Richard E. Meyer was a mathematical physicist who specialized in the physics of fluid motion. His research career began with his doctorate at the Swiss Federal Institute of Technology, followed by a brief period of employment with the English Ministry of Aircraft Production. He then went to the University of Manchester, where he made his first major research contributions. In 1953 he left Manchester for the University of Sydney. By this time he was established as a theoretical supersonic aerodynamicist and he continued with this work as well as assuming the responsibilities of a research group leader. In 1957 he went to the USA and remained there for the rest of his life, essentially abandoning supersonic aerodynamics in favour of water-wave theory. His work was marked by an ability to analyse the approach to limiting conditions, or singularities, in models of physical processes. From the 1970s, he focused increasingly on developing the mathematical aspects of his work.
Born in Berlin of traditional German parents who believed children were to be seen and not heard, Richard Meyer benefited from a very comfortable existence without the abundant affection of more contemporary parents. Born on 23 March 1919, he followed four pre-world war babies and was joined soon by his younger sister Karen. He was perhaps closest intellectually and emotionally to his brother Max Ludwig whose career was also quite impressive, but also to his younger sister whose journalistic career was very notable. It was Karen who later introduced Richard to his beloved wife, Ilse.
Richard excelled at school, becoming fluent in French and English as well as his native German, and well-versed in the classics as he specialized in Greek and Latin in his early studies. (He later also became competent in Italian and Spanish.) He studied at the Collège Français as his father had before him. He later entertained his children with tales of how he fought boredom (playing bridge with pals, for instance) in classes that were evidently not sufficiently fast-paced for his superior abilities.
As Nazi repression of Jews became steadily more severe in the 1930s, Richard’s mother Hedwig and authoritarian father Georg, who specialized in electrical engineering (publishing books and articles on fuses and switch-gears in the early 1900s), left Germany. They settled in England, where other members of the Meyer family ran a successful engineering business, and spent their last years in Slough. Richard’s father wrote him occasional letters and essays through December 1947, and these reveal a love for his son and for his son’s intellect that Georg was not as skilled at sharing in person.
Richard went to Switzerland in October 1937 to pursue his tertiary studies, attending the Eidgenössische Technische Hochschule (ETH, the Swiss Federal Institute of Technology) in Zürich and earning his diploma of mechanical engineering after four years’ study. He subsequently held an appointment asTutorialAssistant to the Chair of Descriptive Geometry before joining the scientific staff of his doctoral adviser, Prof. Dr. Jakob Ackeret, at the Institute forAerodynamics in April 1943. Early in the following year, he was appointed as Assistant at the Institute. His doctoral thesis was entitled ‘Beitrag zur Theorie feststehender Schaufelgitter’ (‘Contribution to the Theory of Fixed Cascades’), the degree of DrScTech being awarded in February 1945. Cascades consist of an array of aerofoils, and fixed (that is, stationary) cascades are usually employed as a means of turning the flow of air passing through them.
Switzerland brought Meyer much joy as he satisfied his thirst for solving mathematical problems deemed to be unsolvable. He was a serious mountain-climber and had been attracted to Zürich by the Swiss mountains as well as by the excellent reputation of the ETH. The mountains offered the challenge of discovering the most elegant manner of reaching a summit as well as the peaceful solitude shared only with a rope and a rocky, often smooth surface (Fig. 1). Meyer was very adept at the most difficult climbs—finding handholds where there were none, agilely balancing as he climbed in a seemingly effortless glide, always further and further. During this time, he published essays on his mountain climbs. Photographs of mountains accompanied him wherever he lived. Mountain surfaces called to him to climb or ski them, depending on the season and snow: New Year’s Eve could find him hurtling down a mountain in the pitch black dark. This love of mountains probably was a trait shared by his father Georg, as the latter artistically captured mountains through compelling woodcut prints.
In a remarkable transition, Meyer left Switzerland early in 1945, a British aircraft carrying him across war-torn France to England. Here he worked for fifteen months as a Junior Scientific Officer in the Ministry of Aircraft Production before being appointed in 1946 as anAssistant Lecturer in Mathematics and Engineering at the University of Manchester. Then from 1947 to 1952 he was an ICI Research Fellow at Manchester. At that time, the University of Manchester was developing its outstanding reputation in fluid dynamics, and Meyer played a part in this by his activities in the growing area of supersonic flow.
In the late 1940s, aircraft speeds were approaching the speed of sound and it was found that this was accompanied by a rapidly increasing drag, together with loss of control of the aircraft. The phenomenon was described as the ‘Sound Barrier’. Developing theory that could cope with this flight regime proved to be extremely difficult and research tended to focus on the more tractable supersonic regime, leaving the transonic regime until supersonic flow was better understood.
The method of characteristics provided a technique for analysis of supersonic flows. Essentially, it involved following mathematically the infinitesimal ‘wavelets’ making up the flow pattern of a supersonic flow. Richard Meyer exploited this technique, using it to tackle a number of problems that broadened understanding of supersonic behaviour. The expertise that he developed in this area was such that he was able to author a review of the method of characteristics as a chapter in the high speed flow version of the standard treatise, Modern Developments in Fluid Dynamics. This work was a partner to a highly successful low speed flow version, of the same title, that was published prior to the Second World War.
The supersonic flow over a convex corner is simple enough, with the ‘wavelets’ or Mach lines making a divergent pattern, but if the corner is concave, the Mach lines converge and focus to a limit line that appears physically as a shock. Meyer discussed this effect in (4), showing how Mach line focusing occurs in other situations; for example, in the formation of cusped limit lines (or shock waves) adjacent to regions of subsonic flow. This also helped to improve understanding of transonic flows.
As Meyer pointed out in (2), the method of characteristics can also be used for unsteady compressible flows, replacing the two co-ordinates of distance by one of distance and the other of time. For instance, this is the basis of the theory of the shock tube, which is used in many branches of science. In (7), he used the method of characteristics to consider the flow patterns produced by two pistons in a tube, one of which is advancing and the other retreating, paying attention to the formation of limit lines.This is applicable to the launching of a projectile in a tube or, by translating into supersonic flow variables, to the flow in a supersonic engine intake.
In (8), he treated the cooling effect of air injection in the boundary layer on a flat plate. This was an early introduction to today’s widely used technique of protecting the blades of the turbine disc of a jet engine. These blades are rows of aerofoils (that is, a cascade) designed to extract energy from the hot flows produced by the combustion chambers of a jet engine, and the more effective the protective cooling of the turbine blades, the higher the temperatures at which the engine can be operated and the greater its efficiency. As part of the research on cascades, this activity can be seen as a brief return to the broad subject of his doctoral thesis.
In (3), he drew attention to some of the effects of focusing of Mach lines on the axis of an axisymmetric supersonic flow. These effects had a decisive influence on the design of nozzles for producing uniform test flows in supersonic wind tunnels, as they implied that the small machining errors in manufacturing the contoured shape of an axisymmetric nozzle would produce disturbances on the axis of the nozzle that were greatly amplified, and thereby produce non-uniformity at the centre of the test flow. Largely for this reason, supersonic wind tunnel nozzles are usually not axisymmetric but consist of two plane, parallel walls, with two contoured nozzle liners mounted between them. But it is worth noting that this is not satisfactory for hypersonic wind tunnels, due to excessive growth of the boundary layer on the plane walls. It is therefore necessary to use axisymmetric nozzles for hypersonic wind tunnels, accepting the attendant difficulty of minimizing the production of flow non-uniformities on the nozzle axis that the focusing effect implies.
Other features associated with axisymmetric supersonic flow were treated in (6). This considers the flow in a supersonic axisymmetric jet discharging from an orifice or nozzle into low-pressure surroundings. In principle, the flow structure could be determined by a numerical method of characteristics computation, but this would have involved laborious calculations at the time this problem was considered and so a simpler, approximate theory was devel-oped.This theory yielded the external shape of the jet, and provided a basis for determining the location of the shock wave that has been observed to occur near the lip of the jet nozzle. The same theoretical approach could be used in dealing with other configurations; for instance, the supersonic flow in an aircraft thrust augmenter or the flow in the neighbourhood of an edge in the surface of a body of revolution. Having drawn attention to the difficulties associated with the focusing effect in axisymmetric nozzle flows, Meyer turned his attention to the factors that would limit the accuracy of a test flow produced by contoured nozzle liners between plane parallel walls—what was becoming the conventional two-dimensional supersonic wind tunnel. The purpose of a supersonic wind tunnel is to produce a parallel and uniform flow in a test zone, which is rhombic in shape. In (5), it is shown that the flow divergence and uniformity can be determined from strategically located pressure measurements within the test rhombus, and that the relation between pressure and flow direction is directly connected to the contour of the nozzle liners. This relationship could be used to govern any corrections to the nozzle liners required to produce parallel and uniform flow.
Invitations to Meyer to share his research abroad resulted in a most delightful consequence. His sister Karen hosted a small dinner party in her London home to celebrate his successful trip to America. The gathering included Ilse Ruth Apt who, with her family, had also escaped from pre-war Germany, and who was now working in London at the publishing house Methuen’s. Visits to London followed. Richard and Ilse became engaged in November 1952 and married a month later, on 18 December 1952, shortly before setting out for Australia.
Richard’s and Ilse’s life adventure together started, most fittingly, on the waves, during the long sea voyage to Sydney, where Richard had been appointed to a Senior Lectureship in Aeronautics at the University of Sydney. Here their first child, their son Peter, was born in 1954. While Richard was intensely committed to his research, it is clear from photographs that he also found pure joy in sharing the flight of a kite or simply spending time with Peter.
At Sydney, Meyer introduced an undergraduate course in supersonic aerodynamics and formed a research group working in that field. This group, centred on the University’s William and Agnes Bennett Laboratory, did both experimental and theoretical research and operated a significant research supersonic wind tunnel. Richard provided leadership to the group, encouraging effective research by members of the group while maintaining his own research activity.
In (16), he went into the question of the departures from uni-directional and uniform test flow in a supersonic wind tunnel in more detail. He treated the particular case of a symmetrical nozzle with two identical nozzle liners, exploring in detail the relation between errors in the shape of the supersonic part of the nozzle liners and deviations from flow uniformity in the test rhombus. The analysis can be applied, for instance, to determining the effect on the velocity distribution of changing the Reynolds numbers on models in the test section by controlling the pressure at which air is supplied to the wind tunnel. Changing the test Reynolds number ensures that the boundary layer behaviour on the model in the test section is a proper representation of behaviour in flight. However, it changes the boundary layer on the nozzle liners and thereby alters the effective nozzle shape. The theory can also be used to determine the effect of tilting the nozzle liners to make small changes in the test section Mach number. The analysis showed that errors were mostly propagated along Mach lines. Errors, in general, accumulate along multiple reflections of Mach lines and tend to be additive if errors of the same sign occur, or cancel if errors of opposite sign occur. The theory also can be applied to unsymmetrical nozzles, either where one nozzle liner is plane, or when liners of different shape are used, a technique that can be employed to allow large changes in Mach number. This theory was tested experimentally in (22), by tilting the two nozzle liners in a symmetrical supersonic wind tunnel. The experiments confirmed that tilting had negligible effect on the subsonic and transonic part of the nozzle flow, and that the supersonic flow perturbations occurred as expected. They also showed that the interaction of the perturbations with other flow disturbances required attention.
Calculation of the effective change in the shape of the nozzle liners of a supersonic wind tunnel caused by the displacement of streamlines by the turbulent boundary layer on the nozzle liners is the subject of (13). The displacement effect in a low-speed, subsonic turbulent boundary layer could be accurately predicted, and the displacement effect in a supersonic boundary layer was obtained by applying a transformation to the subsonic boundary layer, based on the density distribution across the boundary layer, to obtain the displacement effect in a supersonic turbulent boundary layer. Though later work cast doubt on the use of this transformation for turbulent boundary layers, in this case it worked well. It indicated that the displacement effect grew linearly with distance from the beginning of the supersonic flow in the nozzle.
In the design of a supersonic wind tunnel, it is important to correctly represent the flow in the region where it passes from subsonic to supersonic flow; i.e. the transonic portion of the nozzle flow. In (9) and
(14) Meyer addressed this problem. In (9) he briefly considered the question of the shape of the contour to be assumed for the sonic line, along which the flow velocity is equal to the local speed of sound. He pointed out that the method of characteristics implied that the maximum value of the stream direction there was limited, and that this provided a useful check on the assumed shape of the contour. However, in (14), he drew attention to a hodograph method of calculating the flow in this region, due to Cherry, that superseded the methods that had been used to calculate these flows.
With the development of supersonic wind tunnels came the need for methods of measuring the aerodynamic forces experienced by models in the test flow. A widely used method of measuring the drag of aerofoils in subsonic flows involved traversing the wake of the aerofoil, marking measurements of pilot pressure and static pressure. There had been some discussion in the literature about applying this method to supersonic flows, and it had been concluded that possible sources of error were too great for the method to be of general use. In (19), Richard reported a detailed investigation of possible sources of error, concluding that some of these were self-compensating and that others could be reduced by further development of experimental technique. This work was not followed up, as the rapid development of strain gauge techniques for measuring aerodynamic forces in wind tunnels superseded wake traverse techniques.
The power levels that a supersonic wind tunnel requires are such that they are generally achieved by operating the tunnel in the ‘blowdown’mode, in which the required power levels are obtained by pumping air into a high-pressure reservoir over a period of the order of an hour, and releasing the high-pressure air to blow down through the tunnel for the order of a minute. The temperature of the reservoir air falls as the pressure drops, which would lead to a continual change in the properties of the test flow unless a means of controlling the temperature is employed. In (18) Meyer treated an economical means of doing this, involving a large thermal mass through which the air flows at essentially constant pressure. Initially, the air experiences the greatest heat transfer and temperature adjustment in the upstream layers of the thermal mass and emerges from the downstream layers close to the initial temperature of the thermal mass, thereby ensuring that the wind tunnel is supplied with constant-temperature air for the duration of the test period. The unsteady heat regenerator has also found other applications. For example, hypersonic blowdown tunnels require the air to be supplied at an elevated temperature, and this is achieved by pre-heating the large thermal mass prior to a test, ensuring that heated air at the temperature to which the thermal mass is raised will be supplied to the tunnel. Typically, a thermal mass consisting of zirconia pebbles may be used, heated by combustion gases. Another example is the use of the principle of the unsteady heat regenerator in an instrument to measure the mass flow per unit area in the test flow of a hypersonic shock tunnel, with test times of a few milliseconds. A supersonic wind tunnel is an example of supersonic nozzle flow, in which the cross-sectional area increases with downstream distance and the flow accelerates. If the cross-sectional area reduces with downstream distance, the flow decelerates and the pressure increases, to produce a supersonic diffuser flow. Such supersonic diffusers are used, for example, in a supersonic aircraft, in order to increase the pressure at which air is supplied to the engines. If the reduction in cross-sectional area and the associated pressure increase is sufficient, a ‘choking’ phenomenon occurs in which a strong shock wave moves upstream of the diffuser, and the flow in the diffuser is no longer supersonic. It was uncertain whether choking was caused by a boundary layer phenomenon or by the properties of the mainstream supersonic flow. In (10), Meyer showed by analysis that, at the moderate values of Mach number that were of interest at the time, the latter explanation applies, as small mainstream disturbances are trapped in the diffuser, leading to a major mainstream disturbance and choking of the flow. The analysis superseded an earlier attempt by a previous author, which Richard showed to be invalid. At higher values of Mach number, the mechanism analysed by Richard is overtaken by boundary layer phenomena.
In (11), he extended this theory to account for flows in which the disturbances were not so small, examining particularly the structure of the flow pattern that followed the initial wave front. The analysis covered the case of waves with cylindrical symmetry, as for the flow about a slender supersonic missile, and the case of spherical waves, as in the aftermath of an explosion. It showed how waves propagating against the flow direction generate waves that are propagating in the flow direction, which generate waves that are propagating against the flow direction, to strengthen the initial wave. The analysis helps to illuminate the character of cylindrically symmetric, and spherically symmetric, flow situations, which are particularly complicated.
Experience had shown that it was very difficult to produce a shock-free flow about a body such as an aircraft wing as the Mach number approached unity, and this resulted in sharp increases in aircraft drag at these transonic Mach numbers. In (17), Richard offered an approach to this problem, exploring the possibility of producing shock-free flow by conducting an analysis of the equations of inviscid flow using a convergent series of successive approximations based on a small-perturbations analysis. The analysis indicated that smooth, shock-free flow was theoretically possible even when the Mach number exceeded unity only slightly. This analysis was subsequently extended by Richard’s co-author to include weak shocks. Though this theoretical analysis indicated that efficient flow was achievable, the problem of designing aircraft for efficient operation near a Mach number of one was effectively solved by the use of modern computing techniques.
In supersonic or hypersonic flight, the body surface downstream of a sharp leading edge must be curved in order to enclose a finite volume. This implies that the expansion waves produced by the body must cause the shock wave from the leading edge to be curved. A rule of thumb for calculating the distribution of pressure on the body is shock expansion theory, which neglects the shock curvature. In (20), Richard developed an approximate theory that not only yielded the pressure distribution on the body, finding that shock-expansion theory yielded a good approximation, but gave other details of the flow such as the shock-induced entropy gradients and the attendant flow field vorticity.
Meyer’s impressive body of work gave him, in just a few years, a considerable reputation, evidenced by both his election in 1956 as a Fellow of the Australian Academy of Science and his appointment a few months later to a professorship in the Mathematics Department at Brown University, Rhode Island, USA (Fig. 2).
Meyer took up his new appointment in 1957. In the following year, Ilse gave birth to twin daughters, Nicole and Michele. She and Richard were strict parents—how could they not be, having grown up in prewar Germany?—but they communicated a love of learning and exploration to their children and shared a wide range of cultural experiences with them, as well as a passion for music, art, anthropology and literature. Both girls flourished in several languages and later travelled and studied abroad. Michele became an oft-published free-lance journalist, Peter a consultant, author and teacher, and Nicole a professor of French literature.
In 1964, Meyer moved to the Mathematics Department of the University of Wisconsin at Madison, where he stayed until his retirement in 1994. Ilse became one of the fixtures of the fashion world in Madison, and was well known as the owner of the successful Ilse’s Where Else boutiques. However, she considered her main professional passion to be her writing and earlier career working with the president of one of the best-known publishers in London. There she had met and worked with some of the top authors in the country.
On leaving Sydney Meyer began developing new research interests, notably in the dynamics of water waves. However, he did not make the transition suddenly, and he was occasionally to return to a problem in supersonic aerodynamics, or in gas dynamics. One such excursion is apparent in (34) where he considers the behaviour of the shock wave that forms at the tail of an axisymmetric body, such as a slender missile configuration, at supersonic and hypersonic speeds. This conical shock wave appears to originate where the wake of the body ‘necks down’ to a minimum cross-section, to form the vortex of the shock that trails downstream of the body. He analyses the situation where the Reynolds number is large, the boundary layer and wake are therefore very thin, and the Mach number is large but the angle that the shock makes with the free stream is small. Thus, he is not able to use the weak-shock approximation, but he develops an approximate solution that shows that the tail shock flow field in axisymmetric flow fields is different from that in two-dimensional (nonaxisymmetric) flows.
In (24), Meyer made an excursion into gas dynamics, looking at the flow field due to a body rising at supersonic speed through an atmosphere that is stratified (meaning that the density in the atmosphere decays with height). This could apply to the large supersonic jet accompanying the violent eruption of a volcano, or to the flow field associated with a long-range missile rising at supersonic speed through the atmosphere. The shocks and pressure waves associated with the motion of the body are represented by a point source of mass, momentum and energy, and the flow field at large distances is analysed using the acoustic approximation. Such an analysis predicts that, at a considerable height and a large distance from the source, a sudden large upward and outward displacement of the air occurs followed by an oscillatory decay of the displacement. Richard pointed out that this is due to the fact that the acoustic equations involve the mass flux as a dependent variable, rather than the density alone, and the equations show that when the height is great enough for the ambient density to be small compared with the density near zero height, the mass flux remains nearly constant but the velocity and displacements are greater, particularly near the outer skirt of the wave system.
Notwithstanding these excursions, Meyer was to make water waves his major area of research interest after leaving Sydney. This is apparent in (27), (29) and (30).
As water waves approach a beach, the smooth, sinusoidal-like wave surface turns into a wave front characterized by a sudden jump in the height of the surface. This is known as a bore, and is analogous to a shock wave in supersonic flow. In (27), the dynamics of a bore travelling shoreward into water at rest on a beach of uniform slope is investigated, paying particular attention to the level to which the bore climbs up the beach— a feature that is important, for example, in determining the level of damage caused by a tsunami. A connection with the gas dynamics of non-uniform shock propagation is observed, noting the way in which similarity solutions appear to characterize the flow. In this shallow-water problem, the characterization is connected to the shore singularity of the governing equations occasioned by the depth going to zero. An asymptotic approximation for the development of the bore near the shore is developed.
The analysis of (27) covers the case of the bore on a beach of uniform slope, and this is extended to a beach of non-uniform slope in (29). The approximate solution associated with the shore singularity still applies, and features such as the shape of the beach, or the shape of the wave forming the bore, only influence the development of the bore insofar as they determine the basic velocity scale of the bore.
As the bore climbs up the beach, it reduces in height until it collapses. What happens then is considered in (30). Initially a thin sheet of water continues the run-up, to be followed by the development of back-wash of the water. A secondary wave then forms that initially advances up the beach and finally recedes from the shore. The undular jump can be seen as a degenerate bore, combining the smoothly varying height of a water wave with the net increase in surface height of a bore. Studies previous to (39) had assumed that the undular jump involved a steady fluid-dynamic process but researchers had wondered if this was so. By conducting an analysis for small wave height, Richard was able to show that the jump was a dispersive effect, and that the length scale of the jump structure increased with time. The analysis also revealed the structure of the jump.
The equations governing the flow that precedes the formation of a bore on a beach are considered in (32).The two-dimensional motion of water on a beach of small slope, rather close to the shore, is shown to be governed by non-linear shallow-water equations, based on a small seabed slope approximation. This differed from previous work, which was usually based on a small wave amplitude approximation.
Analysis that assumes two-dimensional flow, requiring that the beach be uniform in the transverse (or longshore) direction, is strictly not realistic. Existing beaches generally exhibit a bottom contour and a shoreline both of which vary in the longshore direction.This three-dimensional problem is considered in (35), where the asymptotic equations of surf are derived for shallow beaches with geometry causing longshore variations in the water medium. These shallow-water three-dimensional non-linear equations apply in general, but when the seabed profile parallel to the shore is of gentler slope than the profile normal to the shore, the asymptotic equations are shown to differ only in minor respects from the two-dimensional surf equations. Such weakly three-dimensional surf can be analysed in terms of two-dimensional theory, and the run-up bounds of that theory can be extended accordingly.
The wave patterns further removed from the shore, but where the seabed shape still has an influence, are considered in (42). A wavefront approaching a beach at an angle experiences a reduction in the front velocity as the depth reduces towards the shore, causing the direction in which the front propagates to change and tend towards approaching the shore normal to the shoreline. Thus the reducing depth causes waves approaching a beach to become aligned parallel to the beach, regardless of the direction at which the wave propagates far from the beach. The same effect applies to waves receding from the beach and, due to the increasing depth, a wave propagating in an essentially normal direction as it leaves the shore propagates at an increasing angle, until its propagation direction is parallel to the shoreline. In the case of a beach of uniform slope with zero depth at the shoreline, it follows that waves from the shore turn continuously until they form an envelope (or caustic) where their propagation direction becomes parallel to the shoreline. Thus the waves may be ‘trapped’ in the region between the shore and the caustic, with the wave amplitude decaying with distance beyond the caustic. In (42), a theory based on the smallness of seabed slopes is developed and used to obtain the range of wave frequencies (that is, the spectra) of waves that are trapped, and those that are not trapped, for a variety of shore configurations approximating actual shapes. For example, an island and surrounding seabed is represented by a cone, with its vortex protruding beyond the surface. The wave pattern varies with the seabed slope and shore configuration, but bears no clear relation to waves incident from the ocean.
In (64), Meyer considered the reflection of waves by long undersea ridges and valleys, using an analysis that relied on small seabed slope and/or short wave length compared with the depth. This yielded a wave reflection coefficient for a range of depth distributions.
The trapping of waves implies a potential for resonance phenomena and this is explored in (60), for the case of islands, by extending the analysis to take account of energy leakage. It was found that certain types of trapped wave modes have exponentially small leakage and therefore promote resonance.
The possibility of resonance with trapped waves raised the question of energy absorption at the shore, and this effect was the subject of (65).The decay rate was determined in terms of a shore reflection coefficient for round islands and for a straight beach.This decay rate was liable to be larger than that due to energy leakage to the open sea but was still small, so the possibility of some resonance remained.
Meyer attacked the problem of the shore reflection coefficient in (77). The classical theory for waves on a beach has two solutions, and it had been assumed that only one of them was applicable. This solution predicts total wave reflection at the beach, whereas it is known that beaches are efficient wave absorbers. Meyer’s analysis identified a singularity in the underlying inviscid fluid-motion structure that allowed the difference in the behaviour of incident and reflected waves to appear. In (79), he showed how the theory so developed could be used to elaborate classical theory and thereby produce a unified theory that describes the whole shoaling process, from deep-water waves to the beach.
Meyer’s subsequent work did not involve the development of another major area of the physics of fluids and, though he made occasional excursions into other problems in physical fluid dynamics (for example in (75), (76), (82) and (84)), his later research was largely concerned with further development of mathematical aspects that had been uncovered in his earlier work.
Richard regarded teaching as an important aspect of his career as an academic, and he took pleasure in doing it well. An example of this is provided by the textbook he wrote, Introduction to Mathematical Fluid Dynamics, which also illustrates his approach to physical fluid dynamics. The book was published by Wiley-Interscience and was subsequently republished in a Dover edition. It arose from teaching a graduate course in fluid dynamics to students of mathematics, physics, engineering and meteorology and his recognition that, although the detailed fluid dynamics content varied between these areas, there was benefit in gaining an appreciation of a certain core of knowledge on the subject, concerning particularly the relation between inviscid and viscous fluids. The book shows clear signs of his approach to fluid dynamics, always remaining close to the physical phenomena whilst ensuring that the mathematics was soundly based.
Thus in the opening chapter, he combines mathematical abstraction in defining a fluid with the physical concept that ‘it is impossible to break or cut a fluid’. He then uses the conservation of mass to produce the flow patterns about a number of configurations with an incompressible, ideal (that is, inviscid) fluid, and goes on to discuss vorticity, explaining why migrating birds tend to fly in a ‘V’ formation. In the next chapter he introduces the conservation of momentum, again treating it as a mathematical abstraction before discussing some simple examples of its use. In differential form, this becomes the equation of motion for a fluid. Chapter three begins with an account of an experiment demonstrating the existence of a shear force in a fluid and develops the concept of a viscous, Newtonian fluid. The concept of viscosity thus introduced is also discussed in terms of the kinetic theory of gases. In chapter four the general nature of the flow when the viscosity is very small (as generally applies in flows of air or water) is discussed, leading to the large Reynolds number concept. The boundary layer is introduced by a singular perturbation example, and the solutions for the boundary layer on a flat plate, as well as boundary layers with a pressure gradient, are discussed. Boundary layer separation is also considered. Chapter five treats the motion of a fluid in a rotating frame of reference, which obviously applies to the fluid motion of the Earth’s atmosphere. Finally, chapter six introduces the reader to compressible flows (including supersonic aerodynamics) noting that compressibility give rise to shocks and flow motion in plane waves, features that come about because of the introduction of the thermodynamics of the flow medium. Richard was an excellent lecturer and during his career he presented more than fifty invited lectures in the UK, Europe, the USA and Australia. As an undergraduate lecturer he was stimulating, whilst thorough in the material he presented. As a postgraduate supervisor he was a gifted mentor, encouraging as well as generous with his time and ideas and successfully conveying a sense of excitement about the research. Mei-Chang Shen, who was first student, then colleague and close friend, writes as follows:
Richard was considered as one of the best teachers by many of his graduate students. He seldom used textbooks for his courses, and presented his own notes in a very comprehensible manner. After a student worked on Richard’s lecture notes and assigned references thoroughly, he could catch the deep implications of the course material. Every week Richard assigned only one somewhat difficult homework problem to motivate students to do some research on the topics just studied. In supervising PhD students, Richard always suggested problems of great interest and let them work out the details freely with his to-the-point advice. During discussion sessions, he listened attentively, then pointed out in what directions either as generalizations or specifications of the research problems they should pursue. In a word, he would like his PhD students, the future applied mathematicians, to possess the ability to choose the important research problems of current interest and carry out the research far beyond the original formulation.
As a person, Richard was, Shen writes, ‘honest, straightforward, tenacious, and candid. Certainly some people might not find it easy talking to him. However, you would appreciate very much his genuine comments and true friendship after you knew him well’. Generous and compassionate in nature, he nurtured a profound love and deep respect for Ilse, as she did for him. They died together in Madison on 6 January 2008.
This memoir was originally published in Historical Records of Australian Science, vol.21, no.1, 2010. It was written by:
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