This Part of J.F.M. has been planned as a salutation to Sydney Goldstein on the occasion of his retirement from the Gordon McKay Chair at Harvard. The readiness with which the editors of J.F.M. agreed to set aside this part in this way is a mark of the esteem in which S. G. is held throughout the world of fluid mechanics.

The boundary-layer flow just upstream of the trailing edge of a flat plate is studied when a free streamline is attached to the edge. The separation at the edge occurs with an infinitely favourable pressure gradient and is characterized by a skin friction which is proportional to the inverse eighth power of the distance from the edge. The proportionality factor for the first-order term is independent of the upstream boundary-layer flow. The streamwise velocity profile at separation is non-analytic near the wall Y = 0, and starts with the term $Y^{\frac{2}{3}}$.

The method of matched asymptotic expansions is used to simplify calculations of noise produced by aerodynamic flows involving small perturbations of a stream of non-negligible subsonic Mach number. This technique is restricted to problems for which the dimensionless frequency ε, defined as ωb/a0, is small, ω being the circular frequency, b the typical body dimension, and a0 the speed of sound. By combining Lorentz and Galilean transformations, the problem is transformed to a space in which the approximation appropriate to the inner region is found to be incompressible flow and that appropriate to the outer, classical acoustics. This approximation for the inner region is the unsteady counterpart of the Prandtl-Glauert transformation, but is not identical to use of that transformation in a straightforward quasi-steady manner. For wings in oscillatory motion, it is the same approximation as was given by Miles (1950).

To illustrate the technique, two examples are treated, one involving a pulsating cylinder in a stream, the other the impinging of plane sound waves upon an elliptical wing in a stream.

It is shown that the expansion assumed by Stewartson to describe the flow close to separation in a compressible boundary layer is incomplete. When the wall is cold an infinity of new terms involving log ξ, log log ξ and their products and quotients must be added at each algebraic stage. The skin friction then vanishes like x½ ln x where x is the distance to separation. None of the coefficients of the logarithmic terms are arbitrary and in particular the first two terms in the expansion of the skin friction are known if the heat transfer is given at separation. Convergence is so slow, however, that this is of no practical value.

Initially, this study arose from questions concerning the scattering of tsunamis as they propagate over the irregular topography of the deep waters of the ocean. The mathematical problem to which this led is pertinent to many other phenomena, however, and we direct the analysis, here, to the propagation of gravity waves over an irregular bottom topography and to the lateral oscillations of an elastic string whose ends undergo random longitudinal displacements. Several facets of the mathematical problem are rather fascinating but the results do suggest that scattering is not the most important part of the tsunami propagation.

This paper attempts to emulate the great study by Goldstein (1929) ‘On the vortex wake of a screw propeller’, by looking for a dynamical theory of how another type of propulsion system has evolved towards ever higher performance. An ‘undulatory’ mode of animal propulsion in water is rather common among invertebrates, and this paper offers a preliminary quantitative analysis of how a series of modifications of that basic undulatory mode, found in the vertebrates (and especially in the fishes), tends to improve speed and hydromechanical efficiency.

Posterior lateral compression is the most important of these. It is studied first in ‘pure anguilliform’ (eel-like) motion of fishes whose posterior cross-sections are laterally compressed, although maintaining their depth (while the body tapers) by means of long continuous dorsal and ventral fins all the way to a vertical ‘trailing edge’. Lateral motion of such a cross-section produces a large and immediate exchange of momentum with a considerable ‘virtual mass’ of water near it.

In § 2, ‘elongated-body theory’ (an extended version of inviscid slender-body theory) is developed in detail for pure anguilliform motion and subjected to several careful checks and critical studies. Provided that longitudinal variation of cross-sectional properties is slow on a scale of the cross-sectional depth s (say, if the wavelength of significant harmonic components of that variation exceeds 5s), the basic approach is applicable and lateral water momentum per unit length is closely proportional to the square of the local cross-section depth.

The vertical trailing edge can be thought of as acting with a lateral force on the wake through lateral water momentum shed as the fish moves on. The fish's mean rate of working is the mean product of this lateral force with the lateral component of trailing-edge movement, and is enhanced by the virtual-mass effect, which makes for good correlation between lateral movement and local water momentum. The mean rate of shedding of energy of lateral water motions into the vortex wake represents the wasted element in this mean rate of working, and it is from the difference of these two rates that thrust and efficiency can best be calculated.

Section 3, still from the standpoint of inviscid theory, studies the effect of any development of discrete dorsal and ventral fins, through calculations on vortex sheets shed by fins. A multiplicity of discrete dorsal (or ventral) fins might be thought to destroy the slow variation of cross-sectional properties on which elongated-body theory depends, but the vortex sheets filling the gaps between them are shown to maintain continuity rather effectively, avoiding thrust reduction and permitting a slight decrease in drag.

Further advantage may accrue from a modification of such a system in which (while essentially anguilliform movement is retained) the anterior dorsal and ventral fins become the only prominent ones. Vortex sheets in the gaps between them and the caudal fin may largely be reabsorbed into the caudal-fin boundary layer, without any significant increase in wasted wake energy. The mean rate of working can be improved, however, because the trailing edges of the dorsal and ventral fins do work that is not cancelled at the caudal fin's leading edge, as phase shifts destroy the correlation of that edge's lateral movement with the vortex-sheet momentum reabsorbed there.

Tentative improvements to elongated-body theory through taking into account lateral forces of viscous origin are made in §4. These add to both the momentumandenergyof the water's lateral motions, but mayreduce the efficiencyof anguilliform motion because the extra momentum at the trailing edge, resulting from forces exerted by anterior sections, is badly correlated with that edge's lateral movements. Adoption of the ‘carangiform’ mode, in which the amplitude of the basic undulation grows steeply from almost zero over the first half or even two-thirds of a fish's length to a large value at the caudal fin, avoids this difficulty.

Any movement which a fish attempts to make, however, is liable to be accompanied by ‘recoil’, that is, by extra movements of pure translation and rotation required for overall conservation of momentum and angular momentum. These recoil movements, a potentially serious source of thrust and efficiency loss in carangiform motion, are calculated in § 4, which shows how they are minimized with the right distribution of total inertia (the sum of fish mass and the water's virtual mass). It seems to be no coincidence that carangiform motion goes always with a long anterior region of high depth (possessing a substantial moment of total inertia) and a region of greatly reduced depth just before the caudal fin.

The theory suggests (§5) that reduction of caudal-fin area in relation to depth by development of a caudal fin into a herring-like ‘pair of highly sweptback wings’ should reduce drag without significant loss of thrust. The same effect can be expected (although elongated-body theory ceases to be applicable) from widening of the wing pair (sweepback reduction). That line of development of the carangiform mode in many of the Percomorphi leads towards the lunate tail, a culminating point in the enhancement of speed and propulsive efficiency which has been reached also along some quite different lines of evolution.

A beginning in the analysis of its advantages is made here using a ‘twodimensional’ linearized theory. Movements of any horizontal section of caudal fin, with yaw angle fluctuating in phase with its velocity of lateral translation, are studied for different positions of the yawing axis. The wasted energy in the wake has a sharp minimum when that axis is at the ‘three-quarter-chord point’, but rate of working increases somewhat for axis positions distal to that. Something like an optimum regarding efficiency, thrust and the proportion of thrust derived from suction at the section's rounded leading edge is found when the yawing axis is along the trailing edge.

This leads on the present over-simplified theory to the suggestion that a hydromechanically advantageous configuration has the leading edge bowed forward but the trailing edge straight. Finally, there is a brief discussion of possible future work, taking three-dimensional and non-linear effects into account, that might throw light on the commonness of a trailing edge that is itself slightly bowed forward among the fastest marine animals.

An analytical method of calculating the body motion was given in an earlier paper. Viscosity and surface tension were neglected, and the equations of motion were linearized. It was found that, for a half-immersed horizontal circular cylinder of radius a, the vertical motion at time τ(a/g)½ is described by the functions h1(τ) (for an initial velocity) and h2(τ) (for an initial displacement) where \begin{eqnarray*} h_1(\tau) &=& \frac{1}{2\pi}\int_{-{\infty}}^{\infty}\frac{e^{-iu\tau}du}{1-\frac{1}{4} \pi u^2(1+\Lambda(u))}\\ {\rm and}\qquad\qquad\qquad h_2(\tau) &=& -\frac{1}{8}i\int_{-\infty}^{\infty}\frac{u(1+\Lambda(u))e^{-iu\tau}du}{1-\frac{1}{4}\pi u^2(1+\Lambda (u))}. \end{eqnarray*} The function ∧(u) in these integrals is the force coefficient which describes the action of the fluid on the body in a forced periodic motion of angular frequency u(g/a)½. To determine ∧(u) for any one value of u an infinite system of linear equations must be solved.

In the present paper a numerical study is made of the functions h1(τ) and h2(τ). The integrals defining h1(τ) and h2(τ) are not immediately suitable for numerical integration, for small τ because the integrands decrease slowly as u increases, for large τ because of the oscillatory factor e−iur. It is shown how these difficulties can be overcome by using the properties of ∧(u) in the complex u-plane. It is found that after an initial stage the motion of the body is closely approximated by a damped harmonic oscillatory motion, except during a final stage of decay when the motion is non-oscillatory and the amplitude is very small. It is noteworthy that the motion of the body can be found accurately, although little can be said about the wave motion in the fluid.

Many situations of practical importance both in fluid mechanics and elsewhere are governed by perturbed forms of conservation laws. Generally the perturbations are in the nature of positive dissipation terms in the sense that any initial disturbance from a uniform state ultimately decays to that state. Diverse examples of these are discussed briefly.

A situation in which the perturbation results naturally in a negative dissipation term, in the sense that initial disturbances grow, although not necessarily indefinitely, arises in what has been accepted for a model for the Gunn (1963) effect and other so-called bulk negative resistance effects in semiconductors. The Gunn effect, which is of immense importance in electron-device technology (comparable with transistors), is the appearance of coherent microwave current oscillations in the crystals of a suitable semiconductor, in particular Gallium Arsenide, when they are subjected to a large electric field generally of the order of several kilovolts per centimetre. It now seems to be accepted that the effect is a consequence of the negative resistance (that is the electron drift speed decreases with increasing electric field) properties of the semiconductor crystal.

A typical model with negative resistance properties is described in detail, the resulting perturbed (both singularly and otherwise) non-linear conservation equations ((2.19) and (2.21)) are studied for practical situations of interest and the physical implications discussed in the light of experimental facts. Particular care is given to the shocks or discontinuities that must appear in the solutions when the diffusion is zero. As a result of these comparisons with experiment a simpler model is suggested which should suffice for a large number of practical situations and various quantitative features of this model are given.

Many numerical integrations support Goldstein's theory of the structure of the solution of a laminar boundary layer near the point of separation O when the mainstream is prescribed, and in particular confirm that the solution is singular there. The existence of the singularity, however, implies that the hypotheses of the boundary layer break down in the neighbourhood of O, and it has been suggested that the disturbance to the mainstream near O is sufficient to smooth out the singularity and enable the solution to pass over into another conventional boundary layer downstream of O containing a region of reversed flow. The aim of this paper is to explore this possibility in detail using the methods of the triple-deck, developed by the author and others, which have proved successful in somewhat related problems.

Granted the hypothesis that the interaction between the boundary layer and the mainstream is significant near separation and manifests itself through a triple deck, it is found that its streamwise extent is O(ε2l) where ε−8 is a characteristic Reynolds number, ε [Lt ] 1, and l a characteristic length of the problem. The upper deck is of width O(ε2l), lies entirely outside the boundary layer, and in it the flow is inviscid. The main deck is of width O(ε4l) and constitutes the majority of the boundary layer near O, and the perturbations in the velocity are largely inviscid. Finally, the lower deck is of lateral extent $O(\epsilon^{\frac{9}{2}}l)$ and is controlled by a linear equation of boundary-layer type. The whole structure is found to be consistent provided a certain integro-differential equation can be solved, which takes different forms according as the mainstream is supersonic or subsonic. When the mainstream is subsonic it is found that there is no solution to this equation that is sufficiently smooth on the downstream side of the triple deck. When the mainstream is supersonic it is found that the triple deck can at best postpone the breakdown of the assumed structure which still must occur within a distance O(ε2l) of O.

It is concluded that the singularity is not removable by the methods proposed and it is inferred that the singularity is a real phenomenon terminating the flow which, at high Reynolds number, exists upstream of O.

Goldstein's expansion of the Oseen drag of a sphere in powers of Reynolds number is extended to 24 terms by computer. The convergence is found to be limited by a simple pole at R = − 4·18172. The series is recast using an Euler transformation and other devices to yield accurate results for large R.

In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.