mem-lighthill.qxp 9/23/99 4:02 PM Page 1226

Michael James Lighthill (1924–1998)

David G. Crighton and T. J. Pedley

mathematics almost as a throwaway for those who David G. Crighton like everything formalised, but in fact his style of James Lighthill was acknowledged throughout working was usually the reverse. the world as one of the great mathematical scien- In one of his most celebrated works, his first tists of this century. He was the prototypical ap- paper on “Sound generated aerodynamically” by plied mathematician, immersing himself thor- jet aircraft and the like, he developed the essen- oughly in the essence and even the detail of every tial mathematical structure completely in two engineering, physical, or biological problem he weeks, but felt that the users (aeroengine design- was seeking to illuminate with mathematical ers) would not be able to grasp the implications, description, formulating a sequence of clear math- and so he delayed submission of his manuscript ematical problems and attacking them with a for sixteen months, in which time he worked back- formidable range of techniques completely mas- wards from the conclusions, isolating the mean- tered, or adapted to the particular need, or newly ing at each stage—and refining and simplifying the created for the purpose, and then finally return- mathematics as he did so. ing to the original problem with understanding, pre- He was in no sense simply the deployer of ex- dictions, and advice for action. isting mathematics against a rich range of practi- His published legacy of six books and some cal problems. To be sure, his earliest papers on su- 150 papers (most of them republished in four vol- personic flight already showed brilliant mastery umes in 1997 by Oxford University Press) show at and exploitation of classical techniques. But much every stage a well-nigh perfect correspondence more powerful techniques were needed for prob- between a clearly identified physical process or lems such as those of how waves in fluids are gen- mechanism and its expression and description in erated and propagated, and for this Lighthill made mathematical terms. His papers or lectures often great developments in the theory of Fourier analy- emphasised the physical aspects and gave the sis, generalised functions, and asymptotics—all set out with elegance and economy, and full rigour, David G. Crighton is professor of applied mathematics at in a delightful 1958 book, Introduction to Fourier the University of Cambridge. His e-mail address is dgc@ damtp.cam.ac.uk. This segment is reprinted by permis- Analysis and Generalised Functions. Rather differ- sion of the author from The Independent, where it ap- ent ideas were needed for nonlinear problems, peared on July 22, 1998, following Lighthill’s death four such as the propagation and focusing of sonic days earlier. booms, and here Lighthill provided equally

1226 NOTICES OF THE AMS VOLUME 46, NUMBER 10 mem-lighthill.qxp 9/23/99 4:02 PM Page 1227

original and elegant new techniques, permanent of papers, and ap- and frequently used additions to the armoury. plications include Michael James Lighthill was born in Paris in kidney-stone- 1924 and excelled across the board at Winchester crushing before going up to Trinity College, Cambridge, in lithotripsy ma- 1941 for a two-year wartime B.A. He worked on su- chines and, with personic flight at the National Physical Labora- the same mathe- tory, Teddington, for the rest of the Second World matics, flood War, publishing his first paper before he was waves in rivers twenty. He then went as Senior Lecturer to Man- and traffic flow chester University at the age of twenty-two, before on highways. taking the Beyer Professorship of Applied Mathe- From Manches- matics there, aged twenty-six, in succession to ter, Lighthill went Sydney Goldstein. In his thirteen years at Man- on in 1959 to be- chester (1946–59) Lighthill ran one of the most come director of powerful and inventive fluid dynamics groups ever the Royal Aircraft formed anywhere. Establishment, He had many Ph.D. students who often rose to Farnborough, considerable heights themselves. Indeed, there where his leader- was a period in which no fewer than seventeen of ship extended to his Manchester students held chairs in the UK, the critical exam- Photograph courtesy of University College London. and that at a time when the number of universi- ination of every Sir James Lighthill ties was no more than a third of its present num- report emanating ber. Although prepared to share the credit on a from the RAE. The paper with a colleague, Lighthill almost never al- years 1959 to 1964 saw him again in his element lowed his name to appear as author on any paper (“Wouldn’t change it for anything!”), working on the wirtten by a student. And he was, then and since, aerodynamics of the slender delta wing for Con- tireless in his support for young scientists of any corde, on spacecraft, and on short-haul aircraft. He promise and for scientists working in disadvan- also worked with the post office in developing taged circumstances. commercial use of television and communications During these Manchester years Lighthill worked satellites while managing in unusual detail the extensively on gas dynamics, including effects im- work of the 8,000 RAE staff, of whom 1,400 were portant at very high speed, in his studies of ioni- professional scientists and engineers. Towards the sation processes and the diffraction of shock and end of his RAE time he became dissatisfied with blast waves. He also launched two major new fields the support in national societies for applied math- in fluid mechanics. The first of these, “aero- ematics and founded the Institute of Mathematics acoustics”, or “sound generated aerodynamically”, and Its Applications, of which he was the first was announced in a remarkable paper published president, in 1965–67. From 1964 to 1969 Lighthill by the Royal Society in 1952. Unusually but sig- held a Royal Society research professorship at Im- nificantly, that paper neither contains nor needs perial College, and here he began his great devel- so much as a single reference to any prior work. opment of mathematical biofluiddynamics: the This work has remained for nearly fifty years the quantitative understanding of the flow of blood in progenitor of all subsequent work in the field and mammalian cardiovascular systems, of air in the has been cited in many thousands of later papers. human airways, and of the flying of birds and in- It had immediate implication for noise reduction sects and the swimming of fish. Mastery of biol- in jet aeroengines, motivating the trend begun ogy was, he insisted, the sine qua non for entry into later in the 1950s and still continuing to engines this field. He revelled in lectures, not only in the with higher bypass ratio, greater diameter, and articulation of all the Latin names, but in his abil- lower exhaust speed, as mandated by Lighthill’s fa- ity to perform the appropriate gymnastics to mous eighth power law for jet noise. Remarkably, illustrate certain flying characteristics—in partic- the Lighthill theory was sufficiently versatile to be ular the “clap and fling” mechanism employed by applied also in problems as diverse as the heating the tiny wasp, Encarsia formosa, to endow it with of the Sun’s corona and the noise heard under a lift coefficient far above that obtainable from the water due to breaking surface waves and splash- ordinary aerodynamics in which the component ing drops. parts of the body do not break apart. The second new field, “nonlinear acoustics”, In 1969 he succeeded Paul Dirac, founder of was initiated by a famous 100-page article written much of quantum mechanics, in the Lucasian Pro- in 1956 in honour of the seventieth birthday of an- fessorship of Mathematics at Cambridge—though other great mechanics scientist, Geoffrey Taylor. when he referred to “my predecessor in the chair,” This field is now represented by many thousands one sensed he was thinking primarily of Newton.

NOVEMBER 1999 NOTICES OF THE AMS 1227 mem-lighthill.qxp 9/23/99 4:02 PM Page 1228

Here he taught indefatigably and with enormous gusto six days of the week at nine in the morning. T. J. Pedley He widened his range yet further with work on con- It was during the fifteen years after he left the trol systems; on active control of sound, or anti- Royal Aircraft Establishment (1964–79), first as a sound; more and more on waves; on oceanography Royal Society Research Professor at Imperial Col- and atmospheric dynamics, including monsoon lege and then as Lucasian Professor of Mathemat- prediction and propagation; and on biological me- ics at Cambridge, that Lighthill totally transformed chanics at the microscopic level. the study of biological fluid dynamics. He wrote From 1979 to 1989 Lighthill was provost of major reviews of aquatic animal propulsion (1969), University College London, much engaged in fund- of animal flight (1974 and 1977), and of low raising; in new developments in the college, Reynolds number flagellar hydrodynamics (1976). particularly in the biology and biotechnology sides; Each of these is characterised by an exhaustive sur- and in dramatically improving the representation vey of the animal kingdom to make sure that all of women in senior posts. He still maintained his actual modes of locomotion are covered by the pre- scientific work with studies on the unpredictabil- liminary fluid dynamical analyses that he then ity of large systems, on wave energy extraction de- presents in qualitative if not quantitative forms. vices, and on features of the human auditory sys- (The lectures based on these surveys were always tem. After retirement he took up chairmanship of splendid occasions, in which Lighthill would al- the Special Committee on the International Decade ternate between demonstrating his mastery of the for Natural Disaster Reduction and travelled and relevant taxonomic terminology and making his lectured worldwide. fluid dynamical audience feel at home by incor- His achievements were widely recognised— porating standard terms from other applications, through election as Fellow of the Royal Society at such as—in the flight context—payload, wing-load- the age of twenty-nine, through the award of 24 ing, induced drag, etc.) He himself concurrently honorary doctorates, through foreign membership made major advances in fluid dynamical analysis of the most prestigious academies, through re- in all three areas. ceipt of many medals and prizes, and through In fact, his first analysis of fish swimming (1960) knighthood in 1971. was published while he was director of the RAE, Stories about Lighthill are legion, and no amount and in it he set out the principal features of what of discounting for exaggeration makes them less has become the standard model of fish swimming amusing or less essentially accurate. It is well using body undulations. This small-amplitude, known that he was fined £1 for jumping from a slender- (or elongated-) body theory explains how train as it passed, to his dismay, through Crewe thrust is generated from the reactive (added-mass) without stopping and that on more than one oc- forces experienced by an undulating body as it casion he successfully defended himself on charges gives sideways acceleration to fluid which is mov- of speedy driving, turning the spotlight of his pres- ing backwards relative to the fish at the approxi- ence, charm, and authority on the magistrates as mately steady swimming speed. Moreover, Lighthill he explained how, as Lucasian Professor, he was showed a full appreciation of the difficulties in- fully seized both of the laws of mechanics and of volved with calculating recoil (stemming from the his duty to society not to waste energy, the latter fact that an arbitrary displacement wave will not compelling him to desist from applying the brake in general give rise to instantaneous forces and cou- on any downhill section of road. ples that exactly balance the fish’s transverse and He saw everything as a challenge to his brain, angular momentum fluctuations) and with or to his physique, or to the coordination of the analysing the three-dimensional boundary layer, two. And if no challenge was obviously at hand, he needed to check whether the computed thrust would create one: mastering Portuguese in three does indeed balance the drag at the supposed weeks to the extent that he could give a (long) swimming speed. These features were later ex- after-dinner speech in the language, for example. panded (1970), and the whole theory adapted to He listed his leisure interests as music and swim- large amplitudes (1971). Other aspects were taken ming, to which surely literature, poetry (especially further by visitors to Cambridge in the early 1970s Portuguese), and languages (French, German, Russ- (D. Weihs, M. G. Chopra, T. Kambe). Lighthill ian, Portuguese) should be added. His swimming exploits were legendary—careful T. J. Pedley is G. I. Taylor Professor of Fluid Mechanics at in his homework on tides and local currents, bold the University of Cambridge. His e-mail address is in his ignoring of everything else. On countless oc- [email protected]. This segment appeared casions he came home safely, against the odds. Last originally in Mathematics Today, vol. 34, no. 6, published by the Institute of Mathematics and Its Applications, United Saturday he almost completed a nine-hour swim Kingdom, and it is reprinted with the permission of the round Sark (he was the first ever to do this, at the author and the publisher. An expanded version of this part age of forty-nine) against high winds and huge of the article can be found on the Web at http://www. waves before dying close to the shore. damtp.cam.ac.uk/user/imammb/lighthllbfd.htm.

1228 NOTICES OF THE AMS VOLUME 46, NUMBER 10 mem-lighthill.qxp 9/23/99 4:02 PM Page 1229

himself returned to fish swimming after he had re- adhered to the by now rarely found view that a su- tired from University College, London, through a pervisor’s name should not appear on the papers series of four papers on “Balistiform and gymno- of his students, however much of the work he had tiform locomotion” (1990) and, more recently, with done himself), he instilled his own uncompromis- a fascinating analysis of how a herring can control ing principles for interdisciplinary research. The the motion of its head to enable it to sense pres- motive for doing fluid mechanics in biology is to sure differences across the head of external origin help biologists understand how their systems work. without being swamped by self-generated pres- To this end it is essential “to talk to zoologists and sures (1993). go on talking to them; to read their works and go For flagellar locomotion Lighthill developed a on reading them; to study their collections and go new viscous slender body theory superior to the on studying them” (1973). He did it, and we must customary resistive force theory, in which a long, do it too. narrow, cylindrical flagellum, beating with a long Those of us working in biological fluid dynam- wavelength, is replaced by a distribution of ics know that James Lighthill made huge contri- Stokeslets and dipoles on its centre-line whose butions to many areas of fluid mechanics and ap- strengths are obtained by solving an integral equa- plied mathematics in addition to ours. That merely tion representing the no-slip condition on the cylin- intensifies our admiration for the transformation der surface (1976). In the context of flight Lighthill’s he wrought in our subject. We honour him deeply collaboration with the then professor of zoology and miss him greatly. at Cambridge, T. Weis-Fogh, led to an elegant analy- sis of a newly observed mode of lift generation in small hovering insects, the clap-and-fling (1973). However, in both these areas many of his ideas were put into effect and subsequently taken much fur- ther by Ph.D. students: namely, J. R. Blake, J. J. L. Higdon, and J. M. V. Rayner. Despite these pivotal contributions to external, or zoological, fluid dynamics, Lighthill’s most sig- nificant contribution to the field may have been in internal, or physiological, fluid dynamics—and not through his publications, but through his admin- istrative vision. In the middle 1960s he joined forces with C. G. Caro from St. Thomas’s Hospital to persuade Imperial College to create the Physi- ological Flow Studies Unit. This research group started work, attached initially to the Aeronautics Department, in 1966. Many students, fellows, aca- demic staff, and visitors have worked there over the years (including this writer, from 1968–73), and now, as part of the Department of Biological and Medical Systems, it has a strong international rep- utation, especially in the study of the response of artery walls to haemodynamic stresses, an im- portant part of understanding arterial disease. Lighthill’s direct contributions to physiological fluid dynamics were quite few—though including several masterly, specially written chapters in his 1975 book Mathematical Biofluiddynamics—until towards the end of his career he developed an in- terest in the inner ear. In particular, he demon- strated that the ability of the hair cells of the cochlea to sense different frequencies of sound ac- cording to their distance from the entrance could be associated with a phenomenon of critical layer absorption of the elasto-acoustic waves set up in that organ (1981, 1991, 1992). To the students and colleagues who worked with him, Lighthill gave much more than intellec- tual stimulus. In addition to the generosity with which he shared his ideas and gave his support (he

NOVEMBER 1999 NOTICES OF THE AMS 1229