1.1. From the Preface.

When, over four years ago, I began writing on nonlinear wave interactions and stability, I envisaged a work encompassing a wider variety of physical systems than those treated here. Many ideas and phenomena recur in such apparently diverse fields as rigid-body and fluid mechanics, plasma physics, optics and population dynamics. But it soon became plain that full justice could not be done to all these areas - certainly by me and perhaps by anyone.

Accordingly, I chose to restrict attention to incompressible fluid mechanics, the field that I know best; but I hope that this work will be of interest to those in other disciplines, where similar mathematical problems and analogous physical processes arise.

I owe thanks to many. Philip Drazin and Michael Mclntyre showed me partial drafts of their own monographs prior to publication, so enabling me to avoid undue overlap with their work. My colleague Alan Cairns has instructed me in related matters in plasma physics, which have influenced my views. General advice and encouragement were gratefully received from Brooke Benjamin and the series Editor, George Batchelor.

Various people kindly supplied photographs and drawings and freely gave permission to use their work: all are acknowledged in the text. Other illustrations were prepared by Mr Peter Adamson and colleagues of St Andrews University Photographic Unit and by Mr Robin Gibb, University Cartographer. The bulk of the typing, from pencil manuscript of dubious legibility, was impeccably carried out by Miss Sheila Wilson, with assistance from Miss Pat Dunne.

My wife Liz, who well knows the traumas of authorship, deserves special thanks for all her understanding and tolerance; as do our children Peter and Katie, for their welcome distractions.

Many have instructed and stimulated me by their writing, lecturing and conversation: I hope that this book may do the same for others. I hope, too, that errors and serious omissions are few. But selection of material is a subjective process, and I do not expect to please everyone!

Such writing as this must often be set aside because of other commitments. But for two terms of study leave, granted me by the University of St Andrews, this book would have taken longer to complete. Things were ever so: in 1738, Colin Maclaurin wrote to James Stirling as follows:

... it is my misfortune to get only starts for minding those things and to be often interrupted in the midst of a pursuit. The enquiry, as you say, is rugged and laborious.

St Andrews, September 1984

1.2. From the Publisher.

Wave Interactions and Fluid Flows is a coherent, up-to-date and comprehensive account of theory and experiment on wave-interaction phenomena, both in fluids at rest and in shear flows. On the one hand, this includes water waves, internal waves, their evolution and interaction and associated wave-driven mean flows; on the other, phenomena of nonlinear hydrodynamic stability, especially those leading to the onset of turbulence. Close similarities - and crucial differences - exist between these two classes of phenomena and their treatment in this single study provides a particularly valuable bridge between more specialised, but related, disciplines. As a result, this unique book will appeal to researchers and graduate students of fluid mechanics in its widest sense, including the study of wave-interaction phenomena in such diverse fields as meteorology, aeronautical and hydraulic engineering, optics, solar physics and population dynamics.

1.3. Review by: Triantaphyllos R Akylas.
Science, New Series 235 (4795) (1987), 1522-1523.

The theory of wave interactions was first developed about 25 years ago in the context of nonlinear free-surface waves. Now it is widely recognised that nonlinear wave interactions play an important pan in a variety of wave and instability phenomena, with applications in geophysical fluid dynamics, meteorology, flow instability and transition, and plasma physics. This book focuses attention on wave interactions in fluid flows and discusses both the general underlying ideas and some applications, dealing primarily with surface waves, internal waves, and shear-flow instability.

The book is divided into eight chapters. The main discussion starts in chapter 2, which is devoted to linear concepts and shows how some stratified flow stability phenomena can be understood in terms of linear wave interactions or mode coupling. Chapters 3 and 4 discuss the nonlinear interaction between a finite-amplitude wave train and an underlying mean flow, including the generalised mean Lagrangian approach and the conservation of wave action. Particular examples from free surface and internal waves, including the Craik-Leibovich theory of Langmuir circulations, are mentioned. Chapter 5 deals with resonant triad interactions, the first problem to be tackled in the early '60s, which provided incentive for further research in the area of nonlinear interactions. The basic theory of conservative and non-conservative resonant triads is discussed together with more recent applications on long-short wave interactions and shear-flow instability. Chapter 6 is devoted to the evolution of nonlinear wave packets; particular emphasis is placed on the most important equations in this area, the nonlinear Schrödinger equation and the Korteweg-de Vries equation, which have wide applicability. Chapter 7 presents a discussion of higher-order interactions, quartets in particular, with applications in Taylor-Couette flow between rotating cylinders and Rayleigh-Bénard convection. Finally, in chapter 8, the author indicates some open questions on shear-flow instability and transition to turbulence, together with some of his own thoughts on how these problems should be approached.

This research monograph summarises and reviews the large body of work on wave interactions that has accumulated over 25 years of intensive study. No effort is made to present the material in a pedagogical way. Accordingly, this is definitely not an elementary textbook; it could serve as a reference for researchers working in the field of wave interactions or in an advanced graduate course. I enjoyed reading this book; the discussion of the topics that I am familiar with served as a refreshing, integrated review; the rest of the discussion outlines the important developments and gives enough references to get someone started who is seriously interested in exploring the subject in more detail.

1.4. Review by: David J Benney.
SIAM Review 29 (4) (1987), 648.

This monograph is a worthy addition to the Cambridge series on mechanics and applied mathematics.

While there exist several books on nonlinear waves and instabilities, this one is written for the research worker in fluid mechanics. The field is a vast one and the text relatively short. Many issues are left untouched. The emphasis and choice of material clearly reflects the interests of the author. For example, parallel flow instabilities rather than convection phenomena are stressed. In common with most modern texts, we find extensive references but very little ordering of their relative importance.

Students and faculty alike will find Dr Craik's book to be a useful reference.

1.5. Review by: Kewal K Puri.
Mathematical Reviews MR0952373 (89j:76001).

This monograph deals with the phenomenon of weakly nonlinear wave interaction and the associated topic of weakly nonlinear stability with a focus on water waves and shear flows. It is an attempt to unify the significant available results of topical interest published in various research journals to 1983. Being one of the major contributors to these studies, the author is eminently successful in providing, as he professes in the introduction, "a full account of the state of the art''.

Following the introduction in Chapter One, the book presents a review of the linear theory of wave instability in Chapter Two. The treatment, using modern concepts and based on nearly 140 (mostly) journal and book references, covers an enormous amount of material. The notion of energy, for example, is related to pseudo-energy and instability is shown to result from the coupling of negative and positive energy. In Chapter Three, the author sets out the general methodology required for weakly nonlinear problems. Chapter Four carries discussion of the wave-driven mean flows and wave-mean flow interaction within the framework of channel flows and stratified shear flows. In Chapter Five, the author develops the concept of three-wave resonance and provides discussion of the situation arising out of coupling of waves of differing energy signs. Nonlinear wave trains and the treatment of three- and four-wave interactions form the subject matter of the last two chapters.

The treatment in the book is very representative of the current research and is thought-provoking though rather abbreviated. The reader needs fair background in fluid stability theory to enjoy it. One of the treasured aspects of the book is the thirty pages of references (nearly 800) that a serious reader may use to develop his or her ideas further on any topic covered in the book.

1.6. Review by: Gabriel T Csanady.
Limnology and Oceanography 32 (5) (1987), 1177.

This book describes a fascinating variety of mathematical models relating to the hydrodynamic instability of shear flow and to wavelike motions in stationary and moving fluids, both with and without stratification. Subtle concepts such as critical layers, over-reflection, or generalised Lagrangian mean equations are introduced in a relatively simple manner. Theoretically inclined oceanographers will find much material of interest in this monograph by a well known contributor to the development of geophysical fluid dynamics.

Although the jacket claims that the book is a "coherent account of theory and experiment on wave-interaction phenomena," there is very little reference to experiment and certainly no connection to field observations. To readers of this journal the topic of Langmuir circulations would perhaps be of greatest interest - a topic on which Craik has written several papers, some jointly with Leibovich. This work is briefly summarized in the book, Langmuir circulations being vaguely attributed to the instability of Stokes drift (mean Lagrangian motion associated with surface waves). An old photograph of windrows, originally taken by A Woodcock, shown by Stommel (1951), is included. This figure has the distinction of being the only one in the book depicting a phenomenon observable in nature (there are a few good laboratory photographs reproduced, however). There is no discussion of observed characteristics of windrows, nor of how they might relate to the instability theory.

This is not meant to be criticism, only information for the readers of this journal. The book under review is good theory, but theory only.

1.7. Review by: Vinod Krishan.
Bulletin of the Astronomical Society of India 15 (2) (1987), 166-167.

What are the wild waves saying? Waves scatter and waves gather; waves grow and waves decay; waves induce flows and flows produce waves. It is this intimacy between the fluid flows and wave interactions that is revealed ruthlessly in this book.

The book consists of eight chapters, each more complex than the previous one. The author deals with the Navier-Stokes equation in its entirety. Stability of several configurations of the density and the velocity of the fluid flow under varied boundary conditions is studied Of particular interest are the investigations of the Rayleigh-Taylor and Kelvin-Helmholtz instabilities which play an important role in space and astrophysical situations like in comet tails, extragalactic jets and supernovae atmospheres, though the author does not mention this. It is in chapter four that the wave-driven mean flows which facilitate the transfer of mass and momentum are discussed in the presence of deformable boundaries.

The next four chapters provide a fascinating description of the nonlinear dynamics -of multiple-wave system. The waves come in triads, in quartets and in quintets. Equations for conservative and non-conservative interaction of capillary gravity waves in inviscid and shear flows have been set up and solved in three and four dimensions. Author's contributions to analytic solutions in particular cases where all the three waves are linearly damped or amplified will go a long way in studying related problems in other fields.

Next, one comes face to face with the naked complexities of the interaction of nonlinear surface and interfacial wave-packets. Nonlinear Schrödinger equation with real coefficients describes amplitude modulation in time and two space coordinates. Some of the solutions represent entities called solitons which propagate unchanged in form and survive intact any interaction with other solitons. Nonlinear Schrödinger equation with complex coefficients is amenable to soliton solution, to sideband modulation and to bursting solutions. Discussion of type I and type II instabilities of linearly stable waves in Kelvin-Helmholtz flow can be put to good use in many other physical situations. Not only waves, but rolls in Rayleigh-Bénard convection and vortices in Taylor-Couette flow also undergo triple interactions to the generation of azimuthal and doubly periodic and chaotic flows.

Well, in general wave-wave interactions processes produce amplitude modulation. frequency modulation and spectral transfer. Wave interactions lead to a transfer of energy from unobservable to observable spectral range. This is manifested in many astrophysical situations where the localised longitudinal electron plasma waves undergo three-wave scattering and are transformed into transverse waves detectable with terrestrial telescopes.

It is in this sense that the similarity of the form of equations, describing multi-wave interactions in different fields like laboratory plasmas, fluid mechanics and astrophysical plasmas, can be exploited to use the methodology described so well in this book. Inclusion of examples from these different fields would have stimulated many more workers to take the plunge in. the nonlinear studies. This book serves well only the already initiated which is true of any advanced level monograph for "the books give not wisdom, where none was before, but where some is, there reading makes it more."

2.1. From the Preface.

A few years ago, in the Wren Library of Trinity College, Cambridge, I came across a remarkable but then little-known album of pencil and watercolour portraits. The artist of most (perhaps all) was Thomas Charles Wageman. Created during 1829-1852, these portraits are of pupils of the famous mathematical tutor William Hopkins. Though I knew much about several of the subjects, the names of others were then unknown to me. I was prompted to discover more about them all, and gradually this interest evolved into the present book. The project has expanded naturally to describe the Cambridge educational milieu of the time, the work of William Hopkins, and the later achievements of his pupils and their contemporaries. As I have taught applied mathematics in a British university for forty years, during a time of rapid change, the struggles to implement and to resist reform in mid-nineteenth-century Cambridge struck a chord of recognition. So, too, did debates about academic standards of honours degrees. And my own experiences, as a graduate of a Scottish university who proceeded to Cambridge for postgraduate work, gave me a particular interest in those Scots and Irish students who did much the same more than a hundred years earlier. As a mathematician, I sometimes felt frustrated at having to suppress virtually all of the fine mathematics associated with this period: but to have included such technical material would have made this a very different book. Despite this limitation, I hope that I have managed to convey something of the intellectual ferment and stunning achievements of the age. In the course of researching a work of such wide range, I have benefited much from the writings of others, as the large bibliography attests: my debt to them is obvious. To those who provided more direct assistance or useful comments (sometimes without realising it) I am especially grateful. At my home base of St Andrews University, they are Peter Lindsay, John O'Connor, Eric Priest and Edmund Robertson of the School of Mathematics and Statistics; and the staff, past and present, of the Special Collections Department of St Andrews University Library, especially Robert Smart, Norman Reid, Christine Gascoigne, Moira Mackenzie and Cilla Jackson. ... I am particularly indebted to my wife, Elizabeth Craik, both for her constant support and encouragement, and for nobly reading a complete draft of this work and saving me from many infelicities of expression and arrangement.

2.2. Review by: Karen Hunger Parshall.
Isis 100 (3) (2009), 669-670.

With mathematics effectively serving as the gateway to honours degrees as well as to fellowships at the various Cambridge colleges and to a variety of desirable careers, a high place in the widely publicised rank order of merit - a high wranglership - opened many doors in, especially, mid-century Victorian Britain. It is perhaps ironic, then, that not the college teaching staffs but, rather, private coaches - paid out of pocket and in addition to college fees - served to prepare students for the Mathematical Tripos. Mr Hopkins' Men uses the career of the most successful of those coaches, William Hopkins (1793-1866), as a lens through which to view the spread and development of mathematics in the British Empire in the middle of the nineteenth century. The book is divided into two parts, the first provides a general overview of Cambridge University: its student life; its instructional practices, including the practice of coaching; its curriculum; its emphasis on the Mathematical Tripos ...

In the book's second part, the wranglers themselves - primarily but not exclusively those with some connection to Hopkins - take centre stage. They are loosely arranged according to whether they made their careers in academia or in the clergy as well as according to geography - that is, whether they found themselves in England, Scotland, Ireland, Australia, India, or Africa. While much valuable information on the life stories of these many wranglers is gathered together here for the first time, the exposition ultimately provides very little beyond the purely biographical.

Much work has clearly gone into Mr Hopkins' Men, a book that will serve as a resource for those interested both in the role of mathematics in nineteenth-century Cambridge and in the lives of many of those who so successfully gamed the Cambridge system. Ultimately, though, questions about the real influence of the wranglers as a collective in transforming mathematics education in the British Empire, in shaping the overall research output of Great Britain, and in forming a British mathematical community remain open.

2.3. Review by: Ivor Grattan-Guinness.
The Mathematical Gazette 94 (530) (2010), 358-359.

One of the major features of higher education in the 19th century was the Mathematical Tripos at Cambridge University. In this notoriously hard competition, the student competed in the final examination for a high place in the order of merit of Wranglers (and also a battle among the oarsman for the lowest mark, to gain 'the wooden spoon'). The training was hot-house in the extreme, and it led to the growth of the coach, who ground the theories into the students with relentless efficiency; they were more important to student success than the college non-coaching Fellows.

In his recent book Masters of theory (2003) Andrew Warwick described many features of the workings of the Tripos and some of the researches of the leading Wranglers up to the 1920s. This book is a partial complement, for the author describes the general environment in Cambridge, the teaching and coaching systems, and the later careers of many Wranglers. The common factor is William Hopkins (1793-1866), Fellow of Peterhouse, and much more importantly the leading coach from the late 1820s until his retirement in 1860. His research record is modest (though unusual, focussing upon mathematical geology), but he trained several students who later enjoyed illustrious scientific careers; most notably, Cayley, Adams, Sylvester, Stokes, W Thomson (Kelvin), Tait and Maxwell, and also respectable figures such as Kelland, Todhunter, O'Brien, R L Ellis and a certain Darwin.

The first part of the book comprises three chapters on Cambridge life and then two more introducing Hopkins and his principal Wranglers. A noteworthy feature of the book is an album of 42 quality portraits of high Wranglers that Hopkins commissioned; the author discovered them in Trinity College, and reproduces them all in colour.

The second part of the book starts with four chapters reviewing the careers and other interests (especially religious ones) of Wranglers. A few concentrated on research and/or teaching in mathematics and science at Cambridge; several worked at other universities or military colleges in Britain or in the colonies. Some made their careers elsewhere, especially in the church or in law.

The final two chapters comprise an informative non-technical survey of the very impressive range of achievements in mechanics and mathematical physics due to former Wranglers and figures from other institutions, especially Scottish and Irish ones. Here the unity of the book slips a little, as several of the figures had no Hopkins connection, some not with Cambridge. Further, the remarks about the decay of British universities today in the closing chapter do not seem to fit in. A few misattributions have crept in; for example, a probabilistic element in Legendre's introduction of the least-squares criterion, group velocity to Stokes, and the four colour problem to Cayley.

The author has researched very well, not only in the published primary and secondary literature but also in various archives. His long bibliography is followed by excellent indices; nearly 50 pages of end matter. He has produced an impressive and attractive book - and his publisher has matched him in the quality of the reproductions and indeed in the book as a whole, which even carries a ribbon bookmark. Overall this book and Warwick's shed much light on the Tripos, its context and consequences.

2.4. Review by: Philip D Straffin.
The American Mathematical Monthly 116 (3) (2009), 284-288.

One of the most charming results of Hopkins' interest in his students is an album of pencil and watercolour portraits of 42 of Hopkins' top wranglers from the period 1829-1852, commissioned by Hopkins from the artist Thomas Charles Wageman. Indeed, it was Craik's discovery of this album in the Wren Library of Trinity College that led to his interest in finding out more about the subjects of the portraits, and eventually to the present book. The portraits are reproduced in full colour in the book. They probably contributed to the exorbitant price of the hardcover edition, but they are wonderful: such earnest young men, and such complete individuals. My favourite is the young Arthur Cayley, looking both elegant and cherubic, although Charles Bristed later wrote:

Our Trinity Senior Wrangler ... was a crooked little man, in no respect a beauty, and not in the least a beau. On the day of his triumph, when he was to receive his hard-earned honours in the Senate House, some of his friends combined their energies to dress him, and put him to rights properly, so that his appearance might not be altogether unworthy of his exploits and his College.

Craik also includes a colour engraving of Cayley receiving his degree as senior wrangler from the vice-chancellor in the Senate House, with Hopkins bearing the mace.

Hopkins' students transformed British mathematics and mathematical physics. They also contributed to mathematical teaching at Cambridge. G. G. Stokes assumed the Lucasian Chair in 1849, and John Couch Adams the Lowndian Chair in 1859. (I am proud that both were fellows of Pembroke, my Cambridge College.) Stokes' lectures on fluid mechanics established Cambridge as the pre-eminent centre in the field. Adams was one of the first Cambridge professors to allow women to attend his lectures. They were later joined by James Clerk Maxwell as Cavendish Professor of Experimental Physics, and Cayley as the first Sadlerian Professor of Pure Mathematics in 1863. 1 have to report that Cayley was regarded as a poor lecturer, and "only a few particularly well-motivated students attended his lectures." Cambridge gradually introduced separate Tripos exams in other subjects, and in the 1880s a variety of university lectureships. By the end of the century it was firmly established as Britain's leading provider of advanced education. The tradition of private tutors in mathematics hung on until Hardy's reform in 1910. The success of the methods used by Hopkins and the best other private tutors led to the college tutorial system of individual instruction, which remains a strength of modern Cambridge.

The second half of the book gives short biographies of many of Hopkins' high wranglers, and traces their influence in British science, education, the church, and the British colonies. Green, Adams, and Stokes receive special attention, but there are also accounts of wranglers who became lawyers, politicians, and churchmen. Did their mathematical training prepare them well for these other careers? Perhaps the most colourful case study is J W Colenso (2nd wrangler in 1836), who became Bishop of Natal, where he established a printing press and prepared and published a Zulu-English dictionary, a Zulu grammar, and a revised Zulu translation of the Gospel of Matthew. He also espoused a number of ideas that were revolutionary for the time, some of which might be traced to his mathematical training. For example, he argued against the infallible divine origin of the Pentateuch by calculating that not all species of animals could have fit into Noah's ark. Colenso was eventually deposed from his bishopric and excommunicated for heresy, but he became a master of church legal procedures and was reinstated. His supporter Lord Selbourne evaluated Colenso as "a man of fine presence, a famous Cambridge mathematician, with considerable force of character. These, as far as I know, were his only qualifications for the office of Bishop; though it may be added, to his praise, that, when he filled it, he was zealous for justice to the native races of South Africa."

I should warn you that Craik's book is not uniformly easy to read. The writing is quite dry. There are long lists of positions held by Hopkins' wranglers, and little sense of the personalities of the characters, even Hopkins. The organisation can feel haphazard, with questions which arise naturally in one section possibly answered a hundred pages away. There is no compelling narrative. Nevertheless, there are many riches in the book, even if you are not directly interested in the history of Cambridge University or the rise of British mathematics and science in the 19th century. There are certainly many good stories about good mathematicians. Beyond that, nineteenth-century Cambridge was an extreme case study in pedagogical questions that are still current today. Have you told your students that mathematics is excellent training for the mind, no matter what their ultimate field of interest may be? Do you wonder about the role of examinations in motivating or assessing learning? Do you debate the role of lectures, individual attention, small group work, and problem solving in mathematical learning? Are you interested in the relationship of mathematics and religion? Do you feel that your university is not doing enough for students and is mired in conservative bureaucracy, but still have hope for its ultimate greatness? You will find stimulation here. And you should certainly look at those wonderful pictures of bright young students.

2.5. Review by: Jeremy Gray.
Mathematical Reviews MR2327402 (2008c:01009)

Craik takes an admirably broad view of his theme, mixed with a sharp eye for detail and a deft way of weaving information into a larger argument. His opening chapters sketch the story of the university in the first half of the 19th century. Then we meet Hopkins himself, and his views on education. He was interested in science in its many forms, notably geology, and an advocate of specialised mathematical education rather than the old-style Cambridge instruction that fitted people mostly for the Church. His real gift, however, was for cramming technical mastery into willing students. Then Craik introduces us to some of Hopkins' most conspicuous successes, and in Part II of the book we follow the careers of Hopkins' Wranglers. A few pursued careers in science, notably Adams and Stokes; Green had done his best work before ever going up to Cambridge and was advised by Hopkins but never coached by him. More went into the Church and went out to India, Africa, or Australia as educators; Craik considers a dozen of these. Finally Craik considers the Wranglers' influence on the creation of a research community, not just in England, but in Scotland, Ireland, and across the Empire.

It is evident that Cambridge in the period considered was in transition from one social role to another. Craik is particularly sensitive to the place of the Christian Church in Cambridge life, and thence to the shifting views on the purpose of a Cambridge education. This emerges very naturally from his examination of the careers of so many of Hopkins' men, as well as numerous other figures, and as a result his book is a significant addition to the study of the social history of mathematics. It is also accompanied by a wonderful selection of pictures of people, many in colour.

Other forces were of course at work reforming Cambridge, from individual professors to Royal Commissions. It was recognised that the undergraduate degree did not encourage the idea that research could be done in mathematics, and the Smith's Prize was promoted to meet that need. Many distinguished natural scientists at Cambridge missed being Senior Wrangler, pipped by a future lawyer or bishop, but took the Smith's Prize a year or so later. Tensions developed later in the century between the needs of mathematicians and physicists, and in the early years of the 20th century the system of Wranglers was abolished because it had become seen to be inimical to deep mathematical understanding. This is very well described in Andrew Warwick's book Masters of theory: Cambridge and the rise of mathematical physics [2003] but that book concentrates on the scientists. Craik's account of the first half of the 19th century is original and highly informative on the education of mathematicians.