Superposition.”

  It seems unlikely at first sight that understanding how a plucked string produces that lovely pizzicato sound could be important, but it does lead to what may well be argued as the most powerful method in mathematics and its applications in science. The calculations made by Daniel Bernoulli and his contemporaries showed the way for physicists to follow ever since. I must certainly put calculation 47, strings and Fourier's mathematical poem in my list of significant calculations.

  12.2 TABLES OF BESSEL FUNCTIONS

  It would be a rare person who used mathematics in scientific problems but who did not come across Bessel functions. However, it is likely that most of those persons know little about Friedrich Wilhelm Bessel (1784–1846) himself.

  Bessel was born in Minden, Germany, and went on to become one of the outstanding astronomers of his time. In 1810, at the age of just 25, he was appointed director of the newly founded Königsberg Observatory, and he remained there for the rest of his life despite being offered positions like director of the Berlin Observatory. He made a vast number of observations of stars and processed them to contribute to the catalogues then being assembled. He supplied many of the observational and analysis techniques for that work and published details in 1830 in his great Tabulae Regiomontanae. (See the article on astronomical tables by Norberg and Fricke for a comprehensive biography.)

  Bessel is remembered for two particularly impressive feats. In 1838, he became the first person to accurately find the distance to a star. Using the parallax method with the diameter of the earth's orbit as base, Bessel found a parallax angle of 0.314 arcseconds for the star 61 Cygni, which translates into a distance of 10.3 light years, a result now known to be in error by less than 10 percent. Finally, astronomers had a grasp of the scale of the cosmos. Secondly, Bessel observed variations in the motion of Sirius which could be interpreted as the gravitational effect of another large body, and, in 1844, he announced that there must be a “dark companion” for the star Sirius. It was not until 1862 that Alvan Clark finally observed the white dwarf that shares its motion with Sirius. Bessel was an outstanding mathematician, and it is for his work on the functions that now carry his name that he is best known today.

  12.2.1 What are Bessel Functions?

  Many people first meet Bessel functions through the differential equation

  This equation, or versions of it, turns up in an enormous number of applications of mathematics to physical problems. I will take x and n to be real, although the theory extends into the complex domain. A solution to this equation is y = Jn(x), which is the Bessel function of the first kind and of order n. (For simplicity I will not consider the other kinds here; in fact Jn(x) is the Bessel function most commonly met with in first applications, usually with integer values n = 0 or 1.) Standard methods for solving the differential equation produce series expansions:

  The graphs of J0(x) and J1(x) are shown in figure 12.3. Bessel functions sometimes turn up in integral form such as

  A most useful property of Bessel functions is the recurrence relation:

  Putting n = 1 tells us that we can find the value of J2(x) if we know the vales of J0(x) and J1(x), and then we can move on to J3(x), and so on.

  Figure 12.3. Graphs of J0(x) and J1(x).

  The graphs indicate that these Bessel functions have reducing magnitudes and oscillate in sign, and, in fact, they have an infinite number of zeros. The zeros are not equally spaced (as they are for sines and cosines) although they do tend toward that property for very large values of x as can be deduced from the
asymptotic, large x approximation:

  Bessel made the first systematic study of the functions now bearing his name in 1824, and since then, the literature concerning them has become enormous. (G. N. Watson's comprehensive Treatise on the subject runs to over 800 pages.)

  12.2.2 Using Bessel Functions

  As the use of mathematics in science developed, the need for Bessel functions arose. (The history is covered by Watson in his chapter 1 and in the paper by Dutka.) A few examples will set the scene.

  It was natural for early researchers to pass from a study of the motion of the simple pendulum to the oscillations observed in a hanging heavy rope or chain. What is the displacement y of the rope from the vertical as a function of the distance x along the rope and the time t? See figure 12.4 This problem was solved by Daniel Bernoulli in 1733, and in solving it, he introduced a Bessel function into mathematics, although of course he worked with an infinite series and did not name the function involved. Today we write Daniel Bernoulli's solution as

  The rope is fixed at the top, x = L, and imposing that condition gives us a formula for the oscillation angular frequency ω:

  Thus the possible frequencies of oscillation are related to the roots or zeros of the Bessel function. Bernoulli calculated very accurate values for the first two roots. Thus Daniel Bernoulli could use his superposition technique to solve for the motion of a hanging chain in terms of its harmonics, and the shapes taken up by the chain are given by Bessel functions. Equation (12.6) may be used with Bessel functions for ψn(x).

  Figure 12.4. (a) The hanging heavy rope problem of Daniel Bernoulli. (b) Vibrating circular membrane, or drum, as considered by Euler. Figure created by Annabelle Boag.