1. The form and frequency of the waves passing down the bodies of small free-living nematodes (Panagrellus, Rhabditis and Turbatrix) depend on the nature of the external medium.

  2. Observations of animals moving in such media as syrup, agar gels, and dense suspensions of particles suggest that the relationship between the speed of progression of the animal to the speed of propagation of the waves along the body, depends on the relative resistance exerted by the medium to displacement of the body in directions normal to and tangential to its own surface.

  3. When swimming in water the body of a nematode yaws periodically in a transverse plane and the axis of progression does not coincide with that of the waves; the displacement of an element of the body relative to the medium depends on its position on the body. The envelope of one complete wave exhibits two characteristic nodes.

  4. A suspended particle originally situated near the anterior end of a swimming animal moves tangentially backwards along the surface of the body, but the speed at which it does so is not constant and is always much less than that at which the waves travel relative to the ground. The average speed of a particle close to the surface of the body is about one quarter of that of the waves.

  5. When plotted relative to the ground the path of displacement of a particle exhibits characteristic loops. When viewed from above and in the direction of the animal’s progression, all particles on the left side of the body traverse their loops in a clockwise direction ; all those on the right side move anticlockwise.

  6. There is a highly characteristic pattern of circulation round the body of the animal. Water in the vicinity of a wave crest moves in the opposite direction to that of the propagation of the waves, but in the same direction as the waves when situated in a wave trough. The circulation extends for a considerable distance from the surface of the body.

  7. The flow round the body of a swimming nematode is essentially the same as that in the neighbourhood of an undulating sheet of rubber. Its analysis presents an interesting but difficult hydrodynamical problem.

Apart from the fact that their locomotory activity determines the ability of parasitic forms to penetrate the tissues of their hosts, the movement of nematodes presents certain features which are relevant to the general theory of undulatory propulsion. In the latter respect, two main types of problems arise : (i) To what extent are the form and frequency of the waves passing down the relatively long and cylindrical body determined by the physical nature of the external medium? (ii) To what extent can the speed of progression of an animal be related to the form and frequency of the waves?

That the form of the muscular waves depends on the physical properties of the medium was noted by Looss (1911, p. 420) who found that the larvae of Agchylostoma were only able to progress over the skin of the host and penetrate through a suitable crevice if the skin was moist; when freely immersed in a bulk of water the larvae ‘threshed’ actively from side to side without making much apparent headway and without being able to keep themselves in suspension by their own efforts. On the other hand, when placed on or in a damp and suitably dense suspension of faecal particles the larvae exhibited ‘quick graceful movements ‘leaving behind them well-marked sinusoidal tracks. Precisely similar phenomena have been observed in all the five types of worms used in the present work (PanagreUus, Haemonchus, Rhabditis, Strongylus, Turbatrix); all these exhibited well-marked and graceful creeping or burrowing movements in suitably dense suspensions of particles ; all of them displayed very active movements in water, but only one of them (Turbatrix) was able to support itself by its own activity (see p. 142). More recently, Wallace (1958-60) has published an extensive series of observations on the factors which control the form of the waves and the speed of progression of Heterodera through suspensions of soil particles of varying sizes and in water films of varying depth. Wallace’s observations and those described in the present paper suggest that there is no sharp line of demarcation between creeping and swimming but, in view of the well-marked contrast in the form of an animal when burrowing through a dense suspension of particles and when swimming in water, it is convenient to retain these two well-established terms.

Table 1 shows the marked difference in the form and frequency of the waves (averaged over about twenty individuals in each case) exhibited by animals when creeping—either over the surface of 12I% agar or through a suspension of starch particles—and by individuals swimming in water. The following features are characteristic : (i) The wavelength (λ)*, frequency (f), and speed of conduction over the body (Vw) are all very much less in creeping animals than in those swimming in water.

Table 1.
graphic
graphic

(ii) Although the absolute speed of progression of an animal is higher when swimming than when creeping, the relative distance moved forward during the period of a single wave is very much less, (iii) The ratio of amplitude (b) to wavelength (λ) is substantially the same during creeping and swimming.

(a) Through a suspension of starch grains

When gliding through a relatively densely packed suspension of starch grains the head of a nematode acts as a wedge operated by the forward thrust exerted by the rest of the body. It displaces the grains normally to its own surface and drills a sinusoidal channel through which the rest of the body glides tangentially (Plate 1, figs. 1-4). Under optimum conditions the speed of the animal’s forward progression is equal to the speed at which the waves of muscular contraction pass backwards relative to the head of the animal; the waves remain stationary relative to the ground and the animal leaves behind it a sinusoidal track whose wavelength and amplitude are the same as those of the muscular waves. The mechanics of the movement are identical with those of a snake gliding through a sinusoidal tube with rigid walls (Gray, 1953).

Text-fig. 1.

Haemonchut contortus creeping with zero slip over the surface of an agar gel. Tracings from successive photographs taken at approx, 13 sec. intervals. Note that the waves were stationary relative to the ground.

Text-fig. 1.

Haemonchut contortus creeping with zero slip over the surface of an agar gel. Tracings from successive photographs taken at approx, 13 sec. intervals. Note that the waves were stationary relative to the ground.

If, after their displacement by the anterior end of the animal, the starch grains are not sufficiently closely packed to resist further displacement by the forces which the rest of the body exerts against them in a direction normal to its own surface each element of the body acquires a component of motion (Vn normal to its own surface and no longer simply glides tangentially ; this is equivalent to an increase in the rate of the element’s tangential motion combined with a displacement of the element along the axis of progression ; in other words, the waves are no longer stationary but move backwards relative to the ground. If the waves move backward relative to the ground at speed Ve, the speed of the animal’s forward progression (Vx) is Vw— Vs, where Vw is the speed of the waves relative to the head; the percentage ‘slip’ of the waves relative to the ground is 100 [1-( Vx /Vw)]. As the consistency of a starch suspension is seldom uniform, it is not surprising that the amount of slip varies considerably in different parts of the suspension (Table 2). It may be noted that the wavelength (λw) of the track left behind the moving animal, relative to that of the waves (λw) passing down the body, is determined by the amount of slip ;

Table 2.

Rhabditis gliding between starch grains

Rhabditis gliding between starch grains
Rhabditis gliding between starch grains
formula

Panagrellus, Turbatrix, Haemonchus and Rhabditis all glide readily when placed on the surface of 12I% agar or gelatine. With suitably oblique illumination it can be seen that the animal incises a sinusoidal track on the surface of the jelly; in some cases the wavelength of the track is the same as that of the waves on the body, indicating a total absence of slip (Text-fig. 1). In other cases (see Table 3) appreciable slip is observed. Not infrequently the animals penetrate into the jelly, moving forwards or backwards with equal facility.

Table 3.
graphic
graphic

(c) Movement on a damp rigid surface

When a culture of Panagrellus silusiae is kept in a covered glass container, the animals creep up the walls as long as these are damp, and they display the same type of waves as on the surface of an agar jelly ; the wavelength, amplitude and frequency of the movements appear to be the same in both cases. On the other hand, the amount of slip on the surface of the glass is always very much greater than on the surface of the jelly and no trace of tracks is observed. A comparable amount of slip occurs on a damp sheet of solid anhydrous gelatine and on the surface of mercury. In all such cases there is a marked tendency for a number of animals to adhere together by a film of water and to exhibit a striking degree of co-ordinated movement (Text-fig 2).

Text-fig. 2.

Co-ordinated movement of an aggregate of PanagreUus when on a damp surface of glass. Scale 1 mm.

Text-fig. 2.

Co-ordinated movement of an aggregate of PanagreUus when on a damp surface of glass. Scale 1 mm.

These observations suggest that the ability of Panagrellus or Turbatrix to creep over a damp but smooth surface without appreciable slip only occurs when the animal is able to incise a groove on the surface of the medium. How far this is effected by active displacement of the medium by the head of the animal or how far the body is passively pressed into the medium by the surface tension of the air/water interface is uncertain. That this interface exerts a force against the animal is shown by the fact that animals in a drop of water are unable to move out of it although they can deform the surface of the drop when their heads come into contact with it. It should be noted that other species of nematodes, notably Heterodera, are able to creep over a damp rigid surface with very little slip (Wallace, 1958, 1959). It seems just possible that the failure of Panagrellus and Turbatrix is due to the presence of small traces of surface-active substances—derived from the culture medium—which change the physical properties of the water film surrounding the body.

Forces which determine the amount of slip

As explained elsewhere (Gray & Hancock, 1955) the propulsive force of an undulating organism moving in an aqueous medium is derived from the resistance encountered by each element of the body to displacement in a direction normal to its own surface. Provided the coefficient of resistance to such displacement (CN) is greater than that to displacement tangentially to its surface (CL), a propulsive force is generated. This condition is satisfied when a nematode glides through a suspension of particles since particles in the vicinity of the element have to be displaced relative to each other when the element moves normally to its own surface, but not when it moves tangentially. On theoretical grounds Hancock (1953) showed that for very small organisms operating in a homogeneous aqueous medium the expected value of CN would be twice that of CL; consequently, for waves of the form in such animals as nematodes the expected amount of slip would be about 80 % A corollary of Hancock’s argument is, however, that if CN is not twice CL, the amount of slip depends on the ratio between them. If CN= kCL the expected percentage slip is [100 (B + 1)]/[kB + 1], where B is 2 π2b2/λ2; b being the amplitude and A the wavelength of the propagated waves. For creeping nematodes such as were used in the present study B = 0·5 (approx.) and consequently the percentage slip may be expected to be 100[3/(K + 2)]. Table 4 gives the calculated percentage slip for various values of k.

Table 4.
graphic
graphic

In order to account for the low amount of slip of an animal moving through a dense suspension of particles it would be necessary to assume that the ratio CN/CL was not less than 20.

So far as is known, there is no observational evidence concerning the resistance of relatively dense suspensions to shearing forces of the order of magnitude likely to be generated by very small and slowly moving cylinders comparable with nematodes; but, using thin wires, and assuming that the relative value of the resistance coefficients can be determined by observing the rate at which the wires fall steadily under their own weight when orientated normally and tangentially to their own direction of fall, the following observations are perhaps of some interest.

With the equipment available, it was necessary to use media (glycerine, Golden Syrup) with relatively high viscosity in order to reduce the rates of fall to readily observable values. As shown in Table 5, the calculated ratio CN/CL for wires varying from 0·2 to 1·6 mm. in diameter was found to be remarkably constant at 1-4 to r6 instead of an expected value of 2·0. No appreciable change in the rate of vertical fall could be effected by sharpening the leading end of the wire to a sharp point. When the wires were allowed to fall with their long axes inclined to the vertical there was a marked but constant difference between the axis of the wire and its path of motion during the course of its movements (Text-fig. 3).

Table 5.
graphic
graphic
Text-fig. 3.

Successive positions at 1 min. intervals of a wire (20 mm. long ; wt. 0·075 g.) falling through Golden Syrup when the axis of the wire was inclined to the vertical. Note the marked but constant angle between the axis of the wire and its path of motion.

Text-fig. 3.

Successive positions at 1 min. intervals of a wire (20 mm. long ; wt. 0·075 g.) falling through Golden Syrup when the axis of the wire was inclined to the vertical. Note the marked but constant angle between the axis of the wire and its path of motion.

It may be concluded that changes in the viscosity of the medium are unlikely to induce marked changes in the ratio of propulsive speed to wave speed in the case of such animals as nematodes. This conclusion is confirmed by direct observation of the movements of Panagrellus in a homogeneous medium (‘Golden Syrup’) having a viscosity approximately 1000 times that of water. As shown in Table 6, the average percentage slip was effectively the same as in water although the length, frequency and speed of conduction of the waves were markedly reduced.

Table 6.
graphic
graphic

Just as there is a marked contrast in the behaviour of nematodes in syrup and in agar, so there is a corresponding difference in the movement of iron wires when submerged in these two media. When an iron rod 22 mm. long, 1.4 mm. in diameter and weighing 0·23 g. was gently submerged either vertically or horizontally into a relatively long column of 12% agar, it remained stationary for an indefinite period. On the other hand, a similar rod whose leading end had been sharpened to a point fell vertically through the agar at a steady speed of about 0·33 mm./sec. Further, when such a rod was submerged obliquely in the medium its rate of displacement in a direction normal to its own axis was extremely small, in other words, it glided through the agar with very little slip (Text-fig. 4). If a pointed rod was allowed to fall vertically along the track which had previously been traversed by a similar rod, its rate of fall was greatly increased. A reasonable explanation of these results is available on the assumption that 12 to 1 % agar gel. can be regarded as a network of fibres which yield when subjected to critical shearing forces. These forces are only reached when the effective weight of the rod is concentrated at the sharpened end, which, like the head of a nema-tode, acts as a wedge ; once the structure of the agar has been ruptured some time is required before its original structure is regained. The same principle could be applied to nematodes if the forward thrust exerted by the animal is sufficient to disrupt the structure of the agar.

Text-fig. 4.

Movement of pointed wires through 12% agar. The figure shows the rate and direction of thick wire (2·5 mm. diameter) and a thin wire (1·4 mm. diameter) when falling with their axes vertical and when these are orientated at an angle to the vertical. Note that in the latter case there was only a very slight angle between the axis of the wire and its path of motion.

Text-fig. 4.

Movement of pointed wires through 12% agar. The figure shows the rate and direction of thick wire (2·5 mm. diameter) and a thin wire (1·4 mm. diameter) when falling with their axes vertical and when these are orientated at an angle to the vertical. Note that in the latter case there was only a very slight angle between the axis of the wire and its path of motion.

The general conclusion to be reached from the study of creeping nematodes is that the amount of slip depends on the relative resistance which the medium offers to the shearing forces applied to it by the normal and tangential movements of each part of the animal’s body. Further analysis probably depends on a more precise knowledge of the physical properties of agar gels, and of thin films of water. In the case of nema-todes moving between starch grains the work of Oldroyd (1955) and of Kynch (1956) on the effective viscosity of suspension is of considerable biological interest.

As already mentioned, marked changes occur in the form and frequency of the waves when nematodes are transferred from a dense suspension of particles to a homogeneous aqueous medium ; wavelength, amplitude and frequency are greatly increased, whereas the ratio of propulsive speed to wave speed is greatly reduced (see Table 7). Typical swimming movements are illustrated in Pl. 2.

Table 7.
graphic
graphic

(i) Turbatrix aceti

Turbatrix swims actively to the surface of vinegar when confined in a long vertical tube (Peters, 1928). During the present study, visual observations in a tube 60 cm. long and 0·8 cm. wide yielded an average speed of vertical progression of 15 individuals of 738μ/sec. (max. 1070; min. 220); as determined photographically for horizontal movement over a glass slide the average speed of progression of twenty individuals was 718μ/sec. (max. 1055; min. 220). These figures suggest that the speed of progression is not greatly affected by proximity to a glass surface ; but as neither the form nor the frequency of the waves could be determined when animals were swimming vertically in a bulk of fluid, it is impossible to be sure that these were the same as for similar animals moving horizontally over a slide. As shown by Table 7 the observed ratio of propulsive speed (Vx) to wave-speed (Vw) was sometimes surprisingly high.

A frequent, but not invariable, characteristic of Turbatrix, when swimming in vinegar, is a change in the form of the wave as it passes along the body; the amplitude of the waves increasing as they pass posteriorly along the body. In the individual shown in Text-fig. 5 the amplitude of lateral displacement of the head was about 35/4, whereas that of the tip of the tail was 125μ. As explained elsewhere (Gray, 1958) this characteristic is associated with an ability of the animal to propel itself without yawing from side to side.

Text-fig. 5.

Successive positions at intervals of 110 sec. of a specimen of Turbatrix aceti in which the amplitude of transverse movements of the tail was about four times that of the head.

Text-fig. 5.

Successive positions at intervals of 110 sec. of a specimen of Turbatrix aceti in which the amplitude of transverse movements of the tail was about four times that of the head.

(ii) Haemonchus, Strongylus and Panagrellus

In all these forms the body of a swimming animal displays less than one complete wave during certain phases of each complete cycle, and consequently it is not surprising that its motion relative to the ground should differ from that of a system in which all the transverse force operating against the body are assumed to be in equilibrium (see p. 149).

Relative to the ground, the envelope of a complete cycle of a swimming Haemonchus exhibits two distinct ‘nodes’ (Text-fig. 6) round which the anterior and posterior regions of the body glide tangentially. The existence of these nodes is associated with two characteristic features of the animal’s movement, (i) The whole body is yawing from side to side in the plane of its transverse movement (Text-fig. 76). (ii) Relative to the head of the animal, the waves are accelerating over the anterior end of the body and slowing down over the posterior end. As shown by Text-fig. 6, the motion of an element relative to the surrounding water depends on its position relative to the head of the animal and and is by no means the same for all elements; the motion of the whole body differs very substantially from that which forms the basis of calculation of propulsive speed. On general grounds one might have expected the observed speed of progression of a yawing organism to be less, and not greater, than the calculated value.

Text-fig. 6.

Motion relative to fixed axes of an individual Haemonchus contortus derived from photographs taken at intervals of 110 sec. The tracks of the head and tail are shown by dotted lines. Note the two characteristic nodes.

Text-fig. 6.

Motion relative to fixed axes of an individual Haemonchus contortus derived from photographs taken at intervals of 110 sec. The tracks of the head and tail are shown by dotted lines. Note the two characteristic nodes.

The changes of external bodily form which a nematode produces by its own internal efforts inevitably induces displacement of the surrounding water and a precise knowledge of the nature and extent of this displacement is of very great significance in any theoretical analysis of undulatory propulsion.

The nature of the movement induced in the water by a swimming animal can be determined by observation of the movements of small suspended particles. A series of photographs (at intervals of about 120 sec. and 1160 sec. exposure) of a swimming Panagrellus showed that a starch grain originally situated close to the head on the right-hand side of the body travelled tangentially backwards along the surface of the body until it reached the vicinity of the tail and remained on the right side of the body during the whole of its displacement (Text-fig. 8); relative to the head, the particle moved tangentially over the surface of the body for a distance approximately equal to one wavelength. The photographs showed that during the period required for the particle to travel this distance, four complete waves had passed along the body, and consequently the rate of backward movement of the particle relative to the head was one quarter of the rate of propagation of the waves over the surface of the body.

Text-fig. 7.

The periodic yawing movement of the body of Haemonchui. (a) shows the sinusoidal track along which the animal appeared to glide when a series of successive photographs were superimposed so that the outline of individual waves remained stationary to fixed axes passing through their respective crests. The axis of the sinusoidal track (in a) defines the axis of the waves. (b) shows the periodic yawing of this axis about the axis of the animal’s progression ; the line ab is drawn normal to the axis of progression.

Text-fig. 7.

The periodic yawing movement of the body of Haemonchui. (a) shows the sinusoidal track along which the animal appeared to glide when a series of successive photographs were superimposed so that the outline of individual waves remained stationary to fixed axes passing through their respective crests. The axis of the sinusoidal track (in a) defines the axis of the waves. (b) shows the periodic yawing of this axis about the axis of the animal’s progression ; the line ab is drawn normal to the axis of progression.

Text-fig. 8.

The movement of a particle (*) along the right side of the body. Between position 1 and position 41 the particle moved along the body for a distance of approx, one wavelength. The displacement of the particle relative to the ground is shown by the dotted line ; the particle executed three rather irregular loops traversing the loops in an anticlockwise direction, and during this time four waves passed over the body.

Text-fig. 8.

The movement of a particle (*) along the right side of the body. Between position 1 and position 41 the particle moved along the body for a distance of approx, one wavelength. The displacement of the particle relative to the ground is shown by the dotted line ; the particle executed three rather irregular loops traversing the loops in an anticlockwise direction, and during this time four waves passed over the body.

When the displacements of the head, the tail and the particle shown in Text-fig. 8 are plotted relative to the ground their respective tracks are relatively complex (Text-fig. 9) ; this is partly due to the periodic yawing movements of the whole body in a transverse horizontal plane and partly to the fact that the movements of this animal were less regular than those shown in figs. 5 and 6. As shown in Text-fig. 9 the head and tail of the animal move forward along somewhat irregular sinusoidal tracks whose amplitudes are considerably greater than their wavelengths, the frequency of transverse displacement being the same as that at which the waves are propagated over the body. At first sight the track of the particle appears to have little relationship to that of the head ; it exhibits a series of rather irregular loops with the particle travelling forwards relative to the ground (i.e. in the direction in which the animal is moving forwards) when the particle is in the vicinity of a wave crest, and backwards (i.e. in the direction in which the waves travel relative to the head) when in the trough of a wave. As will be seen in Text-figs. 8 and 10, the average rate at which the particle moves backwards (relative to the ground) is very much less than that at which the waves travel relative to the ground. The whole track of the particle is the resultant of three types of displacement (i) a longitudinal harmonic motion, (ii) a transverse harmonic motion, (iii) a longitudinal displacement at a speed much less than that at which the waves travel relative to the ground.

Text-fig. 9.

Track of the head and tail of the animal shown in Text-fig. 8 and of a suspended particle.

Text-fig. 9.

Track of the head and tail of the animal shown in Text-fig. 8 and of a suspended particle.

Text-fig. 10.

The movement of a particle relative to a stationary wave. If such a filament as that shown in Text-fig. 13 is moved anteriorly along the axis ofthe wave at the same speed (6cm./sec.) as the wave travels posteriorly relative to the head, the waves become stationary relative to the ground and the body glides forward without slip. Both head(←) and particle (•) follow the same track ; in the diagram the positions of the head and particle are shown at successive intervals of 1 sec. In 12 sec. the head travels 113 wavelengths (72 cm.), whilst the particles travel 1 wavelength (54 cm.). In 36 sec. the head completes 4 cycles of transverse displacement, while the particle completes 3 cycles.

Text-fig. 10.

The movement of a particle relative to a stationary wave. If such a filament as that shown in Text-fig. 13 is moved anteriorly along the axis ofthe wave at the same speed (6cm./sec.) as the wave travels posteriorly relative to the head, the waves become stationary relative to the ground and the body glides forward without slip. Both head(←) and particle (•) follow the same track ; in the diagram the positions of the head and particle are shown at successive intervals of 1 sec. In 12 sec. the head travels 113 wavelengths (72 cm.), whilst the particles travel 1 wavelength (54 cm.). In 36 sec. the head completes 4 cycles of transverse displacement, while the particle completes 3 cycles.

Some of the characteristic features of the displacement of a particle can be illustrated by plotting its displacement and that of the head of the animal relative to a stationary wave. For this purpose a series of photographs were mounted so that the outline of individual waves remained stationary in respect to fixed axes passing through their respective crests; in order to do this it is necessary to rotate each photograph in a transverse plane to compensate for the transverse harmonic yawing movements which occur during the passage of each wave (see p. 144). Most of the outline of the whole body then falls on a relatively smooth sinusoidal curve (Text-fig. 7 a) along which the head and all other elements of the body move forward, at the same speed (and with the same frequency and form of transverse displacement) as the waves are propagated along the body. When successive positions of the particle (referred to in the previous paragraph) were plotted along the sinusoidal track followed by the body, the particle moved forward relative to the fixed axes at an average speed about three quarters of that of the head and executed three cycles of transverse movement. The rate of a particle’s forward displacement relative to a stationary wave is, however, not constant; it is greatest when the particle is in the vicinity of a wave crest and least when in the vicinity of a trough—the ratio of the two speeds being of the order of three to one. The movements of the head and particle relative to a stationary wave are shown diagrammatically in Text-fig. 10.

The physical basis of longitudinal harmonic movements of the particle will be considered later, but the fact that fluid surrounding the body exhibits transverse harmonic displacements is illustrated by Pl. 1, figs 5-8, which is a photograph of an animal swimming into a drop of water from one of 50% Golden Syrup ; the body is surrounded by a relatively wide layer of fluid which undergoes transverse movements during the passage of the wave.

For analytical purposes, it is important to know how far the disturbances produced in the water by the movements of the body extend beyond its surface. This is illustrated by Pl. 3, figs. 1-4, which shows four consecutive photographs of Turbatrix swimming in a suspension of starch grains ; the duration of each exposure was approx. 115th sec-and the interval between them 16th sec. ; the frequency of the waves was 3 per second. Starch grains at rest appear sharp, whilst those set in motion appear as streaks. Regions of the body situated near the end of their transverse displacement appear sharp, whilst those in active transverse movement are, as might be expected, blurred. The most striking feature of the photographs is the circulation of particles centred round regions of the body which have reached their maximum transverse displacement, i.e. round the crests and troughs of the waves. The direction of the circulation cannot be determined from the photographs but when the latter are interpreted in the light of the observations of single particles (as in Text-fig 8) there can be very little doubt that the directions of circulation in Pl. 3 is as shown in Text-fig. 11 ; the particles moving forward (i.e. in the direction of the animal’s progression) when in the vicinity of a wave crest and backwards when in a trough. Thus members of any pair of adjacent ‘vortices’ rotate in opposite directions. It will be noted that the circulation extends beyond the surface of the body for a distance equal to ten or fifteen times the diameter of the body. Not the least surprising feature of Pl. 3 is the regularity of the rotational flow patterns despite the relatively complex and somewhat irregular track of individual particles relative to the ground.

Text-fig. 11.

The pattern of the stream lines in the neighbourhood of the body of Turbatrix (see Pl. 3). The centres of rotation move posteriorly relative to the ground at the same speed as the waves.

Text-fig. 11.

The pattern of the stream lines in the neighbourhood of the body of Turbatrix (see Pl. 3). The centres of rotation move posteriorly relative to the ground at the same speed as the waves.

Before discussing the general nature of the flow of water in the vicinity of an undulating body it may be useful to summarize the main features of the movement of particles in the neighbourhood of a swimming nematode as revealed by direct observation: (i) The particles move tangentially along the surface of the body and may remain on the same side of the body during the whole of their displacement, (ii) The rate of tangential displacement is not constant; it is greatest when the particle is situated in the trough of a wave ; it is reversed in direction when situated near a wave crest, (iii) The average speed of posterior displacement relative to the head is about one quarter of that at which the waves are propagated along the body; in order that a particle should be displaced posteriorly for a distance of one wavelength, four complete waves must pass along the body and during this period the particle executes three cycles of transverse movement, (iv) Relative to the ground, particles traverse looped tracks ; particles on the right side of the body move along these loops in an anticlock-wise direction, those on the left side move in a clockwise direction, (v) The flow extends for a considerable distance away from the surface of the body.

As applied to small organisms such as those considered in this paper, the general theory of undulatory propulsion (Taylor, 1951, 1952; Gray & Hancock, 1955) makes a number of assumptions, four of which are of immediate significance: (i) That the disturbance which the moving body induces in the surrounding water is restricted to a very short distance away from the surface of the body (Taylor, 1951). (ii) That the motion of an element of the body relative to the main mass of the undisturbed water is the same for all elements, (iii) That the forces which an element exerts against the surrounding water are the same as those exerted by a corresponding element of a long straight cylinder moving at the same speed and with its surface inclined at the same angle to its direction of motion, (iv) That, at all phases of movement, the transverse forces acting on the body summate to zero ; the body does not yaw from side to side and the axis of progression always coincides with the axis of the waves. None of these assumptions are justified in the case of nematodes and it may be open to doubt how far they can be safely applied to other organisms.

The pattern of flow in the neighbourhood of a swimming nematode is essentially the same as that described some years ago for the flow past the surface of an undulating cylinder of rubber (Gray, 1935). The main factor responsible for the reversed flow in the neighbourhood of the wave crests was revealed when the cylinder was replaced by an undulating rubber sheet 12 in. thick, 4 in. wide and 24 in. long. By attachment to the operating rods of a wave machine the sheet generated waves 30 cm. in wavelength and 5-5 cm. amplitude, thus giving a ratio of amplitude to wavelength approximately the same as that typical of nematodes. The whole apparatus was placed in a large rectangular tank containing sufficient water to submerge the lower half of the sheet—leaving the upper half unsubmerged. A rigid plate was then placed tangentially to two successive wave crests, and a small number of ‘particles ‘(parsnip seeds) were scattered on the surface of the water. The movement of these individual seeds was recorded photographically using exposures of 16/sec. and of 1160 sec. duration. As the whole of the water between two successive wave crests was completely contained within the boundaries of the model and the rigid plate, all particles moved posteriorly in straight lines parallel to the axis of the waves and at a speed equal to that of the waves (see Text-fig. 12.) When photographed with appropriately long exposures, the particles appear as streaks all aligned parallel to the axis of the waves (Pl. 3, fig. 5). As soon as the plate was removed, the pattern of flow of all the particles changed to that illustrated in Pl. 3, fig. 6. The track of individual particles relative to the ground showed characteristic loops. When viewed from above and looking towards the anterior end of the model, all particles passing along the left side of the model traverse their loops in a clockwise direction whilst particles on the right side move anti-clockwise.

Text-fig. 12.

Displacement of particles by a filament undulating within a close fitting rect-angular channel whose width is twice the amplitude of the waves and whose depth is equal to the diameter of the filament. All particles lying between the filament and two successive wave-crests must travel posteriorly at the same speed as the waves and each particle maintain" its position relative to the crests of the waves. Wavelength 54 cm. ; amplitude 6 cm. ; velocity relative to head and ground 6 cm./sec. Period 9 sec. Forward displacement of filament relative to ground nil.

Text-fig. 12.

Displacement of particles by a filament undulating within a close fitting rect-angular channel whose width is twice the amplitude of the waves and whose depth is equal to the diameter of the filament. All particles lying between the filament and two successive wave-crests must travel posteriorly at the same speed as the waves and each particle maintain" its position relative to the crests of the waves. Wavelength 54 cm. ; amplitude 6 cm. ; velocity relative to head and ground 6 cm./sec. Period 9 sec. Forward displacement of filament relative to ground nil.

In the case of the model it seems clear that the looped track of a particle relative to the ground is an expression of the fact that water situated near the leading surface of a wave crest flows anteriorly towards the trailing surface of the crest, i.e. in the opposite direction to that in which the waves are being propagated, while water in the vicinity of a wave trough travels posteriorly, i.e. in the same direction as the waves.

In view of the irregularity of the track of the particle in Text-fig. 8 it may be useful to apply the main conclusions reached concerning a particle’s displacement to an arbitrarily simplified system. This has been done in Text-figs. 13 and 14 by making three assumptions: (i) That the body does not yaw from side to side, (ii) That the form of the wave conforms to that of a sine curve and does not change when propagated along the body, (iii) That, relative to a stationary wave, the speed of a particle at a wave crest is three times that of a particle in a wave trough. In other respects the diagrams incorporate the main characteristics derived from the observational data: (i) That when situated near the body a particle moves tangentially to the latter’s surface at an average speed one quarter of that at which the waves are propagated, (ii) That the frequency of a particle’s transverse displacement is three-quarters of that at which the waves are generated. The arbitrary values selected for the wave characteristics were adopted on the basis of simplicity of geometrical construction; wavelength 54 cm., amplitude 6 cm., frequency one wave in 9 sec. and speed of propagation 6 cm./sec. In using the diagrams it is important to remember that all the particles shown invariably lie close to the surface of the body.

Text-fig. 13.

The movement of a particle (☆) near the right side of an undulating body when the particle moves tangentially along the surface of the body at an average speed one quarter of that at which the waves are generated ; the track of the particle relative to the ground is shown, ---; the dotted line phase 36b shows the track of a corresponding particle on the left side of the body. Wavelength 54 cm., amplitude 6 cm., frequency one wave in 9 sec., speed of waves 6 cm./sec. Phases 0−33 show the position of the particle and of the wave at intervals of 1 sec. The numerals shown on the tracks show the position of the particle at successive intervals of 1 sec.

Text-fig. 13.

The movement of a particle (☆) near the right side of an undulating body when the particle moves tangentially along the surface of the body at an average speed one quarter of that at which the waves are generated ; the track of the particle relative to the ground is shown, ---; the dotted line phase 36b shows the track of a corresponding particle on the left side of the body. Wavelength 54 cm., amplitude 6 cm., frequency one wave in 9 sec., speed of waves 6 cm./sec. Phases 0−33 show the position of the particle and of the wave at intervals of 1 sec. The numerals shown on the tracks show the position of the particle at successive intervals of 1 sec.

Text-fig. 13.

For legend see opposite.

Text-fig. 13.

For legend see opposite.

Text-fig. 14.

The movement of particles relative to the ground and to the surface of the body. All particles on the right side of the body move anticlockwise (a) ; all particles on left side (b) travel clockwise. The length and direction of the arrows show the displacement of the particles relative to the ground and to the surface of the body when the crest of a wave travels 16 cm. (from position, to position) in 1 sec. Compare this diagram with Text-fig. 12.

Text-fig. 14.

The movement of particles relative to the ground and to the surface of the body. All particles on the right side of the body move anticlockwise (a) ; all particles on left side (b) travel clockwise. The length and direction of the arrows show the displacement of the particles relative to the ground and to the surface of the body when the crest of a wave travels 16 cm. (from position, to position) in 1 sec. Compare this diagram with Text-fig. 12.

Text-fig. 13 shows the movement of the waves and of a particle at thirty-six successive intervals of one second. A particle originally situated on the right anterior side of the body opposite a wave crest travels tangentially over the body’s surface through a distance of 54 cm. (one wavelength) in 36 sec. during which time a wave travelled 216 cm. (four wavelengths). In 36 sec. the particle executes three complete cycles of transverse movements whilst the head completes four complete cycles. The diagram also shows that the particle moves anteriorly, relative to both head and ground, when in the vicinity of a wave crest (phases 0/1,11/13, 23/25) and posteriorly at its maximum speed when in a wave trough (phases 5/7,17/19, 29/31) and that it traversed its looped track in an anticlockwise direction. A particle on the left side of the body traversed its track (see phase 3 6 J) in a clockwise direction.

Text-fig. 14 shows the displacement of particles in 1 sec. during which the wave moves 6 cm. In Fig. 14 a all the particles lie on the right side of the body and move along their tracks in an anticlockwise direction; in Fig. 14b the particles lie on the left side of the body and are moving clockwise. The arrows show the displacement of the particles relative to the ground and exhibit a marked contrast with Text-fig. 12.

Text-fig. 15 shows that the track of the particle not only depends on the rate at which the waves move relative to the head but also in the rate at which the whole body moves forward relative to the ground ; in other words, the form of the track depends on the rate at which the waves travel relative to the ground.

Text-fig. 15.

Displacement of a particle relative to the ground when the waves are travelling 6 cm./sec. relative to the head of the animal and the head travelling 1 cm./sec. relative to the ground ; the speed of the waves relative to the ground is 5 cm./sec. Ln la sec. all particles move anteriorly 54 cm. relative to the waves (as in Text-fig. 13), whilst the wave travels 60 cm. posteriorly relative to the ground ; hence, the particles move posteriorly 6 cm. relative to the ground. The positions of the particle after successive intervals of 1 sec. are marked by the numerals along the looped tracks; the relative positions of the head are marked along the dotted line. Note that, as in Text-fig. 13, the particle executes one complete transverse cycle whilst the head executes 112 cycles; also that the track of the particle relative to the ground depends on the speed at which the waves themselves travel relative to the ground.

Text-fig. 15.

Displacement of a particle relative to the ground when the waves are travelling 6 cm./sec. relative to the head of the animal and the head travelling 1 cm./sec. relative to the ground ; the speed of the waves relative to the ground is 5 cm./sec. Ln la sec. all particles move anteriorly 54 cm. relative to the waves (as in Text-fig. 13), whilst the wave travels 60 cm. posteriorly relative to the ground ; hence, the particles move posteriorly 6 cm. relative to the ground. The positions of the particle after successive intervals of 1 sec. are marked by the numerals along the looped tracks; the relative positions of the head are marked along the dotted line. Note that, as in Text-fig. 13, the particle executes one complete transverse cycle whilst the head executes 112 cycles; also that the track of the particle relative to the ground depends on the speed at which the waves themselves travel relative to the ground.

From a theoretical point of view the most significant conclusion to be derived from the present study is that the water in the vicinity of a swimming nematode not only moves backward relative to the ground at a much lower speed than the waves them selves travel relative to the ground, but also exhibits two harmonic motions: (i) Transverse harmonic motion—due to the transverse drag exerted by adjacent elements of the body; the frequency of these movements being less than that of the propagation of the waves along the body, (ii) A longitudinal harmonic motion due to the reversal of the flow over the crests of the waves. It may well be that these characteristics of the flow may be typical of all undulating surfaces.*

It may be noted that whereas Text-fig. 11 and Pl. 3 depict the displacement (during a short period of time of a large number of particles at varying distances from the surface of the body, the dotted lines in Text-fig. 14 show diagrammatically the displacement of individual particles all close to the surface of the body during a relatively long period of time. At first sight it appears strange that the observed paths of displacement (Text-fig. 8) should show great irregularity, whereas Text-fig. 11 shows a very regular pattern. This anomaly would, presumably, disappear if it were legitimate to assume that the flow of water relative to the surface of the body is not affected by yawing movements of the body relative to the ground. A more complete analysis of the flow of water in the neighbourhood of an undulating organism would involve the observation of the movement of individual particles situated at known distances away from the surface of the body. The incorporation of all the data into a generalized theory of undulatory propulsion would appear to present formidable problems in hydrodynamics.

Ewald
,
P. P.
,
Poschl
,
T.
&
Prantl
,
L.
(
1930
).
The Physics of Solids and Fluids
.
London
:
Blackie & Sons
.
Gray
,
J.
(
1935
).
The propulsive powers of the dolphin
J. Exp. Biol
.
13
,
192
9
.
Gray
,
J.
(
1953
).
Undulatory propulsion
Quart. J. Mier. Sci
.
94
,
551
78
.
Gray
,
J.
(
1958
).
The movement of the spermatozoa of the bull
.
J. Exp. Biol
.
35
,
96
108
.
Gray
,
J.
&
Hancock
,
G. J.
(
1955
).
The propulsion of sea-urchin spermatozoa
.
J. Exp. Biol
.
32
,
802
14
.
Hancock
,
G. J.
(
1953
).
The self-propulsion of microscopic organisms through liquids
.
Proc. Roy. Soc. A
.
217
,
96
121
.
Kynch
,
G. J.
(
1956
).
The effective viscosity of suspensions of spherical particles
.
Proc. Roy. Soc. A
,
237
,
90
116
.
Loose
,
A.
(
1911
).
The anatomy and life history of Agchylostoma duodenale Dub. Pt. II
.
Cairo Records Eg. Govt. Sch. Med
.
4
,
167
607
.
Olroyd
,
J. G.
(
1955
).
The effect of interfacial stabilizing films on the elastic and viscous properties of emulsions
.
Proc. Roy. Soc. A
,
232
,
567
77
.
Peters
,
B. G.
(
1928
).
On the bionomics of the vinegar eelworm
.
J. Helminth
.
6
,
1
38
.
Pohl
,
R. W.
(
1932
).
Physical Principles of Mechanics and Acoustics
.
London
:
Blackie & Sons
.
Taylor
,
G. I.
(
1951
)
Analysis of the swimming of microscopic organisms
.
Proc. Roy. Soc. A
,
209
,
447
61
.
Taylor
,
G. I.
(
1952
).
Analysis of the swimming of long and narrow animals
.
Proc. Roy. Soc. A
,
214
,
158
83
.
Wallace
,
H. R.
(
1958
).
Movement of eelworms. I. The influence of pore size and moisture content of the soil on the migration of larvae of the beet eelworm, Heterodera schachtii Schmidt
.
Aim. Appl. Biol
.
46
,
74
85
.
Wallace
,
H. R.
(
1959
).
Movement of eelworms in water films
.
Ann. Appl. Biol
.
41
,
366
370
.
Wallace
,
H. R.
(
1960
).
Movement of eelworms. VI. The influence of soil type, moisture gradients and host plant roots on the migration of the potato root eelworm Heterodera rostochiensis Wollemoeber
.
Ann. Appl. Biol
.
48
,
107
20
.

PLATE 1

Figs. 1-4. Panagrellus silusiae moving through a thin layer of a dense suspension of starch grains ; the animal is moving from right to left. Note the track left behind the body. Scale 550^. Interval between photographs approx. 316 sec.

Figs. 5-8. Panagrellus swimming into a drop of water from one of Golden Syrup. Interval between photographs 18 sec.

PLATE 2

Figs. 1-12. Haemonchus contortus swimming in water. Interval between photographs 115 sec.

PLATE 3

Figs. 1-14. Turbatrix aceti swimming in a dilute suspension of starch grains. Interval between photographs 16 sec. ; exposure time 116 sec.

Fig. 5. Movement of particles when the flow induced by an undulating rubber lamina is restricted by a rigid plate in contact with two wave crests. Exposure rime 110 sec. Note that the particles move along straight lines.

Fig. 6. Flow of particles after removal of plate. Note the circulation round the wave crest on the right of the figure. Exposure time 110 sec.

PLATE 24

Figs. 1—9. Flow of particles induced by an undulating rubber lamina. In fig. 1 the model is at rest. The waves are travelling from right to left. Interval between photographs 13 sec. ; exposure time 17 sec.

*

The wavelength (λ) being closely proportional to the length (l) of the body, it is usually desirable to express the wavelength in terms of body length thereby facilitating comparisons of the wave form in animals of different length.

*

See Ewald, POschl’ Prantl (1930), p. 264, fig. 65, and Pohl (1932), p. 219, figs. 45, 46.