AAS-Biographical memoirs-Mahony

This memoir was prepared with the assistance of many of John Mahony's friends and collaborators. Many quotations have been included in an effort to enliven the prose and give a more penetrating account of the scientific aspects of John's work. In order to balance this essay some written contributions had to be edited; we trust in such a manner that the condensed account is still a valid expression of the contributor's views. The following words of Professor T.B. Benjamin provide an excellent overview of John's academic persona:

I have always had immense respect for John Mahony's mathematical abilities in general, for his mastery of asymptotic methods in particular, and for his general scientific acumen. He was one of the cleverest and most inventive applied mathematicians with whom I have been privileged to collaborate. His comments in mathematical (and other) discussions were delivered in lively fashion and were often fractious and deprecatory; he was not a diplomatic colleague. His comments, however, were always acute and valuable. He was a superb scientist whose great accomplishments have not yet received the wide recognition they deserve.

Introduction

Patrick Francis Mahony emigrated to Melbourne from Dublin with a fierce spirit of independence, a great hatred of social injustice and a very high regard for education. He met Ida May Cain who had emigrated from England, they were married, and their son John was to be their only child. Patrick was an extremely active union and Communist Party member, and no doubt as a result of this found it difficult to find permanent employment, particularly during the depression. He was an insurance agent, labourer, and at the depth of the depression, a wharf labourer. In 1941, by which time the military situation had become very serious for the Allied forces, Patrick joined the AIF. Tragically he was lost in 1942, believed drowned on a Japanese hospital ship.

Even though young John had not yet reached his teens, Patrick's influence on his son was strong, and would remain a driving force throughout John's life. After Patrick's death, John's mother, working as a cleaner, struggled in very difficult economic circumstances to make ends meet and to provide the education she and Patrick had dreamed of for John. Perhaps as a result of these events, Ida set standards for John that were impossible to meet. Ida in fact survived John and continued to maintain critical visual on John's activities right up till his death. Son, like father, would turn into an extraordinarily strong willed and independent character, with uncompromising standards. Accepting other's accounts of a situation (whether academic or social) was not John's way. Such personality gifts/traits, when combined with an extraordinary original and nimble mind, would produce an array of truly original ideas. There were also detrimental effects – much of John's mature life was spent fighting with psychological problems undoubtedly partly brought about by this uncompromising approach to life.

The following lighthearted quote from John's long term collaborator and friend John Philip provides a nice snapshot of John at work:

Like all the best mathematicians John was ever on the alert for soundly based connections between mathematics and real world phenomena and his ability to bring to bear a quick intelligence and deep mathematical knowledge on physically motivated ideas was remarkable. After meeting at Harvard in 1966 we collaborated over a period of decades. Out of his visits to Environmental Mechanics has been generated a legend in the Environmental Mechanics at CSIRO. John and I would frequently have sessions in front of the large blackboard in my office. Quite often I would be convinced that he had missed some physical aspect of a problem; and he, just as often would become outraged at what he saw as my cavalier disregard for proper mathematical rigor. As the people at Environmental Mechanics tell it, the noise level from my corner office would increase as the morning wore on, with my voice remaining deep (but getting louder), and with John's also getting louder, and as well climbing through the octaves. This tended to be punctuated by the peculiar high-pitched braying noise that John emitted at times when he needed to hold the floor while his verbalizing caught up with the high speed of his thoughts. On such occasions the usually most agreeable open planning of the Pye Laboratory became a burden on my colleagues.

From this you may gauge that John was a man of strong convictions and a person whose ability to think on his feet was extraordinary. He was very concerned with all aspects of human behaviour, read broadly and was deeply aware of underlying issues. John's nature was such that he never accepted conventional wisdom – he had to think through the issues himself. As a result of these intense personal investigations, the opinions he expressed were strong and could not be ignored – even if incorrect! It was, in fact, very difficult to convince John about anything and he would never admit to a change of opinion. His loud voice and passionate discussions, and certainty in the correctness of his ideas, very often left his audience floundering. For many, John's arguments and mannerisms were threatening – often because their understanding was superficial. John's idealism coupled with an acute mind led to him espousing educational and environmental causes with great vigour. Unfortunately, his inability to accept second best sometimes led to frustration and despair.

John was not always buried in his work, however; he was a well rounded human being, who had wide interests that included sport, music, gardening and bridge, the last of which led to his meeting his future wife Jocelyn Peters. They were married in February 1957 in Melbourne. They excelled at establishing new homes, in which they entertained their friends most generously. They moved frequently but their children Patricia Helen and Richard John (born 1968) made greater stability of location essential. So at Darlington, a hills suburb of Perth, John and Jocelyn created a rural garden and lovely place for their children to grow up in. John was a most concerned parent, who took an enormous interest in his children's welfare and education; this naturally for John meant involvement with parents and citizens groups and coaching of junior football teams.

When ill health forced his early retirement, he did not vegetate but used his skills in many ways: as a gardener to civilise a large block around the newly built house at Mandurah, where Joc and John finally settled after three moves, as a champion for the improvement of the local environment, as a tutor for children with learning difficulties, and finally in collaborating with Nev Fowkes in writing a pioneering undergraduate text which aimed at teaching mathematical modelling which John always felt to be the cornerstone of his teaching.

Melbourne

The Victorian educational system, without a doubt, played a crucial role in the training of a whole generation of Australian mathematicians. The Leaving Honours Examination (later to be replaced by the Matriculation Examination), gave the school system a challenge, and the opportunity to prepare students for University honours courses in mathematics and in the physical sciences. Under T.M. Cherry's leadership, the mathematics courses at Melbourne University concentrated on a rigorous introduction to classical analysis and its applications to generalised dynamics and continuum mechanics, and proved to lay the foundation for future research careers. The teaching staff at Melbourne was very small in number but well selected, and included for periods people like Keith Bullen, Archie Brown, Felix Behreud, and Russell Love.

John Mahony became fascinated by mathematics during his secondary education, and when he had won entry to the prestigious Melbourne Boy's High School he consistently topped his class in Mathematics and Physics and won a General Exhibition which took him to The University of Melbourne. In spite of this background and the presence of quite a remarkably gifted group of students at Melbourne, John's real interest and enthusiasm lay in achieving excellence in the sporting arena rather than obtaining high marks in his academic work. John's innate ability carried him through the academic requirements of his course but without the distinction one might have expected. Many years later, John would blame his indifferent interest and performance as a student on the narrowness of the courses, but the fact remains that he, with twelve or so others who were undergraduates in the 1940s, later occupied chairs at universities in Australia and abroad without any further undergraduate training. The list of professors would include G.K. Batchelor (Cambridge), R.C.T. Smith (Armidale), G.S. Watson (Princeton), Kevin Westfold (Monash), A.F. Pillow (Toronto and Queensland), Richard Dalitz (Oxford), J.P.O. Silberstein (UWA), C.A. Hurst (Adelaide), H.C. Levey (UWA), J.R.M. Radok (Adelaide), W. Freiberger (Rhode Island), J. Gani (Sheffield, Manchester, California). While Cherry did not wish to establish a research school in his department, he did establish a remarkable relationship with the Aeronautical Research Laboratories (ARL) which enabled most of his best honours students to work usefully and profitably during their vacation – often they found employment there at the end of their course. ARL had been set up as a Division of the Council for Scientific and Industrial Research (CSIR later CSIRO) just before World War II as a scientific and technical back-up for the aeronautical industry which was being fostered with private and public capital in anticipation of the war. The vision of Sir George Julius (Chairman CSIR), L.P. Coombes (Chief of Division), and most importantly H.A. Wills, allowed ARL to grow into a first rate applied mechanics research laboratory, which by the forties was consulted by a wide range of industries as well as by the forces. Because the airplane played such an important role in the war, the development of the jet engine and supersonic flight was thought to be of crucial importance. The field bristled with unsolved problems which could only be solved by combining mathematical analysis (that might be done using Cherry's students) and pioneer engineering (that might be initiated by graduates from Sydney and Perth). Perhaps Cherry's most important work arose out of ARL contact. Using the Legendre transformation, he linearized the compressible flow equations, thereby greatly simplifying the solution process. While this work was not immediately applicable, it stimulated others to take up problems that had been previously regarded as too difficult.

There is no doubt that sport was a lasting passion for John; not just an escape from studies that did not excite him. He had considerable success in a wide range of sports, notably athletics, football, tennis, golf and squash. Excellence in performance was in fact of primary importance to him, and when injury prevented him from competing at a high level he gave up sport to become a dedicated spectator. An inevitable feature of the Mahony household at a later stage was the television tuned to the game of the time with volume sufficiently high to be heard above the conversation, whether technical or non-technical. Somehow John managed to carry on two conversations, one with visitors and one with the television. On such occasions John's conversation was like a patchwork quilt, with detailed technical observations interrupted by OOOh's, AAr's, Stupid!! etc. After a few glasses of beer, the sound level didn't seem to bother, and it all seemed to fit! John's discussions about games, players etc. were every bit as impassioned as those about science; here also he seemed to have an extraordinary ability to dredge up some detail of some game to completely demolish any view contrary to his only presently held opinion about the relative merits of particular player's abilities etc. As a result of his early preoccupation with such important sporting matters, his final honours result was 2A and did not win him an overseas research scholarship. Instead he joined the staff of the Fluid Dynamics Division of ARL and found himself in an environment of enthusiastic young researchers, both theoretical and experimental, with the best facilities available in Australia. Fenton Pillow had recently returned from Cambridge and makes these comments:

I can well remember the first problem I gave John as a vacation student in December 1950. It was the (unforced) damped simple harmonic oscillator with arbitrary initial conditions and very small mass described in non-dimensional form by

Ex[with two dots above] + x[with single dot above] + x = 0

With v = x[with single dot above] the trajectories in (v, x) phase space form a stable node at the origin. I showed him the limiting (E->0) trajectories were initially vertical straight lines which rapidly joined the single straight line v + x = 0, turned sharply, and then headed more sedately for the origin. Given that the exact solution to this problem can be written down immediately I asked him if he could develop an iterative process for determining it asymptotically ab initio (mathematically blindfolded so to speak) without using the exact solution except perhaps for a surreptitious peep now and then. I suggested he use an outer time variable t near v + x = 0, and an inner variable t* = t/E near the initial vertical part of the trajectory, and devise a means for matching the two expansions. These were early days and except for Goldstein's 1936 Proc. Com. Phil. Soc. forgotten paper on the wake nothing much was known on matching. John tried this and, understandably now, made somewhat heavy weather of it, though he sorted out a few terms in both expansions. He complained that the iterates were not uniformly smaller as t -> x, and to cope with the difficulty he eventually introduced a newouter variable t+ = tg(E). A power series expansion for g(E) was obtained by iteratively demanding than no solutions of the homogeneous outer differential equation appear on the R.H.S. as perturbation terms. He thus recovered the +-[square root of] 1-4Et factors that appear in the arguments of the exponentials that occur in the exact solution. The result impressed me immensely and I enthused a lot about it to him. His progress certainly accelerated after that. This was the beginning of John's work on multiple scales. The trouble was that I was too immature to realize the wide ranging applicability of the method and did not sufficiently urge him to apply it immediately. John let it lie fallow too and went off to Manchester.

Manchester

During the war years and immediately after, Sydney Goldstein (who played a major role in the defense research effort, as well as in the development of fluid dynamics) managed to attract most of the best young applied mathematicians in Britain away from the traditional universities to Manchester. Aerodynamics was the exciting area for applied mathematicians to work in at the time, both because of the war effort and because of major advances in understanding. The mathematics developed to handle the problems arising (perturbation methods etc.) was both elegant and quite different to classical techniques. In fact many of today's senior mathematicians (now retiring) had their initial training in this area, where much of the perturbation work was first developed. James (later Sir James) Lighthill initially supervised John. His project was published as 'A Critique of Shock-expansion Theory' and was associated with a theory developed by Fenton Pillow to determine supersonic flow around a body. To determine the strength and path of a shock wave, it is necessary to solve the non-linear compressible fluid equations; a difficult task even today using numerical schemes. Fenton's scheme used a perturbation procedure based on the smallness of the shock strength (essentially M-1, where M is the Mach number of the flow). Based on this simplification, analytic answers can be obtained and important physical parameters can be extracted from higher order expansion terms (in M-1). John examined a situation of a non-uniform bow shock; entropy gradients need to be taken into account in this case (for a uniform shock, the entropy is constant on either side of the shock with a uniform jump across the shock) and found that the theory can't be characterized as simply 'nth order in shock strength' with definite n (e.g. n = 3,4). At this stage Richard Meyer, who had developed a much improved approach to handle such problems, took over John's supervision. John quickly completed the work which was published in two papers in the Philosophical Transactions of the Royal Society of London.

ARL

When John returned to ARL from Manchester, Fenton Pillow was still there and recalls:

After his excellent work on characteristics and accelerating shock waves with Meyer he was still not widely versed in Potential Theory a la Courant Hilbert. Harry (Levey) gave him a lot of help with complex variable theory and John grew to respect his ability. John was good at scaling and model making and I remember many exciting discussions with him about free convection above heated electrical wires and his insight into how the diffusion zone around the cylinder notched into the buoyant plume above, which was known to have self-similar velocity and temperature profiles. He later extended this work to include the first effects of forced convection as well.

This led to the publication 'Heat Transfer at Small Grashof Numbers', Proc. Roy. Soc., Series A, 238 (1956), 412-23. The work was prompted by careful experiments on anemometer wires by Collis and Williams (1), and is significant in John's career in that it probably represents the first occasion in which John worked closely with experimentalists. This pattern of seeking out questions of real physical importance and working with careful experimental backing would be a feature of much of his later work. John's natural bent was analysis but much of his inspiration was derived from practical situations, often prompted by experimental anomalies.

Briefly the background to this problem is as follows: Measurements made by Collis and Williams on the heat transfer from long thin wires revealed something of a paradox for Grashof numbers in the range 10*-3 to 10*-9. As the Grashof number (G = l*2 gbeta(T**b-T-**inf)/v*2) represents the ratio of buoyancy to viscous forces, for such small values of G it appears reasonable to neglect the convective motion entirely, and so simply calculate the heat transfer from the conduction equation alone. This leads to a non-zero heat transfer rate approximately independent of G for small G. The experiments, however, indicated that the heat transfer rate tended to zero as G tended to zero! John showed that, while convection is negligible in comparison with conduction near the body, it becomes as important at distances of the order G*-n from the body, where n varies from 1/3 to 1/2 depending on body shape. By patching together solutions in the two regions, he obtained expressions displaying the effect of body shape on heat transfer rate and thus explained the anomalous results. The results were also presented in a particularly useful practical form.

The often casual collaboration between John and Harry Levey continued until Harry's untimely death, even after John had left ARL and later when they were together in Perth. To quote Fenton again: 'Harry Levey's ability, general intelligence and understanding provided a powerful stabilizing influence for John which he badly needed, and part of John's success must be ascribed to Harry who several times steered him back to saner paths'.

Sydney

In 1958 John joined the Aeronautical Engineering Department at Sydney University as Senior Lecturer. There he started a fruitful collaboration with Bill Wittrick. As Peter Chapman recalls:

John's relatively short sojourn further enlivened the already lively aeronautical engineering department (which had steadily metamorphised into a department of theoretical and experimental mechanics) and had a profound effect on the graduate students in particular. I had started, but had made little headway, on a PhD with Bill Wittrick; I think the latter felt I had need of the intellectual equivalent of the electric cattle prod, and he passed me on to John, who forthwith presented me with the outline of a problem in compressible fluid mechanics involving spherical and cylindrical shocks. I am certain he was confident of the manageabilty of the problem, if not the solution details. He proceeded to educate, coach, coax and bully details of the solution from me over the next two years. He also supervised Tony Watts's investigations of shock dynamics, as well as providing advice (solicited and unsolicited) to the other graduate students at the time; M. Hall, R.J. Stalker, and R.E. Center spring to mind. The effect of several separate streams of verbal advice and/or opinion on matters academic, sporting and political, delivered in more or less simultaneous coffee break conversations, remains an indelible memory. The laboratory technical staff, one W. Jamesson and experimental officer T. Thompson were not exempted from the harangues. They were perhaps even more startled by these harangues than the graduate students, having been more familiar with the somewhat remote and contemplative academic as exemplified by A.V. Stephens.

During this period John developed the idea of multi-scaling, which is now a standard procedure for handling singular perturbation problems; this work is of major importance.

[The original version of this Memoir contains three pages of detailed mathematical explanations at this point. These could not be reproduced due to the limitations of formatting on the World Wide Web.]

Brisbane

After a great deal of argument from Clive Davis, the then Professor of Mathematics at Queensland, the principle of creating multiple chairs in departments at Queensland University was established, and soon after (on 1 January 1961) John was appointed to the first chair of Applied Mathematics in the department. He characteristically threw himself into the task of establishing a pool of adequately equipped graduates and an active research group. A number of these students went on to take up positions in universities and research insitutes around Australia. John resigned on 31 January 1964 to take up an appointment at the University of Western Australia.

Perth

A chair had been created for John at the University of Western Australia (UWA) that gave a promise of fruitful collaboration with Harry Levey, David Hurley, Peter Wynter and Phil Silberstein, all former ARL and Melbourne colleagues. Soon after, Joyce Billings also joined the group. Phil had been enticed away from applied mathematics to pure mathematics by the belief that modern analytic techniques had a crucial part to play in developing new solution methods and their justification. With such a strong group of active applied mathematicians the expectations were great. All told the situation presented a unique opportunity for collaborative classical/functional analytic work. By this time John had established an international name largely as a result of his pioneering singular perturbation work, which was further enhanced by the impression he made on a succession of renowned visitors: Carrier, Goldstein, Meyer, Allendorfer, Oakley, Bondi, Hoyle, Batchelor, Newman, to name a few. At the time Harry was Professor of Applied Mathematics and head of department, and he also enjoyed an international reputation because of his work in gas dynamics. Additionally, another former ARL colleague, Fenton Pillow, returned from Canada to take up John's recently vacated chair in Queensland and strong links were thus established across Australia. These were happy days. There was a regular seminar at which staff and students would discuss their research work, after which all would retire to Steves (the local pub) where the conversation would range across the mathematical spectrum, and also of course more mundane matters such as Australian Rules football and the politics of the time. It was a rich experience for all, but especially for the growing group of post graduate students.

The mathematics department was expanding, but not rapidly enough to cope with the growing number of students. The staffing of the department was a very serious problem which put enormous pressure on the senior academics. The brunt of this pressure was borne by Harry as head of department, who also found himself trying to cope with John's (perhaps unrealistic) expectations. Tragically and unexpectedly, Harry had a heart attack and died. Although logic might suggest that such circumstances were not a strong contributing factor in Harry's death, John was burdened with guilt for the rest of his life, and suffered severe psychological problems that would eventually reach such proportions that he would be forced to retire. The confident and happy days for John at UWA were over. John was almost on the point of resigning, and if it had not been for the imminent prospect of study-leave at Harvard may well have done so.

When John returned after a most productive year in the US he took up his responsibilities, both teaching and administrative with vigour. He served for periods as Head of Department, Dean of Science and took turns on many committees. He took these commitments very seriously but found them very stressful because be could not come to terms with living in an imperfect world.

Nevertheless, the years to follow at UWA were extremely productive. Largely under the guidance of John and David Hurley, many students found themselves working on problems of major importance ranging across the classical applied spectrum. The applied group expanded and many of today's leading Australian applied mathematicians were trained in the department in the subsequent ten years. There was no student in applied that John didn't watch over, even if the students were being supervised by another colleague. Seminars were exciting events with spirited exchanges, with John's incisive comments often dominating. His ability to cut through the detail to expose the underlying issues was legendary, and if there was any flaw in the modelling or analysis it was exposed on the spot. Students were very nervous, and visitors often felt no better. On one such occasion, David Hurley recalls that a distinguished English numerical analyst in his first talk in Australia described work on the numerical solution of Laplace's equation in regions of irregular shape. At the conclusion he posed a problem 'how can the behaviour of the solution near the point where the slope of the boundary is discontinuous be determined?' John and Harry promptly supplied the answer. The visitor was somewhat taken aback; he had not expected such informed comments from Australians! It was, in fact, often the case that problems were solved on the spot.

In the 1960s, one of the major unsolved problems in water wave propagation concerned the energy reflection resulting from bottom topography. If the bottom is flat, then there's no physical mechanism to cause reflection, so that a wave will propagate with undiminished amplitude over such a bottom. In all other cases a reflected wave is to be expected. If there is an abrupt depth change, then mass and momentum conservation matching considerations determine the reflection coefficient. In cases in which the length scale associated with depth changes are large compared with the wave length, the multiscaling methods described earlier can be used to determine amplitude variations in the transmitted wave. However, such techniques do not expose a reflected wave for C*x bottoms, even though obviously there must be one except in very special situations. The difficulty (again) lies with the asymptotics, but can't be overcome using multiscaling modifications. As indicated earlier, asymptotic techniques order terms according to their size in some limit, so that as E -> 0, terms of order e *-k/E cannot be seen relative to to E*n, for any n (however large) and any k (however small). Multiscaling techniques enable one to determine such small terms providing the terms are not exponentially small everywhere. The reflection coefficient is in fact exponential in E (here E denotes the ratio of the wave length to the length scale of bottom variations), so that the amplitude of the reflected wave is exponentially small everywhere; herein lies the difficulty. To get at such uniformly small terms it's necessary to not prematurely take the asymptotic limit. An exact analysis would enable one to do this, but of course the reason for using asymptotic analysis is that an exact solution is not available; so one needs to introduce approximations with care at the right stage of the analysis. Interestingly enough (about 20 years later) this area of exponentially small asymptotics has had a resurgence of interest from physicists, largely because most physical theories represent asymptotic approximations, and the clue to determining the connection between different asymptotic regimes often lies in determining such terms. (Unfortunately, much of the work being presented today was well understood 20 years ago!) The exact analysis required to extract exponentially small terms was in place for known ordinary differential equations for which complex integral representations of the solutions are available, see 'Asymptotic Results for the Solutions of a Certain Differential Equation', J. Aust. Math. Soc., 13 (1962), 147-58. Essentially in such cases one first isolates out the source of the small contribution of interest and then manipulates the integration contour to extract its contribution before taking asymptotic limits. Using complex function methods John, and later John and Peter Chapman, showed that the size of the reflection coefficient in a variable refractive index medium was determined by the location of the nearest singularity associated with the refractive index function when analytically extended into the complex plane – very elegant! The gravity wave problem is much more difficult, there being no available exact solution representation. Garry Fitzgerald, one of the many bright students that had their training at the UWA, was to attempt this question. Here is his account of this work:

After Harry Levey's untimely death in 1966 John took over the supervision of my Honours thesis and subsequently my PhD work. We started examining the effect of bottom topography on the amplitude of tidal waves on the North West coast of Western Australia. These first eighteen months provided me with a glimpse of the brilliance of John Mahony's mathematical insight into a very difficult problem. It was with some trepidation that I would discuss work on the tidal problem with John Mahony when it all seemed so obvious to him what the outcome of any modelling exercise should be. His ability to 'see through' the physics of the problem was emphasized to me in these early days of my PhD program and reinforced the impression gained over several years of undergraduate courses. In spite of this determined effort to understand this complex problem after 18 months we moved on to the more clearly defined problem of determining to effect of bottom topography on the propagation of surface water gravity waves.

The gravity wave problem is normally described by a partial differential equation together with (three) boundary conditions – not a useful form for extracting error estimates. Given the controversial nature of problem at the time it was essential to 'prove' any results obtained (particularly in the small wave length limit), so a major reformulation was necessary. (John and Peter Chapman's work with on light propagation suggested that the reflection coefficient might be exponentially small, but no one really knew!) A major breakthrough in the mathematical description of the model was achieved when the problem was reformulated in terms of a single integro differential equation. Such integral equation formulations are suitable for error estimation purposes. This breakthrough was received by John Mahony in his usual offhand way: he stated simply that progress was being made! Although this first formulation contained all of the main important ideas the representation was in terms of generalized functions at large distances from the major typographical changes. At John's insistence this formulation was modified to be expressed in terms of ordinary functions and this reformulation proved rich in providing both analytical and numerical representations of solutions to the problem (2).

The long wave limit had already been done in both a qualitative and quantitative way but our new representation provided the necessary framework to 'prove' the asymptotic results that had been obtained were indeed correct. More importantly this representation allowed the description, in precise terms, of the short wave limit. The ability to recover the exponentially small reflection terms in a cohesive and routine way was quite unexpected for such a complicated problem as that describing water wave propagation.

Throughout the period of John's supervision I always felt that he already 'knew the answer' and this was more than enough of a spur to keep me one step ahead. The last few years of my PhD work with John were amongst the most stimulating I have ever experienced. I remember John Mahony with great affection. His expectation of perfection in everything attempted remains with me to this very day.

John shared with many great scientists a disdain for second-rate work and expected good ideas without extravagant praise. For John it was the ideas that mattered, not the accolades or who produced them. He literally despised the 'assumption tweeking' type of mathematics that produces endless clutter in the journals. As Jerry Bona reported, 'I know from personal experience that John solved a number of problems that he never even considered writing up for publication because he viewed them as being relatively unimportant exercises. Other distinguished scientists would have been happy to at least write short notes describing the work in question, but John didn't because of his deeply held convictions of what real science was about.' Wherever he went, he would broadcast ideas without expecting any recognition for his contributions. In fact, he refused to have his name on material produced by his students. Given that he normally had effectively solved the problems he handed to them, this was a true gift.

John's work with David Hurley at UWA deserves special attention. As mentioned above, John and David had been colleagues at ARL. David supervised many students and John and David's technical interactions were largely through discussions concerning the progress of their various research students. As well as being a very good classical analyst, David has an exceptional eye for important problems, so the two formed a formidable team. An example of this interaction is Dick Robinson's work on internal gravity waves. This work was again controversial at the time. To put the work in historical context: David commenced supervising Dick as a PhD student in January 1967 while John was on study leave at Harvard, Wisconsin and San Diego. Internal waves are gravitational waves generated in a stably stratified fluid as a result of a disturbance. Such waves are of major importance in oceanography and meteorology. For example velocity fluctuations caused by such waves occur in the wake of air flow over mountains and are a hazard for aircraft flying in the region. The first task David set Dick Robinson was of investigating the reflection of an internal wave in a uniform channel due to a vertical barrier. Soon after this work commenced, Adrian Gill visited the department and indicated that work on this problem had been carried out by Sandstrom (3) at the University of California in his PhD thesis. In the event David obtained a copy and found that the solution was based on 'ray theory' (by analogy with light rays). David observed that the solution obtained was physically unsound in that it did not satisfy the 'radiation condition', the reflected modes consisted of modes transporting energy towards as well as away from the barrier. The problem is a subtle one in that such waves exhibit somewhat counter-intuitive behaviour, and 'ray theory' had an unblemished record of success up to that stage for handling similar problems. Also the analysis is not at all easy; the error, though real, was by no means elementary. The results were also inconsistent with results David and Jorg Imberger (4) (Jorg was David's student at the time) had obtained previously on a related problem. After John returned from leave, David discussed the situation. John suggested that both transmitted and reflected waves should be expanded in terms of only outgoing normal modes whose coefficients could be determined by satisfying the boundary condition on the barrier and continuity conditions above it – typical of John's use of physical insight to cut through mathematical complexity. David passed the advice on to Robinson; it led to the correct solution, see Robinson (5). There was considerable international interest in this work because a number of distinguished mathematicians were (surprisingly) reluctant to discard ray theory for this and related problems.

In September 1966, John Mahony and John Philip shared an office in the Division of Engineering and Applied Mathematics Harvard University – they were visiting at the invitation of George Carrier and Sydney Goldstein. They settled into a steady dialogue and a friendship shared with John's wife Jocelyn that was to last decades. John Philip's pioneering work on water flow through soils would have been well known to John Mahony – in fact few developments in physical science escaped John's notice. In the event, John Mahony was a frequent visitor to the Environmental Mechanics center at CSIRO Canberra, working mainly with John Philip and John Knight. You'll recall John Philip's lighthearted comments on their working sessions at the Pye Laboratory in the introduction. Some of the fallout from the noisy interactions at the Pye Lab is briefly described by John Philip:

Our first collaborative work arose at Harvard in connection with two-particle turbulent dispersion. Our joint study developed a partial differential equation with spatially variable moment generation. The work has affinities with what is now known as the Kramers (6) – Moyal (7) expansions. This was not recognised by us at the time, nor by the reviewers of The Physics of Fluid.

The possibility of momentum exchange across air-water interfaces during unsaturated flow in porous media has been of long-standing interest in the physics of soils and porous media. I developed solutions of two-dimensional Stokes flows (8) which strongly supported my contention that momentum exchange across the small isolated interfaces in such systems is trivially small. John Mahony reinforced my conclusions by extending the study to more realistic three-dimensional Stokes flows. He ingeniously used the vector potential of Benjamin and Mahony to develop means of solving Stokes flow problems in half spaces with mixed boundary conditions; work of major importance in other contexts.

In the late 60s and early 70s considerable progress was made in the analysis of unsteady flow and volume change in swelling soils containing both water and air. A major limitation, however, was that the work was restricted to one-dimensional systems where only the vertical load component was relevant. John Mahony was a pioneer in initiating work on the much more difficult three-dimensional problem. He asked what a tensionmeter measures under anisotropic loading. His conclusion was that essentially it responds to the trace of the load tensor. This work anticipated by some years the work on colloid pastes now to be described.

The primary unresolved problem of three-dimensional flow and volume change in swelling soils concerns the constitutive relations connecting the stress and strain tensors. In the early 80s we decided that insight and guidance on how we might develop a well founded phenomenological approach could follow from analysing the mechanics of colloidal pastes. The analysis would involve solving the Poisson-Boltzman electrical double-layer equation in appropriate arrays of colloidal particles, and interpreting the results (on both microscopic and macroscopic scales) with the aid of Gibbsian thermodynamics, see Philip (9). John Mahony's most important contribution to this work was to show us that a variational principle applies to the Poisson-Boltzman solutions. This leads to a clarification of the theory, and cuts through the cumbersome and convoluted mode of argument customary among colloid scientists (10).

Essex

In late 1970, John took leave from UWA, which he spent mainly at the newly created Fluid Mechanics Research Institute (FMRI) at The University of Essex near Colchester England. Brooke Benjamin had won a large grant from the Science Research Council (later to be called the SERC) to found the Institute, which was planned to ally advanced mathematical and experimental research in fluid mechanics and related subjects. The grant provided for five post-doctoral research assistants, a Laboratory technician and a number of senior visitors each of whom would spend one year at the Institute. John was the first of these visitors. He became closely involved in all the research activity done by the FMRI during that year, which was to be probably the most productive period of his life. Notably, the young research assistants included 'theoreticians' Ron Smith and Jerry Bona, and 'experimentalists' Barney Barnard and Bill Pritchard. Those of us that work closely with experimentalists will know that a gifted experimentalist is not common. Certainly Bill is one of these. He and John would continue working together for some 15 years producing a sparkling array of original results mainly concerning gravity water waves. Some of this work will be described later. It should be pointed out that the area of wave propagation is especially difficult experimentally. Even producing a wave of prescribed wave length travelling down a long channel is fraught with dangers. To do this, a paddle at one end of the channel is moved at the required frequency. One finds however, that at certain frequencies one ends up with a choppy mess, basically because energy feeds from the propagating mode into modes travelling across the channel (called 'cross waves'). John in fact identified the coupling between the modes and made theoretical predictions about the dependence of the onset (frequency range) of such instabilities on the physical characteristics of the channel. Not only is it difficult to devise reproducible experiments, with all sorts of unexpected effects intervening, but it is also often hard to know what to measure that will shed light on the process of interest. It will be seen, in fact, that much of the beautiful theoretical work John did in this area arose out of experiments that 'went wrong'.

By early January Bill and Barney joined Ron, Jerry, Brooke and John, and the laboratory began to take shape. The combination proved to be electric. One interesting problem after another surfaced and was attacked by a moving mosaic of the faculty and graduate students. Some indication of the excitement generated. is contained in Ron's account:

The young research staff there had quite disparate academic backgrounds and needed to be shown how to interact. With this in mind Brooke and John would develop research ideas openly at morning coffee. A pattern developed in which Brooke would carefully dismantle those parts of the previous day's speculative construction which were unsupportable or downright wrong. Thus a quick firing, highly animated response by John, with initially hesitant involvement of the young researchers, would build up a new edifice of sparkling ideas mixed with misunderstandings to be discriminated between by Brooke the next day. The reputation of these sessions spread. Researchers from Cambridge (80 kms away along winding country roads) would sometime come day after day to participate. The sessions lost one of the essential components when John returned to Western Australia, but the job of teaching young researchers was well done.

Now for an account of the work of the group as recorded mainly by Bill Pritchard:

As Barney Barnard and I arrived to join the Institute group JJM was putting the finishing touches on his theory for the generation of cross waves by a wave-maker in an infinitely long channel, and Brooke was busy finishing off an important paper on the stability of solitary waves (11). A theory for cross waves in a channel of finite length had previously been developed by Chris Garrett, but those methods would not work in an open-ended channel and John had to develop new techniques, involving bifurcation from a continuous spectrum, to handle the unbound domain. The basic ideas of his analysis are explained nicely in a model calculation presented in his crosswaves paper, along with the analysis of the cross-wave instability itself. This theory finally resolved a problem first raised by Faraday in the 1830's. In view of these calculations of John's it was decided that Barney and I should make some laboratory experiments to check the marginal stability curve predicted by theory. John's theory gave a very good description of the experimental results. There were to be many further discoveries concerning the generation and development of modes and their interaction, by a variety of authors (Jones, Lichter, Schemer and Miles to mention some of the names), but the origins of all this derives from the Mahony and the Barnard-Pritchard (12) papers, which were published as a back to back pair in JFM.

Much of the work of the Institute was done during prolonged coffee sessions that took place in the unusually shaped, triangular room adjacent to the laboratory. One of the problems under consideration at the time arose out of a yen Brooke had of proving existence of solutions to the Korteweg-de Vries (KdV) equation. The KdV equation

u**t + u**x + uu**x = 0

arises in connection with the propagation of water waves. To elaborate:

Propagating waves have a tendency to steepen due to nonlinear flow effects, and smooth out due to dispersion effects. Depending of which of these competing effects wins, the wave will eventually break (often seen at a shore line), propagate as a simple oscillatory wave, or a balance may be struck, in which case an unchanging form may propagate. The third situation is of course most interesting and is seen when a hydraulic bore propagates under appropriate tidal conditions. The full fluid dynamics equations are analytically and numerically intractable, and even if they were 'solvable' the results would probably give little insight concerning the physics – the dependence on initial conditions is strong and the possible scenarios seem endless and complex. The KdV equation represents a simple approximation to the full equation designed to model the undirectional propagation of waves with wavelength long compared with the water depth; u(x,t) is the fluid particle velocity (uniform with depth in this theory), u**xxx models dispersive effects, and the uu**x term nonlinear effects. In the absence of these competing effects the KdV predicts that the wave will propagate with unchanged form (as described by u**x + u**t = 0); so that the equation has all the appropriate behaviour. Furthermore, special exact solutions are available that exhibit bore like behaviour. For all these reasons, the KdV equation plays a central role in wave studies and questions concerning the existence, uniqueness and stability of particular solutions, as well as the validity of the approximation, are of major concern. Interestingly enough, the KdV equation also has solutions of a 'solitary wave' type. Such waves are called this because they consist of a single hump (rather than a series of humps) that the KdV equation predicts will travel unchanged in form with constant velocity in fixed depth of water. This interesting (mathematical) wave aroused much curiosity, particularly since such waves are not normally encountered. (Scott Russell had reported such waves in 1844, but few believed that such a wave form could be sustained in practice.) Was the wave an artifice of the approximation? Was it 'not observed' because the conditions required to set it up were hard to achieve? Was it unstable so that although it could be set up it could not be sustained for long? The paper Brooke was writing (13) when the FMRI was set up answered some of these questions.

Returning to Bill's account:

John and Brooke had been beating their heads against the existence problem for some time and so it was one day that the discussion came round to the same old chestnut again. Jerry had given a series of tutorial lectures on distribution theory, ending with a very weak theory for the initial value problem. Both John and Brooke were unhappy with what Jerry described and John went off to think about it. He came back in a day or two with the very interesting observation that the u**xxx term in the KdV equation could be replaced by -u**xxt without changing the formal level of the approximation. Thus arose the celebrated equation

u**t + u**x + uu**x -u**xxxt = 0,

now referred to as the BBM equation after its originators. Brooke greeted this remark with interest since his student Howell Pregrine had made the same point several years earlier in his PhD work, but had done very little with it. He had, in fact, found that the KdV equation difficult to numerically integrate and so resorted to the above modification to circumvent the numerical problems. The penny dropped! Although the two equations were equally valid formal approximations, their stability behaviour was very different! Whereas the KdV equation was such that small wave length waves were amplified as time increased, such waves decayed for the BBM equation. A cursory examination of the approximate dispersion relations indicated this to be the case, but proof was essential. The three began work in earnest and established that the BBM solutions to the initial value problem were remarkably well behaved. For example any singularities (for example a discontinuity in u**xx(x,0)) simply do not propagate! ((u(x,t) – u(0,t) is C*x and analytic in t.) The KdV equation, on the other hand, requires comparative stringent conditions on u(x,0) for its solution to be meaningful. The resulting paper, a genuine collaboration between all three authors, appeared in the Phil. Trans. and sold well. Even before it appeared in print it was controversial – not surprising given the effort that had been devoted to the KdV. In fact there was no serious rivalry between the proponents of the two models; both are equally valid long wave approximations but the BBM is much to be preferred for numerical work, and so is the better basis for comparison with experiments. The paper has been influential well beyond the initial context and has spawned several, somewhat disparate lines of research. As is typical of John he gave the work little thought after the initial foray, being correctly of the opinion that all the essential ideas were already in the first paper.

One very interesting outcome of the coffee sessions was the theory of Mahony and Smith for the so-called spatial resonance phenomenon. It all started in the laboratory. There was this graduate student Ian Huntley for whom we were trying to find a good PhD project. Brooke wanted him to study acoustical damping properties of bubbles in a liquid. In trying to set up a nice wave field in a glass beaker by pushing on the wall of the cylinder at a point (at a frequency of around 2 kHz) it was found that waves spontaneously arose on the surface of the liquid in the cylinder. This phenomenon, which was annoying in the extreme in the context of the original experiment, greatly fascinated John because of the tremendous disparity in the applied acoustic frequency and the generated gravity wave frequency (two orders of magnitude!). A similar phenomenon was observed by R. Franklin at Oxford (14). Conventional wisdom at the time (based on observations and coupled oscillator works etc.) was that in order to get an energy feed from one mode to another mode there must be a simple integral relationship (1 to 1, 2 to 1 etc.) between the forcing frequency and the large response frequency. Under such 'parametric resonance' circumstances even small linear or nonlinear couplings between the modes can result in a transfer of energy. Conventional wisdom would be changed by this work. It was also observed that the spatial resonance phenomenon was not at all sensitive to frequency ratio; not characteristic of parametric resonance. John and Ron in fact showed that energy could transfer from the acoustic mode into a surface mode with a similar spatial distribution via the surface boundary condition, which is why the term spatial resonance is used. They suggested that such a phenomenon is not confined to the interaction between water waves and acoustic fields, but may occur generally in systems having modes with related spatial patterns but greatly different frequencies. Clearly this one is a sitter!

At the end of April on a train trip to Manchester John put together the analysis for what was to be the basis of the paper by Barnard, myself and John on 'The excitation of waves near a cut off frequency'. The reason for interest in this problem was that it provided an explicit situation in which one could cheek empirically the calculational methods used in weakly non-linear analyses of water waves. These methods had been (and still are) widely used in the analysis of water wave problems, but there had been virtually no quantitative checks of how well the methods actually work. With this in mind John specifically designed his theory around a problem that was realizable in the laboratory and Barney and I undertook experiments relating directly to the Mahony calculations. We must have repeated that silly experiment at least ten times over and it was only when I was able to prove that the inviscid solutions that had been guiding our thoughts could not possibly describe the experimental results that a real breakthrough was made. John and I completed most of the theoretical work on this paper during my visit to Perth in 1975 and finally appeared as an FMRI report in September 1976, and as a Phil.Trans. paper in June 1977. I think the full value of this work has not been as widely appreciated as it merits. It is, to my knowledge, the most thorough examination of the calculational methods for weakly nonlinear waves that has been published to date and, although the qualitative agreement is very good, the quantitative agreement is wanting somewhat. Interestingly the work showed how very crucial even seemingly small viscous effects could be on the outcome of the solutions and hence on the observed waveforms.

Although slightly out of historic context, we'll describe further work arising out of the very productive Pritchard/Mahony combination, as recalled by Bill:

John and I published two other joint papers, both of which stemmed directly from my experimental work. One was entitled 'Wave reflexion from beaches', and the other was called 'Withdrawal from a reservoir of stratified fluid'. The research for the latter paper was finished well before that of the former, but it encountered incredible obstacles in getting published and so appeared much later than the paper on wave reflexion.

Our work on wave reflexion was sparked by a comment I made to John one day that if one runs waves of extremely small amplitude onto a beach they do not break. This aroused his curiosity and led us to look at a boundary-layer theory for waves over sloping beaches. The modelling leads one directly to a parameter gamma = (vw*3)*1/2 /g alpha*2, which measures the importance of viscous damping on the wave absorption process. Here v is the kinematic viscosity or, in the oceans, should be represented by an eddy viscosity, w is the wave frequency, g is the gravity constant, and alpha is the beach slope. The central role played by the parameter gamma in determining the importance of bottom friction seems not to have been appreciated before our paper, but I believe it to be a very basic parameter that someday will become part of the standard nomenclature. It is amazing to me that, until our paper, the standard folklore would have it that most of the wave energy over a plane beach is absorbed by breaking, but in fact this couldn't be further from the truth. Thus, most of the breaking we see on beaches occurs over some local structure, such as a sand bar, which induces the wave to steepen rapidly and break. When writing the paper we decided to keep it very simple and presented the theory only for the plane-beach ease, even though John had done the calculations for a beach of arbitrary profile. The paper is very disarming in its simplicity, both theoretically and experimentally; once the basic balances reflected in the parameter gamma have been identified, the mathematics is straight-forward, though the interpretation is somewhat more subtle and quite a bit of the paper is devoted to that. The experiments, while seemingly trivial, broke new ground in that no one had been able to do reliable experiments in this area before. One of the problems lay in achieving 'good' conditions at the shoreline, another was that people previously had not done the experiment at small enough amplitudes. I should also mention that the work for this paper was a genuine interplay between experiment and theory. One of the reasons, I guess, why people had never been game enough to publish such a simple theory was that they didn't have the answers available in the form of reliable experiments, and it was the application that really framed our thinking.

The problem of determining the flow induced by the withdrawal of fluid from a stratified reservoir is one of major practical importance. As one would expect, (heavier) more saline layers of water are found at greater depth in dammed water, so that by extracting water from a particular depth one can control the salinity (quality) of the extracted water. For human consumption lower salinity levels are desirable. In densely populated dry countries it's not so easy to maintain acceptable water quality and it's necessary to 'cocktail' water from different dams. To do this one needs to know the salinity of water extracted from a particular depth in the dam. Of course one might anticipate the salinity to be the same as that in the dam at that level before the pumps are turned on, but this won't be quite true, with different outcomes depending on stratification levels, pumping rates etc. The problem is also theoretically interesting, and controversial. John and controversy were uncomfortable bedfellows throughout his professional life. In Bill Pritchard's words:

The work John and I did on the withdrawal problem was at once the most interesting and the most challenging of all the scientific problems on which we worked. I had started work in the area in the early 1970s when I realised that the similarity theories on which our understanding of withdrawal was based could not be applied to the experimental situation.

A lot of these experiments were completed by 1976 when John visited Essex and Bill needed help with interpreting the results. Again in Bill's words:

I clearly recall how we made the important breakthrough in our thinking on this problem. I had been working with the experimental data late one Saturday evening and I was having trouble making sense of the results – nothing seemed to fit together sensibly, so the whole family drove around to the house the Mahonys were renting in Marks Tey. John was in bed with his dressing gown on: we spent the next four or five hours in that room trying to understand withdrawal, and it was at this stage we discovered the scalings that gave such an incredibly good correlation of the data. The scalings reflected balances we had not anticipated at all and, even more unusually, they reflected balances that were not actually struck in the final steady motion. However, no other scaling would work for my experiments, and the one that did work was remarkably good; so it must mean something. Exactly what we weren't sure; this remains a challenging open problem.

The work was finally published in Proc. Roy. Soc. These remarks might be further clarified by a quotation from the summary of the paper:

For flows initiated in a uniform tank by suddenly opening a valve in the outlet line, the width of the withdrawal layer seemed to be uniquely determined from terms that are negligible once the steady flow has been established. By placing suitable obstructions in the tank it was possible to obtain similar flows, but with various widths.

Perth again

In 1974, John was elected as a Fellow of the Australian Academy of Sciences, the first applied mathematician to be so honoured for a long period of time. John was appointed in 1975 as Founding Editor of the Journal of the Australian Mathematical Society, Series B. (Applied Mechanics). He had by this time developed a reputation of being a difficult customer, so Vince Hart (an associate editor) must have been just a little apprehensive but found no difficulty at all working with John on publication matters. However, John was very pessimistic about the journal's finance future and predicted bankruptcy within 18 months. John resigned in despair; the publication survives to this day.

John had long been encouraged and thought seriously about writing a book on singular perturbation theory, and who better to write the definitive book? He delayed writing because he felt good books had already been written describing the basic ideas (notably Cole, van Dyke, Nayfeh) and the area was not yet well enough understood to write a definitive work. Finally, in the 1980s he felt the time was ripe and he set about writing; the book would not be just a concise summary of known results but would contain much original material: He obtained an Australian Research Committee grant to provide for research assistance for this major task and, in due time, John Shepherd arrived to take up the appointment in 1980. The task turned out to be a much greater one than John expected and his health was declining (perhaps partly because of his growing realization of the magnitude of the task). The book was not completed, but major gaps in understanding were filled.

[The original version of this Memoir contains two pages of detailed mathematical explanations at this point. These could not be reproduced due to the limitations of formatting on the World Wide Web.]

Yuriko Renardy was one of the last of John's students. By this time John's health had deteriorated to such an extent that he could barely function effectively for extended periods of time. The slightest disagreement would topple him into a psychological darkness, and he'd end up in hospital receiving medication and counselling. A few weeks later he would return to work, but the pattern would then be repeated. The following account of her collaboration with John provides a snapshot of John at the height of his mathematical abilities, but struggling to maintain balance.

My collaboration with John from 1977 to 1980: Somewhere along the line, John never had the hang of going at an even pace. His enthusiasm for research was infectious, and his pace so frantic, that it was next to impossible to pin him down and become specific at a mathematical level. His gift lay in the huge reservoir of knowledge about techniques, his experience with many problems in perturbation theory and fluid dynamics, and the gusto with which he applied this knowledge.

When I arrived in Perth in 1977 to study under his supervision, he was ready and willing to take on a student with a blank slate. He arranged for me to learn how to program and develop computational skills, and proceeded to do his very best to introduce me to the thesis topic. The area of research was pretty clear to him. Inviscid long-wave theory had been used to show that plane periodic mono-chromatic water waves can resonate over a cylindrical island submerged in an ocean. This linear theory yielded the resonant frequencies, see Longuet-Higgins (15). On the other hand, experimental data indicated the absence of resonance, see Barnard, Pritchard and Provis (16). John wanted me to check if the discrepancy was due to viscous dissipation and nonlinear effects; words which meant very little to me at the start. John made obvious desperate attempts to tell me how I ought to attack the problem, but I didn't 'get it' for a long time.

The pattern of my work with him was as follows. I would come up with questions and go and see him on a regular basis. He responded with tremendous enthusiasm, and energetically took over the conversations, but rarely answering my original questions. He had numerous ideas, many of which were not quite right, and if I could prove to him that this was the case, it prompted him to reply 'I didn't say that'. I learned not to begin a sentence with 'I think ...' but to put everything into a third person, e.g., 'It appears that ...'; otherwise, he would immediately interrupt and tell me 'You're wrong.' It took me a while to figure out these tactics, but it helped that his response was so predictable .

During my second year, he arranged for Dr William Pritchard, then at the University of Essex, to visit for several weeks. This gave my thesis a huge boost, clarifying the direction to take. During that happy time, the full linear theory, without the long-wave assumption and including viscous effects, took shape, see Renardy (17). This boost exemplified the way John supported and fostered my work to the hilt: he was absolutely dependable.

After another year, we focussed on theoretical aspects of the nonlinear problem. He kept giving me lots of ideas to try. In the end, I 'got it' and he did his best to make me feel like I did it all by myself, see Renardy (18). He topped it off by sending me to a postdoctoral position at the Mathematics Research Center, University of Wisconsin-Madison, which opened the door to future opportunities. The fact that he could arrange this was proof of his powerful, highly respected standing on the international scene.

John followed with great interest Yuriko's career since leaving WA.

Industrial work

In the late 60s, Alan Tayler and John Ockendon set up the Oxford Study Group with Industry at the Mathematics Institute in Oxford, UK. They did this in response to a perceived need in UK for basic research on industrial problems. Also, the hope was that a new breed of applied mathematicians might be trained to handle the diverse and different problems that arise out of industry. The program they set up has been running ever since and has spawned similar activities around the globe. Many graduates have passed through the school and an enthusiastic stream of international visitors have eagerly participated in this useful and exciting activity. Certainly the program has been a major success academically, and industry in the UK and elsewhere is now much more aware of the important role mathematics can play. John Mahony watched this development with great interest, and his extraordinary ability to see through the clutter normally surrounding the questions raised in such study groups gave him prima donna status at the meetings he attended.

Australian industry was also sadly ill-informed in industrial mathematics, and claims of industrial ineptitude were (and still are) undeniably valid. One of John's former students, Noel Barton, was to play a major role in promoting industrial mathematics in Australia. Noel had completed a PhD under Peter Chapman on a topic suggested by John, and had subsequently worked in Cambridge, Brisbane and the University of New South Wales. Noel joined CSIRO in 1981, in time to participate in the important industrial developments experienced by CSIRO in the mid-eighties.

Senior CSIRO management decided to set up a Mathematics-in-Industry Study Group in Australia in 1984. John and Hilary Ockendon of Oxford were invited to Australia to set up the first meeting, and in this, they were assisted by staff of CSIRO and Siromath. Needless to say, John was there at the first meeting, and regularly attended and dominated meetings until his health made this impossible. Even then, his influence was still felt through comments and criticisms transmitted to colleagues through letters.

The MISG concept has continued in Australia to this day. CSIRO's applied mathematicians, with Noel as their head, championed the concept over a period of nine years, whilst recently the concept has been transferred under the aegis of the University of Melbourne. John's influence was still felt through comments and criticisms transmitted to colleagues through letters. Noel's comments:

John was clearly instrumental in bringing the concept of industrial applied mathematics into Australia. He wasn't alone in this; Alan Head (who also had strong Oxford connections) might have been more significant. Nevertheless with John's support the Study Group was set up, and I believe has filled a valuable role ever since. John's exceptionally sharp intelligence and exceptionally forthright mannerisms made him a unique participant. He was mostly right about things, occasionally wrong, and, never ever a shrinking violet. Over the years I decided that foolhardy bravado was the best tactic to adopt with John, always making sure to preserve a means of escape if he disagreed with me. Looking back, I find it remarkable that someone could possess intelligence which was simultaneously razor sharp and yet wielded like a sledgehammer.

Retirement

After retirement from the University in 1986, John and Jocelyn moved residence to Mandurah and immediately became involved in local affairs. He became one of the pillars of the local Wildflower Society (gardening was a consuming passion for both John and Joc), and became intensely involved in local environmental matters. Throughout much of his life he was a committed conservationist, bringing both his professional knowledge and energy to bear on local and global problems. In fact, he strongly believed in the principles of humanism, reverence for life, and individuality underlying Quakerism, and his concern for the environment was just one manifestation of his commitment to these noble principles. John had joined the Society of Friends in 1985.

Immediately after retirement, research mathematics had no appeal and was avoided. However, John could not be inactive for long. At considerable private expense, he expanded his computer facilities; not just for his personal use to cope with his financial records and to control the irrigation of the extensive garden he was establishing, but also he commenced setting up a large computer base which he planned to use to provide useful information for societal decisions. Thus, for example, he collected and collated information on world forests, population patterns etc. with the greenhouse effect in mind. He offered his insights to the Labor Party in which he hoped to be able to play a more active role as a branch member. However, when the State Government failed to live up to expectations on environmental matters and the Federal Government did not measure up to what he believed were the party's ideals during the Iraqi crisis, he resigned in disgust.

We have hinted that John was always concerned with community issues and had done more than just serve on committees even before he retired. He had found great satisfaction guiding a high school student with a severe hearing defect through her mathematics studies so she could pass at the highest level. When an opportunity arose after the Mahonys had settled in Mandurah for John to help at a Montessori secondary school, which had been set up on the site of a home originally built for migrant boys, he grasped the opportunity eagerly and turned himself into a remedial mathematics tutor. This work was not undertaken in a casual way, but work with every student was carefully planned and constantly reviewed. This work started in 1987 and, apart from interruptions for health reasons and travel, was continued on a regular two day a week basis until the gravity of his health prevented him from continuing.

Another contribution John made to community education was to participate in the Mandurah activities of the University of the Third Age by being in charge of the group studying the environment; John was never at a loss for an occupation. Inevitably, however, his mathematical ability would assert itself and he recommenced working with Peg-Foo Siew on a consultancy problem associated with the magnetic detection of diamond pipes. The presence of laterite layers obscures the magnetic signature of diamond pipes; how can one see through the magnetic clutter? The work is mathematically very elegant, and provides a very clear picture of how to practically interpret the observed magnetic anomalies. Halfway through the exercise of rewriting the paper, John decided not to proceed with the project but encouraged Peg-Foo to continue while providing advice. As was typical, John refused to let Peg-Foo put him as a co-author of the paper subsequently published in QJMAM (19).

In late 1989, John, with Nev Fowkes, embarked on a book aimed at training students in the skills of mathematical modelling, particularly in the context of industrial problems. John had a great interest in teaching in schools and university and fought hard by all means at his disposal to try to improve the teaching of mathematics. Gifted students found his lectures (often based on his current research) inspirational. He also attempted to achieve his lofty educational aims by serving on a variety of committees, notably the Mathematics Syllabus Committee in WA (for a prolonged and somewhat frustrating period), where he tried to improve the standard of secondary school mathematics.

The book entitled An Introduction to Mathematical Modelling was to be published by Wiley and, like all of John's academic undertakings, was to be radical in approach. It was John's belief that mathematical modelling skills will be greatly in demand in tomorrow's world, and that these skills need to be developed early in a student's career. In fact, he perceived a world in which mathematical modelling would achieve the status of a profession in its own right. In a world in which powerful computation becomes cheaper, and all else seems to become more expensive, this scenario seems inevitable. For industry, in particular, mathematical modelling represents by far the cheapest way to carry out the necessary design experiments. For a long time this has been the case in the aircraft industry (where experiments are very expensive to set up and difficult to interpret); it's just that such an approach is now also required in more everyday areas, such as steel casting, and even in such mundane areas as the design of simple items like furniture. The artistic skills of modelling need to be developed and an ability to see features in common between problems from entirely different contexts is required. With this in mind, the book presents universal mathematical and physical patterns in different practical contexts, and techniques are introduced when required, and judged within context. The book was to be an ambitious undertaking, an extremely broad sweep of ideas was to be presented in the context of specific (largely industrial) problems that John and Neville had encountered in recent years. The Oxford Industrial Mathematics group work and Australian Mathematics-in-Industry meetings run by CSIRO provided much of the source material. Other contextual material was developed specifically for the book.

The purely technical mathematical skills that plague students and professionals alike were avoided in the book by using an algebraic package, freeing the student to see the broader picture. Algebraic packages were first introduced to John by Grant Keady and immediately John embraced the idea, as Grant's account indicates:

I had enjoyed computing since my first Fortran programming in 1965. In 1979 I began efforts in Computer Algebra (CA), with REDUCE. In those early days of CA, I think most people thought of it as just a research tool. I had enthused about CA for perturbation methods to John, and he embraced its use. To my amazement, and horror, John started using REDUCE in 2nd year applied maths teaching; in 1985 I think. The experiment is best described as only partially successful at the time. I had myself used CA for advanced student projects and advanced classes since 1980 but I didn't really think it was a tool to be used earlier. I now believe I was wrong and John was right. Later (1987) Maple was introduced for the Macintosh at UWA (the University had lots of Macs by then) and I ran courses for Maths staff through 1988, and sought, and obtained approval to join with Murdoch and Curtin in use of Maple in 2nd year teaching, and I began this in 1989. Maple is now a standard part of the mathematics program from first year.

John could see clearly that, not only are such algebraic packages useful to escape tedium, but also the whole methodology of applied mathematics would change as a result of their introduction.

A first crude draft of the book was sent to Wiley in early 1991, and surprisingly was well received (there were many major errors and the material was marginally readable). Based on reports received from referees, Wiley decided to go ahead and the contract was sent. Soon after (April), John received word that he had cancer cells in his liver and that the statistics were such that his life expectancy was less than one year, but could be as little as a few months. Eighteen months earlier, John had surgery for colon cancer, and the subsequent medical reports were encouraging until then. Such secondary occurrences are much more difficult to treat and medical treatment for liver cancer is unsuccessful; the situation seemed hopeless. I (NF) recall vividly John's phone call on the night he received the news. In a matter of fact way he conveyed the news, indicated that his primary responsibility was for the needs of his family, and that thoughts of all other matters (including the book) would remain in abeyance. A week later we spoke. He would not allow the cancer to defeat him! In a reply to a long term friend's (Douglas Rogers) enquiry about his health John wrote:

It was good to receive your letter with its wishes that my health problems can be contained. The secondary cancers seem to have formed only on the liver and for the moment do not cause any major difficulties. The medical profession offers no effective treatment and merely indicated a median expectancy of six months. But the distribution on which this is based has a range of a few days, for those that give in. to twenty plus years. They have no understanding of the factors which produce this variation; so my life expectancy is finite, as I always knew. All I can do is to make the most of the unknown period of time which is left while attempting to join those who occur in the far tail of the distribution.

In the meantime life appears rich and satisfying and there is no temptation to join those that do not feel it is worth the effort to keep fighting for more of it. The amount of time I devote to working on the book has been reduced and I am giving priority to those parts where my own insights have most to offer. For the remainder I am much closer to my family and friends, who have been very helpful in the support they have proffered. I have become aware that they value me rather more than I had ever given myself credit for. One of the surprising features for someone with my long history of completely debilitating bouts of depression, is that there have been absolutely no indications of any problems under stressful circumstances far worse than those which have triggered previous attacks. Can it be that I have at last set the depression attacks behind me?

I had a very supportive letter from Ed Hewett and have written to him in what I hope is a like vein. His letter suggested that he has found acceptance of those things which cannot be changed and is working on those things which might be. Both he and I are among the very fortunate ones in that life has offered us so much to enjoy; 60 that when fortune demands that some of what we have should be taken away, there is still much of value left to savour.

For several months the tests seemed to indicate an improvement, but a persistent cough troubled John and tests revealed that the cancer had spread to the lung. Chemotheraphy was indicated, and was undertaken. The side effects of the therapy were marked. For a week after each treatment John remained in hospital, recovering. During the following week he could work in short bursts. I (NF) recall very distinctly a few days before the results of tests (indicating the effectiveness of the treatment) were available. On that particular day the final contract with Wiley was drawn up by a lawyer and, as we sat together waiting for the typed copy to appear for signing, John leaned across and asked that I delay sending off the contract until we knew the results of the tests. It was the first time that I heard John even admit the possibility of defeat, and this he offered only because of his concern that the task I faced in completing the book alone in the specified time was impossible. Significantly, Jocelyn was outside waiting in the car. His dad used to say to him 'the game is never finished until the final whistle sounds'. John lived with this 'never give up' principle throughout his life. Next day John informed me the tests were not encouraging, but I forwarded the contract. During the remaining months I would work with John one or two days a week in between treatments. His mind was as sharp as ever, but there was little energy left. We would work for a couple of hours, and then John would seemingly 'hit a wall', and then we would meditate together. Then John would sleep. Jocelyn, who wasn't well herself, somehow managed to hold things together. Two weeks before John died, we spent the morning working together but I could see something had happened. Although to an outside observer nothing had changed, that very special mind was no more, and I could see that he was aware of the loss. In despair, I returned home hoping that the end would come quickly. Two days later Jocelyn rang to tell me that he had fallen and was taken to hospital. He was close to death when I arrived and told me that he would not survive the night. Miraculously he survived the night, and the next two weeks. He would not let go. In subtle ways I indicated to him that it was allright to let go, but he desperately held on. In the afternoon of June 30th Jocelyn rang to say that he had died.

P.S. The book was completed one year later and was published by Wiley in April 1994 [with N.D. Fowkes, Mathematical Modelling].

Epilogue

In these pages an attempt is made to give an insight into the manner in which the mathematical scientist, John Mahony, worked and in outline what he achieved, and a hint at how others benefited by his efforts. Because, on the personal level, crises and difficulties tend to excite the readers attention, it needs to be said to balance the account that John was a warm and sociable person who had many firm friends in his lifetime. He enjoyed company and was blessed with a wonderfully supportive wife, Jocelyn. who fulfilled her role magnificently 'in sickness and in health', providing nursing support in bad times and being a wonderful hostess when all was well. John developed a close relationship with his children which outlasted their teens and which was a significant support in his final illness. He stimulated his friends, scientific colleagues and many others who came in contact with him, and all of us who knew him during his lifetime were enriched as a result.

Notes

(1) D.C. Collis and M.J. Williams (1954), 'Two-dimensional convection from heated wires at low Rynolds numbers', J. Fluid Mech. 6 357-84.

(2) G. Fitzgerald (1976), Phil. Trans. Roy. Soc. A., 284, 96-134.

(3) H. Sangstrom (1966), 'The importance of topography in generation of internal waves', PhD Thesis, University of California, San Diego.

(4) D.G. Hurley and J. Imberger (1969), 'Surface and internal Waves in a liquid of variable depth', Bull. Aust. Math. Soc., 1, 29-46.

(5) R.M. Robinson (1969), 'The effects of a vertical barrier on internal waves', Deep Sea Research, 16, 421-29, & R.R. Robinson (1970), 'Internal Waves', PhD Thesis, University of Western Australia.

(6) H.A. Kramers (1940), Physica, 7, 284.

(7) J.E. Moyal (1949), J. R. Stat. Soc., 11, 151.

(8) J.R. Philip ZAMP, 23, 353 and 23, 960.

(9) J.R. Philip (1973), 'Theoretical and Applied Mechanics', Springer 3-31, & (1972), 'Mechanics of Swelling', Springer 3-31.

(10) J.R. Philip (1970), J. Chem. Phys., 52, 1387.

(11) T.B. Benjamin (1972), 'The stability of solitary waves', Proc. Roy. Soc. A., 328, 153-183.

(12) B. Barnard and W.G.D. Pritchard (1972), 'Cross Waves II Experiments', J. Fluid Mech., 55, 244-65.

(13) T.B. Benjamin (1972), 'The stability of solitary waves', Proc. Roy. Soc. A., 328, 153-183.

(14) M.E. Franklin, M. Price D.C. and Williams (1973), J. Fluid Mech., 57, 257-68.

(15) M.S. Longuet-Higgins (1967), 'On the trapping of wave energy round islands', J. Fluid Mech., 29, 781-821.

(16)J.S. Barnard, W.G.D. Pritchard and G. Provis (1983), 'Experiments on wave trapping by a submerged cylindrical island', Geophys. Astrophys. Fluid Dynamics, 24, 23-48.

(17) Y. Renardy (1983), 'Trapping of water waves above a round sill', J. Fluid Mech., 132, 105-18.

(18) Y. Renardy (1983), 'Weakly nonlinear interactions and wave trapping', J. Fluid Mech., 130, 27-39.

(19) P.S. Siew (1990), 'Magnetic anomalies of cylinders', J. Mech. Appl. Math., 43 (3), 373-85.

N. Fowkes and J.P.O. Silberstein, Department of Mathematics, University of Western Australia.

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