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.