boundary conditions are unpleasant, there are singularities in the field and the relevant functions are multivalued. The ingenious method devised by Lighthill to overcome these hurdles and to handle solid boundaries was fully exploited for design purposes only many years later, when powerful computers became available. When an aircraft flies at supersonic speed, it overtakes the disturbances it generates and these pile up behind it as an approximately conical sonic boom. Whitham and Lighthill showed that, although the magnitudes of the disturbances were correctly given by linear acoustic theory, the characteristics along which the disturbances propagate were not and had to be correctly accounted for in the nonlinear theory. This they achieved using the method of strained coordinates (also known as the PLK method), a technique they invented. Lighthill successfully applied this method, normally applicable only to hyperbolic wave problems, to a variety of other problems.

In the fifties and sixties, one of the most serious problems facing aircraft jet engine manufacturers was the problem of jet noise. The noise from a highly turbulent jet is caused by the turbulent fluctuations in the jet. How could this be estimated when so little was known about turbulence itself? In a pair of papers of great insight, Lighthill showed, by a skilful recasting of the governing equations and making a number of reasonable assumptions about the nature of the turbulent field in the jet, that the noise from a subsonic jet increases in proportion to the 8th power of the jet Mach number. This is the celebrated ‘Lighthill’s 8th power law’. I should add that whereas MJL left the area after writing the seminal papers, a legion of less well-endowed colleagues made their bread and butter for years out of the area that had been opened up by him.

In the sixties and the seventies Lighthill, working closely with physiologists and biologists, made many fundamental contributions to biofluid dynamics. These include his work on animal locomotion in general and the swimming of fish in particular and on the flow of energy in the cochlea. My own favourite is his elegant work on the Weiss–Fogh mechanism of lift generation. The Cambridge zoologist Weiss-Fogh had made in 1973 the remarkable discovery that the wasp Encarsia formosa generated unusually high lift coefficients by a non-classical mechanism. Lighthill analysed this lift mechanism using an inviscid, two-dimensional model and showed exactly how the lift was generated. A fruitful interaction between a biologist and a mathematician.

As a final example of Lighthill’s work let me just mention his analysis of a device designed and manufactured by Vickers Ltd to generate energy from the ocean. Their basic idea was to use submerged resonant ducts, nice robust devices, to extract the energy from the pressure head associated with the more regular ocean swell rather than from the more variable waves. Lighthill was able to cleverly model the device and process and, as in so much of his other work, use complex analysis to pull out the relevant answers.

Like many other great scientists, his success was due to his uncanny ability to take a difficult, messy problem, distil out the essence and then sort this out rapidly and skilfully. Apart from his great physical intuition, he combined in himself a mastery of classical mathematical analysis with a remarkable capacity to carry out difficult calculations rapidly and effortlessly. He was also an excellent speaker who could hold his audiences spellbound with his erudition, enthusiasm and wit; he was greatly in demand at conferences.

For us in India, there was something special about Sir James Lighthill because, I believe, he had a soft corner for

India and Indian science. He has graciously written the preface for a number of books written by Indian scientists. I have always had the impression that he was especially generous in making helpful suggestions, in providing encouragement, and even in the matter of acceptance when he acted as editor for a journal; I have to ruefully admit that I have, myself, been a beneficiary of such generosity. But it has not all been one way. His collected papers have been edited by M. Y. Hussaini, now settled abroad but originally from Chennai, who was an advisee there of Prof.
Abdurahiman, who was himself an early student of Lighthill at Manchester! Lighthill had visited India on a number of occasions. He first came in 1961 to give an invited talk at a conference organized in Bangalore by Dr. P. Nilakantan, the first Director of NAL. His visit in 1989 was in connection with the Golden Jubilee of

CSIR; he had been invited to give talks at a number of venues around the country. All of us who met him at the time were charmed not just by his scientific talks but also by his modesty and graciousness.

P. N. SHANKAR

Bangalore 560 017, India