## SUMMARY

The dynamics of pectoral fin rowing in the threespine stickleback are investigated by measuring the instantaneous force balance on freely swimming fish throughout the stroke cycle and comparing the measured forces with fin motions and an unsteady, blade-element model of pectoral fin propulsion. Both measured and modeled forces suggest that attached vortex and circulatory forces and not inertial (added mass) forces dominate the force balance. Peak forces occur at midstrokes. There is no evidence for large force peaks at the stroke transitions due to either rapid fin rotation (supination) or rapid fin closure against the body. The energetics of pectoral fin rowing are estimated using the unsteady blade-element model and an indirect method based on the center of mass dynamics. The results indicate that the mechanical efficiency of pectoral fin rowing is low (0.1–0.3) relative to a flapping mechanism and possibly relative to axial undulation at comparable speeds.

- locomotion
- unsteady fluid dynamics
- energetics
- blade-element analysis
- circulatory force
- acceleration reaction
- mechanical power
- mechanical efficiency

## Introduction

Pectoral fin motions for propulsion and maneuvering are highly variable among fishes, but at least some of this variation can be summarized by an axis in which fore–aft rowing characterizes one extreme while dorso-ventral flapping characterizes the opposite extreme (Breder, 1926; Lindsey, 1978; Webb and Blake, 1985; Walker and Westneat, 2002a,b). The geometry of pectoral fin motion has important dynamic consequences (the magnitude of lift and thrust generated throughout the stroke cycle and how much energy is wasted in the generation of thrust) that should affect how a fish exploits resources in its environment (Walker and Westneat, 2000). In order to explore the performance consequences of pectoral fin motion in more detail, we need good comparative measures of how different pectoral fin designs generate propulsive forces.

There have been many previous attempts to infer the dynamics of pectoral fin propulsion in free-swimming fishes from an analysis of either the pectoral fin kinematics (Webb, 1973; Blake, 1979, 1980; Geerlink, 1983; Archer and Johnston, 1989; Gibb et al., 1994; Arreolla and Westneat, 1996; Gordon et al., 1996; Lauder and Jayne, 1996; Drucker and Jensen, 1997; Walker and Westneat, 1997, 2002a; Hove et al., 2001; Ramamurti et al., 2002) or the pectoral fin's wake geometry (Drucker and Lauder, 1999, 2000, 2001, 2003). Investigations of pectoral fin energetics are few and have relied on either oxygen consumption measures (Webb, 1974; Gordon et al., 1989; Korsmeyer et al., 2002) or quasi-steady blade-element models of real (Blake, 1979, 1980) or theoretical fish (Walker and Westneat, 2000).

Blake's model of the dynamics and energetics of pectoral fin rowing (Blake, 1979, 1980) has been influential in the aquatic locomotion literature. The major conclusions of the model are that (1) while the acceleration reaction contributes to the work budget, its positive and negative contributions to the mean thrust cancel and (2) the overall mechanical efficiency, η, of the fin is low relative to the efficiency of body-and-caudal (BCF) swimming at preferred swimming speeds but is, perhaps, higher than that of BCF swimming at slow swimming speeds (no estimate of η for BCF swimming at slow speeds was given so this last assertion cannot be evaluated). Unfortunately, there was no attempt to verify the predictions of the model with empirical data and, consequently, the validity of the results remains in question.

The dynamics of pectoral fin propulsion have been inferred indirectly by
analysis of the center of mass kinematics. A qualitative analysis of body
displacement relative to stroke cycle position in the flapping stroke of the
shiner surfperch, *Cymatogaster aggregata*, indicated that thrust is
characteristic of both down- and upstrokes and that positive and negative lift
alternate between strokes (Webb,
1973). The mean acceleration over each halfstroke was used to
estimate the net downstroke thrust and upstroke thrust in the queen coris,
*Coris frerei* (Geerlink,
1983). A net negative thrust was found for the downstroke and a
net positive thrust was found for the upstroke. The instantaneous
accelerations of the body in the bird wrasse, *Gomphosus varius*, were
used to infer the downstroke and upstroke forces
(Walker and Westneat, 1997).
At all speeds, thrust was predominantly generated during the upstroke but some
thrust (between 12% and 22%) was generated during the downstroke. By contrast,
positive lift was generated during the downstroke at all speeds but only at
slow speeds during the upstroke.

Recently, digital particle image velocimetry (DPIV) has been used to
measure the net force over each halfstroke in the bluegill, *Lepomis
macrochirus*, and the black surfperch, *Embiotoca jacksoni*
(Drucker and Lauder, 1999,
2000). In contrast to
previously described results for other fishes, the sunfish at low speeds and
black surfperch at moderate to high speeds generated most of the thrust during
the downstroke. Similar to previous results, lift in both species was positive
during the downstroke and negative (or absent) during the upstroke.

These alternative methods for inferring the dynamics of pectoral fin
propulsion (body displacement and DPIV) have only been applied to fishes that
present more of flapping stroke than rowing stroke (with the exception,
perhaps, of *L. macrochirus*, whose stroke is difficult to place along
a rowing–flapping axis). Similarly, all work on the energetics of
pectoral fin propulsion using oxygen consumption methods has been applied only
to fishes that present more of a flapping stroke
(Webb, 1974;
Gordon et al., 1989;
Korsmeyer et al., 2002).
Consequently, our only data on the dynamics and energetics of the rowing
stroke of fishes are from a blade-element model whose results were not
verified with any empirically measured data, such as center of mass
displacement, oxygen consumption or wake geometry. The dynamics of pectoral
fin rowing were recently investigated with a non-flexing, motor-driven fin
resembling the planform of the pectoral fin of the centrarchid fish,
*Micropterus salmoides* (Kato,
1999). Application of these results to the fins of teleost fishes
is limited, since teleost fins are highly flexible and deform as a consequence
of both elastic and fluid dynamic stresses
(Geerlink, 1983;
Archer and Johnston, 1989;
Gibb et al., 1994;
Lauder and Jayne, 1996;
Westneat, 1996;
Drucker and Jensen, 1997;
Westneat and Walker,
1997).

To rectify this major gap in our understanding of pectoral fin function in
fishes, the dynamics and energetics of a pectoral fin rower, the threespine
stickleback, *Gasterosteus aculeatus*, are presented. This work has
three goals. First, to measure empirically the instantaneous lift and thrust
balance on the body throughout a complete stroke cycle. Second, to apply a
hydrodynamic model to measured fin kinematics in order to estimate the various
contributions and timings of circulatory and added mass forces on net lift and
thrust. The validity of the model is checked by comparing the modeled and
measured estimates of lift and thrust. Third, to estimate the economic
effectiveness of the rowing stroke by estimating its mechanical efficiency,
which is the ratio of the useful to the total work done by the fin on the
water.

Indirect measures of instantaneous thrust and lift generated by the fins are estimated from a force-balance model using the digitized displacement of the center of mass (Walker and Westneat, 1997, 2002a). This indirect measurement relies on few assumptions (see Materials and methods) and allows the measurement of lift and thrust in freely swimming fish. An indirect measure of lift and thrust could be estimated more accurately by tethering an individual to a force transducer, an experimental technique that is common in insects but has never been applied to fishes. Many of the assumptions (e.g. pectoral fin dynamics alone balance weight and drag) of the indirect force measurement method used in this study also apply to direct force measurements from tethered individuals. Similarly, the decomposition of the net lift and thrust into circulatory and added mass components requires a (virtual or physical) model. The major limitations to tethering experiments are (1) simulating a specific speed and (2) inducing the animal to activate the same kinematic patterns at this simulated speed as it would if moving freely at this speed. Direct measurements of circulatory forces on a fin would involve either instrumenting the fin with a series of pressure transducers or measuring the distribution of fluid velocities around the fin using quantitative flow visualization, such as DPIV. Neither method is technologically mature enough to apply to the small, highly deformable fins of the stickleback. DPIV is, with few exceptions (Anderson et al., 2001), restricted to the wake behind a fin and can, consequently, only give a summary (such as the mean lift and thrust over a stroke) of the fluid dynamics of the fin stroke. While qualitative flow imaging (Srygley and Thomas, 2002) has been successful in identifying key fluid dynamic features at a fluid–wing boundary (such as leading edge vortices), one cannot estimate instantaneous forces with this technique. One possible solution that might allow the estimate of the instantaneous force balance on a flexible, pectoral fin is the recently developed defocusing DPIV system used to measure the velocity distribution throughout a volume of fluid surrounding a deforming object (Pereira and Gharib, 2002).

To estimate the contribution of circulatory and added mass forces and to explore the timings of these components, a previously developed, unsteady blade-element model (Walker and Westneat, 2000; Walker, 2002b) is further generalized to allow its application to the stickleback kinematics. The chief advantage of the model is its trivial computational burden, allowing its rapid application to a diverse array of fin movements. Despite its computational simplicity, the model has been remarkably effective at recovering most of the dynamic patterns identified by either robotic models or by more sophisticated virtual models. While motor-driven robotic fins offer an elegant method for investigating lift and thrust generation on an oscillating plate, current robotic models are not adequate for modeling stickleback fins because the actuation mechanism necessary for the types of fin motions presented by a stickleback pectoral fin (active control of multiple joints) is far more complex than the mechanisms found in current robotic fins and wings (active control of a single joint).

A model for indirectly measuring mechanical power and efficiency from
measured body accelerations and wing kinematics
(Pennycuick et al., 2000) is
generalized and applied to the stickleback data. These measured estimates of
power and efficiency are compared to modeled estimates of power and efficiency
computed from the blade-element model. Using a model of oxycalorific
equivalents and swimming muscle efficiency, one can convert measures of
O_{2} consumption to estimates of the mean mechanical power over a
stroke cycle or compute an estimate of mechanical efficiency. Ward et al.
(2001) and Schultz and Webb
(2002) have critically
reviewed some of the assumptions with this type of modeling. Even if we had
good estimates of muscle efficiency for stickleback, respirometry for
individual fish the size of sticklebacks swimming at uniform speeds is not a
viable option because of constraints on the design of water tunnel
respirometers (J. Herskin and J. F. Steffenson, personal communication). One
advantage of the indirect measure and modeled measure of mechanical power and
efficiency is the ability to apply these methods to smaller fish.

## Materials and methods

Sticklebacks (*Gasterosteus aculeatus* Linnaeus 1758) are small fish
that inhabit coastal marine and freshwater throughout much of the temperate
northern hemisphere (Bell and Foster,
1994). The sticklebacks in this study are anadromous individuals
that were captured in Rabbit Slough, near Wasilla, AK, USA. Fish were filmed
swimming in freshwater in a flow tank designed after Vogel and LaBarbera
(1978). A centimeter grid was
placed on the rear side of the tank to calibrate video images. Water
temperature in the flow tank was maintained at 21±1°C. Fin stroke
sequences were filmed in lateral view at 250 Hz using a RedLake Motionscope
high-speed digital camera. Digital sequences were saved as QuickTime®
files and digitized using a modification of the public domain NIH Image
program (developed at the US National Institutes of Health and available on
the Internet at
http://rsb.info.nih.gov/nih-image/)
for the Apple Macintosh (the modification, which is available from the author
upon request). Eighteen sequences from six individuals swimming at one of
three speeds were archived for further analysis. Following filming, fish were
sacrificed in MS-222, measured and weighed
(Table 1). Fin aspect ratio and
standardized first through third moments of fin area
(Ellington, 1984a) were
measured on the preserved specimens.

### Kinematics

Stroke frequency (*n*), averaged over multiple (5–10) beats,
stroke angle (Φ) and stroke plane angle (β) were estimated from all
18 sequences (methods following Walker and
Westneat, 1997).

Instantaneous fin geometry was measured for six sequences. In order to justify the methods for measuring this geometry it is necessary to describe the stickleback fin stroke qualitatively. The largely fore–aft stroke is described and illustrated from a left lateral view (Fig. 1). QuickTime® videos of selected sequences are available at http://www.usm.maine.edu/~walker/movies.html.

Immediately prior to the recovery stroke, the fin rotates counterclockwise along the body. The first part of the recovery stroke is characterized by the fin rays peeling off the body starting with the leading edge ray and proceeding, ray by ray, to the trailing edge ray. As the leading edge rays peel off the body, the trailing rays continue to rotate counterclockwise along the body. The point marked by the red arrow in Fig. 1A illustrates the break point along the distal edge separating the hydrodynamically active leading edge region from the inactive trailing region. The trailing edge ray peels off the body at about the time it reaches the same dorso-ventral position as the leading edge ray. The second part of the recovery stroke is characterized by the fin translating anteriorly with the plane of the fan pitched slightly ventrally (Fig. 1B). About the time the leading edge ray reaches its most forward position, the fin rays distinctly spread out, forming a larger surface area.

The power stroke begins with a rapid posterodorsal translation of the leading edge ray. The dorsal translation of the leading edge causes a clockwise rotation of the anterior fin surface into a broadside orientation (Fig. 1C). A wave of fin ray rotation passes posteriorly as the leading edge rays translate posteriorly. During the posterior translation of the leading edge surface, the trailing rays stop translating anteriorly but may move slightly ventrally. The red arrow in Fig. 1C illustrates the break point on the distal edge separating the leading edge region, which is rapidly translating posteriorly, from the trailing edge region, which has largely stopped translating. During this time, the fin is sharply curved at this break point. It is important to note that the fin does not rigidly rotate into a broadside orientation but instead resembles the peeling of a carpet off a floor by pulling one end back and up. Following rotation, all fin rays simultaneously translated back toward the body (Fig. 1D). The backstroke ended when the largely posteriorly directed fin rays closed against the body.

To digitize the fin geometry throughout fin motion, the positions of the dorsal fin base, ventral fin base, leading edge tip and trailing edge tip were digitized in each frame. An additional landmark was digitized to mark the break point along the distal edge separating the active from inactive regions of the fin during the first part of the recovery and power strokes. Only the active part of the fin is modeled. During the second part of each stroke, the entire fin is effectively active.

A blade-element (or strip) method was used to infer the geometry of the fin from base to distal edge. The implemented model assumes that the orientation of the fin base does not itself rotate during the stroke, that the pitch of the fin varies linearly from the fin base to the distal edge (that is, it twists down its span), that the span of the fin is constant (that is, the bony rays do not bend due to fluid dynamic loading) and that there is no camber along a chord. The first three of these assumptions are easily relaxed but would require more detailed kinematic measures to account for the variation. While the assumption of a constant fin base angle is met for the stickleback because of the relative immobility of the joints within its shoulder plate, this assumption (which, again, can be relaxed if the appropriate kinematics are measured) is certainly violated in some fishes (Drucker and Lauder, 2003). The last assumption (zero camber) could be relaxed only with the appropriate empirical force coefficients.

The fin stroke cycle begins with fin abduction and ends when the fin closes
against the body (any pause phase with the fin against the body is not
modeled). The stroke cycle has a period (τ) that was divided into
*N*=250/τ frames each of time Δτ=1/250 s. Time was
standardized not across the entire cycle but within each stroke. Following
this standardization, the total time for each stroke is 0.5 and the
standardized period () is 1.
Finally, the fin, with span *R*, was arbitrarily divided along its span
into 11 elements with equal width, Δ*r*=*R*/11. The
length-specific radial position is
*r̂=r/R*, where *r* is distance
from the fin base. In the following, the bracketed subscripts indicate that a
variable is a function of time (*t*) and/or radial position along the
span (*r*).

The position of the fin, γ_{(t)}, was estimated as the angle
between the leading edge ray, projected onto the stroke plane, and a unit
vector directed back along the *x*-axis. In this coordinate system,γ
_{(t)} is 0° when the leading edge is back against the body
and 90° when perpendicular to the body axis. The fin articulates at its
base with an angle, θ_{b}, relative to the horizontal and
oscillates about a flapping axis with an angle, θ_{f}, relative
to the horizontal (θ_{f} is normal to the stroke plane). While
often modeled as the same angle, θ_{b} and θ_{f}
differ in the stickleback. The difference between the angles,θ
_{b}–θ_{f}, is δ. As the fin
translates, it twists down its span. The pitch, α_{(r,t)}, of
the distal edge of the fin was estimated as the angle between the distal edge
chord and the fin base chord following the projection of both chords into the
sagittal plane. The distal edge chord was measured for the active part of the
fin only (from the leading edge to the break point) and α_{R},
therefore, reflects the pitch of the functional portion of the fin. The pitch,α
_{(r,t)}, at the radial position
*r̂* along the span is
*r̂*α_{(r,t)} while the
geometric angle of attack (angle relative to free stream),α
_{g(r,t)}, is:
1

Because the data were digitized from a two-dimensional lateral view (the
*xz* plane), the coordinates of the third (*y*) dimension had to
be reconstructed using the known lengths of the fin rays. This method assumes
that spanwise deformations of the fin rays are small relative to their length.
Because the reconstruction error is confined to the *y*-axis, estimates
of α_{(r,t)} and γ_{(t)} will be largely confined
to that part of the stroke when the fin is near its maximally adducted
position (back against the body). The accuracy of this pseudo-3-D method has
previously been tested using a data set in which the 3-D coordinates were
measured. The median absolute difference in the estimate of the stroke angle
between the measured-3-D and pseudo-3-D coordinates was 3.5°
(Walker and Westneat,
2002b).

### Net force balance

In a fish swimming at a steady speed, lift and thrust must balance weight
and drag. I assume that the lift and thrust generated by control surfaces
other than the pectoral fins are trivial relative to that generated by the
oscillating pectoral fins, and, therefore, the instantaneous thrust and lift
acting on the body is effectively equal to that generated by the fins (this
assumption is discussed further in the Discussion). The instantaneous force on
the body of a freely swimming fish cannot be directly measured, but its
fore–aft (*F*_{x}) and up–down
(*F*_{z}) components can be estimated by simply multiplying
either the fore–aft or up–down acceleration component by the mass
of the accelerating system. I refer to this estimate of the instantaneous
force as the measured force (as opposed to the modeled force, which is
estimated from the unsteady, blade-element model – see below). From the
measured force components, measured lift and thrust were estimated by:
2
3
where *M*_{F} is mass of the fish, *M*_{Ax} and
*M*_{Az} are the added masses in the direction of thrust
(*x*-axis) or lift (*z*-axis), d*U*/d*t* is the
acceleration of the body in the direction of thrust or lift, *D* is the
dead-drag on the body, and *W* is the weight of the fish in water. The
added mass of the body associated with fore–aft or up–down
acceleration was not directly measured but estimated using empirically derived
coefficients measured for bodies of revolution with similar dimensions
(Brennen, 1982):
4
where ρ is the density of freshwater at 20°C, *l* is standard
length, *d*_{max} is maximum body depth,
*b*_{max} is maximum body breadth, and the added mass
coefficients (μ) in the fore–aft (*x*) or up–down
(*z*) direction are 0.405 and 0.9255, respectively
(Brennen, 1982). The added
mass of the median and caudal fins, which would contribute little in the
*x* and *z* directions, was not included in this model.
*D* was estimated for a body of revolution using
(Hoerner, 1965):
5
where *U* is the free stream speed, *A* is the wetted area of
the body according to:
6
*C*_{D} is the dead-drag coefficient of the body:
7
and *f* is the fineness ratio,
.

Body acceleration was estimated by using numerical differentiation. To
measure displacement of the body, a landmark near the approximate center of
mass was located and digitized frame-by-frame. Center of mass displacement in
18 sequences from six individuals was digitized. Each sequence consisted of
two stroke cycles and began and ended with the fin maximally abducted (end of
the recovery stroke). While the entire two-cycle sequence was digitized and
fit with a spline function (see below), only the central fin beat, beginning
with fin abduction and ending with the fin closing against the body, was
compared among sequences. In order to determine the error in the estimate of
body displacement, three arbitrarily chosen sequences were each digitized
three times. The grand mean deviation (averaged over all points and sequences)
was 0.0039 cm (0.00054 *SL*) for the *x* (anteroposterior) axis
and 0.0047 cm (0.00065 *SL*) for the *z* (dorsoventral)
axis.

The displacement data were smoothed and twice differentiated using a
quintic spline function. The optimal smoothing parameter for the spline was
estimated using the true predicted mean-squared error (MD=3) criterion
(Woltring, 1985,
1986). In a large simulation
study, the MSE quintic spline algorithm performed well compared with other
published numerical differentiation algorithms
(Walker, 1998) and is
available in the software QuickSAND (available from the author upon request).
The mean standard deviation for the three sequences digitized three times each
was 0.363 pixels on the *x*-axis and 0.41 pixels on the
*y*-axis. In addition to digitizing error, measurement error includes
the component due to the transformation to a discrete (pixel) space. This
component has a maximum error of 0.5 pixels; a reasonable assumption of its
average is 0.25 pixels. The total variance, which is the sum of the squares of
these two components, was used as the predicted MSE.

### Unsteady model of fin dynamics and energetics

The dynamics of unsteady, oscillating foils can be modeled with reasonable
accuracy using a simple, unsteady blade-element model
(Walker and Westneat, 2000;
Sane and Dickinson, 2002;
Walker, 2002b). A
blade-element model allowing for both unsteady circulatory and added mass
forces for a limb oscillating about its root was developed previously
(Walker and Westneat, 2000).
Accuracy of the model (tested by comparison with robotic oscillating plates)
is discussed in the original paper and, more thoroughly, in Walker
(2002b). The kinematics of the
model are similar to that of Fung
(1993), but allowed the
flapping axis to be arbitrary (not necessarily 0°), and to that of
DeLaurier (1993), but allowed
for large amplitude motions. Becauseθ
_{f}≠θ_{b} in the stickleback, the model is
further generalized here.

The normal, ν_{n(r,t)}, and chordwise, ν_{x(r,t)},
flow due to fin translation and rotation are:
8
9
The first component of equations
8 and
9 is due to the fin element
translating with a speed,
,
due to fin oscillation. The second component is due to the fish translating
through the water at speed *U*, where the component of *U*
normal to the leading edge is:
10

The third component (in equation
8) is due to the fin rotating around a spanwise axis located
from the leading edge and a chordwise center of incident flow located
from the leading edge, where
or
is a percent
distance along the chord and *c*_{(r,t)} is chord length. The
first component of equations 8,
9 gives rise to the translational
circulatory force, and the third component gives rise to the rotational
circulatory force (Ellington,
1984b; Dickinson et al.,
1999); the second component is absent from the hovering situations
considered by Ellington and Dickinson.

The (hydrodynamic) angle of attack, or angle of incidence,
, is±
tan^{-1}(ν_{n(r,t)}/ν_{x(r,t)}) where
the ± takes the sign of ν_{x(r,t)}. This angle is used to
estimate the lift and drag coefficients (see below) and the components of the
combined translational and rotational forces normal to and parallel with the
fin chord:
11
12
(d*F*_{n(r,t)} is positive out from the medial surface of the
fin, while d*F*_{x(r,t)} is positive toward the leading edge.)
and
, the components
of the translational and rotational circulatory forces normal to and parallel
with the local stream, are:
13
14
where
15Δ
*r* is the width of the chord element, and
and
are the lift and drag
coefficients (see below). The combined translational and rotational
circulatory lift and thrust on a blade element are:
16
17

The added mass force normal to the fin element is:
18
where μ_{n} is the added mass coefficient of the fin section (a
value of 1.0 was used). The section lift and thrust components of
d*F*_{a(r,t)} are:
19
20

The sectional power needed to oscillate the fins against the water is: 21 and the power need to accelerate the water is: 22

Sectional lift, thrust and power are summed along the span and multiplied
by two to give the total modeled lift [*L*_{modeled(t)}],
thrust [*T*_{modeled(t)}] and power
[*P*_{modeled(t)}] for the pair of fins. Note that while the
total modeled force includes a lateral component, this component is not
reported because there is no measured lateral component for comparison. The
mechanical efficiency, η, is a measure of the percentage of total work
done by the fin on the water that is useful (that is, contributes to thrust;
another way to think of this is the percentage of the total work that is not
wasted). The quasi-steady estimate of the mechanical efficiency is:
23

### Coefficient model

In a previous blade-element model, lift and drag coefficients derived from
a robotic wing oscillating at a Reynold's number (*Re*) of 192
(Dickinson et al., 1999) were
used to model the dynamics of rowing and flapping propulsion because these
were the only ones available that accounted for the augmented effect of an
attached leading edge or trailing edge vortex
(Walker and Westneat, 2000).
These unsteady coefficients additionally include the effects of an unmeasured
induced velocity component. Scale (*Re*) has only a small effect on
*C*_{L(r,t)} and *C*_{D(r,t)} in the range
10^{2}<*Re*<10^{5}
(Usherwood and Ellington,
2002), which suggests that the robotic wing coefficients should
give good estimates of force magnitudes for any oscillating airfoil in this
range. Because of time delays in the generation of force production on
impulsively started plates, *C*_{L(r,t)} and
*C*_{D(r,t)} were reduced by the Wagner function
(Fung, 1993):
24
where τ_{(r,t)} is the number of chord lengths traveled following
stroke reversal.

### A measured force model of power and efficiency

The quasi-steady model of pectoral fin energetics requires accurate
estimates of detailed fin kinematics and assumes that circulatory and added
mass forces dominate the force balance and that these forces can be accurately
estimated with quasi-steady coefficients. Pennycuick et al.
(2000) made the novel
suggestion that measured forces (specifically *L*_{measured})
be used in place of the modeled forces to compute mean work and power. Their
method assumed a vertical stroke plane (θ_{b} andθ
_{f}=0), a constant pitch of 0° down the span throughout
the stroke, and a constant *C*_{L} down the span at any one
time in the stroke cycle (Pennycuick et
al., 2000). Note that the equal *C*_{L} assumption
is only compatible with the two kinematic assumptions if the local stream
vector is dominated by either the free stream component or a flapping
component (that is, gliding or hovering). The kinematic assumptions, however,
are easily relaxed, and the relevant normal force coefficient,
*C*_{n}, can be computed by rearranging equations given above.
Such an exercise would prove fruitless for the stickleback stroke since
*C*_{n} must change radically along the fin's span, at least
during the recovery stroke and the stroke transitions.

Given the detailed kinematics of a fin or wing, however, it is possible to drop the equal coefficient assumption as well and collapse the problem of finding the normal force at the spanwise center of force. The total force on a fin at any point in the stroke cycle is: 25 where 26 27

The normal force coefficient on an airfoil is
,
hence:
28
where *k*_{(t)} is an unknown proportionality factor that is
constant along the span but varies over time. Substituting
*F*_{measured(t)} for *F*_{N(t)} and
rearranging, this proportionality factor becomes:
29

The spanwise center of force (or mean moment) is: 30

Using the standardized center of force,
*r̂*_{F(t)} =
*r*_{F(t)}/R, the `measured' power, summed over both fins, is:
31

The `measured' mechanical efficiency is: 32 where the means are taken over the stroke cycle.

### Standardization

Measured and modeled forces are compared among sequences by standardizing
using:
33
where *A* is the summed area of both pectoral fins, and
*r*_{2} is the radial second moment of area.

## Results

### Morphometrics

The pectoral fin of the *Gasterosteus aculeatus* articulates with
the body at a relatively steep angle
(θ_{b}=69.0±3.0°), has a modest aspect ratio for fish
(AR=3.0±0.2) and high size-standardized radial moments of area
(,
,
)
(Table 1).

### Kinematics

The stroke of the threespine stickleback is qualitatively described and
illustrated above (see Materials and methods/Kinematics). Additionally,
animated GIF and QuickTime® videos of selected sequences are available at
http://www.usm.maine.edu/~walker/movies.html.
As speed increases from 1.4 *L* s^{-1} to 2.8 *L*
s^{-1}, the stroke plane angle from the vertical (β), which is
numerically equivalent to the flapping angle (θ_{f}), decreases
from 61.3° to 54.5° (*P*=0.04), the stroke angle increases from
94.7° to 104° (*P*=0.04), and the frequency increases from 4.3
Hz to 4.8 Hz (*P*=0.05) (Table
2). These kinematic changes are modest, which should not be
surprising given that the top speed measured in this study is about half the
pectoral fin powered critical swimming speed measured for this species.

The geometric angle of attack, α_{g}, at the distal chord
decreases rapidly from 90° to ∼15° during the first part of the
recovery stroke, decreases to about –15° near the end of the
recovery stroke, and rapidly rises to near 90° during the power stroke
(Fig. 2). The values above
90° at the end of the power stroke indicate that the distal chord is
positively twisted (positive α).

The angle of incidence,
, at the distal chord
rapidly decreases to a plateau of about 5° during the first part of the
recovery stroke (Fig. 2). The
transition from the recovery to the power stroke is characterized by the
at the distal
segment decreasing to about –10° near the end of the recovery
stroke, rapidly increasing to about +10° at maximum abduction and rapidly
decreasing to about –80° at the beginning of the power stroke. Note
that in
Fig. 2 slightly differs from
previously published estimates from the same data
(Walker and Westneat, 2002a)
because the previous data were based on normal and chordwise flow estimates
(equations 8,
9 above) that failed to include
the generalization for the angled fin base
[δsinγ_{(t)}].

Inspection of the accelerations for the three sequences digitized three
times each indicates the robustness of the acceleration estimates
(Fig. 3). Acceleration
estimates were converted to measured force coefficients for 18 sequences at
three different speeds (12.6 cm s^{-1}, 18 cm s^{-1} and 23.4
cm s^{-1}). These speeds will be referred to as low, medium and high,
respectively. Each *C*_{T} and *C*_{L} curve was
interpolated to 21 points at
increments using a cubic spline. Using these binned values,
is the grand mean *C*_{T} or
*C*_{L}, averaged over all sequences, for any single time
increment, while is the group mean
*C*_{T} or *C*_{L}, averaged over all sequences
within one speed class, for any single time increment (where the group is low,
medium or high speed).

Force coefficients are illustrated in
Fig. 4. Measured thrust
coefficients, *C*_{T}, are negative and small (peak
) throughout the recovery stroke.
Immediately following the stroke transition, *C*_{T} rises to
large, positive values (peak ). The peak
at high speeds is significantly greater than the
peaks at low and medium speeds (Tukey HSD, *P*<0.05). The power
stroke peak occurs at
. While peak
occurs at
at the two lower speeds, it
occurs at at high speeds.
Measured *C*_{L} gradually rises to moderately positive values
(peak ) during the recovery stroke and
falls to very small negative values (peak )
during the power stroke. There is no difference in peak
between speeds during the recovery stroke, but
the minimum during the power stroke of the high
speeds is more positive than those for the low and middle speeds (Tukey HSD,
*P*<0.05). The recovery stroke peak
occurs at while the power
stroke minimum occurs at
. The recovery stroke peak
occurs at
for both the low and high
speeds but at for the middle
speeds. The power stroke minimum occurs at
for both low and middle speeds
and at for the high
speeds.

The quasi-steady model was applied to six sequences, three each at speeds
of 12.6 cm s^{-1} and 18 cm s^{-1}
(Fig. 5). Both the modeled and
measured *C*_{T} and *C*_{L} curves for the six
sequences were interpolated to 21 points at
increments using a cubic
spline. Mean coefficients for each time increment were computed for the
measured and modeled coefficients. Values for the two speeds were pooled. Note
that the measured mean coefficients differ from the grand mean coefficients
above because they include the values of only the six sequences that were
modeled.

The peak negative during the recovery
stroke is significantly more negative than the peak negative
(*t*-test, *P*=0.05).
Similarly, the peak during the power
stroke is significantly greater than the peak
(*t*-test, *P*=0.002).
The modeled thrust during the recovery stroke is dominated by the circulatory
component (Fig. 5). During the
power stroke, the broad modeled thrust peak is due to a large, positive added
mass component that peaks at
and a large, circulatory component that peaks at
(Fig. 5). Both the average and
the peak circulatory thrust are larger than the average and peak added mass
thrust (Table 3). The timing of
measured peak thrust during the power stroke does not differ from the timing
of the modeled and circulatory peak thrust but is significantly later than the
timing of the added mass peak thrust (Table
3).

While the peak does not differ from
(*P*=0.07), the modeled lift
curve presents two recovery stroke peaks, in contrast to the measured lift
curve, which presents a single peak (Fig.
5). The two peaks in the modeled curve result from the interaction
between the single, positive peak of the circulatory lift curve and the
single, negative peak of the added mass lift curve
(Fig. 5). The timing of the
measured peak lift during the recovery stroke does not differ from the timing
of the modeled, circulatory or added mass peak lift
(Table 3), although the mean
timing of the peak lift for the added mass model reflects the maximum positive
lift occurring either much earlier (two sequences) or later (four sequences)
than the measured peak (Fig.
5).

The input power estimated from the quasi-steady and measured force models
have similar shapes but the quasi-steady model produces power maxima that are
over twice those of the measured force model
(Fig. 6). Mechanical efficiency
estimated from the quasi-steady model (η_{modeled}) ranges from
0.10 to 0.22, while that estimated by the measured force model
(η_{measured}) ranges from 0.13 to 0.29
(Table 4).

## Discussion

### Notes on the method for estimating the force balance

Lift and thrust balances were estimated for free-swimming sticklebacks. The
accuracy of these estimates is a function of the accuracy of several
components: measured body accelerations in the anterior-posterior and
dorso-ventral directions, measured body mass, measured weight in water,
modeled added mass and modeled parasite drag. Measured acceleration error
affects both the magnitude of the lift and thrust and the shape of the lift
and thrust curves. The error in the measured mass and added mass of the fish
affects the scale of the curve but not its shape. The error in estimating the
parasite drag and the weight (in water) of the fish affects only the position
of the curve along the *y*-axis. The error due to estimating the mass
of the fish is trivial, while the error in the estimation of the added mass is
unknown. The errors in estimating the weight in water and the parasite drag
are unknown but the fact that the thrust and lift coefficients are near zero
when the fin is against the body suggests that any error in the estimate of
weight and drag is small.

By far the largest error component is that due to estimating accelerations, and this error can influence both the shape of the curve and the magnitude of the peaks. The error in the acceleration estimates has two sources: precision, or the ability to repeatedly measure the same value, and accuracy, or the ability to measure the true value. The precision of the acceleration estimates can be estimated using the three sequences that were measured three times each. Within any sequence, there are three estimates of the maximum forward acceleration during the power stroke and the maximum upward acceleration during the recovery stroke. The percent deviation of any one estimate from the mean of the three estimates is . The maximum percent deviations for the maximum forward acceleration were 6%, 9% and 11% for the three different sequences. The maximum percent deviations for the maximum upward acceleration were 9%, 14% and 33% for the three different sequences.

Precise estimates suggest accurate estimates, but this may not be the case
if there is some unknown factor that consistently biases the numerical
differentiation method. The MSE quintic spline algorithm has a consistent
downward bias in the estimate of maximum accelerations
(Walker, 1998). This bias
could be corrected if there were a known relationship between bias and some
measurable parameter summarizing the data. One potential parameter is the
relative noise, ν (or roughness parameter), in the data
(Corradini et al., 1993):
34
where σ_{signal} is the standard deviation of the smoothed
displacement (after removing any net translation over the stroke cycle) andσ
_{noise} is the standard deviation of the difference between
the raw and smoothed values (again, after removing the net translation). Theν
for the six, modeled sequences ranged from 18% to 36% in the
*x*-direction and from 58% to 122% in the *z*-direction.
Unfortunately, the relationship between ϵ and ν at these levels of
error has not been explored but an extrapolation of the data from Walker
(1998) suggests that the bias
for the MSE quintic spline algorithm in this range of ν lies between–
25% and –50%.

Clearly, there is a great need for further exploration in this area if we are going to advance our knowledge of locomotor control from studies of freely moving animals. Nevertheless, the reasonable consistency between repeated estimates of accelerations from the same sequence, between estimates of accelerations among sequences and between accelerations and the quasi-steady model suggests the method should be exploited more often.

### Dynamics of pectoral fin rowing

Recent work on robotic insect wings has elegantly shown the importance of
an attached vortex augmenting the circulatory force during the translational
and, in *Drosophila*, the rotational phases of the stroke cycle
(pronation and supination in *Drosophila* are confined to short
intervals bounding the stroke transitions;
Ellington et al., 1996;
Dickinson et al., 1999). The
absence of a peak in the measured force curves
(Fig. 4) near the transition
from recovery to power stroke suggests that circulatory forces at this point
in the stroke cycle are trivial. This result may seem surprising given the
rapid change in the angle of attack during this part of the stroke
(Fig. 2). Again, as described
in the Materials and methods, this change in attack angle is only occurring at
the extreme leading edge region of the fin (a relatively small area).
Following this rotation of the leading edge, the subsequent rays are simply
translated posteriorly.

The quasi-steady circulatory forces based on the unsteady coefficients are
significantly greater than the measured forces. The power stroke force maxima
is generated by a fin that is translating with the distal half at an attack
angle, α′, of ∼70°
(Fig. 2). The equilibrium drag
coefficient for a flat plate at 70° to the flow at the *Re*
relevant to stickleback swimming is 1.1
(Hoerner, 1965) while the
corresponding unsteady coefficient is 3.0. The high maximum
, relative to
, raises the possibility that the
unsteady coefficients are too high to accurately model the dynamic environment
of the stickleback fin, perhaps because the unsteady coefficients were
measured at a smaller *Re*. The range of noise–signal ratios,ν
, discussed above suggests that, alternatively, the measured force
estimates are too low because the quintic spline algorithm has a consistent
downward bias in the estimation of maximum second derivatives
(Walker, 1998).

At the *Re* of pectoral fin rowing in the stickleback, the
acceleration reaction should have a large influence on the force balance. To
optimize the acceleration reaction, the fin should oscillate along a
horizontal stroke plane with the fin surface oriented normal to its motion
along its entire span throughout the stroke. For the six digitized fin
sequences, the acceleration reaction component of the thrust balance would
have an optimal peak *C*_{T,AddedMass} of about 1.8 occurring
at the stroke transition ()
(Fig. 6). By contrast,
at
does not differ from 0, which
suggests that the stickleback's fin motion is not optimizing the acceleration
reaction. Indeed, the estimated acceleration reaction curve has a much smaller
(*C*_{T,AddedMass}=0.88) and later
() peak
(Fig. 6). Instead of dominating
the force balance, the acceleration reaction contributes about half as much to
the thrust balance as the circulatory force
(Table 3). The estimated
acceleration reaction at is
much smaller than the optimal acceleration reaction at
because the fin is beautifully
feathered at this point in the stroke cycle and simply cannot accelerate a
large volume of fluid with this orientation. While the modeled acceleration
reaction is significantly less than the modeled circulatory force, the shape
of the *C*_{T} and, especially, *C*_{L} curves
suggests that the influence of the acceleration reaction may be even less than
indicated by the model.

The acceleration reaction model presented for the stickleback differs
radically from that presented for the angelfish, *Pteryphylum emekei*
(Blake, 1979). Blake
(1979) argued that the positive
contribution to the thrust balance at the beginning of the power stroke
canceled the negative contribution to the thrust balance at the end of the
power stroke with the net result of zero contribution of the acceleration
reaction to the thrust balance. Given the kinematics used by Blake
(1979), in which the fin was
apparently oriented broadside to its motion throughout the entire power
stroke, the acceleration reaction during the power stroke should have
resembled the second half of the optimal acceleration reaction curve in
Fig. 6.

### Reduced recovery stroke drag

A major influence on the mechanical efficiency of the rowing fin is the recovery stroke geometry (Walker and Westneat, 2000; Walker, 2002a). For efficient rowing, the recovery stroke should generate little drag or lift. Reduced loading can be achieved by minimizing fin speed, fin area or fin angle of attack. Animals with jointed limbs typically minimize average limb speed by flexing the limbs (actively or passively) during the recovery stroke (Walker, 2002a; this works because speed is a function of both angular velocity and radial distance from the limb base). Limb area is reduced in some animals by collapsing webbed limbs or swimming hairs (Hughes, 1958; Nachtigall, 1974; Koehl, 1993). Limb angle of attack is reduced in larger animals by feathering the appendage (Walker, 2002a).

Stickleback fins are supported by bony fin rays that lack movable joints, except at the fin base, and the rays are too stiff to allow the passive flexion necessary for substantial speed reduction. While adducting the fin rays can reduce fin area, sticklebacks do not exploit this mechanism to reduce loading during the recovery stroke. The primary kinematic mechanism used by sticklebacks to reduce loading during the recovery stroke is angle of attack reduction by fin feathering. Feathering requires the stickleback to twist its fin down its span. While the kinematic model used to estimate forces on the recovery stroke assumed that the fin twisted linearly along the span, the much smaller measured drag relative to modeled drag suggests that the stickleback is able to feather its fin more effectively than expected by the model.

Feathering the appendage presents a second obstacle to efficient swimming because the fin must be rotated into the feathered orientation. Any form of stiff rotation would generate substantial drag. As an alternative, the stickleback peels its fin off its body. By rotating the trailing portion of the fin dorsally along the body while the leading region peels off in a feathered orientation, the fin generates little drag.

The final obstacle presented by a feathered appendage is the optimal rotation back into a broadside orientation to start the power stroke. Instead of stiffly rotating into a broadside orientation, a wave of re-orientation passes from leading edge to trailing edge. This kinematic mechanism results in a resultant force that is largely confined to the frontal plane, producing thrust and lateral forces with very little lift. By contrast, a rapid, stiff rotation about a spanwise axis lying posterior to the chordwise center of pressure would generate thrust and a large, negative lift.

While the net drag during the recovery stroke is small, substantial lift is
generated. This lift balances the weight of the fish but it is unclear if the
stickleback maintains negative buoyancy to balance the lift necessarily
generated by a fin that cannot feather more optimally (because of the higherα
_{m} toward the fin base) or if the fin is generating lift in
order to balance an obligately negatively buoyant body.

### Energetics

The mean η_{modeled} (0.16) and η_{measured} (0.2)
lie in between the mean optimal η of twisted (0.08) and perfectly
feathered (0.24) rowing fins oscillating at equivalent reduced frequencies.
Stickleback rowing efficiency is expected to be better than that of the model
twisted fin for several reasons. First, the stickleback fins expand distally,
a shape that has been shown to optimize rowing performance
(Blake, 1981). Second, the
stickleback fin articulates with the body at an angle of 69° compared with
an angle of 90° for the model fin. The average angle of attack along the
span in the stickleback fin should be less than that for the model twisted
fin. Finally, the dynamic data discussed above, and possibly these energetic
data, suggest that the stickleback may be able to feather its fin along its
span better than expected by a linearly twisted model.

The mean efficiency estimated by the quasi-steady blade-element model is exactly that estimated by a quasi-steady model for the rowing fin of the angelfish (Blake, 1979, 1980). As the models differ in the geometry of the angle of attack, the source of the empirical force coefficients and, importantly, the model of drag on the fins (Walker and Westneat, 2000), this similarity is partly coincidence. Blake (1979) used a dead-drag measure of a fish with its pectoral fins extended out from its body as the measure of parasite drag. But the parasite drag that a rowing fish has to work against does not include the pressure drag produced by extended pectoral fins because these have a time-averaged pressure distribution that results in net thrust, not drag on the `vehicle'. Consequently, the fins only need to work against the parasite drag of the body and any viscous drag on the pectoral fins (the viscous drag on the pectoral fins was not included in the model of mechanical efficiency because of the difficulty of estimating this parameter for a deforming, oscillating body). A similar argument was made for fishes that power swimming by body and caudal fin undulations, although these authors argue that the body does not even work against its own viscous drag and, consequently, the concept of efficiency (for a self-propelled, undulating fish) is meaningless (Schultz and Webb, 2002).

The measured efficiencies in this study are supported by the only other
comparable data collected by an independent method: an optimal efficiency of
0.15 measured from the motor-controlled rowing stroke of a stiff fin modeled
on the pectoral fin of the largemouth bass, *Micropterus salmoides*
(Kato, 1999). Combined, the
quasi-steady model, the measured-force model and the motor-driven physical
model all indicate that rowing is a relatively inefficient means of transport,
at least relative to a flapping-fin mechanism
(Walker and Westneat,
2000).

This conclusion, that rowing is an inefficient propulsive mechanism, raises an interesting paradox. Marine sticklebacks are anadromous fishes, migrating hundreds of kilometers between the open ocean and spawning sites in either estuaries or freshwater streams (Wootton, 1976; Cowen et al., 1991). Marine sticklebacks power these steady cruising behaviors using only pectoral fin rowing. Indeed, they lack the band of slow, oxidative (red) muscle fibers in their axial musculature (te Kronnie et al., 1983) that is necessary for powering steady, BCF locomotion (Jayne and Lauder, 1994).

Why don't sticklebacks have the high aspect ratio, tapered, flapping
pectoral fins common to fishes that swim with greater endurance (Walker and
Westneat, 2000,
2002a;
Bellwood and Wainwright, 2001;
Fulton et al., 2001)? The
design of the stickleback shoulder and fin could reflect a trade-off between
optimal designs for continuous swimming and other behaviors such as low-speed
maneuvering or nest fanning. Alternatively, rowing could reflect an
ontogenetic constraint due to the low *Re* experienced by the fins of
juvenile sticklebacks. Anadromous juvenile sticklebacks have been found
hundreds of kilometers offshore, and available data indicate that these
juveniles have the endurance to actively swim long distances
(Stevens, 1993). The mean
*Re* for a 20 mm stickleback swimming from 1 *BL* s^{-1}
to 3 *BL* s^{-1} is approximately 50 to 60. Available published
data suggest that no aquatic animals swim by flapping appendages below
*Re*≈80–100 (Walker,
2002a), which supports the hypothesis that juvenile sticklebacks
must row to swim effectively. While viscous forces on a fin are strongly
influencing fin performance at *Re*<100, a simulated comparison of
rowing and flapping fins showed that rowing is not expected to outperform
flapping until an *Re* less than ∼20 is reached
(Walker, 2002a). Furthermore,
rowing fins at these low *Re* (<50) must use a combination of
reduced fin area and recovery stroke speed (see above) in order to outperform
flapping because feathering is not an effective mechanism in this *Re*
range (Walker, 2002a). If
juveniles have the same recovery stroke kinematics as found in adults, the
efficiency of the juvenile fin stroke should be much lower than if it were
oscillating with a flapping geometry. This suggests that active swimming at
*Re*<100 is not a constraint on the design of the stickleback
fin.

### Conclusions

The threespine stickleback swims at sub-burst speeds by generating thrust from paired pectoral fins that present a stereotypical rowing stroke. While the rowing stroke is inefficient relative to a flapping stroke (Walker and Westneat, 2000), the design of the stickleback fin, with its multiple, independently actuated bony struts supporting a thin, flexible membrane, results in less wasted energy than would occur if the fin were rowing as a stiff, flat plate. This design feature, which is common to actinopterygian fishes (Lauder and Drucker, in press; Westneat et al., in press), presents a difficult challenge for constructing detailed models of pectoral fin function (specifically) or the performance consequences of pectoral fin design variation (more generally). Addressing either this specific or more general question will require a joint research effort combining the state-of-the-art computational, visual and physical modeling tools (Gharib et al., 2002; Lauder and Drucker, in press; Mittal, in press; Triantafyllou et al., in press) with the more traditional methods exploited here.

**List of symbols**

- A
- total pectoral fin area
*A*_{wet}- wetted area of fish body
*b*_{max}- maximum body breadth
- mean chord
*c*_{(r)}- chord length
- group mean force
- grand mean force
*C*_{D}- parasite drag coefficient on body
- sectional drag coefficient
*C*_{L}- mean lift coefficient
- sectional lift coefficient
*C*_{T}- mean thrust coefficient
- D
- parasite drag of body
- d
*D*_{a(r,t)} - aft component of added mass force
- d
*D*_{c(r,t)} - aft component of circulatory force
- circulatory drag
- d
*F*_{a(r,t)} - added mass force
- d
*F*_{n(r,t)} - normal component of circulatory force
- d
*F*_{x(r,t)} - chordwise component of circulatory force
- d
*L*_{a(r,t)} - upward component of added mass force
- d
*L*_{c(r,t)} - upward component of circulatory force
- circulatory lift
*d*_{max}- maximum body depth
- d
*P*_{a(r,t)} - sectional inertial power
- d
*P*_{c(r,t)} - sectional circulatory power
- d
*T*_{a(r,t)} - added mass thrust
- d
*T*_{c(r,t)} - circulatory thrust
- f
- fineness ratio
*F*_{measured}- measured force on fin
*F*_{N(t)}- normal force on fin
*F*_{x}- fore–aft component of measured force on body
*F*_{z}- dorso-ventral component of measured force on body
*h*_{(r,t)}- heaving velocity
*k*_{(t)}- proportionality coefficient
- l
- standard length of fish body
*L*_{measured(t)}- indirectly measured lift on body
*L*_{modeled(t)}- modeled lift on body
*M*_{A}- fish added mass
*M*_{F}- fish mass
- n
- stroke frequency
*P*_{modeled(t)}- modeled power on body
- R
- fin length
- r̂
- non-dimensional radial position along fin span
- r
- radial position along fin span
*r*_{2}- radial second moment of area
- Re
- Reynold's number
*r*_{F(t)}- radial center of force along fin span
- T
- period of stroke cycle
- t
- time in stroke cycle
*T*_{measured(t)}- indirectly measured thrust on body
*T*_{modeled(t)}- modeled thrust on body
- U
- free stream speed of flow
*U*_{n(t)}- normal component of free stream
- ν(r,t)
- velocity of fin element
- νn(r,t)
- normal velocity of fin element
- νx(r,t)
- chordwise velocity of fin element
- W
- weight of body in water
*x̂*_{i}- non-dimensional chordwise location of center of incident flow
*x̂*_{r(r)}- non-dimensional chordwise location of center of rotation
- Δr
- width of fin blade element
- Φ
- stroke angle
- α(r,t)
- fin pitch
- αg(t)
- geometric angle of attack
- αR
- pitch of the functional portion of the fin
- angle of incidence
- β
- stroke plane angle (relative to vertical)
- γ(t)
- azimuthal position of fin
- δ
- angle between fin base and flapping axis
- η
- mechanical efficiency
- μ
- added mass coefficient
- θb
- angle of fin base relative to horizontal
- θf
- flapping axis (normal to stroke plane)
- ρ
- water density
- τ
- stroke period
- &̂tau;
- standardized stroke period

## ACKNOWLEDGEMENTS

Ideas presented here greatly benefited from spirited discussions with Tom Daniel, Michael Dickinson, Frank Fish, George Lauder, Paul Webb, Mark Westneat and Brad Wright. Two anonymous reviewers are gratefully acknowledged for improving the manuscript. This work was supported by a National Science Foundation Postdoctoral Fellowship in the Biosciences Related to the Environment, Office of Naval Research N00014-01-1-0506, Office of Naval Research grant N00014-99-0184 to M. Westneat and J. Walker, and the Bioscience Research Institute of Southern Maine.

- © The Company of Biologists Limited 2004