Hydrodynamics of surface swimming in leopard frogs (Rana pipiens)
1 Dept of Theoretical Ecology, Lund University, Ecology Building, SE-223 62
Lund, Sweden
2 Dept of Organismic and Evolutionary Biology, Harvard University, 26 Oxford
Street, Cambridge, MA 02138, USA
* Author for correspondence (e-mail: christoffer.johansson{at}teorekol.lu.se)
Accepted 23 August 2004
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: frog, Rana pipiens, anuran, amphibian, DPIV, digital particle image velocimetry, paddling, kicking, swimming, locomotion, kinematics, hydrodynamics, vortex ring, limb, foot
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Both jumping and swimming are important escape behaviors in frogs, and some
studies have suggested the possibility of co-adaptation between jumping and
swimming such that locomotion in any one environment is a compromise resulting
from the requirement that frogs should be able to move effectively in both
environments (Emerson and De Jongh,
1980; Nauwelaerts and Aerts,
2002
). Some recent work has suggested differences in the kinematic
pattern between jumping and swimming in frogs
(Nauwelaerts and Aerts, 2003
),
indicating the possibility of adaptive behaviors associated specifically with
swimming. One basis for differences between jumping and swimming performance
may be found in the different physical properties of the ground and the water
as platforms for force production. While most terrestrial substrates are
effectively not deformable and are almost unaffected by forces exerted on
them, water yields when pushed upon, resulting in a different resistance to
the limb motion. In water, the force produced, and thereby the resistance to
the motion of the feet, depends strongly on the actual kinematics of the
kicking feet. To get an understanding of the force production during limb
motion in water and to understand the effect of previously recorded muscle
activation patterns, it is necessary to study the hydrodynamic basis of the
force production and to measure directly forces produced by swimming animals.
In other words, it is necessary to study the flow around the feet and in the
wake of the swimming frogs to better understand the mechanisms of force
production in the water to provide a basis for comparisons with terrestrial
locomotion.
Although swimming frogs may utilize two distinct gaits, asynchronous and
synchronous kicking (Calow and Alexander,
1973; Nauwelaerts and Aerts,
2002
), most previous research has focused on the more powerful
synchronous kicks. This focus has been motivated by the desire to compare
swimming and jumping in frogs as both legs move together during a terrestrial
jump. However, the use of two gaits suggests the possibility of different
hydrodynamic mechanisms being used for asynchronous and synchronous kicks, and
one goal of the present study is to evaluate experimentally and compare the
hydrodynamics of both types of kicks in leopard frogs swimming in the water.
In addition, previous work on aquatic locomotion in frogs has suggested
specific hypotheses that are amenable to test using experimental fluid
mechanical analyses of frog swimming. Gal and Blake
(1988b
) suggested that the
first part of the propulsive kick is based on drag (fluid resistance to
motion) and acceleration reaction (fluid resistance to acceleration) and is
followed by a phase when a jet is produced as the feet come together at the
end of the power stroke. This hypothesis was proposed because of a mismatch
between the forces required to accelerate the frog and the result of a
blade-element analysis of the drag and acceleration reaction forces on the
feet. An alternative hypothesis, which could explain the observed acceleration
pattern, is that the feet function as suggested for the feet of swimming birds
(Johansson and Norberg, 2003
).
In this case, the feet could initially produce thrust using a drag- and
acceleration-reaction-based mechanism followed by a lift-based (force
generation perpendicular to the motion of the feet) phase. Predictions from
this hypothesis include a relatively early shedding of a start vortex and
relatively large vortex rings being generated. This mechanism was suggested
only recently for birds and has not yet been examined in frogs.
The hydrodynamics of swimming frogs have previously been studied
indirectly, using kinematic information to infer the mechanics of the force
production (Gal and Blake,
1988a,b
).
Here, we use digital particle image velocimetry (DPIV), a technique that we
have previously applied to study locomotor hydrodynamics in fishes (e.g.
Drucker and Lauder, 1999
,
2002
;
Lauder and Drucker, 2002
;
Tytell and Lauder, 2004
),
which makes it possible to measure quantitatively the flow around the kicking
feet and in the wake of swimming frogs. Our aim is thus to test the
hydrodynamic hypotheses outlined above and to describe the wake and vortex
generation by surface-swimming leopard frogs using the two gaits.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Kinematics
Three-dimensional kinematics of swimming frogs were quantified by
digitizing (using Scion Image; Scioncorp,
www.scioncorp.com)
morphological marks on the feet of frogs from high-speed video images of frogs
swimming in a flow tank. The flow was kept at a constant speed of
approximately 0.076 m s1. The filming was conducted using
two synchronous high-speed video cameras (Redlake Motionscope PCI 500;
Redlake, San Diego, CA, USA) operating at 250 frames s1. The
video sequences were taken with one of the cameras filming, though a mirror,
the ventral view of the frog while the other camera filmed, synchronously,
either the lateral view or, through a mirror placed downstream at 45° in
the flowtank, the caudal view. The points digitized were the tips of all toes,
the heels and the ankles. Kinematic data from the two views were then
corrected for any parallax error using direct linear transformation (DLT).
Owing to a slight shift in the rear-view mirror between the calibration images
and the kinematics images, the rear-view calibration image was shifted to
compensate for the misalignment. The actual shift was determined by the visual
shift between the images as well as by a minimization of the residual error of
a number of test points. The median residual error in the data after the DLT
was 0.85 mm, which includes digitization error.
In addition to the detailed kinematics of the motion of the feet, the average coordinates of three marks on the body of the frogs and the tip of the third toe, from a more zoomed-out view, were used to calculate swimming velocity and foot velocity from representative sequences. Body and foot velocity were calculated, using Matlab 6.5 (MathWorks Inc., Natick, MA, USA), as the first derivative of a smoothing spline of the timeposition curve of the average body coordinate and the tip of the third toe, respectively.
Digital particle image velocimetry (DPIV)
DPIV is a technique where the motion of small, reflective particles,
suspended in a fluid (for example air or water), is estimated by analyzing the
changes in the intensity pattern of small sub-samples of, in our case,
successive digital images (e.g. Drucker and
Lauder, 1999; Lauder
2000
; Lauder et al.,
2002
; Nauen and Lauder,
2002
). The result is presented as velocity vectors distributed
homogeneously over the analyzed image. Based on these vectors, it is then
possible to calculate relevant parameters, such as vorticity and circulation.
We used hollow, silver-coated, glass beads as reflective particles, as in
previous research. To keep the particles suspended and to increase the time
the frogs spent within the field of view, we used a flow tank (mean flow
velocity of 0.093 m s1 for the analyzed sequences). The
particles were illuminated using an 8 W continuous-wave argon-ion laser
(Coherent Inc., Santa Clara, CA, USA) focused into a thin light sheet.
Preliminary analyses indicated high flow velocities close to the feet and in
the vortex ring jets. Therefore, for all experiments we used a high-resolution
(1280x1024 pixels at 8-bit gray scale), high-speed video system (NAC
Hi-Dcam II, NAC Image Technology, Simi Valley, CA, USA) filming at 500 frames
s1 to obtain high-resolution sequences of flow around the
feet. Despite the high frame rate, displacement between images was sometimes
suboptimal, resulting in underestimates of the force generated. To avoid
blurring, the shutter speed was set as fast as the available light allowed
(1/1000 s). As a consequence of the high film rate and short shutter speeds,
the light available was barely enough for filming, and the camera's electronic
gain was turned up. This, however, resulted in noticeable static image noise.
The noise was removed before the PIV analysis by subtracting the average image
intensity of hundreds of frames. The resulting image was then adjusted to
obtain an image intensity where the particles were clearly visible. All image
processing was performed in custom programs written in Matlab.
The main source of propulsive force is from the kicking of the hind limbs
(Peters et al., 1996), and we
therefore concentrated our study on the flow around the feet of the frogs. We
chose to study only the propulsive kick of swimming frogs, as this phase has
previously been shown to be the determinant of maximum and average swimming
speed (Nauwelaerts et al.,
2001
). Previous studies have also indicated two different modes of
locomotion in swimming frogs: asynchronous and synchronous kicking
(Calow and Alexander, 1973
;
Nauwelaerts and Aerts, 2002
).
We studied both types of locomotion. To obtain an understanding of the
three-dimensional structure of the flow behind the frogs, we ran two sets of
experiments: one with a horizontal light sheet to image the xz plane
(filming from the ventral view) and one with a vertical light sheet to image
the xy plane (filming from the lateral view); see
Fig. 1. To minimize image
disturbance, due to reflections of light at the surface, only sequences where
the vortex ring did not interact significantly with the surface were used for
the vertical light sheet. The same control was not possible for the horizontal
light sheet, but the light sheet was positioned well below the surface. Thus,
interaction between the feet and the surface was minimized to the extent
possible experimentally, although we cannot rule out some possible surface
effects on the initial formation of the vortex rings. In addition to the
camera used for the DPIV, we used a Redlake camera to simultaneously film a
zoomed-out ventral view during DPIV experiments. This camera was directed
further upstream to allow us to determine the swimming velocity of the frogs
for most of the sequences. For the DPIV sequences where a ventral view from
the Redlake camera was available, we calculated the maximum swimming velocity
at the end of the power stroke. The velocity was calculated from the slope of
the timeposition curve of a well-defined mark on the body over four
images, starting when the feet reached their most caudal position relative to
the body. To calculate the actual swimming velocity, we then added the flow
velocity of the tank.
|
The PIV analysis was conducted using a code written by E. Tytell for Matlab
v.6.5 (Tytell and Lauder,
2004). The analysis was done by cross-correlation of successive
images following the procedures of Hart
(2000
), with an initial search
window size of 64x64 pixels and a final window size of 32x32
pixels with a 50% overlap. This resulted in 5120 vectors per image. The
vectors were smoothed using an adaptive Gaussian smoothing routine, based on
an optimal window size calculated according to Agui and Jimenez
(1987
). Ten image pairs, from
after the completion of the kick, from each sequence were analyzed, resulting
in 10 vector fields per sequence. The foot or feet were often still in the
image, and sometimes the DPIV analysis picked up the motion of the feet,
generating vectors going opposite to what would be expected from the flow.
While this proved useful in allowing us to directly calculate the velocities
of different parts of the feet during the kick using DPIV, smoothing such
images resulted in artefactual vector patterns for nearby water flows.
Therefore, the vector field (of the 10 available for each sequence) showing
the least amount of disturbances was chosen for further analysis. For the
figures shown in this paper, a number of sequences were analyzed in full to
illustrate the flow field close to the feet and the development of the flow
during the kick.
To calculate the propulsive force, we determined the size and circulation
of the vortex rings produced by the feet. The location and circulation of each
vortex center was measured, after subtracting the mean flow vector from the
vector field [in Tecplot 9.1 (Tecplot Inc., Bellevue, WA, USA) using
custom-written software]. The area over which the vorticity was integrated to
estimate the circulation was determined by finding the area that maximized the
circulation within a limit set by the researcher. Assuming a circular vortex
ring structure, with a diameter determined by the distance between the vortex
centers, we calculated the momentum (circulation multiplied by the area of the
ring multiplied by the density of the fluid) of the flow generated by the feet
(see Drucker and Lauder, 1999).
A circular vortex structure is a reasonable estimate, as the radius of the
vortex ring in the vertical and horizontal views did not differ for either
asynchronous or synchronous kicks [general linear model (GLM);
Palt=0.471, Psync=0.939]. To be able
to compare our results with those in the literature, we calculated the
mass-specific momentum produced during the kick. The mean force during the
kick was calculated by dividing the momentum by the duration of the kick (e.g.
Drucker and Lauder, 1999
). Jet
velocity was calculated as the mean velocity of the vectors in an area between
the vortex centers. Jet angle relative to ring angle was calculated as the
difference between the mean direction of the jet and the direction of the ring
plane, determined from the position of the vortex centers. Ring angle relative
to forward swimming direction was determined according to
Fig. 1.
Statistical analyses were conducted in SPSS 12.0. We used GLM on loge-transformed variables, with `individual' as a random factor to account for individual variation. Variables were tested for equality of variance using Levene's test, and the residuals were tested for normality using the KolmogorovSmirnov Z-test.
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
|
|
|
The velocity profile of the body and the tip of the third toe from an asynchronous and a synchronous kick are shown in Fig. 5. The average maximum swimming speed differs between gaits and, for the sequences used for DPIV, the speed is approximately 3.5 times as high during the synchronous kicks as during the asynchronous kicks. During synchronous kicks, the speed of the frog increases steeply during the first portion of the kick. However, shortly after the feet start sweeping medially, at the last part of the kick, the forward velocity starts to decrease. At this point, the momentary force produced is less than the inertial and hydrodynamical resistance of the body. During asynchronous kicks, the point at which the body starts decelerating occurs earlier during the kick, suggesting that the force produced is less relative to the resistance than during synchronous kicks. This is reasonable as during asynchronous kicks the alternate leg is in recovery phase (and hence moving forward) after the previous kick and thereby increases the resistance of the frog.
|
DPIV data from asynchronous kicks are presented as vector plots on top of the original video images at four different times during the kick as seen from the ventral (Fig. 6) and lateral (Fig. 7) views. In the initial phase of the kick, as seen from the ventral view, water is sucked along with the foot, and a shear layer builds up. As the kick progresses, the shear layer rolls up into an attached vortex around the edges of the feet. When the feet start sweeping medially, the distal part of this vortex is shed and, shortly thereafter, the medial part of the ring becomes visible and is shed. After the completion of the kick, each foot has produced individual vortex rings that, in the horizontal plane, are almost perpendicular to the swimming direction (Fig. 6; Table 1). In lateral view, the pattern is similar but there is a strong tendency for the lower vortex center to be shed before the upper vortex center. This results in a tilted vortex ring, with a downward-directed jet and thus an upward-directed force acting on the frog.
|
|
DPIV data for synchronous kicks are presented as vectors on top of the calculated vorticity of the flow at three different stages of representative kicks as seen from the ventral view (Fig. 8) and the lateral view (Fig. 9). There is also a video available online as supplementary material showing the vectors superimposed on the original images of a synchronous kick, with the vectors color coded by vorticity. The buildup and shedding of the vorticity follow essentially the same patterns as for the asynchronous kick. There are two areas with counter-clockwise and clockwise vorticity linked by an area of high rearward velocity in between, indicating the generation of separate vortex rings for each foot. In the ventral view, the feet are still in the image (Fig. 8). No central rearward-directed jet between the feet is visible in the ventral view (Fig. 8). A velocity profile through the wake (Fig. 10) clearly shows two distinct jets, one for each foot; there is no single central jet. In fact, the velocity profile through the two vortex rings suggests a forward flow between the feet, not a rearward flow (Fig. 10).
|
|
|
Thrust (force in x-direction; Fx), but not mass-specific momentum (M), shows a significant positive relationship with swim speed (GLM with velocity as covariate, PFx<0.005, PM=0.479; Fig. 11A,B). Furthermore, speed (U) and force, but not mass-specific momentum, are significantly different between asynchronous and synchronous kicks, with higher values for synchronous kicks (GLM, PU<0.001, PF<0.05, PM=0.320). Mass-specific momentum shows no correlation with the duration of the kick (GLM with stroke duration as covariate, PM=0.931; Fig. 11C).
|
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The frogs in our study appear to follow a different acceleration pattern
for synchronous kicks than the frogs described by Gal and Blake
(1988a,b
).
In their study, they suggest that frogs accelerate throughout the entire kick
(power stroke). This could be due to interspecific differences in locomotor
patterns as they studied Hymenochirus boettgeri while our study
focused on Rana pipiens. However, examination of the plots of speed
versus time in Gal and Blake
(1988a
,b
)
suggests that a slight deceleration is present at the end of the kick in
Hymenochirus. More importantly, however, is that in
Hymenochirus the velocity of the feet, at mid power stroke, was
forwards relative to the still water, resulting in negative drag-based thrust
(Gal and Blake,
1988a
,b
).
This is different from what is seen in our frogs, as reflected in the data
shown in Fig. 5. Based on a
blade element analysis of the feet of Hymenochirus, Gal and Blake
(1988b
) concluded that the
thrust produced by drag and accelerative forces on the feet was insufficient
to explain the observed acceleration pattern of the frog and suggested a
squeezing jet being produced at the end of the kick. Although the velocity of
the feet in our study is backwards relative to the water during the entire
acceleration phase of the frog, our results suggest that a jet produced at the
end of the kick may not be a reasonable explanation for the observed
acceleration pattern in Hymenochirus. On the other hand, the
kinematics of the fully aquatic frog Hymenochirus, presented by Gal
and Blake (1988b
), suggest the
possibility of a more substantial use of a lift-based mechanism than do our
results. In Hymenochirus there is a continuous change of the angle of
the foot relative to the swimming direction during the kick and a relatively
short duration of a rearward motion of the feet relative to the water (less
than 50% of the power stroke) while the feet continue to sweep medially
throughout the kick, suggesting a larger dependency on lift-based forces. The
foot shape in Hymenochirus is more symmetrical than the feet of
R. pipiens, which are more skewed. The feet of Hymenochirus
are also held in a more lateral position that makes them more symmetrical
relative to the motion of the feet (L.C.J. and G.V.L., personal observations).
Although this triangular shape of the feet is relatively efficient for
drag-based propulsion (Blake,
1981
), it also suggests the possibility of a delta wing mechanism
(Johansson and Norberg, 2003
).
The use of a lift-based mechanism in Hymenochirus thus seems more
reasonable than the formation of a jet at the end of the kick.
Gal and Blake (1988b) also
suggested an interaction between the feet at the end of the kick as an
alternative to the central jet hypothesis but did not further explain how such
an interaction would function hydrodynamically. We do find some support for a
possible interaction between the vortex rings produced by the two feet, as we
observed significant positive (forward) velocities in the region between the
two vortex rings (Fig. 10). This could be due to fluid dynamic interactions between the vortex rings
resulting from vorticity from the ring produced by one foot enhancing the
opposite sign vorticity of the ring produced by the other foot. The forward
velocity between the two vortex rings is higher than the forward velocity on
the outside of the two rings (Fig.
10). However, this may alternatively be an effect of the
interaction between the drag wake of the frog body and the two vortex rings
from the feet. Either way, this may affect the development of the rings as
well as the strength of the rings and thereby the force produced. The
consequence of such an interaction is not clear at this point, but we have not
observed any merging of the vortex rings, and the rings produced by the feet
are approximately one ring diameter apart when they are shed into the
wake.
The kinematics of swimming frogs have been used to estimate forces during
swimming, and a few issues regarding these kinematic data are worth noting. A
different leg configuration between terrestrial and fully aquatic frog species
has been noticed, with a more lateral knee position observed among aquatic
frogs (Nauwelaerts and Aerts,
2003). The aquatic position is mimicked by semi-aquatic frogs when
swimming (also described by Peters et al.,
1996
), suggesting some hydrodynamic significance. One reasonable
explanation is that changing knee position reduces pitching moments, making
the frog swim along a straighter path. This lateral positioning of the legs
with the leg segments in approximately the same plane has prompted the
suggestion that the top view alone is adequate for studies of limb kinematics
(Peters et al., 1996
).
However, our three-dimensional kinematic analysis of the feet
(Fig. 4) shows that additional
views of foot and toe motion are very helpful in obtaining a more detailed
understanding of what happens with the hindlimb and foot kinematics. Another
kinematic feature worth noting is that the legs straighten more during
swimming than during jumping (Peters et
al., 1996
), although the angular joint excursions are generally
similar. The ankle joint is different, however, and goes through a larger
angular excursion during swimming. Also, Kamel et al.
(1996
) showed that limb EMG
patterns suggest a longer activation of the M. plantaris longus, the muscle
responsible for the straightening of the ankle joint. Together with our
results, this suggests that the sweeping motion of the feet at the end of the
kick is of hydrodynamic importance. At least two different reasons for this
are plausible. First, bringing the feet into a streamlined position behind the
body of the animal reduces drag during the glide phase. Second, the sweeping
motion may facilitate the shedding of the attached vortex and reduce adverse
effects of decelerating the feet at the end of the kick.
Several studies have compared swimming and jumping in frogs and found that
the forces produced during swimming are less than the forces produced during
jumping (Calow and Alexander,
1973; Kamel et al.,
1996
; Nauwelaerts and Aerts,
2003
). However, these results are based on calculations of the
forces from the movements of the bodies of the frogs. The data for jumping
have been validated using force plates
(Nauwelaerts and Aerts, 2003
),
and our data can be used to validate the calculations for swimming. The
mass-specific momentum of synchronous kicks calculated here is slightly lower
than those previously calculated from kinematics of fully submerged swimming
frogs (Fig. 11C) and clearly
lower than the values for jumping
(Nauwelaerts and Aerts, 2003
).
Our result thus supports these previous comparisons of force magnitudes of
aquatic and terrestrial limb motion. Also, electromyogram (EMG) data indicate
more powerful kicks during jumping, with twice as strong EMG signals for
jumping compared with swimming, although no correlation between EMG intensity
and swim velocity or jumping distance could be found
(Kamel et al., 1996
).
Nauwelaerts and Aerts (2003
)
set up a number of hypotheses to explain the difference in momentum generated
during jumping and swimming. One of their hypotheses suggests large lateral
forces during swimming, but our results indicate that the lateral forces are
small during swimming (Table 1) as foot vortex rings generate minimal side forces (in contrast to data from
fish fins in which side forces can be larger than thrust forces;
Drucker and Lauder, 1999
).
In agreement with previous results, we find that asynchronous kicking
results in a significantly lower swimming speed than synchronous kicking
(Ostry et al., 1991;
Nauwelaerts and Aerts, 2002
).
Here, we propose, based on the physical properties of water and the resistance
to body movement close to the surface, that the non-overlapping swim speeds
between the two gaits may, in surface swimming, be a consequence of wave
resistance. When swimming at the surface, drag is increased substantially due
to the formation of waves. The wave resistance is a function of the Froude
number, which is a measure of the speed (U) relative to the gravity
(g) and the length (l) of an object
(Hoerner, 1965
). In our study,
the frogs swam below hull speed (the speed where the wave length of the bow
wave is the same as the length of the hull, which is often seen as the
`maximum' speed of a displacement hull) when using asynchronous kicks.
Swimming speed during synchronous kicks was substantially higher than the hull
speed, calculated from the outstretched length of the frog at the end of the
kick. Furthermore, the Froude number calculated according to Hoerner
(1965
)
[Fr=U/(gl)0.5] suggests that the frogs
swim at such a high speed that waves do not have time to form, resulting in a
lower wave drag than at speeds close to hull speed
(Fig. 12). Thus, the fast
acceleration and high speed resulting from the synchronous kick enable frogs
to make a faster escape than would have otherwise been the case.
|
Although we have used traditional, steady-state, fluid-dynamic terminology
to illustrate different hypotheses for the force generation, the force
generation in swimming frogs does not seem to be fully explainable in these
terms. Instead, the force is, as in many other biological systems, generated
by attached vortices and/or vorticity centers in close proximity to the
propulsive appendage (e.g. Ellington et
al., 1996; Birch and Dickinson,
2001
; Usherwood and Ellington,
2002
). The exact consequence of limb kinematics on the generation
and efficiency of the vortex-based propulsive system in frogs is still unclear
and needs to be tested using mechanical or computational models that permit
relatively easy manipulation of kinematic parameters. This is especially true
for systems such as frog limbs that involve strong acceleration and
deceleration within the time period of the force generation.
![]() |
Acknowledgments |
---|
![]() |
Footnotes |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Agui, J. C. and Jimenez, J. (1987). On the performance of particle tracking. J. Fluid Mech. 185,447 -468.
Birch, J. M. and Dickinson, M. H. (2001). Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412,729 -733.[CrossRef][Medline]
Blake, R. W. (1981). Influence of pectoral fin shape on thrust and drag in labriform locomotion. J. Zool. Lond. 194,53 -66.
Calow, L. J. and Alexander, R. McN. (1973). A mechanical analysis of a hind leg of a frog (Rana temporaria). J. Zool. Lond. 171,293 -321.
Daniel, T. L. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24,121 -134.
Drucker, E. G. and Lauder, G. V. (1999).
Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics
quantified using digital particle image velocimetry. J. Exp.
Biol. 202,2393
-2412.
Drucker, E. G. and Lauder, G. V. (2002). Experimental hydrodynamics of fish locomotion: functional insights from wake visualization. Int. Comp. Biol. 42,243 -257.
Ellington, C. P., van den Berg, C., Willmott, A. P. and Thomas, A. L. R. (1996). Leading-edge vortices in insect flight. Nature 384,626 -630.[CrossRef]
Emerson, S. B. and De Jongh, H. J. (1980). Muscle activity at the ilio-sacral articulation of frogs. J. Morphol. 166,129 -144.
Gal, J. M. and Blake, R. W. (1988a). Biomechanics of frog swimming. I. Estimation of the force generated by Hymenochirus boettgeri. J. Exp. Biol. 138,399 -411.
Gal, J. M. and Blake, R. W. (1988b). Biomechanics of frog swimming. II. Mechanics of the limb-beat cycle in Hymenochirus boettgeri. J. Exp. Biol. 138,413 -429.
Gillis, G. B. and Biewener, A. A. (2000).
Hindlimb extensor muscle function during jumping and swimming in the toad
(Bufo marinus). J. Exp. Biol.
203,3547
-3563.
Gillis, G. B. and Blob, R. W. (2001). How muscles accommodate movement in different physical environments: aquatic vs. terrestrial locomotion in vertebrates. Comp. Biochem. Physiol. A 131,61 -75.
Hart, D. P. (2000). Super-resolution PIV by recursive local-correlation. J. Visual. 3, 187-194.
Hoerner, S. F. (1965). Fluid-Dynamic Drag. Bakersfield, CA: Hoerner Fluid Dynamics.
Johansson, L. C. and Norberg, R. Å. (2003). Delta-wing function of webbed feet gives hydrodynamic lift for swimming propulsion in birds. Nature 424, 65-68.[CrossRef][Medline]
Kamel, L. T., Peters, S. E. and Bashor, D. P. (1996). Hopping and swimming in the Leopard frog, Rana pipiens: II. A comparison of muscle activities. J. Morphol. 230,17 -31.[CrossRef][Medline]
Kargo, W. J., Nelson, F. and Rome, L. C.
(2002a). Jumping in frogs: assessing the design of the skeletal
system by anatomically realistic modeling and forward dynamic simulation.
J. Exp. Biol. 205,1683
-1702.
Kargo, W. J. and Rome, L. C. (2002b). Functional morphology of proximal hindlimb muscles in the frog Rana pipiens. J. Exp. Biol. 205,1987 -2004.[Medline]
Lauder, G. V. (2000). Function of the caudal fin during locomotion in fishes: kinematics, flow visualization, and evolutionary patterns. Am. Zool. 40,101 -122.
Lauder, G. V. and Drucker, E. (2002). Forces, fishes, and fluids: hydrodynamic mechanisms of aquatic locomotion. News Physiol. Sci. 17,235 -240.[Medline]
Lauder, G. V., Nauen, J. and Drucker, E. G. (2002). Experimental hydrodynamics and evolution: function of median fins in ray-finned fishes. Int. Comp. Biol. 42,1009 -1017.
Lutz, G. J. and Rome, L. C. (1994). Built for jumping: the design of the frog muscular system. Science 263,370 -372.[Medline]
Marsh, R. L. (1994). Jumping ability of anuran amphibians. Adv. Vet. Sci. Comp. Med. 38B,51 -111.
Marsh, R. L. and John-Alder, H. B. (1994).
Jumping performance of hylid frogs measured with high-speed cine film.
J. Exp. Biol. 188,131
-141.
Nauen, J. C. and Lauder, G. V. (2002).
Hydrodynamics of caudal fin locomotion by chub mackerel, Scomber
japonicus (Scombridae). J. Exp. Biol.
205,1709
-1724.
Nauwelaerts, S. and Aerts, P. (2002). Two distinct gait types in swimming frogs. J. Zool. Lond. 258,183 -188.
Nauwelaerts, S. and Aerts, P. (2003).
Propulsive impulse as a covarying performance measure in the comparison of the
kinematics of swimming and jumping in frogs. J. Exp.
Biol. 206,4341
-4351.
Nauwelaerts, S., Aerts, P. and D'Août, K. (2001). Speed modulation in swimming frogs. J. Motor Behav. 33,265 -272.
Ostry, D. J., Feldman, A. G. and Flanagan, J. R.
(1991). Kinematics and control of frog hindlimb movements.
J. Neurophys. 65,547
-562.
Peters, S. E., Kamel, L. T. and Bashor, D. P. (1996). Hopping and swimming in the Leopard frog, Rana pipiens: I. Step cycles and kinematics. J. Morphol. 230,1 -16.[CrossRef][Medline]
Tytell, E. D. and Lauder, G. V. (2004).
Hydrodynamics of eel swimming. I. Wake structure. J. Exp.
Biol. 207,1825
-1841.
Usherwood, J. R. and Ellington, C. P. (2002).
The aerodynamics of revolving wings. II. Propeller force coefficients from
mayfly to quail. J. Exp. Biol.
205,1565
-1576.