School of Mathematics and Statistics

Dr Peter Chapman - A Mathematical Legacy

Peter Chapman photo

 My colleague and good friend Peter Chapman passed away on the 8th February 2008.  After completing his Ph. D. at the Department of Aeronautical Engineering at the University of Sydney he worked at the University of Toronto for 9 years, and then he shifted to UWA where he remained until he retired. He continued working until his death. He left behind a mathematical legacy of published and unpublished material. More details of Peter's professional life can be found in his (academic) obituary below, and a list of his published work is presented in `Publications'.

Peter truly enjoyed mathematics, science and engineering for their own sake. He read broadly and always felt the need to establish or verify results in his own idiosyncratic way. He was not at all concerned with publication, so much of his academic legacy is not publicly available.  In todays academic world this would be regarded as an unacceptable luxury.  In any case this has meant that much of his work remains unknown to the broader mathematical community.  His widow, Connie, and I have attempted to make his material generally available by collecting together his files.

Peter's primary mathematical interest was in asymptotics and he frequently transformed problems or specific equations arising out of engineering and science in a way that enabled him to use these techniques to extract results and thus expose the physics. In many cases the results he obtained were not new and so were not of primary research interest, but his approach certainly led to new insights. As an example of this, Peter explored the quantum mechanics/classical mechanics interface in Quantum Theory and Classical Mechanics, displaying the result that classical mechanics represented an asymptotic limit of the quantum mechanics equations. This work had been done earlier by pioneers in the area but not in the way that Peter did it. His (extensive) notes on this are interesting and of real educational value.In other cases the work certainly was of primary research interest but, as indicated above, Peter's concern was not with publications but with understanding. He felt happy that he now understood the problem in his own way and so moved on.  I put his important work Asymptotics and the Helmholtz Equation work in this category. We have briefly described the material here. Peter was a generous character and was in no way possessive about his contributions, however, his family and I would ask colleagues who make use of this unpublished work to acknowledge his contribution. 

 

Nev Fowkes
School of Maths and Stats,  UWA
Stirling Highway, Crawley
WA Australia 6160
fowkes@maths.uwa.edu.au

October 2010. 

 

File List:

 

ASYMPTOTIC SOLUTION OF THE HELMHOLTZ EQUATION


The Helmholtz equation describes the amplitude variations that occur in wave propagation contexts. In the light propagation context the wave length is typically very short compared with the length scale of refractive index variations and in this case asymptotic methods can be used to extract useful solutions. The classical theory (geometric optics) fails near shadow zones and caustics where the normal procedures fail. These phenomena are of major interest. In this clever and important paper Peter manages to obtain a uniformly valid solution description that extends into such shadow regions by approximating the relevant exact integrals. Previous results have patched together known results in the two regions, a less than satisfactory result.


QUANTUM MECHANICS CLASSICAL MECHANICS CONNECTIONS


Commencing with the Schrodinger Wave Equation for a charged particle in an electromagnetic field, Peter uses multiscaling ideas to show that in the classical (small h) limit two equations arise. The first of these equations determines possible trajectories in space time, and coincides with the Hamilton-Jacobi equations of classical mechanics. The second of these, which determines the phase, is known as the eikonal equation and arises classically when describing wave propagation. Thus his procedure exposes the particle/wave connection that is central to an understanding of the nature of quantum mechanics. This very interesting and important result is unfortunately not new, although Peter's procedure for exposing this connection is undoubtedly his own.


A FAMILY OF SOLUTIONS OF THE RIKITAKE EQUATIONS


Peter examines the stability of solutions of the Rikitake dynamo equations which model the metal flow within the Earth's inner core that is responsible for the Earth's magnetic field.  It is thought that field reversals are described by the Rikitake equation, so that understanding these transitions is essentia
This is an earlier version of Peter's 1989 Math Proc Camb Phil Soc publication.


A NOTE ON THE MOUNTAIN WAVE PROBLEM: 2D STEADY FLOW OVER A BARRIER


An exponential air density distribution, decaying with height above the ground is assumed. This is an incomplete attempt to handle a singularity in the solution for the stream function at infinity. (The relevant reference is Yih's book: Dynamics of Nonhomogeneous Fluids). 

A ROTATIONAL INVISCID FLOW


Whilst complex function theory can be used to obtain exact solutions for irrotational inviscid flow in 2D no such results are available for rotational inviscid flow. Peter obtains new exact solutions for flow in a wedge. These solutions exhibit singularities along the wedge edge so it is not clear if the results are just a mathematical curiosity or of practical importance. It is possible that the solutions have applications for the design of ducts for turning flow through an angle. Also the solutions may serve for testing numerical simulations of rotational flow.  

AN APPROXIMATE SOLUTION TO A PERTURBATION PROBLEM OF THE SINH-GORDON EQUATION

 
Peter obtains slowly modulated traveling wave solutions of the Sinh Gordon equation, using the averaging technique of Kuzmac.  The equation is generally thought to be of fundamental importance in the study of solitons (finite waves that travel with almost unchanging shape) and arises out of the study of water waves and particle physics. 

APPROXIMATE SOLUTIONS OF PERTURBATION PROBLEMS FOR THE KORTEWEG-de VRIES EQUATION BY THE METHOD OF KOGELMAN AND KELLER


Peter examines two alternative asymptotics formulations for obtaining wave solution approximations for the Korteweg de Vries equation which describes nonlinear water waves. 

EXTENDING THE APPLICABILITY OF THE LIOUVILLE-GREEN APPROXIMATION


The Liouville Green approximation enables one to determine the effect of a small modification in the restoring force on a harmonic oscillation. Such effects are important in the study of non-linear oscillations or propagating waves, where relatively small but persistent effects can greatly affect the long term response; for example the amplitude of the oscillation. Using multiscaling techniques Peter extends the classical results to deal with more critical situations. 

SOME FURTHER COMMENTS ON LIGHTHILL'S METHOD OF STRAINED COORDINATES


Peter clears up a misunderstanding about the applicability of Lighthill's technique for determining the behaviour of singular differential equation systems. The technique is an asymptotic technique that overcomes difficulties for correctly positioning the singularity (the characteristics in a wave propagation system or the singularity on an ordinary differential equation). He does this by considering a simple example.

ON THE METHOD OF STEEPEST DESCENTS


The steepest descent technique is an asymptotic procedure for approximately evaluating an integral in the complex plane. Peter examines a range of cases not handled by the classical theory.  

REDUCTION AND PROPERTIES OF THE GAUSE-LOTKA-VOLTERRA SYSTEM (n = 3)


The Lotka Voltarra system is a standard ordinary differential equation system used to describe the population evolution of a number of interacting species in a biological system. The same system arises in some fluid dynamics studies. Typically the outcome is a (limit) cyclic situation in which the population levels oscillate between levels determined by parameters describing the interaction. A general understanding of the behaviour of such systems would b e invaluable but the complexity increases greatly with the species number n. Peter uses various clever transformations/observations to reduce the  n=3 case to a study of a 1D problem, which he examines using known results. 

SIMPLE SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS


Here Peter displays examples in which he obtains useful approximate results for the solution behaviour of a second order ordinary differential equation system (nonlinear) near a singularity. The suggestion is that a reasonably general theory may be possible using the approach he adopts for handling a range of specific situations. 

THE COMPLEX LORENZ EQUATIONS


The Lorenz equations (which come out of a meteorological context) are the most studied equations in dynamical systems.  The three coupled equations exhibit Hopf bifurcation when one of the equation parameters passes through a critical value. Typically the system destabilizes and oscillations appear. Under certain circumstances modifications of this standard behaviour occur. Peter characterizes a range of such situations.  


THERMAL IGNITION IN A REACTIVE SLAB WITH UNSYMMETRIC BOUNDARY TEMPERATURES (with K. K. Tam)


 By either removing or supplying heat through the boundaries thermal ignition in a material may be either suppressed or encouraged.  Peter examines a classical heat conduction Arrhenius reaction model of this situation and identifies parameter ranges associated with thermal runaway. 

UNIFORMLY APPLICABLE APPROXIMATIONS TO A CLASS OF INTEGRALS OCCURRING IN STUDIES OF SLOW RESONANCE


Oscillatory systems excited by external forces at the natural frequency will result in growing amplitude oscillations (resonance). Usually, however, the resonant forcing is not sustained and the challenge is to determine the amplitude growth that occurs under intermittent natural frequency forcing. Peter points out (by example) that in many circumstances the solution can be represented in integral form so that powerful complex variable and asymptotic methods can be used to extract results. These is an earlier account of Peter's J Aust Math Soc B vols 34, 35  1994 publications. 



PETER'S (ACADEMIC) OBITUARY


Dr Peter Chapman (1934 - 2008)


Peter Chapman's first two degrees were in Aeronautical Engineering and Science at the University of Sydney in the mid 1950s.  He graduated with first class honours and a University Medal.  He subsequently continued at USyd under the supervision of John Mahony, completing a PhD entitled Some Problems in Wave Propagation.  The thesis was on the theoretical behaviour of shock waves in cylindrical and spherical geometries.  Peter had a wicked sense of humour.  As John's first student, Peter used to say that John viewed him as an historical curiosity, or perhaps an item of memorabilia.  They were of course dear friends.

After completing his PhD, he learned many practical skills whilst working in the Australian Government Aircraft Factory in Melbourne.  He soon accepted an invitation to apply for a position at the University of Toronto, won the job, and stayed there nine years.

He then joined the University of Western Australia in 1969.  It was a time when mathematics, especially applied mathematics, flourished at UWA.  John Mahony was of course there, at the full expression of his great talent.  Colleagues at the time included Joyce Billings, Neville Fowkes, Malcolm Hood, Dave Hurley, Ron List and Peter Wynter.  Peter supervised many students at various levels, but just one PhD student, Noel Barton, from 1970-73. 

Undergraduate students were fond of Peter and his quirks of style.  Displaying much more bark than bite, there was sometimes an explosion of grumpiness and salty language, which always evaporated quickly leaving his residual good nature.  He never lectured from notes -- everything went directly from head to board, regarded by most as a courageous practice, and was presented in his neat handwriting without errors.

Peter had broad interests and an intense curiosity in all aspects of life, including history, the arts, politics, the sciences, aviation, golf and squash in his earlier days, and of course horse racing.  He had a distinctive, very individual approach to matters academic and otherwise.  It was part dogged, part self deprecating, part cynical.  He loved to snort a characteristic grawfoorrrrr when he exposed a new truth. 

Although Peter worked on a broad range of mathematical problems, his special passion was reserved for really big ideas.  He loved to grasp the essence of such ideas, even if they'd been around a long time and were no longer actively researched.  He needed to express them in his own particular way, striving always for a better exposition.  As an example, in the early 70's he laboured mightily on ergodic theory.  As a still current example, Peter recently finished a manuscript which exposes further connections between classical and quantum mechanics.  Another recent example was new important material on geometric optics.  These last two major contributions were accomplished since his retirement in 1994.

Peter died on 8 February 2008 after a short illness.  He is survived by his wife, Constance, and daughters, Sophia and Katherine.  He was a dear friend and highly respected colleague of all who worked with him.

Dr Noel Barton (Sunoba Pty, Ltd) and Dr Neville Fowkes (UWA).
 

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