STUDIES IN ANIMAL LOCOMOTION
I. THE MOVEMENT OF FISH WITH SPECIAL
REFERENCE TO THE EEL
BY J. GRAY.
King's College, Cambridge.
(From the Laboratory of Experimental Zoology, Cambridge.)
(Received 20th October, 1932.)
(With Four Plates and Eleven Text-figures)
a body is moving in water it encounters a resistance in the direction of its
motion, and consequently the body must be supplied with energy if motion is to
occur at a uniform speed. A study of the mechanism of propulsion of a fish falls
therefore into two parts, (1) a study of the forces resisting motion through the water,
and (2) a study of the mechanism whereby the fish utilises the energy liberated by
its muscles for overcoming the forces of resistance. To some extent these two
aspects of the problem are interdependent and involve considerable hydrodynamical
difficulties, but in the present paper an attempt will be made to show that the movements of a fish's body1 are such as to generate forces capable of opposing the forces
of resistance whatever be the nature or magnitude of the latter. The problem was
attacked two centuries ago by Borelli and by Pettigrew in 1873; since then comparatively little attention has been devoted to the subject except by Breder (1926),
whose results will be considered later.
Since all propellers operate by driving astern a volume of water, the reaction
from which compensates the surface resistance of the moving object, the initial
problem of the fish's movements consists in demonstrating that the fish moves its
body in such a way as to drive water away from its surface in a backward direction.
All inanimate propellers belong to one of three types: (1) The jet propellers—as
exemplified by all reaction turbines which project a current of water from a nozzle.
The reaction caused by the water moves the nozzle in a direction opposite to that of
the movements of the water. (2) Paddles—whereby a backward thrust is exerted on
the water, parallel to the direction of motion of the paddle and at right angles to the
surface of the paddle. The paddle can only be submerged during one-half of its
complete movement, or it must be capable of being rotated about an axis at right
angles to its line of motion, in order that no appreciable thrust is exerted during the
period which follows the effective phase of the movement. (3) Screws—the theory
WHEN
1
The present paper deals only with the propulsive properties of the bodies of a selected number of
fish whose appendages play little or no part in the propulsion when the fish are moving at reasonable
speeds. The propulsive properties of the caudal fin will be considered in a subsequent paper.
Studies in Animal Locomotion
89
of screw propulsion is essentially that of an inclined plate—which, by motion
through the water, generates a force at right angles to its surface (see Fig. 1). This
force (P) has a component (T) at right angles to the direction of motion of the plate,
which tends to move the plate along a line at right angles to its original direction of
motion.
The mechanism of propulsion of a typical fish does not conform to the design of
a jet or a paddle, and since all screws operate by means of a true rotary movement,
the possibility of a screw is, at first sight, excluded. The object of this paper is to
consider the motion of a fish's body and to compare the underlying mechanism with
that of a typical screw propeller. During the whole of the work, an attempt has been
made to record the form and position of the fish at known intervals of time by photo-
Fig. 1. AB is a cross-section of the blade of a screw moving along cd in the direction of the large
arrow. A force P is generated at right angles to AB. This has a component T which tends to move the
screw in the direction of a.
graphic means. An experimental tank was set up in the field of a timed cinematograph camera, so that the position of the fish could be determined by means of a
graduated field placed immediately underneath the fish. The method of recording the
interval between successive photographs has been described elsewhere (Gray, 1930).
I am greatly indebted to Mr J. E. Harris for his valuable help in the preparation
of these photographic records.
I. OBSERVATIONS ON THE MOVEMENTS OF FISH.
As observed by the human eye, the motions of various types of fish appear to
vary considerably from one species to another. The most conspicuous features of a
moving eel—as are seen in the photographs taken by Marey (1894)—are the waves
of curvature which pass along the length of the body from head to tail. In the
dogfish (and, still more, the mackerel and whiting), the presence of such waves is
less obvious, and the visible movements appear to be due to transverse strokes
90
J. GRAY
executed by the posterior end of the body across the axis of motion. It can be seen
from the photographs reproduced in Figs. 2-10 (Pis. I—IV), however, that in all these
cases waves of curvature pass along the body alternately on the two sides, but that they
differ in the various fish in certain important characteristics. Firstly, their speed of
propagation along the body varies greatly. In the examples illustrated the approximate speeds of the waves and the rates of movement of the fish are as shown in
Table I. Secondly, the form of the waves differs. In the reversing eel (Fig. 10) the
amplitude of the waves is very large, and is of approximately the same value as
their wave-length. In Ammodytes (Fig. 7) and the mackerel (Fig. 5), the relative
amplitude is very much smaller, while the dogfish, glass-eel, butterfish, and
rockling occupy intermediate positions. Thirdly, when the fish are swimming at a
steady rate, the frequency of the waves per second varies in the different species. In
the examples illustrated, the approximate number of waves passing down each side
of the body are shown in Table II. Fourthly, the amplitude of the waves is always
greatest at the posterior end of the body, but the variation between the amplitude of
the head and tail varies very greatly in different types. In the small eel the amplitude
of the movements of the head is relatively very much greater than those of the
mackerel or whiting.
Table I.
Velocity of wave
cm. per sec.
Gla9s eel (Anguilla vulgaris)
Butterfish (Centronotus gunnellus)
Whiting (Gadus merlangtit)
Dogfish (Acantlrias vulgaris)
Mackerel (Scot/tber tcombrus)
Ammodytes (A. lanceolatut)
6-2
I7-S
25-0
55
77
160
Velocity o
cm. per
4-0
117
168
29
42-5
80
Table II.
Waves per min.
Glass-eel
Butterfish
Whiting
Dogfish
Mackerel
Ammodytes
93
120
120
54
170
120
The movement of the muscular waves along an eel's body was recorded
photographically by Marey (1894), who made no attempt to define the mechanical
principles which are responsible for the forward movement of the whole fish. These
principles have been considered by Breder (1926), whose description of "anguilliform" movement is as follows: "The forward motion is certainly attained by the
pressure of the fish's body against the water in the following manner. The mechanical forces brought to bear on the water are diagonally backwards (from the posterior
surfaces of each of the curves of the body). As these are distributed symetrically
about the line of progression, a forward resultant of reaction follows, for pressure
Studies in Animal Locomotion
91
from a moving plane is always at right angles to its surface." That the fish's body
exerts a pressure on the water at right angles to its own surface is in accordance with
the analysis given later in this paper, but Breder goes on to state that " It might be
objected that as the eel is moving ahead there is likewise adverse pressure diagonally
forward from the anterior sides of these backwardly moving waves. The truth of
this is evident and it simply makes it necessary for the fish to pass these waves
posteriorly at a rate considerably faster than it expects to move forward
The
speed of the waves moving backward must exceed that of the forward motion of the
animal as a whole. If the two speeds just equalled each other it would mean that any
point on a wave such as its crest would be stationary with reference to the seabottom; but as one is dependent on the other this is obviously impossible." The
mechanical principles involved by this explanation are by no means clear, for it
is certain that the propulsive thrust of the moving body is due to the fact that
each part of the body is executing a series of transverse movements. Although
these movements can be expressed in terms of longitudinally moving waves of
contraction, the principles of propulsion of a fish are much more readily derived
from a study of the transverse movements of each section of the body than from
a direct investigation of the propagated waves of contraction. In the present paper
an attempt will be made to investigate the propulsive effect of those transverse
movements which are induced in the various parts of the body by a series of
muscular contractions which are of such a nature as to produce the phenomenon of
a propagated wave.
The movements of the body can be considered in two ways. Firstly, it is possible,
from a series of instantaneous photographs taken at equal and known intervals of
time, to plot the position in space of any particular point on the surface of the body.
Secondly, it is possible to consider the movements executed by one part of the
body relative to any other part, and not to fixed points in the environment of the
fish. By combining these two sets of observations it is possible to form an idea of
the way in which the contractions of the muscles induce changes in the relative
position of the parts of the body, one to another, which are such as enable the fish,
as a whole, to transmit to the water a backward momentum equal and opposite to
that of the frictional forces which oppose the motion of the fish through the water.
Owing to the well-defined nature of its muscular waves, attention may be concentrated on the small glass-eel (Anguilla vulgaris), about 7 cm. in length, shown in
Fig. 2; the same type of analysis can be applied to other forms, but for various
reasons it is convenient to defer this until later.
Fig. 12 A-C shows the track of the head, the middle point of the body, and the
tip of the tail of an Anguilla whose form during motion is shown in Fig. 12 D. It
will be noticed that the successive positions of each point lie along a sinusoidal curve
whose " pitch " or wave-length is the same in all cases, namely 3-2 cm., and that this
is less than the "pitch" or wave-length of the waves which characterise the body
itself, viz. 4-7 cm. It can also be seen that the amplitude (to) of the waves is least at
the head and greatest at the tip of the tail. The axis of motion (ab) of the fish is
shown in the figure, and it can be seen that if a line (cd) is drawn at right angles to
92
J. GRAY
this axis at points where the track of the point on the body cuts the axis of motion,
then the angle (8°) between the track of the fish and this line cd (the transverse axis
of movement) becomes progressively less as the tip of the tail of the fish is approached. If we now examine (Fig. 13) the angle (6m) made by any part of the body
of the fish and the line cd as this particular part crosses the line ab (i.e. crosses the
axis of forward movement), it can be seen from Fig. 13 that this angle also decreases
Head
" Mid Point
"Tip of Tail
A
B
C
D
Fig. 12. A-C. The paths followed by (i) the head, (ii) the middle of the body, (iii) the tip of the tail
of a young Anguilla (glass-eel). Note that the amplitude is greatest at the posterior end of the body,
and that the wave-length of each track (A) is less than that of the curve of the animal's body (D).
from the head to the tail; since the pitch of the body is greater than the pitch of the
curve of movement, it follows that the angle (8m) made by the body with the
transverse axis (cd) must be greater than the angle (80) between the path of motion
and the transverse axis of movement. The difference between these two angles
(8m — 8p) is of fundamental importance and will be called the angle of attack; it is
designated by the symbol a. Similar curves to those shown in Figs. 12 and 13 can
be constructed for other types of fish with similar results except that in most fish the
amplitude of the transverse movements of the head are very small compared with
Studies in Animal Locomotion
93
those of the tail. It can be seen in Figs. 2-10 that not only are the wave crests
travelling along the body of the fish but they are also travelling backwards with
reference to the environment.
To define the movements of a point on the body relative to other parts of the
body it is necessary to adopt two fixed axes of reference. One of these is provided
by the axis of forward movement (ab), for this is also the axis about which each point
of the body is moving in a transverse direction relative to any other point. It is not
so easy to obtain a fixed transverse axis. The ideal procedure would be to plot the
position of each point on the body with reference to a transverse axis which is
<a
Fig. 13. Tracings from enlarged photographs of Anguilla showing that the angle between the body
and the axis of forward movement (ab) decreases from the anterior to the posterior end of the body.
moving forward with the fish at a velocity equal to the average forward velocity of
the fish. This can be done within small limits of error if successive photographs are
enlarged and then superimposed on each other in such a way that the tip of the head
lies along the same transverse axis and if the longitudinal axis of motion (ab) of each
photograph is superimposed on that of the others. This has been done in Figs. 14
and 15, which represent a fish whose waves are moving down the body in the normal
way, but whose body is unable to progress forwards. It can be seen that during each
phase of its motion, any given point forms part of a segment1 of the body (Fig. 14)
which is inclined with its leading surface (i.e. the surface towards the direction of
1
The term "segment" is not used in its strict morphological sense.
fl
J. GRAY
transverse movement) turned towards the hinder end of the body. Thus in Fig. 14
the segment XY is travelling from the right side of the axis of movement towards the
left side and its leading surface is facing backwards and towards the left. Con-
\a
Fig- ISFig. 14. Enlarged drawings of a young Anguilla arranged to show the movements of short segments
of the body during the passage of the complete wave past the segments. Note that the segment XY
is travelling from right to left and is directed backwards and to the left. The segment X1Y, is travelling
from left to right and is directed backward and towards the right. Note also that the tip of the tail is
moving in a figure of 8 curve.
Fig. 15 A. Tracings of left side of a butterfish showing the passage of a wave, and the corresponding
positions of the tail (1—7). The dotted line shows the longitudinal axis of motion.
Fig .15 B. Shows the relative transverse velocity of the tail at different phases of its motion. Note
that the velocity is greatest when the tail is crossing the axis of forward movement.
versely, the segment X1 Yy shows the corresponding movement of a segment from
left to right, and the leading surface is facing backwards and towards the right. It
will be noted that as the segment XY is passing from right to left, the segment forms
part of a wave whose crest is travelling down the right side of the body, and that as
Studies in Animal Locomotion
95
X1 y x moves from left to right it is part of a wave travelling down the left side of the
body.
Although the movements executed by each segment of the fish closely resemble
the movements of the blade of an oar when sculled from the back of a boat, the
body of the fish exhibits certain peculiar features of considerable theoretical
importance.
(1) The speed at which a segment moves along its transverse path is not
uniform. When displaced to its maximum extent from the longitudinal axis of
movement (ab), the segment is moving very slowly; as it crosses the axis of longitudinal movement it is travelling at its maximum speed. Fig. 15 B shows the position
of a segment at equal periods of time, and it can be seen that the speed of its
movement varies inversely with its displacement from the longitudinal axis of
movement; during the phase of movement towards this axis the segment is
accelerating, and during the phase of movement away from this axis the segment
is decelerating.
(2) During each phase of movement the angle (9) between the body segment and
the longitudinal axis of movement is changing. It is greatest at the extreme positions
and is least as the segment is crossing the line (ab). Towards the end of each phase,
the segment is parallel to the axis of longitudinal movement, but as it begins to move
towards this axis, the segment becomes more and more inclined backwards—after
crossing the axis the process is reversed until the segment again points directly
forward; finally it becomes inclined in the opposite direction as it begins to move
backwards towards the axis (ab). The important point to notice is that the angle 6
is least when the segment is crossing the line ab—i.e. when it is travelling at its
maximum transverse speed.
(3) It is only when the segment is near to the axis ab that its leading surface is
bounded by a plane—in all other positions the surface is curved. As the segment
approaches the axis ab the leading surface is bounded by a curve which is concave
towards the direction of movement; after passing the axis (ab) the leading surface is
convex towards the direction of movement.
(4) If a point is marked on the surface (e.g. the base of the tail-fin in Fig. 15 B), it
is found to travel in a figure of 8 curve relative to the head. The transverse axis of
the figure of 8 is at right angles to the axis ab. The greater the amplitudes of transverse movement relative to the pitch or wave-length of the curve of the body, the
more marked are the figures of 8 (see footnote, p. 97). The figure of 8 movement
can be expressed in another way, namely, when the whole fish is travelling forward
at a constant average velocity, the forward velocity of segments at the extreme
positions of transverse displacement is rather greater than the average velocity of
the whole fish and the forward velocity of segments which are crossing the axis of
motion is rather less than this average value.
When the fish is in motion, the path traced out in space by any given point on
the body represents, of course, the locus of a point travelling along a figure of
8 curve which is endowed with a forward velocity equal to the average forward speed
of the fish. The angle between the body of the fish and its path of motion is of
96
J. GRAY
vital significance when we consider the propulsive properties of the body; the effect
of the figure of 8 motion is, as is seen in Fig. 16, to increase this angle when a
segment of the body is crossing the axis of longitudinal motion, and to decrease it as
it approaches the extreme positions of its transverse displacement.
The figure of 8 motion of the tail of a slowly moving sturgeon was observed by
Pettigrew (1873), who implied that the movement is a physiological adaptation to
efficient propulsion. It can, however, be shown that it is the inevitable result of the
propagation of a wave of curvature along an inextensible body.
12
Fig. 16. If AB is a segment travelling transversely along the dotted line a,^ and endowed with a
velocity of 1 cm. per unit of time towards the right, then the track of the mid-point of the segment
is shown by the dotted curve to the right of the figure. If AB travels along the figure of 8 (0,0,0,
ft, 6,6,) and is endowed with the same transverse velocity as before, then the track of the mid-point is
shown by the full curve on the right of the figure. Note that the angle ot, is greater than the angle
at,. The larger the amplitude of the figure of 8 the greater is the increase effected in ot at the midpoint of the transverse movement. [The reverse is the case at the extreme positions.]
Fig. 17 shows an inextensible string along which is passing a wave of curvature
of constant form. If one end of the string be allowed to move along the axis (cd),
and the position of any given point on the string be marked in respect to this axis
and the longitudinal axis (ab), it can be seen that except at stated intervals (of halfwave-lengths from the constrained end of the string) the path traced out by a fixed
point along the string is a figure of 8.
The greater be the relative amplitude to the wave-length of the waves the more
marked is the figure of 8 movement. Since the relative amplitude at the tail end of
Studies in Animal Locomotion
97
a fish is greater than at any other point, the tail exhibits a more marked figure of
8 than does any other point1.
The observed movements of the fish's body can be summarised as follows:
(1) All parts of the fish's body which are in transverse motion have their leading
surfaces directed backwards and towards the direction of transverse movement, but
the angle of inclination is most pronounced when the segment is crossing the axis of
longitudinal motion, and at this point the segment of the body is travelling at its
maximum speed.
(2) Each point on the body is travelling along a figure of 8 curve relative to a
transverse line which is moving forward at the average forward velocity of the whole
fish. In other words, a segment when moving across the axis of longitudinal motion
is travelling backwards relative to a segment which has reached the extreme
position of lateral displacement. The track of any point on the body (relative to the
earth) is a sinusoidal curve whose pitch or wave-length is less than that of a curve
1
Cr
2
.3
4.
5
6
7
8
9
Fig. 17. The dotted lines show the loci of points situated at i, 5, 10, 175 cm. from the end of an
inextensible string along which a sine wave is moving. Note that each point travels on a figure of 8
curve unless it is situated at or about one-half of a wave-length from the front end of the string.
The larger numerals show successive positions of the crest of wave; the smaller numerals show the
corresponding positions of the selected points.
which defines the body of the fish. There is therefore a definite angle between the
surface of the fish and its path of motion.
The movements executed by the body of a fish are closely analogous to those
exhibited by aflexiblebut elastic body, one end of which is made to vibrate along a
transverse axis. The validity of this analogy will be discussed elsewhere.
1
It might be objected (see Fig. 17) that if the tail were to lie at a point equal to a multiple of half
a wave-length from the head, it should travel approximately in a straight line. It must be remembered,
however, that this is only the case under two purely artificial conditions, (a) when the amplitude of
the movements is the same along the whole length of the body, (b) when the axis of reference is such
that the head is moving along a straight transverse line. If it were possible to refer the movements to
an axis which is moving forward at the same average forward velocity of the whole fish (i.e. if the
propagation of the wave involved no displacement of the centre of gravity of the system), every point
of the body would appear to travel in a figure of 8, the horizontal amplitude of which would vary
directly with a power of the amplitude of the metachronal wave. If the wave have the form of a sine
curve and the amplitude be small, it can be shown that the longitudinal amplitude of the figure of
8 curve is —r-, where OJ is the amplitude of the sine wave and A is the wave-length.
98
J. GRAY
It now remains to be shown that (i) the movements of the body are such as to
drive the fish forwards through the water, and (2) the movements of the body are the
direct effect of a series of waves of muscular contraction which start at the anterior
end of the body and pass backwards towards the tail. The latter consideration will
be discussed in a later paper.
II. THE BODY AS A PROPELLER.
Towards the extreme positions of each transverse cycle the velocity, form and
inclination of the body are rapidly changing and although it is possible to see in an
empirical way their general propulsive significance, these changes render impossible
an analysis of the effect of a complete cycle on the distribution of the surrounding
water. This distribution may be affected by the acceleration as well as by the
velocity of the body, and for this reason it is convenient to consider in thefirstinstance
the forces acting on a segment as the latter passes the axis of forward movement,
i.e. when it is inclined to this axis to its maximum extent, and when the velocity of
its transverse movement is greatest and reasonably constant.
H
Fig. 18 a.
Fig. 186.
Fig. 18 a. The dotted line shows the locus of the point O on the segment AB when the latter is
travelling transversely from left to right. The position and inclination of the segment at various
phases of its movement are shown at 1-7. ab, longitudinal axis of movement, cd transverse axis of
movement. The fish is depicted as stationary.
Fig. 186. Diagram showing the position of the segment AB of a stationary fish in respect to its own
direction of motion and to the longitudinal axis ab. The segment is travelling along OE with a
velocity equal to OE. The angle BOE = a, and is the angle of attack. The angle BOd is the angle
(6m) at which the segment is inclined to the axis cd. The movement of water relative to the segment
is at first EO; after meeting the body, a finite volume of water flows along OA with a velocity OH.
The effect of the water on the body is to endow the latter with a velocity GO relative to the water.
Studies in Animal Locomotion
99
Symbols used.
a = the angle between the surface of the body and its direction of motion.
9 = the angle between the surface of the body and a line drawn at right angles to
the longitudinal axis of movement (ab) of the fish.
9m = the value of 9 for any particular point on the body when this point is crossing
the axis ab.
ab = the longitudinal axis movement of the fish.
cd = an axis at right angles to the longitudinal axis of movement, i.e. an axis of
transverse movement.
A*= the pitch or wave-length of the muscular waves on the body.
OJ = the amplitude of the muscular waves on the body.
P = the pressure normal to the surface of the body.
F = the factional force tangential to the surface of the body.
T •= the forward component of the force P.
D = the transverse component of the force P.
X = the transverse component of the force F.
Y = the forward component of the force F.
V = the velocity of forward movement of the
fish.
Vg = the transverse velocity of any given
segment.
VR = the resultant velocity of any given segment and the surrounding water.
As long as the fish does not move forward,
the direction of movement (relative to the
earth) is to the side and backwards (Fig. 18);
but when the fish is travelling forwards the
direction of motion of AB is to the side and
forwards. During this movement water is
displaced, and the water displaced at the
leading surface (AB in Fig. 19) must either
flow along the surface of the body or over the
dorsal and under the ventral fins. Since the
body presents a relatively flat surface to the
water, some at least of the water must be
deflected along the surface of the body in a
backward direction, i.e. in the direction OA.
Let the mass of water so deflected be m gm.
^ Z
per sec. If the original velocity of this water axis ab. The segment AB is moving along the
(relative
to the body)
(
y) is Frcm. pper sec. ((= OE, ^^
^^
?J%
^^ ^ ^t ^t t h e ?J%
Fig. 19), this Can be resolved into two com- inclination (0m) is FOd. The normal pressure
ponents, one (OF) being parallel to the body ° f D the ^ J °" t h e w.atfer is proportional to
r
'
v
'
1
, 1 1
OP and the fnctional force is proportional
and the other (EF) normal to the body. to OF-OH.
ioo
J. GRAY
After encountering the body, the component EF is lost in respect to m gm. of
water per sec., hence there has been a loss of normal momentum by the water of
m , EF gm./cm. per sec.: this momentum must be gained by the body and it
represents the pressure of the water on the body in a direction at right angles to
its surface. Now EF = Vr sin a (where a. is the angle between the body and
its direction of motion), hence the normal pressure of the body (P) on the
water is m Vr sin a and the component of this force along the axis of longitudinal
A
/A
K
N
Jo?'
\1
N
Jh
M
N
A'
N
/
/
/
/
A
M
4
K
/
t
a/'
L
I,
KN
1
A
L
M
5
A
L
M
6
Fig. 20. Showing the effect of an increase in forward speed on the value of the angle a. r, fish
stationary; the forward velocity of the segment has a negative value of MN. 2, forward velocity of
fish = backward velocity of segment. 3 and 4, forward velocity giving reduced but positive values
for a. 5, forward velocity giving zero value for a. 6, forward velocity giving negative value for a,
giving negative thrust. AIN —- forward velocity of segment; LAf -= trans, velocity of segment;
MK = the pitch of the segment which is a function of the wave length of the muscular waves;
cd = trans, axis of movement.
movement is the corresponding propulsive thrust. As long as the body moves at an
angle to its own path of motion, there must therefore be a tendency for the fish to
move bodily through the water. As soon as the fish begins to move, however, two
events occur: (i) the angle a diminishes, and (2) frictional forces are generated at its
surface. The diminution of the angle a is seen in Fig. 20, where it can be seen that
there is a value for the rate of forward progression which is such that a = 0°, and at
this point the propulsive thrust must be zero; at the moment it may be noted that
the faster is the forward speed of the fish the smaller is the angle of attack and the
smaller is the normal pressure of the fish against the water.
Studies in Animal Locomotion
101
As long as the fish is in motion, its movements will be resisted by factional
forces which are due to the disturbances set up in the water in the neighbourhood
of the body. These forces act along all surfaces and their direction is tangential to
the direction of motion of the surface, so that when water is moving past the body
of the fish, the velocity of the water is reduced. Thus in Fig. 19 if FO be the
relative tangential velocity of m gm. of water struck by the fish per sec., and if OH
be the relative velocity of this water after passing over the segment, then the water
has lost momentum equivalent to m (FO-OH) gm./cm. per sec. in the direction of
OH. This momentum is gained by the body and represents the frictional force (F)
acting in the same direction. Thus the net effect of moving a segment of the body
through the water at an angle to its own direction of motion is to impress on the
body two forces—one normal to the surface (P)1 and the other tangential to the
surface (F). The longitudinal resultant of these two forces represents, if the present
K
analysis is correct, the net propulsive thrust which drives the fish against the
resistance of the water.
If forward motion at a uniform velocity is to take place, the resultant forces
acting on the segment of the body when measured in any direction must be zero.
The conditions under which this will occur can be seen by resolving the forces P
and F along the longitudinal (ab) and transverse (cd) axes of movement respectively.
Thus in Fig. 21 let LM = Vo = the velocity at which the segment AB is travelling,
along the' transverse axis of movement. Let MN = V = the velocity of forward
movement of the fish. Then the segment inclined at an angle 9 to LM is travelling
along LN with a velocity LN = Vr — VF o a + V2,* and the angle between the body
1
The existence of a force normal to the surface of the body, and the reduction of the longitudinal
component produced by an increase in the angle of inclination (0) of the body, was pointed out by
Breder (1926).
• This is the flow of the water relative to the body if the water is stationary in respect to the
earth.
102
J. GRAY
and its path of motion is a. The force P (which depends on the value of a, see
p. ioo) and the force F can both be resolved along ab and cd. The propulsive thrust
along ab is T - Y and the resisting force along cd is D + X. Now if these forces are
to have no resultant they must be compensated by equal and opposite forces
operating on the segment. These latter forces are (i) the force exerted by the
muscles, (ii) the resistance exerted by other parts of the body. Whether or not a
structure (e.g. the dorsal and ventral fins, the skull, etc.) is exerting a propulsive
thrust there will always be a frictional force at its surface, and as such structures may
be moving in any forward direction it follows that the frictional forces can always be
resolved into components acting along ab or cd. Uniform motion will ensue, therefore, when the forces along cd are collectively equal and opposite to the force
exerted by the muscles, and when the forces acting along ab are equal and opposite
to the longitudinal components of all the forces developed by other parts of the fish
which resist the forward motion of the segment. It follows that when a fish (which
is initially at rest) begins to move its body in such a way that the leading surface is
inclined backwards at an angle to the axis (ab) it will move forward with increasing
velocity until the angle (a) between the leading surfaces and their direction of
motion is reduced to such a value that the net propulsive thrust is exactly equal
and opposite to the effect of the frictional forces acting on the body.
It will be remembered that the value of 6 varies for different phases of the
movement of a single segment (see Fig. 14) and for different regions of the body
(Fig. 14). For regions lying towards the middle of the body of the fish shown in
Fig. 14 the value of 8m (when the segment is passing the longitudinal axis of movement) is about 500. As the segment moves away from this axis the value increases
to 900, and as this increase occurs, the transverse velocity falls. It is clear from
Fig. 21 that a rapid fall probably occurs in the thrust and in the work done as the
value of 8 increases—and for high values, a very weak and very inefficient thrust
remains. It has also been shown that the value of 8 varies for different regions along
the body of the fish. The thrust and therefore the useful work done by a segment
thus depend on its position in the body as well as on the particular phase of its own
cycle, so that the total thrust exerted by the whole fish represents the sum of the
thrusts exerted by all the segments of the body, all of which have different values
of 8 and may have different values of a. Before examining these phenomena in
greater detail it is convenient to consider the relationship which exists between
the values of the angles 8 and a on the one hand and the form and properties of the
muscular waves, which pass from one segment to another, on the other. This will be
done in a subsequent paper.
The analysis given above assumes that the relative velocity of the body and the
surrounding water is the resultant of the transverse velocity of the body and its
forward velocity through the water; in other words, that the water which encounters
the body is at rest relative to the earth. It is unlikely that this condition is
strictly fulfilled, since the anterior regions of a fish such as that of a mackerel may
influence the rate of flow of the water past the segments lying more posteriorly; in
this case the velocity with which these segments encounter the water may not be
Studies in Animal Locomotion
103
simply the resultant of the transverse and longitudinal motions of the fish itself. At
present there is no means of determining the exact flow of water past the fish, and it
is necessary to assume that disturbances of this type are comparatively small.
III. SUMMARY.
1. The waves of muscular contraction which pass along the body of a swimming
eel occur also in other fish. The waves vary greatly in speed of propagation,
amplitude and frequency. The speed of propagation of the waves is too low to be
controlled by the rate of conduction of a simple nervous impulse.
2. The movements executed by a localised area on the surface of the body are
such that each area moves in a direction transverse to the line of forward movement.
During these movements the leading surface of the body is inclined backwards
towards the tail and at an angle to the path of motion of the area concerned. The
angle of inclination and the angle made with the path of motion vary with (a) different regions of the body, and (b) with different phases in the motion of each region.
3. Each point on the body travels in a horizontalfigureof 8 relative to a transverse
axis which is moving forward at the same average velocity as the whole fish. A segment of the body at the mid-point of its transverse motion is travelling forwards at a
rate slightly less than that of a segment at the extreme position of its transverse movements. These movements are the mechanical result of the inextensibility of the body,
and they effect significant changes in the angle between the surface of the body and
its direction of movement.
4. The movements of each part of the body are shown to be such as to generate
a forward thrust which drives the fish forwards against the resistance of the water.
The magnitude of the forward thrust depends among other things on (a) the angle
which the surface of the fish makes with its own path of motion, and (b) on the
angle between the surface of the fish and the axis of forward movement of the whole
fish, (c) on the velocity of transverse movement of the body.
5. The propulsive properties of each segment of the body are greatest as the
segment is crossing the axis of forward movement.
REFERENCES.
BREDER, C. M. (1926). Zoologica, 4, 159.
GRAY, J. (1930). Proc. Roy. Soc. B, 107, 313.
MAREY, E. J. (1894). he Movement. Paris.
PETTIGREW, J. B. (1873). Animal Locomotion. London.
EXPLANATION OF PLATES.
PLATE I.
Fig. 2. Successive positions of a young eel (Anguilla vulgaris) (7 cm. long) during a period of 1 sec.
The photographs were taken at 009 sec. intervals. The side of each square is 1 in. The passage of
the muscular waves is marked by black dots and crosses. The dark line represents the pigmented
mid-dorsal line of the transparent animal. Note the well-defined curvature of the body.
104
J-
GRAY
Fig. 3. Successive positions of a butterfish (C. gunnelltu) in i sec. The photographs were taken at
0-05 sec. intervals. The side of each square ia 3 in. Note the 9mall amplitude of transverse movement
of the head.
Fig.4. Successive positions of a butterfish in J sec. The photographs were taken at 005 sec. intervals.
The side of each square is 1 in. The passage of the waves is marked by dots or by crosses. Note
that the tail is almost at right angles to the path of motion of the fish when it is crossing the
longitudinal axis in photographs 1 and 10.
PLATE II.
Fig. 5. Successive positions of a mackerel (Scomber tcombrui) within a period of 0035 sec. The
interval between each photograph was 005 sec. and the grid shown has 3 in. squares. The grid has
been inked over in the photograph—note the disturbance of the water in the neighbourhood of the
fish. Note also the rapid rate of propagation of the muscular waves and the high forward speed of
the fish.
Fig. 6. Successive positions of a whiting (Gadus merlangus) within a period of 05 sec. Interval
between each photograph 005 sec. Scale 3 in. The wave crests are marked by dots. Note that the
pitch angle of the tail is greater than that of the butterfish (Fig. 4), but less than that of the mackerel
(Fig. 5) or of Ammodytes (Fig. 7).
Fig. 7. Successive positions of a sand-eel (Ammodytes lanccolatus) within a period of 0 5 sec. The
interval between each photograph is approx. 005 sec. Scale 3 in. Note the relatively high pitch of the
body as compared to the eel shown in Fig. 2. Note also the much greater forward velocity in comparison to Fig. 2. The forked appearance of the tail is due to the shadow cast on the bottom of the
tank.
PLATE III.
Fig. 8. Dogfish. Note the large amplitude of the movements of the body and tail. A wave crest is
marked by a black dot in photographs 3-8. Interval between the photographs 010 sec. Scale 3 in.
Fig. 9. Rockling (Onos). Note that the transverse movements are almost completely confined to the
tail. Note that the angle of inclination (9) of the body is distinctly steeper than in the dogfish and
that the frequency of the movements is higher. Interval between the photographs 005 sec. Scale
3 in.
PLATE IV.
Fig. 10. An eel moving backwards. In 10 sec. the fish has moved back about 3 in. Note the passage
of the waves from the tail towards the head of the fish: note also the large amplitude of the waves.
Interval between each photograph approx. o-1 sec.
Fig. 11. A young glass-eel which is stationary and yet exhibits curvature of the body. Compare with
photograph 4 in Fig. 2. The form of the waves is approximately the same as when the waves are
moving and the fish is in motion. Total period 05 sec. Scale 1 in.
PLATE I.
JOURNAL OF EXPERIMENTAL ISIOLOGY, X, i-
1
2
31
41
51
61
71
8
9
10'
11
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1
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5
i
6 ii 7
i 8 ,
9
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Fig. 4»
GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).
11
JOURNAL OF EXPERIMENTAL RIOLOGY, X, i.
Fig. 7.
GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).
PLATE II.
JOURNAL OF EXPERIMENTAL BIOLOGY, X, i
PLATE III.
12
00
GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).
JOURNAL OF EXPERIMENTAL BIOLOGY, X, i.
PLATE IV.
Fig. 10.
iV
d
3
•
•
4
i
5
6
7 .
i
Fig.
Ii.
GRAY—STUDIES IN ANIMAL LOCOMOTION (pp. 88—104).