Friday, September 24, 2010

Adrien-Marie Legendre

On page 181 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce Legendre polynomials. The Legendre polynomial P2(cos(θ)) arises naturally when calculating the extracellular potential in a volume conductor at a position far from an active nerve axon. We include the footnote “You can learn more about Legendre polynomials in texts on differential equations or, for example, in Harris and Stocker (1998).” On page 184, we list the first four Legendre polynomials (and have another footnote referring to Harris and Stocker). Any physics student should memorize at least the first three of these polynomials:

P0(x) = 1
P1(x) = x
P2(x) = (3 x2 – 1)/2 .

Legendre polynomials have many interesting properties. They are a solution of Legendre’s differential equation

(1-x2) d2Pn/dx2 – 2 x dPn/dx + n(n+1) Pn = 0 .

You can calculate any Legendre polynomial using Rodrigues formula

Pn(x) = 1/(2n n!) dn((x2-1)n)/dxn .

They form an orthogonal set of functions for x over the range from -1 to 1, which is rather too technical of a property to explain in this blog entry, but is very important.

The astute reader might note that Legendre’s differential equation is second order, so there should be two solutions. That is right, but the other solution—called a Legendre function of the second kind, Qn—is rarely used, and tends to be poorly behaved at x = 1 and x = -1. For instance

Q0(x) = ½ ln((1+x)/(1-x)) .

A definitive source for information about Legendre polynomials is the Handbook of Mathematical Functions, by Milton Abramowitz and Irene Stegun.

When do Legendre’s polynomials appear in physics? You often find them when working in spherical coordinates, especially when (to use an analogy with the earth) a function depends on latitude but not longitude (axisymmetry). For instance, the general axisymmetric solution to Laplace’s equation in spherical coordinates is a series of powers of the radius r multiplied by Legendre polynomials with x = cos(θ), where θ is measured from the z-axis (or, to use the earth analogy again, from the north pole). Take an introductory class in Electricity and Magnetism (from, say, the book by Griffiths), and you will use Legendre polynomials all the time.

Why do I bring up Legendre polynomials today? Regular readers of this blog may recall my recent obsession with all things French. Adrien-Marie Legendre (1752-1833) was a French mathematician. Details of his life are given in A Short Account of the History of Mathematics, by Rouse Ball.
Adrian Marie Legendre was born at Toulouse on September 18, 1752, and died at Paris on January 10, 1833. The leading events of his life are very simple and may be summed up briefly. He was educated at the Mazarin College in Paris, appointed professor at the military school in Paris in 1777, was a member of the Anglo-French commission of 1787 to connect Greenwich and Paris geodetically; served on several of the public commissions from 1792 to 1810; was made a professor at the Normal school in 1795; and subsequently held a few minor government appointments. The influence of Laplace was steadily exerted against his obtaining office or public recognition, and Legendre, who was a timid student, accepted the obscurity to which the hostility of his colleague condemned him.

Legendre's analysis is of a high order of excellence, and is second only to that produced by Lagrange and Laplace, though it is not so original. His chief works are his Géométrie, his Théorie des nombres, his Exercices de calcul intégral, and his Fonctions elliptiques. These include the results of his various papers on these subjects. Besides these he wrote a treatise which gave the rule for the method of least squares, and two groups of memoirs, one on the theory of attractions, and the other on geodetical operations."

Friday, September 17, 2010

Augustin-Jean Fresnel

Apparently, dear reader, I am still obsessed by my trip to Paris last summer, because this will be the third week in a row that this blog has been about a famous French scientist. I hope you enjoy it.

Diffraction is a fundamental topic in physical optics that receives scant attention in the 4th edition of Intermediate Physics for Medicine and Biology. In fact, the index contains no entry for diffraction. (By the way, Russ Hobbie and I worked hard to make the index as complete and useful as possible.) However, a search for the term "diffraction" yields many appearances. Often it shows up as part of the term “X-ray diffraction,” but I have already addressed that technique in this blog a few weeks ago. A footnote on page 327, in Chapter 12 about images, mentions interference and diffraction in the context of coherence, and diffraction appears several times when discussing point-spread functions in that chapter. In Chapter 13 on ultrasound, diffraction is mentioned again as representing a limit to our ability to obtain an image. In Chapter 14, diffraction is discussed as a factor limiting our visual acuity.

The study of diffraction has a fascinating history, going back to the fundamental work of the French physicist Augustin-Jean Fresnel (1788-1827). Fresnel makes only one brief appearance in Intermediate Physics for Medicine and Biology, when discussing diffraction effects and the “Fresnel Zone” produced by an ultrasound transducer. To try and make up for Fresnel’s absence from our book, I will provide here some of the highlights of his short life (he died at age 39). Incidentally, I’m not the only blogger interested in Fresnel.

I first came to appreciate Fresnel’s contributions when reading the books of physicist Mark Silverman. In particular, I enjoyed Silverman’s Waves and Grains: Reflections on Light and Learning. He writes
“Fresnel, as the reader will discover (if it is not already obvious), is a central figure and something of a hero in this book. Pathetically all too human in his desperate desire to distinguish himself in the world of science, his ambitions are the ambitions of all of us who do research, write papers, and seek recognition. As a young man trained in engineering, he first turned his attention to industrial chemistry but learned to his chagrin that what he thought was original work was anticipated by others. Disappointed, he later immersed himself in the wave theory of light, guided and encouraged by Francois Arago—one of very few wave enthusiasts in the Paris Academy—who helped publicize his work both in France and abroad. […]

In 1817 the Paris Academy launched a competition for the essay best accounting for the diffraction of light. With the exception of Arago, the committee responsible for the event consisted exclusively of partisans, like Laplace and Biot, of the particle hypothesis [of light…]. Fresnel, as one might imagine, was not initially enthusiastic about entering—his whole direction of research having apparently already been ruled out by the wording. Nevertheless, urged on again by Arago, he composed a lengthy paper summarizing his philosophical approach, his methods, and his results. It is an amusing irony of history that Simeon-Denis Poisson—another graduate of the Polytechnique noted for his broad theoretical contributions to physics and mathematics, and a staunch advocate of the corpuscular theory—noted a glaring inconsistency in Fresnel’s theory. Applying this theory to an opaque circular screen, Poisson deduced the (to him) ludicrous result that the center of the shadow (doit) etre aussi eclaire que si l’ecran n’existait pas (must be as brightly illuminated as if the screen did not exist). Arago performed the experiment in advance of the committee’s decision, and the bright center—which history records as Poisson’s spot—showed up as predicted.

Fresnel, his relentless efforts finally recognized, received the prize—but Biot, Poisson, and other remained unshaken in their particle convictions.”
If you get a copy of Silverman’s book, don’t miss the last chapters on Science and Learning.

Living here in Michigan, surrounded by the Great Lakes, I have become fond of lighthouses, and particularly with the spectacular Fresnel Lenses that you can find in many of them. Click here to see pictures of some, and here to see information about Fresnel lenses found in Michigan. It is another of Fresnel’s many contributions to science.

Friday, September 10, 2010

Joseph Fourier

The August 2010 issue of Physics Today, published by the American Institute of Physics, contains an article by T. N. Narasimhan about Thermal Conductivity Through the 19th Century. A large part of the article deals with Joseph Fourier (1768-1830), the French physicist and mathematician. Russ Hobbie and I discuss Fourier’s mathematical technique of representing a periodic function as a sum of sines and cosines of different frequencies in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology. Interestingly, this far-reaching mathematical idea grew out of Fourier’s study of heat conduction and thermal conductivity. Russ and I introduce thermal conductivity in homework problem 15 of Chapter 4 about diffusion. This is not as odd as it sounds because, as shown in the problem, heat conduction and diffusion are both governed by the same partial differential equation, typically called the diffusion equation (Eq. 4.24). The concept of heat conduction is crucial when developing the bioheat equation (Chapter 14), which has important medical applications in tissue heating and ablation.

Narasimhan’s article provides some interesting insights into Fourier and his times.
“In 1802, upon his return to France from Napoleon’s Egyptian campaign, Fourier was appointed perfect of the department of Isere. Despite heavy administrative responsibilities, Fourier found time to study heat diffusion. He was inspired by deep curiosity about Earth and such phenomena as the attenuation of seasonal temperature variations in Earth’s subsurface, oceanic and atmospheric variations in Earth’s subsurface, oceanic and atmospheric circulation driven by solar heat, and the background temperature of deep space. […]

Thermal conductivity, appropriate for characterizing the internal conduction, was defined by Fourier as the quantity of heat per unit time passing through a unit cross-section divided by the temperature difference of two constant-temperature surfaces separated by unit distance […] Fourier presented his ideas in an unpublished 1807 paper submitted to the Institut de France.

Fourier was not satisfied with the 1807 work. It took him an additional three years to go beyond the discrete finite-difference description of flow between constant-temperature surfaces and to express heat flow across an infinitesimally thin surface segment in terms of the temperature gradient.

When Fourier presented his mathematical theory, the nature of heat was unknown […] Fourier considered mathematical laws governing the effects of heat to be independent of all hypotheses about the nature of heat. […] No method was available to measure flowing heat. Consequently, in order to demonstrate that his mathematical theory was physically credible, Fourier had to devise suitable experiments and methods to measure thermal conductivity.

It is not widely recognized that in his unpublished 1807 manuscript and in the prize essay he submitted to the Institut de France in 1811, Fourier provided results from transient and steady-state experiments and outlined methods to invert exponential data to estimate thermal conductivity. For some reason, he decided to restrict his 1822 masterpiece, The Analytical Theory of Heat, to mathematics and omit experimental results.”
For more insight on Fourier’s life and times, see Keston’s article Jospeh Fourier: Policitian and Scientist. It begins
“The life of Baron Jean Baptiste Joseph Fourier (1768 - 1830) the mathematical physicist has to be seen in the context of the French Revolution and its reverberations. One might say his career followed the peaks and troughs of the political wave. He was in turns: a teacher; a secret policeman; a political prisoner; governor of Egypt; prefect of Isère and Rhône; friend of Napoleon; and secretary of the Académie des Sciences. His major work, The Analytic Theory of Heat, (Théorie analytique de la chaleur) changed the way scientists think about functions and successfully stated the equations governing heat transfer in solids. His life spanned the eruption and aftermath of the Revolution; Napoleon's rise to power, defeat and brief return (the so-called Hundred Days); and the Restoration of the Bourbon Kings.”

Friday, September 3, 2010

Jean Leonard Marie Poiseuille

Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology contains an analysis of the flow of a viscous fluid through a pipe. Russ Hobbie and I show that the fluid flow is proportional to the fourth power of the pipe radius. We then state that
“This relationship was determined experimentally in painstaking detail by a French physician, Jean Leonard Marie Poiseuille, in 1835. He wanted to understand the flow of blood through capillaries. His work and knowledge of blood circulation at that time have been described by Herrick (1942).”
The paper by Herrick appeared in my favorite journal, the American Journal of Physics (J. F. Herrick, Poiseuille’s observations on blood flow lead to a new law in hydrodynamics, Volume 10, pages 33–39, 1942.). The key paragraph in the paper is quoted below.
“The important role which the physical sciences have played in the progress of the biological sciences has eclipsed, more or less, the contributions which biologists have made to the physical sciences. Some of these contributions have become such an integral part of the physical sciences that their origin seems to have been forgotten. An outstanding example of such a contribution is that by Jean Leonard Marie Poiseuille (1799-1869). About 100 years ago Poiseuille brought a fundamental law to that division of physics known as hydrodynamics—which is a branch of rheology, according to more recent terminology. This law resulted indirectly from his observations on the capillary circulation of certain animals. Most physicists, chemists and mathematicians associate the name of Poiseuille with the phenomenon of viscosity because the cgs absolute unit for the viscosity coefficient has been named the poise in his honor. Few know the story leading up to the discovery of the law which bears his name. This law had more fundamental significance than Poiseuille himself realized. It established an excellent experimental method for the measurement of viscosity coefficients of liquids. The underlying principle of this method is in use today. Since Poiseuille’s law was based entirely on experiment, it was purely empirical. However, the law can be obtained theoretically. Those who are familiar with only the theoretical development are generally surprised to learn that the law was originally determined experimentally—and still more surprised to know that Poiseuille got his idea from studying the character of the flow of blood in the capillaries of certain animals.”
More about Poiseuille and his law can be found in a paper by Pfitzner (Poiseuille and his law, Anaesthesia, Volume 31, Pages 273-275, 1976)
"Jean Leonard Marie Poiseuille (1791-1869) was born and died in Paris. Remarkably little seems to be known about his life. He studied medicine for a considerable time and submitted a thesis for his Doctorate in 1828 (aged 30-31 years). Where he carried out his early experiments studies, and how they were financed, is obscure.

His published work includes […] “experimental studies on movement of liquids in tubes of very small diameter” (his most famous paper, completed in 1842 and published in 1846). For his work “On the causes of the movement of the blood in the capillaries” he was awarded the Paris Academie des Sciences prize for experimental physiology. In later life he became a foundation member of the Academie de Medecine of Paris.”
My biggest question about Poiseuille is the pronounciation of his name. I gather that it is pronounced pwah-zweez. The unit of the Poiseuille has been proposed for a pascal second (or, newton second per square meter), but is not commonly used.