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Douglas Samuel Jones MBE. 10 January 1922
−− 26 November 2013
B. D. Sleeman and I. D. Abrahams
Biogr. Mems Fell. R. Soc. 2015 61, 203-224, published 24 June 2015
originally published online June 24, 2015
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DOUGLAS SAMUEL JONES MBE
10 January 1922 — 26 November 2013
Biogr. Mems Fell. R. Soc. 61, 203–224 (2015)
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DOUGLAS SAMUEL JONES MBE
10 January 1922 — 26 November 2013
Elected FRS 1968
By B. D. Sleeman1 and I. D. Abrahams2
1School
of Mathematics, University of Leeds, Leeds LS2 9JT, UK
and
Division of Mathematics, University of Dundee, Dundee DD1 4HN, UK
2School of Mathematics, University of Manchester, Manchester M13 9PL, UK
Douglas Jones was an extremely creative and influential mathematician. His contributions
to the theory of electromagnetic and acoustic waves and his development of original and
exceptionally powerful mathematical techniques with which to study them has led to the
solution of problems of both practical and social importance. His work is fundamental to
the design and performance of radar antennae wherein it is necessary to optimize their
transmitting and receiving characteristics. Jones also investigated the manner in which
electromagnetic waves interact with objects having sharp edges. These studies are basic
to the construction of ‘stealth’ aircraft, in which the geometrical shape is designed to
minimize the aircraft’s signature. When supersonic airliner capability was realized in the
development of Concorde there was considerable public concern regarding the excessive
noise created during take-off and landing and the impact of ‘sonic boom’ on built-up
areas. This prompted investigations into the noise levels experienced on the ground due
to a moving acoustic source. This inspired Douglas to develop a mathematical theory of
noise shielding. To address these difficult problems he developed powerful techniques
of analysis: these included the asymptotic expansion of multidimensional integrals and
the generalization of the method of stationary phase; the solution of integral equations
arising in diffraction and obstacle scattering theory; the development of multidimensional
generalized functions; uniform asymptotics and Stokes’ phenomenon; the Wiener–Hopf
technique; and powerful numerical techniques to solve integral equations arising in
electromagnetic wave theory.
Douglas Jones was a very private man, not given to small talk, but once engaged was
stimulating and amusing company and always happy to engage in the exchange of ideas. He
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© 2015 The Author(s)
Published by the Royal Society
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was an important guiding light to young staff and research students, many of whom have gone
on to distinguished careers.
He and his wife Ivy, who predeceased him, were a devoted and mutually supportive team.
They were both very active in the work of Tenovus Scotland and the World Wildlife Fund for
Nature. Douglas Jones was a fine man, a friend and mentor and is greatly missed. He was
survived by his sisters Dot and Joyce (sadly Joyce died in early 2014) and two children Helen
and Philip.
Early life and education
Douglas Samuel Jones was born on 10 January 1922 in Corby, Northamptonshire, the son of
Jesse Dewis Jones (1887–1932) and Bessie Jones (1900–92; née Streather). At that time Corby
was a small mainly farming village of two streets. At an early age Douglas was introduced to
the things that a farmer had to know: scything and drying grass, milking cows and building
haycocks. Although iron ore had been known to exist in Corby since Roman times, this was
not exploited until the 1930s with the result that the farming community was destroyed and
Corby was converted to an industrial town, the process being accelerated by the steel tube
manufacturer Stewarts & Lloyds moving the inhabitants of a whole Scottish town down to
Corby. The last time Douglas visited there, many years ago, a main road had been driven
through the fields and lanes where he used to herd cattle and hitch up the pony and cart.
Douglas was the eldest of four children; he had one brother, Jimy (Gerald), and two sisters,
Dot (Doris) and Joyce. Douglas’s father Jesse became a general manager for Tarmac (which
merged with Lafarge to become Lafarge Tarmac in 2013) when Douglas was born. This
position, however, brought with it the demand to move with the work. So, after a short move
to Redcar, the family finally settled in Bilston, where Douglas spent his childhood.
Douglas’s father was a great sportsman. He captained the village cricket team and played
soccer for Nottingham Forest. He was also a keen gambler, playing cards (at which he
invariably won) and attending greyhound race meetings (where he frequently lost). He was
also fond of horse racing, which he managed to attend more often than might be imagined
because of his habit of working every day of the week including Sunday; this allowed him
some flexibility in arranging time off. Although severely wounded in World War I and
suffering therefrom to the end of his life, he maintained his interest in sport and was manager
of the works sports teams in addition to his other duties. Whenever possible the Jones
family was expected to accompany him on all these activities. Douglas’s mother was a keen
sportswoman and she became the club tennis champion. It is fair to say that the Jones children
probably saw more greyhounds, horses, football matches, etc., and, on Sundays, slag heaps
(with the occasional side trip to the seaside) than many adults of that time saw in their whole
lives. It was inevitable that such a hectic and busy life would take its toll on Jesse. Although
he survived one bout of pneumonia in 1930, two years later he died of double pneumonia just
two days after Douglas’s tenth birthday.
Jesse left behind the memory of a warm, loving, vigorous and kindly man, a house covered by
an insurance policy, but no money; it had all gone on gambling but none on drink, for he was a
strict teetotaller. Although he had little formal education, Jesse was of high intellectual capacity
and taught himself shorthand at the age of 12 years. He had a great belief in education and left
one injunction on Bessie, namely to see that the children had the best education possible.
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Douglas Samuel Jones207
With no money coming into the Jones household, Bessie was in dire circumstances because
there was no widow’s pension. Jesse’s brothers and sisters offered to help by each taking one
of the four children into their homes and bringing them up, leaving Bessie to look after herself.
However, she had the courage and love to reject this offer; Douglas’s life might otherwise have
been very different.
Douglas’s education began in 1927 at Ettingshall Primary School, known locally as
the ‘The Tin’ because it was constructed from corrugated iron. From there he attended
Wolverhampton Grammar School (1931–40). Among its alumni were A. E. H. Love (1863–
1940; FRS 1894), famous for A treatise on the mathematical theory of elasticity (Love 1920)
and the development of surface waves, now known as ‘Love waves’; A. Goldie (Professor
of Pure Mathematics, University of Leeds 1963–86), known for ‘Goldie’s theorem’ in ring
theory; and Mervyn King (now Baron King of Lothbury), Governor of the Bank of England
and chairman of its Monetary Policy Committee from 2003 to 2013.
Douglas had been at Wolverhampton Grammar School as a fee-paying pupil for only a
year when his father died and so the only hope of remaining there was to win a scholarship.
This he managed to do; moreover, the scholarship included free lunches, free travel and
a contribution towards the cost of clothing. His sisters and brother secured places at the
Royal Wolverhampton School as boarders as soon as they were old enough to do so, and
the expenses of keeping them during term time were therefore minimal. The school also had
some notable alumni, including Eric Idle of ‘Monty Python’ fame, and Gilbert Harding, the
journalist and radio and television personality of the 1950s. The company that had employed
Douglas’s father contributed a small pension. To supplement the family income Douglas’s
mother learned shorthand and became a shorthand teacher in the evenings, teaching local
office girls at 6d. (2½p) an hour. So the family was able to survive financially, although when
funds were at a low ebb it was necessary to take in lodgers. In these circumstances Douglas
had to do his homework in the kitchen and in so doing acquired the ability to concentrate,
which was so valuable in later life. Although the Joneses led a frugal life it was a harmonious
family, thanks to the unfailing cheerfulness of Bessie. Douglas remembered Christmas as
a glorious time—mixing the cake and pudding from carefully hoarded ingredients, making
mince pies and preparing banana cream. Bessie’s sisters, not much better off than the Joneses,
all managed to send the family Christmas presents and, on occasion, one of Jesse’s sisters
would provide a Melton Mowbray pork pie together with a string of sausages.
At this time Douglas was in the classics stream at school, concentrating on Latin and Greek
on the advice of his form master. In this stream he took the School Certificate, achieving
second place among all entrants from the school in that year. Having obtained his School
Certificate, Douglas thought about getting a job. However, it was decided that he should go
into the sixth form and aim for the Executive Class of the Civil Service. The family doctor
advised mortgaging the house to permit Douglas to train for a medical career, but that avenue
was quite out of the question with three other children to think about. Furthermore, Latin
and Greek offered few prospects for employment should Douglas miss out in the fierce
competition for the Executive Class. So he decided to switch to mathematics in the hope that
it would give him more opportunities in the employment scene. Douglas confessed that at this
stage he never really liked the subject but was greatly influenced by his mother, who while at
Kettering High School had developed a love for geometry. (Always mentally active, Bessie
at the age of 70 years decided it was high time she learnt German.) For Douglas, school was
six days a week with a large chunk of homework set for Sunday. Two half-days were devoted
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to sport, of which Douglas took full advantage, eventually representing the school at chess
(captain), football, cricket (captain) and fives (Eton and Rugby). Another school activity was
the Officers’ Training Corps (OTC), in which he rose to the rank of company quartermaster
sergeant. In such a system there was sufficient flexibility for a switch of subjects to be feasible,
if rare, provided that one ‘put one’s back into it’. He was also Senior Prefect, Victor Ludorum
and House Captain.
Douglas never took the Civil Service examinations because recruitment was stopped by
World War II. Bessie now thought that a job was both possible and essential for her. She
became a book-keeping clerk, a job that gave her much pleasure because she always enjoyed
arithmetic. In addition there was more money coming into the household, so she contemplated
the possibility of Douglas’s going to university. After her discussions with the headmaster,
Douglas found himself taking the examinations for the Mathematical Scholarship at Corpus
Christi College, Oxford. Douglas was awarded the second of these scholarships, the first going
to R. H. Tuck (latterly Professor of Agricultural Economics at the University of Reading,
1965–82). However, F. B. Pidduck, Fellow of Corpus Christi College, at the time remarked
that Douglas’s knowledge of mechanics was abysmal.
There was then the problem of finding enough money to supplement the scholarship,
which was inadequate on its own. Fortunately, this was resolved by Douglas’s winning a
Staffordshire Major Scholarship and by the allocation of a Kitchener Scholarship. Together
these were sufficient to relieve his mother of the cost of supporting him. In fact, he was able to
pay her a small weekly sum during the vacations. So armed with a Higher School Certificate
in mathematics, physics and English together with a Mathematical Scholarship at Oxford,
Douglas was able to give up gas-mask making and digging air-raid shelters, which he had been
doing at home while his mother became a fire watcher and air-raid warden.
World War II service, 1942–45
Before entering active service, Douglas intended to train as an electrical engineer under a
special scheme just begun at the Clarendon Laboratory (Oxford). Shortly afterwards he joined
the Royal Air Force Volunteer Reserve (figure 1) and was appointed to the Signals Branch
(Radio Detection Finding), later renamed the Signals Radar Branch. Although Douglas’s
training had equipped him with some basic principles he had little knowledge of the practical
realities of engineering, so the latter had to be absorbed rapidly under rather pressing wartime
conditions.
The elder of Douglas’s sisters, Dot, joined the Women’s Auxiliary Air Force as soon as she
was eligible; later, his brother, Jimy, joined the Tank Corps but his young life was cut short by
Hodgkin’s disease. His younger sister, Joyce, was at school for the duration of the war.
The first military unit that Douglas served with was 85 Squadron, which was equipped with
a modified version of the Douglas Havoc. The front of the Havoc was sliced off and replaced
by a searchlight (Turbinlite; figures 2 and 3) powered by banks of accumulators distributed
throughout the aircraft. The idea was that the Havoc would use its A.I. (Airborne Interception
radar—an early form of radar) to find a German aircraft, then catch it in the spotlight and
let a normal day fighter (such as a Hawker Hurricane; figure 4) attack it. The difficulties of
coordinating such an attack at night can well be imagined, and the whole project was finally
abandoned in early 1943, although Douglas had left the squadron by then.
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Douglas Samuel Jones209
Figure 1. Douglas on joining the Royal Air Force Volunteer Reserve, December 1941.
(Source: Bennett Clark Ltd, Wolverhampton.)
After a short spell with a conventional night fighter squadron Douglas was posted to a
squadron equipped with Boulton Paul Defiants (figure 5), which, because of their cumbersome
turret and vulnerability, had been assigned the job of being the first airborne countermeasures
group. These aircraft were installed with the exotically named ‘Moonshine’ and ‘Mandrel’
spoofer/jammer devices used to defeat German radar. This technology was developed at the
Telecommunications Research Establishment, where Douglas came into contact with Martin
(later Sir Martin) Ryle (1918–84 FRS 1952), the radio astronomer and later Nobel laureate,
and William Cochrane. Douglas recalls flying many nights in the gun turret of the Defiant,
making crucial measurements of German radar frequencies for jamming purposes. Although
the turret allowed the Defiant to defend itself from more angles of attack than a conventional
fighter, it also inhibited manoeuvrability. As a result the Defiant was a ‘sitting duck’ when
attacked from below.
From then on, Douglas was always involved with countermeasures, arriving eventually
with the formation of the 90 Group. Douglas remained a serving officer with an operational
squadron with the rank of flight lieutenant. His war service was recognized by the award
of an MBE and by being Mentioned in Dispatches. Douglas recalled playing in 1945 for a
services cricket XI against an overseas services XI. In the latter was an Australian (very) fast
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Figure 2. The Turbinlite. (Source: photograph MH5711, from the collections of
the Imperial War Museums.)
Figure 3. Douglas Havoc Turbinlite. (Source: photograph MH5710
from the collections of the Imperial War Museums.)
Figure 4. Hawker Hurricane. (Online version in colour.)
Figure 5. Boulton Paul Defiant. (Copyright © RAF Museum, Hendon; reproduced with permission.)
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Douglas Samuel Jones211
soon-to-be-test-match bowler Keith Miller. Douglas said that as far as he was concerned every
ball from Miller was to be a quick single! At the end of the war Douglas had just been posted
to the Far East when word came that he was to be demobilized and should return to Oxford
immediately.
Oxford; Massachusetts Institute of Technology, 1945–48
At Oxford, Douglas turned to the matter of becoming a mathematician, with an actuarial
career vaguely in mind. His tutor at Corpus Christi College was F. B. Pidduck, one of the
kindest men as far as his students were concerned but somewhat at loggerheads with the other
Fellows, so that he never graced the Senior Common Room with his presence. Jones recalled
that Pidduck was not a particularly good lecturer but was not the worst. That accolade went
to J. H. C. Whitehead (1904–60; FRS 1944), who totally changed his notation and started the
course afresh at every lecture. For Douglas the best lecturer was U. S. Haslam-Jones. During
his time at Oxford, Douglas took part actively in field sports and also in squash and chess.
In his final year (1947) Pidduck suggested that Jones widen his experience by going to the
USA by trying for a Commonwealth Fund Scholarship. He was fortunate in his application.
The other successful candidate that year was Freeman Dyson (FRS 1952), an already wellestablished theoretical physicist. It was as an undergraduate that Douglas wrote his first
paper (1)*. This little note pointed out pitfalls in the use of complex conjugate functions as a
means of determining equipotentials of an electrified curve as a result of the possible presence
of singularities. After graduating with a 1st in Moderations and Finals (specializing in
applied mathematics), Douglas went to Massachusetts Institute of Technology (MIT) with the
intention of doing research under the supervision of the electrical engineer J. A. Stratton, noted
for his classic text Electromagnetic theory (Stratton 1941). However, it was very clear soon
after Douglas’s arrival that Stratton was fully occupied in his pursuit of the presidency of MIT
(1959–66) and therefore had little intention of supervising any research students. All callers
were discouraged by having to get past three secretaries in sequence before setting foot in his
office. It was a technique that ensured that research students really undertook independent
research. Undaunted, Douglas took the opportunity to learn physics from Victor Weisskopf
(1908–2002), H. Feshbach (1917–2000) and R. D. Evans (1907–95). Douglas was particularly
inspired by Evans’s lectures on radiation and its medical applications. Indeed, it was through
the good offices of Evans that Douglas was able to participate in the first experiments on
the absorption of calcium by bone, using tracer methods. Apart from helping with blood
samples and ultracentrifuges, his main contribution to the research was in devising a formula
governing the absorption. It also brought into sharp contrast the differences between research
in the biomedical sciences and that in the physical sciences and engineering. At the end of
this year Douglas went on a grand tour of the USA, visiting about 40 states in the company
of John Westcott (1920–2014; FRS 1983). It was a wonderful experience and he would have
been tempted to seek a position had it not been for his desire to return to see his mother and
the rest of the family. In addition, Douglas had secured an appointment as assistant lecturer at
Manchester University.
*Numbers in this form refer to the bibliography at the end of the text.
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Manchester, 1948–57
The Mathematics Department of Manchester University at the time of Douglas’s appointment
was led by Sydney Goldstein (1903–89; FRS 1937), Beyer Professor of Applied Mathematics
(1945–50), Max Newman (1897–1984; FRS 1939), Fielden Professor of Pure Mathematics,
and James (later Sir James) Lighthill (1924–98; FRS 1953), who had recently been appointed
to a senior lectureship (1946) and who succeeded Goldstein in 1950 to the Beyer Chair. Alan
Turing (1912–54; FRS 1951) was appointed to a readership in 1948 before becoming Deputy
Director of the Computing Laboratory and working on the software for the Manchester Ferranti
Mark 1 computer. With his future secure, Douglas married Ivy Styles on 23 September 1950 in
Bilston, Wolverhampton. It was a most happy, loving and companionable marriage.
Douglas’s first researches concerned acoustic and electromagnetic diffraction by bodies
with sharp edges. With F. B. Pidduck (2) he developed a perturbation method, valid for
high conductivity, to find an approximate solution to the problem of diffraction by a highly
conductive wedge. Between 1950 and 1952 (3–5) Douglas set down a sound theoretical basis
for obtaining rigorous solutions to diffraction problems involving edges. In particular he
established sufficient conditions at edges and corners to ensure uniqueness of solutions. At
that time no such conditions had been rigorously established, although their need had been
noted by C. J. Bouwkamp in 1946. Jones also demonstrated (5) that some of the solutions
then available did not satisfy these conditions. He also demonstrated (6) that the usual method
of reducing an electromagnetic scattering problem, when the scattering object is of infinite
extent, to an integral equation produced a field that failed to satisfy the radiation condition.
His alternative formulation avoided this difficulty.
Having established the theoretical basis for the correct formulation of scattering problems,
Douglas developed powerful new methods to handle a variety of scattering problems
involving parallel planes, and cylinders of semi-infinite or finite length. In particular, Douglas
modified and extended the Wiener–Hopf technique so that it was directly applicable to such
problems; for bodies of finite length his is the established approach taken to this day. In
the 1950s the standard procedure to employing the Wiener–Hopf technique was first to use
Green’s theorem, or a similar identity, to construct an integral equation (or a system of such
equations) defined on a half-line in physical space. This could then be transformed into a
functional equation in a suitable complex-variable plane. Such an approach was used by J.
Schwinger in the USA and by E. T. Copson in the UK; however, it was cumbersome and often
required the construction of complicated Green functions. Douglas (7) showed that the process
could be very significantly simplified by directly applying the two-sided Laplace transform
(or equivalently a Fourier transform) to the governing equation and boundary conditions and
thence relatively easily obtaining the associated Wiener–Hopf functional equation holding
in a strip in the Laplace transform complex plane. This is now the standard approach for
formulating diffraction problems via the Wiener–Hopf technique. Douglas’s implementation
of the Wiener–Hopf technique is referred to as ‘Jones’s method’.
As an aside, Ben Noble’s book (Noble 1958) entitled Methods based on the Wiener–Hopf
technique, first published in 1958, remains to this day the definitive text on the subject. It was
Ian Sneddon (1919–2000; FRS 1983), in his capacity as editor of the International Series of
Monographs in Pure and Applied Mathematics, who suggested to Noble (1922–2006) that he
write a book on the subject. It is believed that Douglas supported Ben Noble in the writing
of the book, which was mostly based on Douglas’s own notes, articles and bibliography list.
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Douglas Samuel Jones213
Douglas Jones turned his attention in the mid 1950s to the solution of scattering of acoustic
and electromagnetic waves by a circular disc. The scattering problem for an acoustically soft
disc can be reduced to finding the solution of a Fredholm integral equation of the first kind
with kernel
2
2
2π
ð
2
eik x + y −2 xy cosθ
∫0 x 2 + y 2 − 2 xy cosθ dθ .
He then made the extraordinary observation (8) that by writing the Fredholm equation as a
pair of integral equations of Volterra type it was fairly easy to invert the integral equation with
kernel
ð2π
2
cos k x 2 + y 2 − 2 xy cos θ
∫0 x 2 + y 2 − 2 xy cosθ dθ ,
which leads to a more effective way of solving the problem. Indeed, the final result of a
series of complicated manipulations leads to the solution of the original problem in terms of a
Fredholm integral equation of the second kind with kernel
sin k ( x − y )
.
x− y
This equation is especially well suited to numerical evaluation in the low-frequency limit.
In 1956 Douglas held a visiting professorship at the Courant Institute of Mathematical
Sciences in New York, where he collaborated with Morris Kline (1908–92) on the asymptotic
behaviour of integrals. This led to their seminal paper (11) on the asymptotics of twodimensional integrals of the form
∫∫ g ( x, y )e
ikf ( x , y )
dx dy ,
which occur frequently in physical applications and are of intrinsic mathematical interest.
Although there had been earlier work on the topic, this paper pioneered techniques and results
that are now standard textbook material. Furthermore, it gave powerful impetus to the use of
generalized functions (or distributions) in addressing the problem of the asymptotic behaviour
of integrals with oscillatory integrands. Before the final draft of the paper was completed,
Jones and Kline entered into a flurry of correspondence across the Atlantic. In a letter to
Douglas dated 19 April 1956, Morris wrote:
I attended a meeting on electromagnetic problems about ten days ago and Prof. Erdélyi [1908–77;
FRS 1975] was there. I discussed the idea of this paper with him and in fact gave him a verifaxed
copy whose content is exactly the same as the enclosed except that the figures were not ready
at the time. As you know from previous correspondence I have been a little concerned as to the
justification of this paper in view of Focke’s results [Focke 1954]. I gave some arguments in the
text which I hope you will comment on if you feel more or less should be said. I discussed this
point with Erdélyi and he believes that the method of this paper is simpler and more direct. He
feels that Focke’s method is rather clumsy.
In response Douglas wrote back on 5 July:
I agree with Erdélyi that the simpler the method the better, but I do not see how we can get
a simpler approach than we have already. The basic ideas are in a sense almost trivial, that of
transforming a double integral to a single Fourier integral. The actual labour of evaluating the
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Biographical Memoirs
expression can scarcely be avoided in view of complicated form that it takes. Having made that
remark, I shall, of course discover a paper in which everything is put quite trivially.
On returning to Manchester, Douglas was promoted to senior lecturer but was soon to move
to the University of Keele as head of department. During his time at Manchester, Douglas
supervised several research students, some of whom went on to significant mathematical
careers. Most notably these included W. E. Williams (Professor of Mathematics, University of
Surrey), T. B. A. Senior (Radiation Laboratory, University of Michigan) and A. Sharples (New
Mexico Institute of Mining and Technology, 1968–2000).
Keele, 1957–64
During his visit to New York in 1956 Douglas was thinking about his future and had applied
to the University of Glasgow for the position of head of a new Department of Applied
Mathematics and Simson Professor of Mathematics. He travelled to Glasgow to hear about
their plans for the new department, having been one of two candidates on the shortlist. When
he got there he found that the department was intended to teach engineering students, would
have no undergraduates of its own and would not be able to take on any mathematical research
students. Douglas was unsure how he could form a department along these lines. The matter
was resolved by the appointment of the other candidate, Ian Sneddon. Sneddon had been the
first holder of the Chair of Mathematics at the University of Keele. So as Ian Sneddon headed
north to Glasgow, Douglas and Ivy set off for Keele. It was during the early years in Keele
that their daughter, Helen Elizabeth, was born on 28 May 1958, followed by their son, Philip
Andrew, on 9 February 1960.
High-frequency scattering was the subject that Douglas Jones researched on arrival in
Keele. His work on scattering by convex bodies already contained the ingredients that were so
brilliantly brought to fruition in Joe Keller’s theory of geometrical diffraction (Keller 1962).
To give the flavour of Jones’s arguments (9, 10), consider the scattering of a plane wave by a
perfectly conducting convex obstacle. It is assumed that the radii of curvature at all points of
the body are large compared with the wavelength so that the high-frequency approximation
can be used. The points of glancing incidence will form a curve D (the shadow boundary) on
the obstacle. At points on the illuminated side, not near D, the field is given by the geometrical
acoustics approximation and here the coefficient σ, defined as the total energy flux outwards
from the obstacle in the scattered wave divided by the energy flux in the beam of the incident
wave that falls on the obstacle will be 2. In the neighbourhood of a point P of D, D will be well
approximated by a cylinder whose axis is parallel to the tangent to D at P and whose radius
is the radius of curvature R of the obstacle in a plane through P perpendicular to the tangent.
Here Douglas ignored a small correction due to the variation of the radius of curvature in the
penumbra. Then, if the tangent at P makes an angle π/2 − β with the direction of propagation
of the incident wave, Douglas showed that the energy scattered per unit length of D, for an
incident wave carrying unit energy per unit area, is
b0k −2/3 R1/3 cos1/3 β ,
where k is the incident wavenumber and b0 = 0.9962 in the acoustically sound hard case and
−0.8640 in the sound soft case. The scattering coefficient of the obstacle can then be shown
to be given by
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Douglas Samuel Jones215
σ= 2 +
b
0
2/3
k S0
∫R
1/3
cos1/3β ds ,
D
where S0 is the projected area of the body on a plane normal to the direction of propagation
of the incident wave and s is the arc length of D. Douglas then went on to determine explicit
values of σ for the sphere and the spheroid and extended the ideas to high-frequency scattering
by electromagnetic waves.
It was at Keele that Douglas first showed his remarkable ability to balance outstanding
leadership and commitment to administrative duties with mathematical research and
scholarship. For a while as the head of department busily engaged in building on the work
done by Ian Sneddon to secure a lively and active department, and as Dean of Science from
1959 to 1962, he continued his researches into high-frequency scattering and wrote his
monumental 807-page book The theory of electromagnetism (12). This book alone would
have cemented Douglas’s reputation as world leader. It was an ambitious work with the aim,
as Douglas wrote in the preface, ‘to provide a text which will take the student from a first
acquaintance with Maxwell’s equations to within striking distance of modern research.’
In 1962–63 Douglas made a return visit to the Courant Institute of Mathematical
Sciences, New York, and embarked on his remarkable contributions (13, 14) to the problem
of high-frequency diffraction by a circular disc. Here he showed that the governing integral
equation for scattering by a sound soft circular disc could be reduced to a Fredholm equation
with kernel
eik ( x − y ) eik ( x + y )
+
.
x− y
x+ y
He then recast this equation into an equation of the second kind that could be solved very
effectively at high frequency.
At about the time of his return from New York, Queen’s College in the University of
St Andrews was wishing to found a Department of Applied Mathematics, and Douglas and
Ivy turned their thoughts to a future career in Dundee.
Dundee, 1965–92
At the time of Douglas Jones’s appointment as Ivory Professor of Applied Mathematics, the
Department of Mathematics was small and headed by W. N. (Norrie) Everitt (1924–2011),
who had been appointed to the Baxter Chair of Mathematics in 1963. Douglas’s appointment
came amid the result of the Robbins Report on higher education, which recommended that at
least one and perhaps two new universities be founded in Scotland. As a consequence, Queen’s
College became the University of Dundee in 1967. Douglas wasted no time in setting about
building Applied Mathematics with several appointments. One of us (B.D.S.) was indeed
fortunate to have been one of his earliest appointees. In addition to recruiting young applied
mathematicians, Douglas also had the foresight to see that numerical analysis had to be a vital
component in the development of applied mathematics. So it was that a chair in numerical
analysis was founded in 1967. The first holder of this chair was the inspirational A. R. (Ron)
Mitchell (1921–2007). Under their leadership the department grew rapidly in both number
and prestige, and its influence was felt not only in Scotland but also throughout the UK and
internationally. The late 1960s and early 1970s saw the initiation of the biennial conferences
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devoted to numerical analysis held in the odd years and to the theory of differential equations
held in the even years. The Numerical Analysis Conference continues to this day and is now
hosted by the University of Strathclyde. In addition, the Department of Mathematics initiated
two Master of Science courses: one in numerical analysis run by Ron Mitchell, and one in
functional analysis and differential equations, run by Norrie Everitt. The research environment
in the department also benefited from a constant flow of visitors, research students and
postdoctoral fellows, many of whom have gone on to distinguished careers. Among the more
notable applied mathematicians are W. G. C. Boyd (Bristol), P. Davies OBE (Strathclyde), R.
Fletcher (Dundee; FRS 2003), P. Grindrod CBE (Oxford), the late D. Morgan, A. D. Rawlins
(Brunel), B. P. Rynne (Heriot-Watt), N. H. Scott (East Anglia), P. D. Smith (Macquarie,
Australia), D. Wall (Canterbury, New Zealand) and G. E. Tupholme (Bradford).
In his early years in Dundee, Douglas embarked on a programme of developing existing
and new mathematical techniques with which to rigorously address many problems arising in
acoustic and electromagnetic wave theory. Perhaps as a result of his work on the asymptotic
behaviour of integrals (11) he worked on multidimensional theory of generalized functions,
a subject dealt with earlier in the one-dimensional case by Lighthill and G. Temple. In
this regard, it is fitting to note the following remark of Lighthill relating to the theory of
generalized functions made at a conference in 1992 at Dundee University to mark Douglas’s
70th birthday; it concerned Douglas’s book The theory of generalised functions (21):
I have moreover been overjoyed that my tiny 80-page Introduction to Fourier analysis and
generalised functions [Lighthill 1958], which concentrates on functions of just one variable, has
proved to be a suitable appetite-whetting ‘starter’, as it were, leading up to Douglas’s superbly
concocted ‘main dish’ in 540 pages which extends all the results in a comprehensive fashion and
includes the corresponding properties of functions of many variables.
The method of geometrical diffraction is an important technique for studying the
propagation of sound at high frequencies. It is based on an assumption of locally plane
wave behaviour, except in certain regions where properties change abruptly. In these
latter regions, coefficients are determined from certain canonical problems that enable the
calculation of the transition from one kind of local behaviour to another. Douglas brought
a new insight into diffraction by considering the problem of reflection and transmission at
high frequencies of an acoustic wave by a curved interface between two media of differing
refractive indices. Contrary to what would be predicted by geometrical acoustics, total
reflection does not completely annihilate the transmission of energy. Rather, propagating
waves emerge from a region of evanescence and are responsible for an energy flow. The
rays that give rise to this flow were called ‘tunnelling rays’ by Douglas (19). There is a
connection here with the concept of ‘electromagnetic wormholes’ that arise in the study of
‘cloaking devices’ (Greenleaf et al. 2009). Although the energy transmitted by tunnelling
rays on a single reflection may not be large, there is the possibility, in situations where
multiple reflections are important, of accumulation with a significant flux in directions
where it may not be expected. Tunnelling rays do not occur with plane interfaces but are a
feature of the effects of curvature.
Another topic in which Douglas’s insight has led to significant advances is in the solution
of the problem of scattering of waves by an obstacle. For simplicity we consider the twodimensional acoustic sound-hard scattering problem. If D is the exterior of a simple closed
curve C and the function G is a solution of Helmholtz’s equation that has the property of
behaving like a source on the curve C and satisfies the Sommerfeld radiation condition, then
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Douglas Samuel Jones217
the solution ϕ of the scattering problem when restricted to the curve C must be a solution
of the integral equation
1
∂
− φ ( p) + ∫ φ ( q )
G ( p, q) ds( q) =
f ( p),
2
∂nq
C
where
∂φ ( q)
= f ( q),
∂nq
for all points q on the bounding curve C.
If, as is frequently done, we choose G to be the free-space Green function, then it is
well known that the uniqueness of the solution fails at those values of the wavenumber that
coincide with an eigenvalue of the interior sound soft problem for C. This difficulty can be
overcome by a variety of devices but at the price of introducing other complications, which
lead to increased computer time and storage. Fritz Ursell (1923–2012; FRS 1972) suggested
(Ursell 1973) a beautiful modification that was relatively simple to apply and was potentially
useful in computations. It did have the practical drawback of requiring the computation of an
infinite series. To address this point, Douglas (16) proposed a further modification in which
an arbitrary limit is imposed on the size of the wavenumber k. His analysis then led to the
replacement of the above integral equation by


∂
1
− µ ( p ) + ∫  µ ( q)
G ( p, q) + χ ( p, q)  ds( q) =
f ( p),
∂n p
2

C

to be solved for the unknown function μ, where G is the free space Green function and
χ represents a suitable finite series of radiating surface harmonics. The solution ϕ to the
scattering problem is then given by
=
φ (P )
∫ [ µ (q)G(P, q) + χ (P, q) ] ds(q).
C
The solution obtained in this manner overcomes the non-uniqueness inherent in the classical
approach.
With the growth of air travel in the 1970s, accompanied by the introduction of largecapacity jet airlines, the noise generated on the ground by such aircraft became a real concern.
The noise produced by the supersonic airliner Concorde at take off and landing and also flying
at speeds in excess of Mach 1 over land proved, in particular, a major issue. The problem
of noise pollution from aircraft remains to this day, especially for people living in densely
populated areas in close proximity to airports.
Armed with his considerable array of modelling and mathematical techniques, Douglas
investigated a wide range of problems in which the occurrence of noise and the means to limit
it was of major interest. In particular he was concerned with acoustic noise generated by sharp
edges such as the leading and trailing edges of wings, and the noise created by gaps in wings
when flaps were lowered in the process of landing. Another problem was to assess the noise
created on the ground when jet engines were either situated above or below aircraft wings.
Douglas (17) developed rigorous theories of such situations and at the same time built on
asymptotic ray methods of Keller and Lighthill’s theory of aerodynamic noise (Lighthill 1952).
To illustrate Douglas’s contributions with a concrete example, he sought to understand
the way in which sound waves are refracted or scattered by jets, which is relevant to
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Biographical Memoirs
turbomachinery noise (including the humble hand-dryer) and also in aeroacoustics. The
simplest model that can be contemplated is the situation of a plane vortex sheet, separating
a still medium from one moving with uniform velocity and illuminated by acoustic radiation
from a line source. This model was examined by Douglas and others in several articles (for
example (15), where earlier references will be found). Perhaps surprisingly for such a simple
problem, difficulties were found to arise when ensuring that there was no sound field before
the source was switched on (that is, satisfying causality), being resolved ultimately by working
with what Douglas described as ‘rather abstruse entities’ known as ultradistributions. The
physical interpretation of these obscure generalized functions is not obvious and so doubt was
cast on the adequacy of the model.
The trouble with the investigation of the vortex sheet stemmed from the presence of a
certain non-real pole in complex wavenumber space. To attempt a deeper understanding
of the issue, Douglas resolved to obtain an exact analytical solution of the sound scattered
by a simple shear layer. In this model the flow increases linearly, from 0 below the layer
to a constant speed U, say, above it. Suppose the layer is of height h and a monochromatic
acoustic line source irradiates the layer with radian frequency ω so that a Strouhal number
can be defined as ωh/U. This work was reported in a ‘tour-de-force’ article of 42 pages in
Philosophical Transactions A (18). In it Douglas displayed masterful powers of analysis,
extensive knowledge of special functions (Airy and Whittaker functions) and their properties,
very delicate asymptotics including ray theory, and profound physical interpretation of the
results. This allowed him to make a very careful examination of the singularity structure for
arbitrary values of the Strouhal number, from which he discovered that there was an infinite
number of poles, some of which lie on the real axis. The multiplicity of solutions for the
shear layer is thereby far worse than that for the vortex sheet. However, it was found that as
the Strouhal number was reduced, two of the real zeros came into coincidence and shifted
off the real axis. The observation turned out to be crucial in resolving some of the nonuniqueness, especially because most of the poles ultimately gave an insignificant contribution
to the field. The final step that disposed of all questions of uniqueness was causality, and this
Douglas showed could be complied with by conventional functions; that is, there is no need
to employ ultradistributions in the solution. Physically, Douglas deduced that for a Strouhal
number below a critical value (when the poles have coalesced) the field contains a (standard)
Helmholtz instability wave in the shear layer, whereas above this value there is no instability
wave.
Perhaps stimulated by his war experiences, Douglas also made significant contributions to
antenna theory. In (20) he wrote:
The concept of an antenna as a piece of wire or portion of dielectric which radiates electromagnetic
energy is simple enough in principle, but the derivation of quantitative results of value for design
purposes is fraught with difficulties. Even when the isolated antenna can be described as a
straightforward boundary-value problem, it can rarely be solved with any ease. In fact the antenna,
to be of any use as an element of a communication system, must be coupled with a transmission
line or waveguide, and coupling forms an important but complicated part of any real system. For
these reasons a substantial amount of analysis has been devoted to antennas, not always with
success. The advent of large computers has made it possible to generate numerical answers to
problems which had hitherto defied solution.
It must be confessed, however, that the mathematical detail has often obscured the physical
principles involved leaving the engineer up in the air when both analysis and computer fail. For
example, to keep computer requirements reasonable, some type of symmetry is often assumed but
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Douglas Samuel Jones219
the symmetry is usually lost as soon as a transmission line is connected. While it is our purpose
to enumerate some of the analytical and numerical techniques that have been tried, it is hoped not
to lose sight completely of physical principles which may be helpful.
These principles are clearly adhered to in Douglas’s contributions to antenna theory. On the
one hand he provided a rigorous and complete analysis of problems, and on the other he gave
clear formulations of the underlying principles and a simple exposition of formulae necessary
for computational modelling.
Douglas returned to his studies of Wiener–Hopf problems in the early 1980s, motivated
by new scattering problems involving multiple bodies or barriers. These led to coupled
systems of such equations, for which no general constructive method of solution has yet
been found. It remains to this day an important open problem in mathematics. However,
Douglas obtained elegant solutions in some special cases of physical interest, and also
offered the first class of ‘large’ coupled Wiener–Hopf systems that yields an exact solution
(23), an extension of the Khrapkov–Daniele commutative (2 × 2) matrix form. Interested
readers may wish to refer to Lawrie & Abrahams (2007) for details and an extensive
bibliography.
During the 1970s and 1980s mathematicians began to direct their attention to the potential
of exploiting mathematical ideas to address problems arising in the biological and medical
sciences. This initiative arose, in part, from the development of the groundbreaking work done
by Alan Turing on biological pattern formation and carried forward by J. D. Murray (FRS
1985) and others.
As a forward-thinking mathematician and scientist, Douglas realized that the new and
rapidly evolving subject of ‘mathematical biology’ should be made accessible to undergraduate
students. This led in 1983 to his co-authored book Differential equations and mathematical
biology (26, 27). Mathematical biology is now recognized as a major field of applied
mathematical research, and most universities in the UK and worldwide offer mathematical
biology courses to students.
Notwithstanding all this inspiring and creative mathematical activity, Douglas was also
an outstanding administrator and mentor. In addition to serving as head of department on
several occasions he was also Dean of the Faculty of Science from 1976 to 1979. He was also
highly regarded by the wider community for his wisdom and sound advice, serving on several
influential boards and committees. For example, Douglas served on the Council of the Royal
Society (1973–74), the Propagation Aerials and Waveguides Committee of the Electronics
Research Council (1970–76), the Noise Research Committee of the Aeronautical Research
Council (1971–79) and the Computer Board.
Douglas Jones was a tireless champion and campaigner for the promotion of mathematics
and of the professional mathematician. He was appointed a member of the Mathematics
Subcommittee of the University Grants Committee (UGC) in 1970, succeeding W. H.
Cockcroft as its chairman in 1976 together with membership of the main committee. In 1981
he published the controversial report on behalf of the UGC entitled Whither mathematics?.
The report highlighted the serious problems caused by the bulge in the group of academic staff
35–45 years of age that was reflected in the boom in recruitment in the 1960s as a consequence
of the Robbins Report on university expansion. With a predicted fall by 36% in mathematically
trained students, it was recommended that these staff in mid-career be compulsorily retired. As
a result of both public and academic pressure, no government action was taken. As his fellow
committee member B. G. (Brian) Gowenlock recalled (Gowenlock 1993):
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Biographical Memoirs
Figure 6. Celebrating the 25th anniversary of the foundation of the Institute of Mathematics and its Applications,
1988. (Source: IMA; reproduced with permission.) (Online version in colour.)
Those were the days when smoking was still permitted in meetings and I was therefore treated to
a sustained monthly dose of indirect smoking from Douglas. He was always content to say that
the statistical chances of serious illness for himself were not sufficiently large to cause him to alter
his habits and it was therefore left to me to assume that my own health was not at any serious risk.
Brian further recalled that Douglas was a delightful person to know and had the ‘gift of
wisdom’.
As chairman of the UGC Mathematics Subcommittee, Douglas was also responsible for
overseeing the development of computer science. In typical Douglas Jones style he began
an address to the Inter-University Committee on Computing’s Colloquium in 1983 (22) as
follows:
I must open with a disclaimer. Any views which I express must be treated as personal opinions
and should not be attributed to the UGC Delphic oracle.
Secondly, when your kind invitation to address you was extended to me, the suggestion was
made that I should attempt to assess where we were and peer into the murky bowl of the future
to indicate the path we are about to tread. In the present foul political weather this is rather like
batting when you can’t see the other wicket let alone the bowler and since my only claim to
distinction in computing is that I am one of the few people in this room who learned it from
Turing, I am none too sure whether it is the game of cricket or tennis which engages us.
Douglas then went on to try to outline his views on the course of the universities over the
following decades.
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Douglas Samuel Jones221
Alan Turing must have had a significant effect on Douglas, so it should not come entirely
as a surprise that he was also able to make a useful contribution to computing with his books
on 80x86 assembly programming (24, 25). At the heart of nearly all major desktop computers
today is an Intel or Intel-compatible central processing unit running in ‘virtual 80x86’ mode.
Douglas was a founding Fellow of the Institute of Mathematics and its Applications (IMA),
served on its Council and was appointed President in 1988 (figure 6). It was during his
presidency that he led the negotiations with the Privy Council that resulted in the IMA’s being
incorporated by Royal Charter and subsequently being granted the right to award Chartered
Mathematician status.
Personal thoughts (B.D.S.)
I was most fortunate to begin my academic career by being among Douglas’s early
appointments at Queen’s College, Dundee. On my arrival I was immediately aware that
teaching and scholarship in its widest form were of the highest priority. There was also no
pressure on colleagues to write grant proposals but rather to pursue research for its own sake
and to make original contributions. Douglas never directed the research of young staff but
was always there to give encouragement and offer ideas. In my case, after a couple of years,
Douglas told me that he thought it would be a good idea if I spent a year at the Courant Institute
of Mathematical Sciences at New York University. In particular he arranged for me to work in
Joe Keller’s group. So my wife, Julie, and I, together with two very young children, headed
off to the ‘Big Apple’ and spent what for me was a momentous and exciting year, which had a
fundamental influence on my career. There I met R. (Richard) Courant and enjoyed seminars
by P. D. Lax, L. Nirenberg, J. Stoker and E. (Eugene) Isaacson as well as Joe Keller.
With regard to teaching back in Dundee, Douglas assigned lecturing duties that on the one
hand one would enjoy and on the other he thought would be ‘good for the soul’. On my return
to Dundee he assigned to me a new course on approximation theory, which was being offered
to the first graduate students on the new Numerical Analysis and Programming Masters
Course. I knew absolutely nothing about approximation theory and thought that Douglas had
made a mistake with the assignment. So, plucking up courage I decided to go and discuss
the matter with him. After knocking on his door and waiting for the red light to turn blue,
indicating entry, I was ushered in. ‘Professor Jones,’ I said, ‘you have assigned the NAP course
on approximation theory to me, but I know nothing about the subject.’ His response was firm
and short, ‘Well you will do when you have given the course.’ Such was Douglas’s approach
to many things, throwing out challenges and widening one’s horizons. In writing our book
on Differential equations and mathematical biology, Douglas and I began by deciding what
the book should contain and the audience for which we were writing, and we agreed on how
the work should be shared in assigning chapters. From then on we worked independently and
when one of us had finished writing a chapter it was passed to the other for comment. In this
way we were both free to express our own ideas. Furthermore, the book has evolved over each
edition, with new appraisals of content being assessed in each new writing.
Douglas was a great friend and mentor; he and Ivy (figure 7) were friends to our family.
He will be greatly missed by all.
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Biographical Memoirs
Figure 7. Douglas and Ivy. (Online version in colour.)
Honours and awards
1943
1945
1964
1967
1968
1971–73
1975
1980
1981
1986
1989
2013
Mentioned in Dispatches
MBE (Military)
Fellow of the Institute of Mathematics and its Applications
Fellow of the Royal Society of Edinburgh
Fellow of the Royal Society
Keith Prize of the Royal Society of Edinburgh
Honorary DSc of the University of Strathclyde
Marconi Prize of the Institute of Electrical Engineers
Honorary Fellow of Corpus Christi College Oxford
The Balthasar van der Pol Gold Medal of the International Union of Radio
Science (figure 8)
Naylor Prize and Lectureship of the London Mathematical Society
Fellow of the Institution of Electrical Engineers
Life Member of the Institute of Electrical and Electronics Engineers
Acknowledgements
We are grateful for access to Douglas’s autobiographical notes deposited with the Royal Society. One of us (B.D.S.)
is indebted to Dot and Joyce for allowing access to Douglas’s library and permitting us to consult some of his
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Douglas Samuel Jones223
Figure 8. Celebrating the award of The Balthasar van der Pol Gold Medal, of the International Union of Radio Science,
to Douglas in 1981. Left to right: Professor Jack Lambert, Douglas, Professor Ron Mitchell, Professor Norrie
Everitt, in the Mathematics Department, Dundee University.
personal correspondence, and to Peter Grosvenor and Jane Pyzniuk for helping sort through the archive of relevant
papers. Special thanks go to Paul Martin for his help in preparing an up-to-date list of Douglas’s publications. We
are also grateful to many people who have shared their reminiscences with us: Brian Gowenlock, Barbara and Greg
Kriegsmann, Jack Lambert, Tony Rawlins, Norman Riley, Paul Smith, David Thomas and Alistair Watson. Thanks
are also due to Sophie Abrahams, Mark Chaplain, Bill Horspool, David Youdan and Peter Grosvenor for help with
photographs. Finally we thank David Sleeman for his technical assistance, and Peter Grosvenor, Paul Martin, Juliet
Sleeman and David Colton for their careful reading of several previous drafts and for their invaluable support.
The frontispiece photograph was taken by Godfrey Argent and is reproduced with permission.
References to other authors
Focke, J. 1954 Asymptotische Entwicklungen mittels der Methode der stationären Phase. Ber. Verh. Sächs. Akad.
Wiss. Leipzig 101(3), 1–48.
Gowenlock, B. G. 1993 University Grants Committee. Bull. Inst. Math. Applic. 29, 99.
Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. 2009 Cloaking devices, electromagnetic wormholes and
transformation optics. SIAM Rev. 51, 3–33.
Keller, J. B. 1962 Geometrical theory of diffraction. J. Opt. Soc. Am. 52, 116–130.
Lawrie, J. B. & Abrahams, I. D. 2007 A brief historical perspective of the Wiener–Hopf technique. J. Engng Math.
59, 351–358.
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211, 564–587.
Lighthill, M. J. 1958 An introduction to Fourier analysis and generalised functions. Cambridge University Press.
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Love, A. E. H. 1920 A treatise on the mathematical theory of elasticity. Cambridge University Press.
Noble, B. 1958 Methods based on the Wiener–Hopf technique for the solution of partial differential equations.
London: Pergamon Press.
Stratton, J. A. 1941 Electromagnetic theory. New York: McGraw-Hill.
Ursell, F. 1973 On the exterior problems of acoustics. Proc. Camb. Phil. Soc. 74, 117–125.
Bibliography
The following publications are those referred to directly in the text. A full bibliography is
available as electronic supplementary material at http://dx.doi.org/10.1098/rsbm.2015.0005
or via http://rsbm.royalsocietypublishing.org.
(1) 1948 Note on an electrostatic problem. Math. Gaz. 32, 84–85.
(2) 1950 (With F. B. Pidduck) Diffraction by a metal wedge at large angles. Q. J. Math. 1, 229–237.
(3)
Note on diffraction by an edge. Q. J. Mech. Appl. Math. 3, 420–434.
(4) 1952 Diffraction by an edge and by a corner. Q. J. Mech. Appl. Math. 5, 363–378.
(5)
The behaviour of the intensity due to a surface distribution of charge near an edge. Proc. Lond. Math.
Soc. 2, 440–454.
(6)
The removal of an inconsistency in the theory of diffraction. Proc. Camb. Phil. Soc. 48, 733–741.
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A simplifying technique in the solution of a class of diffraction problems. Q. J. Math. 3, 189–196.
(8) 1956 A new method of calculating scattering with particular reference to the circular disc. Commun. Pure
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(9) 1957 Approximate methods in high frequency scattering. Proc. R. Soc. Lond. A 239, 338–348.
(10)
High-frequency scattering of electromagnetic waves. Proc. R. Soc. Lond. A 240, 206–213.
(11) 1958 (With M. Kline) Asymptotic expansions of multiple integrals and the method of stationary phase. J.
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(12)1964 The theory of electromagnetism. Oxford: Pergamon Press.
(13) 1965 Diffraction at high frequencies by a circular disc. Proc. Camb. Phil. Soc. 61, 223–245.
(14)
Diffraction of a high-frequency plane electromagnetic wave by a perfectly conducting circular disc.
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(15) 1972 (With J. D. Morgan) The instability of a vortex sheet on a subsonic stream under acoustic radiation.
Proc. Camb. Phil. Soc. 72, 465–488.
(16) 1974 Integral equations for the exterior acoustic problem. Q. J. Mech. Appl. Math. 27, 129–142.
(17) 1977 The mathematical theory of noise shielding. Prog. Aerospace Sci. 17, 149–229.
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(19) 1978 Acoustic tunnelling. Proc. R. Soc. Edinb. A 81, 1–21.
(20)1979 Methods in electromagnetic wave propagation. Oxford University Press.
(21)1982 The theory of generalised functions, 2nd edn. Cambridge University Press.
(22) 1983 Future prospects. IUCC Bull. 5, 113–117.
(23) 1984 Commutative Wiener–Hopf factorization of a matrix. Proc. R. Soc. Lond. A 393, 185–192.
(24)1988 Assembly programming and the 8086 microprocessor. Oxford University Press.
(25)1991 80x86 assembly programming. Oxford University Press.
(26) 2003 (With B. D. Sleeman) Differential equations and mathematical biology, 2nd edn. Boca Raton:
Chapman & Hall/CRC.
(27) 2010 (With M. J. Plank & B. D. Sleeman) Differential equations and mathematical biology, 3rd edn. Boca
Raton: Chapman & Hall/CRC.