J. exp. Biol. 163, 119-137 (1992) \\

120 C. E. JORDAN

largely as a balance of pressure and inertial forces, while the balance of pressureand viscous forces dominates the realm of small slow swimmers.

A convenient way to distinguish between these two physical realms is with theReynolds number (Re), the dimensionless ratio of inertial to viscous forces givenby Ul/v, where U is a characteristic velocity, / is the characteristic length and v isthe kinematic viscosity of the fluid. For high Re (>1000), the majority of thrustarises from organisms doing work against a fluid's mass, while at low Re (

A model of undulatory locomotion 121

h(x,t)

Fig. 1. (A) Dorsal view of an adult Sagitta elegans (after Hyman, 1959). The specimenis 20 mm in length, with a maximum body width of 1.8 mm. In the model, the region ofthe body without fins is represented as a cylinder with sectional surface area Sb=2jrrdxand projected area /lb=2rdjc. The region with fins is represented as a cylinder with flatplates of height/, such that the sectional surface area is Sf=2(jtr+2f)dx and projectedarea Af=2(r+f)dx. In the model, the fins are located on the body as depicted in thisdrawing. (B) Lateral view of an adult chaetognath. The animal is shown in normalswimming posture, moving at speed U with propulsive wave speed c. The body waveamplitude grows from head to tail and can be described by its local displacement h(x,t)and its limit e(x). The enlarged section shows 8, the angle a body section makes withthe instantaneous direction of travel.

realm are very poorly understood. This study directly addresses the effect ofvariation in body morphology and kinematics on undulatory locomotion anddefines a good measure of locomotory performance at this physical scale. Previouswork on undulatory locomotion at intermediate Re focused on the work requiredor the energetic efficiency of two swimming modes without discussing thecorresponding variability in locomotory performance (Vlymen, 1974; Weihs,1974).

I address the question of locomotion at intermediate Re using the chaetognathSagitta elegans (Fig. 1A). These animals swim in rapid 100 ms bursts of dorso-ventral undulatory propulsive waves that pass from head to tail (Fig. 2).Chaetognaths generate this mode of locomotion by rapid contractions of the bodywall musculature that runs nearly from head to tail without opposing circular or

122 C. E.JORDAN

0.00 s

0.04 s

0.06s

0.08 s

0.10s

Fig. 2. Tracings made from high-speed 16mm cine' films at 200framess L.fourth frame is shown. The top tracing shows the frames superimposed.

Every

A model of undulatory locomotion 123

radial muscles (Duvert and Salat, 1980). It is not known whether the longitudinalmuscles contract locally to produce the body's undulations (Duvert and Savineau,1986) or whether some type of moving buckling wave (C. E. Jordan, personalobservation) is produced as a result of opposition to the muscles by a fiber-woundhydrostatic skeleton (Ahnelt, 1984).

A thorough understanding of the physical environment at intermediate Reyn-olds numbers is necessary in order to evaluate its role in constraining themorphology and kinematics of aquatic organisms at this scale. For example, theentire phylum Chaetognatha shows remarkable morphological similarity (Hyman,1959). All 65 or so species of chaetognaths are similar to the S. elegans depicted inFig. 1A, with a long slender body, a tail fin, one or two pairs of lateral fins andgrasping spines on the head for which the phylum is named.

Materials and methodsSpecimens

Chaetognaths were collected in San Juan Channel (San Juan Island, Wash-ington) at dusk from a depth of 75 m with a 1-m 635-jam mesh plankton net with asealed cod end. To avoid damaging the specimens the net was pulled vertically atless than 0.5 ms"1 . Healthy Sagitta elegans, roughly 20 mm in length (mean=20.4mm, s.D.=2.2mm, yV=ll), were separated from other zooplankters andmaintained in 41 jars immersed in running sea water (12-15°C) at densities of oneanimal per liter. Individuals were used within 48 h of capture.

Kinematics

All filming was carried out with a Red Lake Locam (51003 with Nikon 55 mm3.5f macro lens) at 200±10framess~1, using Ektachrome 7250 color film(ASA 400). Animals were filmed swimming in a 11 rectangular container cooledwith a recirculating water jacket to keep the chamber temperature between 10 and12°C. Illumination was provided by a Nikon MKII fiber light.

Chaetognaths were made to swim by tapping on the chamber or by directcontact with a pipet. Each swimming event, or sequence, was filmed for roughlyIs . The powered swimming phase lasted less than 250ms. Eleven sequences,representing seven individuals, were used in the analysis. Sequences were chosenin which the animals started from rest and remained in the plane of focus and fieldof view while swimming in a fairly straight path.

Sequential frame analysis was used to determine the three important kinematicvariables: wave number, wave speed and wave amplitude. The first 50 frames(250 ms) of each sequence were digitized by drawing a line down the center of theimage projected onto a digitizing pad (SummaGraphics). A 128-point cubic splineapplied to these digitized data yielded a continuous curve describing body positionin each frame. Each digitized frame was fitted with a principal componentregression line, whose slope gave the instantaneous direction of travel and whose

124 C. E. JORDAN

grand mean gave instantaneous coordinates of the center of mass. Distancetraveled, velocity and acceleration were thus determined by tracking the grandmean (center of mass) from frame to frame. The speed, number and amplitude ofpropulsive waves were determined by recording the maximum of each half-waveas it passed along the body.

Theory

A mathematical model of chaetognath locomotion was constructed by combin-ing the existing high Reynolds number reactive theories of Lighthill (1975) withthe low Reynolds number resistive theories of Gray and Hancock (1955). Thefundamental idea underlying Lighthill's theories is that propulsive forces, pro-portional to lateral body accelerations, arise from reactions to fluid accelerationsnormal to the surface of the animal. The resistive theories applied to undulatoryswimming, as developed by Gray and Hancock (1955), assume that propulsiveforces arise from viscous stresses generated by rearward components of the motiondue to whole-body undulations.

Since resistive theories do not account for fluid inertial forces, and the reactivetheories contain no viscous terms, any model of locomotion at intermediate Remust combine both theories in order to include the situation in which both viscousand inertial forces are of comparable magnitude. The model uses as a basisNewton's law, Ft=MdU/dt, where M is the animal's mass, U is the forward speedand F, is the sum of forces generated by the body acting on the fluid. In this case,force is calculated sectionally for both the inertial and viscous components, andthen summed along the length of the body at each instant in time. Calculated inthis manner, force can then be used to predict the body speed at the next instant intime by applying Newton's law and solving the force balance equation nu-merically.

Reactive forces

Reactive theory for elongated body locomotion describes a body of length / thatis displaced laterally by h(x,t)=r(t)e(x)s\n{a)[t-(x/c)]}, where x is the positionalong the length, a> is the angular frequency, c is the wave speed, T{i) generates thegrowth in amplitude during the first 20 ms of swimming and e(x) is a function thatdescribes the growth in amplitude along the length, or amplitude envelope(Fig. IB; symbols for these and all following equations are defined in theAppendix). Thus, the lateral velocity V of any section of a body moving at aforward speed U is given by:

v=dh/dt+u(t)dh/dx. (i)

The sectional force (Fs) generated normal to the body at x is the product of themass and lateral acceleration:

Fs = [9/3* + U(t)d/dx][m(x)V(x,t)], (2)

A model of undulatory locomotion 125

where m(x) is the sectional added mass (see for example Lighthill, 1975, chapter4). Equations 1 and 2 give the instantaneous force in the direction of travel:

FXi) = f' sin0(Fs)dx= f1 sine[m(x)d2h/dt2 +

Jo Jo

2m(x)U(t)d2h/dxdt + U(t)Vdm/dx +

m(x) U(t)2d2h/3x2 + m(x)(dU/dt)(Bh/dx)]dx, (3)

where 8 is the angle between segments of the body and the instantaneous directionof movement (Fig. IB).

Resistive forces

For steady-state flows, the two components of drag on a section are given by:

Dn = kpACdnV2, (4)

D, = ip5Cdly,2, (5)

where S is sectional surface area, A is projected area, the subscripts n and 1represent normal and tangential, Cdn and Cd! are the sectional drag coefficients fornormal and tangential flows and Vn (=VcosO) and Vi (=Vsin#) are the normal andtangential components of velocity. Once the resistive terms are resolved into thedirection of body translation, thrust and drag forces differ only in sign. Thus,equations 3, 4 and 5 can be combined to give the total instantaneous forcegenerated by the body:

Ft(t)= J1 {s\ne[d/dt+U(t)d/dx][m(x)V(x,t)] +

hp(ACdnVn2 + SCd[V?)}dx. (6)

Equation 6 is numerically integrated along the body with m(x) constant withineach body section. Also, since the angle that the body makes locally with thedirection of travel is small (6>

126 C. E. JORDAN

terms, m, Cd, A and S of equation 9. In cross flow, the body approximates acylinder of constant radius r or a rectangular flat plate of height 2(f+r) in theregions with fins or at the tail. For longitudinal flows, I treat the drag as purelyfrictional; thus, the body is represented by sectional flat plate coefficients withareas Sb in the regions without fins and Sf in the regions with lateral or tail fins (seeFig. 1 for a description of sectional geometry and location of fins). The expressionsfor describing the local cross-sectional shape-dependent terms are as follows. Inthe region without fins (curves fitted for cylinders in cross flow and flat plates instreamwise flow from White, 1974, and Vogel, 1981):

Cdn = 1 + 10(7?enr2/3 , Ren = 2rVn/v, (10)

Cdl = 0.64(foxl)-1/2, Rexl = xVl/v. (11)

In the region with fins (curves fitted to data for flat plates in cross flow fromHoerner, 1965):

Cdn = l0l^28-0.8090oSRen)+0.13A(iogRef] ^ ^ = 2(r+f)VjV, (12)

l ) -1 / 2 . (13)

The only drawback to this choice of force coefficients is that as the fin height goesto zero the two expressions for Cdn, with and without fins, do not converge. At /=0these two expressions differ by a factor of 1.8, primarily as a result of the inherentdifferences between flows around a cylinder and those around the sharp edges of aflat plate. Ideally, one should use the values of Cdn for a cylinder with appendedflat plates; to my knowledge, however, no such geometry exists in the literature.

Equation 9 was rearranged to give dU/dt as the dependent variable. In thisform, it is numerically integrated with respect to time by a fourth-orderRunga-Kutta method to yield predictions of instantaneous body speed. There-fore, for a given set of kinematic variables, or initial conditions, the model predictsthe forces generated by the propulsive wave and uses these to calculate bodytranslations.

Results

Kinematics

The duration of the swimming bursts ranged from 65 to 240 ms (mean=128 ms,s.D.=65ms, N=ll), during which time the individuals traveled from 3 to 17mm(mean=8.19mm, s.D.=4.78mm, 7V=11). To standardize for variation in time ofpowered swimming between individuals, I examined the kinematics only over65 ms, the longest swimming duration common to all seven (Table 1). Allmeasures of kinematic variables are scaled to the individual's body length.

Over the first 65 ms, the animals swam at an average speed of 2.99 bodylengthss"1 (BLs- 1) , through an average distance of 4.1mm (Fig. 3). Despite thevariability in distance traveled, the kinematics of the propulsive wave were fairlywell conserved, with an average wave speed of 12.0BLs~\ approximately four

A model of undulatory locomotion 127

Table 1. The observed kinematic variables for the first 65 ms of 11 swimming events

Trialnumber

0102030405060708091011

MeanS.D.

Size(mm)

21.521.518.222.522.518.015.822.122.119.919.9

20.42.2

Speed(BLs-1)

2.0421.1992.2213.6502.2085.4922.3292.4662.5413.8484.966

2.9971.325

Distance(mm)

2.91.72.75.33.46.82.43.73.85.16.7

4.051.71

Waveamplitude

(BL)

0.0300.0270.0380.0500.0360.0370.0340.0350.0420.0370.028

0.0360.065

Wavespeed

(BLs"1)

9.311.512.58.8

15.315.010.011.010.310.618.0

12.02.9

Wavenumber

1.21.41.41.21.41.31.31.31.41.31.3

1.30.1

All values are scaled to the body length given in column 2.Speed is calculated as the net displacement divided by 65 ms, and is thus an underestimate of

final speed. All other values are taken directly from digitized high-speed cine film frames.The amplitude height is the a term in the exponential wave amplitude envelope for each

swimming event (see Fig. 4) and is calculated as the average wave height over the central 60 %of the body.

times U, and an average wave number of 1.3. The variable that showed thegreatest variation between individuals, and the one that theory predicts canaccount for the variation in distance traveled, is the amplitude of the propulsivewave. In all cases the wave grew as it passed from head to tail (Fig. 4), but tendedto be larger in those individuals that traveled farther (Kendall's ranked corre-lation, r=0.51, P

128 C. E.JORDAN

1.20

0.90

0.60

0.30

0.0020 40 60

Time (ms)

80 100

Fig. 3. Observed distance traveled with time during 11 swimming events. All sevenindividuals started from rest, and swam when stimulated. No individual was used morethan twice.

exponential function of the form e(x)=a+l3eyx (Fig. IB) to generate an amplitudeenvelope with fairly constant height a and a sharp increase at the tail. Todetermine the coefficients a, /3 and y I fitted this function with a least-squaresregression routine to the absolute value of the observed amplitude distribution(Fig. 4), ignoring the displacement of the head since it contributes negligibly tothrust (Lighthill, 1975). The amplitude measures (a) specific to each swimmingevent, given in Table 1, were determined by averaging the wave heights measuredover the central 60 % of the body. This method does not include the large waveamplitude observed at the head. I chose to ignore the contribution to thrustgenerated by the larger amplitude over the first 20 % of the body on experimentaland theoretical grounds. In particular, Lighthill (1975) has demonstrated that thethrust generated by an undulating body at high Re is determined primarily by theamplitude at the trailing edge, where momentum is shed into the wake. In theresistive case, Lighthill (1975) showed that thrust depends on the mean squarecosine of 6. This implies that resistive thrust is a function of the average waveheight along the entire body and is less sensitive to localized peaks, such as in thehead and tail regions. Additionally, the average wave height over the first 20 % ofthe body differs by only 10 % from a, the average over the center 60 % of thebody.

Using the mean values for a, /3, y, A, /, c and a>, the model generated a predictedbody translation (Fig. 5A) that was well within the range of observed values

A model of undulatory locomotion 129

CL IP I

0.25

0.20

•2J2 0.15

"5.E<

0.10

0.05

0.000.0

Head0.4 0.6 0.1

Position along body (body lengths)

1.0

Tail

Fig. 4. Distribution of the absolute value of propulsive wave maxima for sevenindividuals during 11 swimming events, scaled to body length. Each circle representsthe peak of a wave on the body during the powered phase of swimming. Values weretaken from digitized high-speed cin£ film frames shot at 200 frames s"1. The heavy lineis e, the exponential curve fitted to represent the amplitude envelope in h{x,t).

(Fig. 3). When the function e(x) is fitted to the values for the individuals with thelargest and smallest amplitude distribution, roughly corresponding to the steepestand shallowest curves in Fig. 3, comparing predicted body translations with thosemeasured for these two individuals, very good agreement was obtained betweenthe observed and predicted distances traveled (Fig. 5B,C). Furthermore, as thepropulsive wave amplitude is varied, the model continues to be a good predictor ofdistance traveled in that there is no trend in the goodness of fit with amplitudeheight (Kendal's ranked correlation of residuals vs observed, P>0.5).

Discussion

Because predicted and measured displacements are well correlated (Wilcoxonsigned rank test, P=0.02), the model developed here gives reasonable estimates ofaverage propulsive forces. To test the role of animal behavior and morphology ingenerating forces, I define distance traveled over a short interval of time as themost appropriate performance criterion for the chaetognath Sagitta elegans.However, whenever a performance measure increases without being limited bychanges in behavior or morphology, as is the case here with increasing propulsivewave amplitude, limitations to force production must be considered in order togenerate physiologically feasible motion (Daniel and Meyhofer, 1989; Meyhofer

130 C. E. JORDAN

20 40 60

Time (ms)

80

Fig. 5. Model predictions of distance traveled with time. (A) For this simulation,predictions were made based on the average kinematics shown in Table 1. (B) Thebold line is the measured swimming performance of the individual with the largestpropulsive wave amplitude (no. 4, Table 1). The solid line is the predicted swimmingperformance based on the kinematics of this individual. (C) A comparison of observedand predicted swimming performance for the individual with the smallest waveamplitude (no. 2, Table 1).

and Daniel, 1990). The model was originally developed to predict average forcesgenerated by the body, but it does so by integrating sectional instantaneous forcesand, thus, is a predictor of local instantaneous stresses generated by the body wallmusculature. Amidst the scaling and evolutionary arguments, emerge rules forundulatory locomotion at intermediate Re that apply broadly to many aquaticorganisms.

Limitations to the model

I have made several assumptions to simplify the hydrodynamic analysis. Whilethe reactive terms are fully unsteady and account for fluid accelerations, theresistive terms are calculated in a quasi-steady manner, ignoring the timedependence of their development. The coefficients of drag, both normal andlongitudinal, are taken from published values for similar shapes moving through afluid at a constant velocity (White, 1974; Hoerner, 1965). In reality, developmentof flow patterns, wake formation and separation are all time-dependent, and may

A model of undulatory locomotion 131

affect the magnitude of the drag and added mass coefficients (Schlichting, 1979;Daniel, 1984). Additionally, fin-tail and fin-body interactions were neglected byassuming that fin wakes do not affect relevant flows about downstream structures.However, recent experimental work shows little difference between quasi-steadypredictions and measured forces on an oscillating appendage at low and intermedi-ate Re (Williams, 1990).

The choice of distance traveled over a short interval of time as the best criterionof performance stems from the assumption that chaetognaths swim rapidly eitherto avoid predation or to attack motile prey. There are few published observationsof predation on chaetognaths by fishes or other zooplankters, though it has beennoted that many chaetognath species are cannibalistic (Ghirardelli, 1968).S. elegans swims when disturbed either by direct contact or in response to anapproaching object (C. E. Jordan, personal observation), indicating that they arecapable of an escape response. Velocities of particles in the flow field of suction-feeding fish (Lauder and Clark, 1984) are similar to those observed for chaetog-naths on the same time scale, implying that S. elegans is able to escape fishpredation. There have been several studies of predation by chaetognaths focusingon prey detection (Feigenbaum and Reeve, 1977; Newbury, 1972), but with noreference to the whole-body kinematics of feeding. Until observations showotherwise, it is reasonable to assume that S. elegans swims in rapid bursts in orderto escape predation or to capture prey. A rapid start-up would then be directlycorrelated to survival and, in an adaptationist sense, the behavior and morphologyof 5. elegans could be selected to optimize this measure of swimming performance.

Correlates of force, morphology and behavior

Despite any limitations and assumptions, the model, constructed by summingthe instantaneous forces along the body, is a reasonable predictor of bodytranslation as a function of the kinematics of the propulsive wave. The modelaccurately predicts the acceleration of a body started from rest by numericallyextending existing constant body velocity theory into a time-dependent form. Themodel also predicts an increase in average thrust produced by the body withincreases in wave amplitude corresponding to the increased performancemeasured in individuals with increasing wave amplitude.

The experimental observations of chaetognaths show a large variation in thedistance traveled in the first 65 ms of swimming and, hence, a large variation inperformance. What then are the mechanical determinants of locomotion such thata variation in performance can be correlated with a change in behavior ormorphology? At low Re, thrust is correlated positively with the square of waveamplitude and number (Gray and Hancock, 1955), while at high Re, thrust iscorrelated with the square of wave amplitude (Lighthill, 1975; Daniel, 1988).However, the mechanical determinants of performance at intermediate Re are notnecessarily a linear combination of these results. Average forces scale as theamplitude squared (see Fig. 8A,B), but viscous forces can predominate instan-taneously (see Fig. 7). Fig. 6 shows the model's prediction for distance traveled

132 C. E. JORDAN

0.50

400 600

Time (ms)

800 1000

Fig. 6. Comparison of the model (bold line) with a purely resistive (solid line) and apurely reactive (dashed line) case. Both A and B represent the same simulation; A is adetail of the first 100ms. In this simulation, the model is run only with the averagekinematic values (Table 1, Fig. 5A) and compared with both a run in which only theviscous terms (see equations 1, 5 and 6) are present (the resistive limit) and a run inwhich only the inertial terms (see equation 4) are present (the reactive limit). Note thatthe two limiting cases do not sum to the case in which all terms are present.

with time compared with the two limiting cases, either purely viscous or purelyinertial. Two results are striking. (1) During the early phase of swimming(Fig. 6A), the situation can accurately be represented solely by inertial terms, butlater (Fig. 6B), when the start-up force transients have decayed, the viscous modelbecomes a good approximation. (2) At all times the combination of viscous andinertial terms predicts more force than either the purely viscous, or purely inertial,limit.

The model predicts that resistive and reactive forces are of similar magnitude,although they have markedly different time histories during swimming (Fig. 7).Though the average force is a good predictor of distance traveled, the existence oftransient peaks much higher than the temporal average suggests importantmechanical limitations to force production.

A model of undulatory locomotion 133

150.0

75.0

-75.0

-150.0

Time (ms)

Fig. 7. Prediction of instantaneous force production over time, plotted by component.The bold line is the total force, the solid line is the resistive force component and thedashed line is the reactive force component. The model was run with the averagekinematic values (Table 1, Fig. 5A). All three values are calculated simultaneously anddisplayed individually such that the solid and dashed lines are the instantaneouscontributions of the resistive and reactive terms, respectively, and hence sum to thetotal (bold) instantaneous predicted force.

There have been many studies of the kinematics of fast-start performance in fish(Webb, 1976, 1978, 1983), but the accompanying theory (Weihs, 1973) does notaccount for viscous flows and cannot be applied at this scale. The only othertheoretical work on burst swimming at intermediate Reynolds numbers is that ofVlymen (1974) and Weihs (1974, 1980), whose formulations are in terms of theenergetics of larval fish during locomotion. Vlymen's model was unique in takinginto account both inertial and viscous terms in undulatory locomotion, but he didnot address the problem of force generation acting to accelerate the body. Thework of Vlymen, therefore, is descriptive and not predictive. His formulationcannot be used to explore the mechanical determinants of locomotion. The workof Weihs addresses the relative efficiency of burst and glide versus continuousswimming through ontogenetic size and shape changes. His method does notinvolve calculating instantaneous force distributions along the body and, hence,cannot predict forward speed.

The theory presented here can be used as a means to explore the effect ofmorphological and behavioral variation on performance. For example, isometri-cally scaling a chaetognath to twice its original linear dimension, and prescribing

134 C. E. JORDAN

similarly scaled kinematics, will generate much greater forces and hence higherperformance. Such morphological variation is well within the biological range, butthe limiting factors may be the stress generated by muscle contractions. Althoughsuch stresses have not been measured for chaetognath body wall musculature, themaximum isometric stress of muscle is known and is fairly conserved across allmuscle types examined (Prosser, 1973). The model generates instantaneous forcesthat can be equated with this isometric stress by dividing the maximum instan-taneous force within the first 65 ms of swimming by the local cross-sectional area ofmuscle. Such a calculation yields a predicted stress of 0.32 MPa, comparable to thevalues published for striated muscle from organisms in other phyla (0.8-12.0 MPa;Prosser, 1973). This mechanical limitation of the chaetognath musculatureprovides a realistic boundary to the range of feasible morphologies and kinema-tics.

Fig. 8 represents level surfaces generated by simulating morphological variationto predict the effects on performance. For an array of body lengths and fin heights,wave amplitude (a) is increased until the mechanical boundary of the maximumisometric stress is reached. The corresponding distance traveled is recorded asperformance (Fig. 8A). Fig. 8B represents the same result as Fig. 8A, but shows

3.0-,

4.0lyle"s

A model of undulatory locomotion 135

instead the wave amplitude at which the mechanical boundary was reached. Thesurfaces show a sharp tuning of the wave kinematics and maximum distancetraveled that corresponds to a body length of roughly 2cm. Fig. 8A also showsthat, for a given body size, increasing fin height results in lowered performancewith increasing amplitude. The general shape of Fig. 8A can be explained by thefollowing scaling argument. Comparing Figs 8A and 8B, it is apparent thatdistance traveled during the initial 65 ms scales with the square of the waveamplitude, as predicted by the linear theory of Lighthill (1975). The loweredperformance with increased fin height follows because the added mass coefficientscales linearly with body depth. For a given muscle stress, as fin height increases,sectional force production also increases. To remain below the maximum stresslevel in the muscle, the wave amplitude must decrease to conserve force. Anyincrease in thrust due to the presence of increased fin size is exceeded by thedecrease in thrust due to reduced amplitude.

The prediction that increased fin height decreases performance seems counter-intuitive in the light of the extreme morphological conservation exhibited by thephylum, which includes a tail fin and at least one pair of lateral fins on all species(Hyman, 1959). However, the optimum of no fins may be explained byconsideration of the body architecture. The longitudinal body wall musculature isarrayed in four discrete bundles just beneath the epithelium (Duvert and Salat,1980), one in each quadrant of a cross section, such that the dorsal pair acts inopposition to the ventral pair. With the fins only in the frontal plane midwaybetween the dorsal and ventral muscle masses, they would act to damp out anyrotational components to the body motion introduced by a phase difference incontraction between the two muscle bundles within either the dorsal or ventralmuscle masses. The fins would act to stabilize the swimming motion at the expenseof performance because of their increased drag and added mass. Ultrastructuralstudies of 5. hispida show the presence of transverse body wall musculaturecapable of deflecting the fins, raising the possibility that the fins could play anactive role in directional control (Shinn, 1989). There may be an optimal fin sizeand height predictable from a species' behavioral pattern based on its need forstability versus maneuverability. To elucidate the role of fins of a given size, finremoval studies similar to those done on fish (Webb, 1977) could be performed.The model predicts increased performance with fin removal, and observation mayshow a corresponding loss of control.

Appendix

Ab, A( Sectional projected area of the body and of thebody with fins

c Propulsive wave speedCd\, Can Drag coefficients of longitudinal and normal

flowD\, Dn Sectional longitudinal and normal drag force

136 C. E. JORDAN

/ Fin and tail height^t> Fi> Fs Total, inertial and sectional forceh(x,t) Lateral body displacement/ Body lengthm(x)=pit(r+f)2 Sectional added mass; /=0 in areas without finsM Mass of chaetognath (1.006p^r2/)r Radius of the body, constant along the lengthRe Reynolds number5b, S{ Sectional surface area of the body and of the

body with finst TimeU Forward body velocityV, Vn, V\ Lateral velocity; normal and longitudinal com-

ponentsx Position along the bodya, fl, y Coefficients of the propulsive wave amplitude

envelopee(x) Propulsive wave amplitude enveloper(t)=herfc(tan{ji{(t/ b)-(l/2))}\ Propulsive wave amplitude growth during b,

the start-up phaseA Propulsive wave lengthv Kinematic viscosity of sea waterp Mass density of sea water6 Angle made by a body section with the direc-

tion of travela> Angular frequency of the propulsive wave

My research has benefited greatly from the advice and guidance of T. L. Daniel.I thank A. O. D. Willows for providing laboratory space and support at the FridayHarbor Laboratories. This manuscript has been vastly improved by criticalreadings by T. L. Daniel, G. Nevitt and D. Grunbaum. P. Karieva, A. J. Kohn,G. M. Odell and three anonymous reviewers have provided many helpfulcomments. This work was supported by grants from the Whitaker Foundation andthe National Science Foundation (grant DCB-8711654), both to T. L. Daniel.

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