© 2002 by The Society for Integrative and Comparative Biology
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Dynamics of Dolphin Porpoising Revisited1
1 Faculty of Aerospace Engineering Technion, Haifa 32000, Israel
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Porpoising is the popular name for the high-speed surface piercing motion of dolphins and other species, in which long, ballistic jumps are alternated with sections of swimming close to the surface. The first analysis of this behavior (Au and Weihs, 1980
) showed that above a certain "crossover" speed this behavior is energetically advantageous, as the reduction in drag due to movement in the air becomes greater than the added cost of leaping. Since that publication several studies documented porpoising behavior at high speeds. The observations indicated that the behavior was more complex than previously assumed. The leaps were interspersed with relatively long swimming bouts, of about twice the leap length. In the present paper, the possibility of dolphins using a combination of leaping and burst and coast swimming is examined. A three-phase model is proposed, in which the dolphin leaps out of the water at a speed Uf, which is the final speed obtained at the end of the burst phase of burst and coast swimming. The leap is at constant speed and so the animal returns to the water at Uf, goes to a shallow depth and starts horizontal coasting while losing speed, till it reaches Ui. At that point it starts active swimming, accelerating to Uf. It then starts the next leap. Ranges of speeds for which this three-stage swimming is advantageous are calculated as a function of animal and physical parameters.
Notation
C—Constant defined in equation (12)
CD—Coasting drag coefficient
D—Drag
g—Gravitational acceleration
H—Height of jump
J—Energy required for jump
k—Ratio of swim length to jump length
l—Distance
L—Total distance (eq. 28)
m—Added mass
M—Animal mass
M1—Total mass
r—Coefficient defined in eq. (22)
R—Ratio of energies, for three-phase swimming
R2—Ratio of energies, for burst and coast swimming
t—Time
T—Thrust
U—Speed
V—Body volume
W—Weight
—Emergence (=return) angle
ß—Swim / coast drag penalty ratio
—Surface effects drag ratio
—Density of seawater and cetacean.
Subscripts
a—air
av—Average
b—Burst phase
c—Coast phase
e—Reference (maximal) thrust
f—Final, at end of burst
i—Initial, at start of burst
j—Jump phase
n—Nominal reference thrust
o—Optimal
s—Surface swimming
w—Water
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INTRODUCTION |
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TOP
SYNOPSISINTRODUCTIONANALYSISRESULTS AND DISCUSSIONReferences |
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Porpoising is the popular name for the high-speed motion of dolphin schools, in which long, ballistic jumps are alternated with sections of swimming close to the surface. The first mechanically based analysis of this behavior appeared in 1980 (Au and Weihs) who showed that above a certain speed (which they called the "crossover" speed) this behavior is energetically advantageous. This results from the fact that the reduction in drag due to movement in the air becomes greater than the added cost of leaping. Since that publication several studies (Au and Perryman, 1982
; Hui, 1989) documented porpoising at high speeds. The observations indicated that there were two phenomena involved which were not dealt with in the original Au and Weihs (1980) analysis. First, the leaps were interspersed with relatively long swimming bouts, of about twice the leap length (Au et al., 1988; Hui, 1989). Second, the angle at which leaping began was around 39°, i.e., lower than the optimal 45°. The second point may well be a compromise between maximal jump length which requires the leap angle to be 45° and the maximal horizontal speed which requires an exit angle of 30° (Gordon, 1980). The first point indicates that a more complex behavior pattern may be involved and Au et al. (1988) surmise that this may be the result of a "variant of the energy saving power and glide swimming...where the glide phase begins at the top of the leap," based on the burst and coast swimming model of Weihs (1974). Azuma (1992) followed up on this suggestion with an analysis including burst and coast effects. However, the effectiveness of his analysis is limited by the need to obtain various parameters not presented there. As a result, his final results are only given in general terms, for limiting cases. Also, the glide phase can more reasonably start at the moment of full emergence, the ballistic trajectory allowing for constant speed, Uf. In the present paper, this possibility of dolphins using a combination of leaping and burst and coast swimming is reexamined. A three-phase model is proposed, in which the dolphin leaps out of the water at a speed Uf, which is the final speed obtained at the end of the burst phase of burst and coast swimming. The leap is at constant speed and so the animal returns to the water at Uf, goes to a shallow depth and starts horizontal coasting while losing speed, till it reaches Ui; at that point it starts active swimming, accelerating to Uf, at which point it starts the next leap. Thus the three-phase swimming model is actually a combination of the Au and Weihs, (1980) model somewhat modified as a result of new data, and the burst and coast mode described by Weihs (1974). In this analysis, nondimensional ratios are obtained, for which all parameters are known, enabling numerical results for the effectiveness of porpoising.
For burst and coast swimming to be effective, the drag penalty ß due to swimming, resulting from induced drag and the body motions, must be larger than unity. In various fish species this penalty increases the drag by a factor of up to 4 (Lighthill, 1971). Carangiform swimmers are at the low end of this range, at about 1.5. Hui (1989) assumed this factor to be unity, but in a later review (Fish and Hui, 1991) show that power output estimates were 4–16 times larger than the expected drag on a rigid (non-swimming) body. While at least part of this (up to a factor of 5) is due to surface proximity effects (Au and Weihs, 1980) the high upper limit indicates that ß > 1 for swimming cetaceans. This ties in nicely with data for Tursiops truncatus that indicate that ß can be up to 5 (Skrovan et al., 1999).
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ANALYSIS |
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SYNOPSISINTRODUCTIONANALYSISRESULTS AND DISCUSSIONReferences |
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According to the proposed model, porpoising has three distinct phases
- A jump phase—described by a ballistic trajectory in the air above sea level.
- A glide phase—during which no swimming motions take place, starting upon penetration of the sea surface.
- An acceleration phase—characterized by high-thrust swimming motions at constant depth, except for the final moments, at which a climb and ballistic emergence angle are achieved.
As mentioned in the Introduction this three-phase model is actually a sequential combination of two types of behavior analyzed separately in previous work. These are porpoising as defined by Au and Weihs (1980) for the first phase, and burst and coast swimming as defined by Weihs (1974) for phases 2 and 3. As Azuma (1992) and Webb and Weihs (1983) among other books include these models, only an abridged version of the development of the two separate models will be presented here.
First, we define the energy required for constant speed swimming at the water surface. This is the reference energy usage to which we later compare the cost of the three-phase motion. The energy required in order to traverse a distance L, while swimming at the surface is,
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Which at uniform speed is equal to DL, as the thrust is equal to drag at uniform speed. The drag can be written as (Au and Weihs, 1980)
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The drag coefficient for active oscillatory swimming for a body close to the surface is
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Where ß is the ratio of drag when performing swimming motions, to drag of the stretched-out body. ß has been calculated to be up to 5 for strenuously swimming Bottlenose dolphins (Skrovan et al., 1999). In our calculations we limit ß 
3, which seems a good estimate for cyclical behavior. is the ratio of drag swimming close to the water surface, relative to swimming at depth. This ratio can reach values of 5 depending on the depth (Hertel, 1966; Lang and Daybell, 1963). These large increases in the drag penalty are the basis for the advantage of the three-phase motion.
We continue the analysis of the three-phase motion by examining the jump part. We assume a calm, current-free sea-state. The effects of any existing surface wave or currents are neglected, so that the analysis is general. Next, we take the aerial trajectory of the animal to be ballistic, i.e., air resistance is neglected also. This will be shown, below, to have an effect of less than 0.01% on the energy budget. The drag in air is, from (2)
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the equivalent drag in the water, while swimming is given by (2). Dividing, we obtain
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for standard conditions a/w 
1/800; ß 3; 4.5 so that Da/Ds 1/10800. Thus from (1), the ratio of energy lost by viscous drag in air is less than 0.01% compared to an animal swimming close to the surface and about 0.1% for a coasting cetacean in deep water, (where ß = 1).
Performance during a ballistic leap can be calculated from the laws of mechanics. The horizontal distance traversed from the point of exit, during which the center of mass of the animal is out of the water is, from Au and Weihs (1980):
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the time spent out of the water is
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the extra energy required for the jump J can be estimated from the height achieved
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where the added mass coefficient m is roughly 0.2 for a coasting animal, (Webb, 1975) and 0.2ß for a rapidly swimming animal. J however is not the energy lost during the jump, as the cetacean's body, moving at equal speed when leaving, and returning to the water surface, retains the energy of the body. Thus energetic losses are only incurred due to the water exit and entry splashes, neglecting viscous drag in the air. The splash energy can be roughly estimated from the added mass of the animal, which is defined as the amount of water moving with the animal at an equal speed. Thus, as the animal leaves the water while performing swimming motions, and returns in a coasting configuration, the energy carried by the exiting accompanying water is lost, while additional energy is invested in accelerating water upon entry. These two contributions can be written as
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Au and Weihs (1980) used a different, less accurate, estimate of Ej = MU2(1 + m)/4. Azuma (1992) just defined a splash energy, without specifying. Recalling that m 0.2, the present estimate of splash energy is somewhat different (up to ±33%, depending on the value of ß) from that of Au and Weihs (1980).
Next the coast phase is studied. From Weihs (1974), the differential equation describing the decelerating motion during coasting is, in present notation:
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here CD is the drag coefficient (3) of a coasting, deeply submerged animal. We assume here that reentry is at angle and the animal changes to horizontal motion after penetration of the water surface. We now denote
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i.e., eq (11) is now
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The boundary conditions for this differential equation are
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Where Uf is the speed of penetration and Ui the lower speed attained after time tc spent coasting. This equation is solved in Weihs (1974), leading to the expressions
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for the length of the coast phase, and
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for the time spent coasting. Actually only one boundary condition, (14), is needed for the solution, as (13) is a first order ordinary differential equation. Equation (15) is then applied to obtain specific values.
When the decreasing speed reaches the value Ui, the animal starts accelerating at constant, maximal thrust Te (This was shown in Weihs, 1974, to be the optimal strategy). The differential equation for the burst phase is
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we define the amount of time tb spent accelerating, including pitching up and leaving the water, at speed Uf. This emergence speed is specified, as the ballistic jump phase has equal emergence and reentry speeds. Eq. (18) can be solved using (19). The distance traversed during the burst phase is obtained from (20) as
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and Ue is a reference velocity obtained from
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i.e., Ue is the maximal possible speed, obtained from (18) as the speed, where no further acceleration is possible so that the derivative in the first term vanishes. Ue can be calculated from (23) if the maximal thrust Te is known. Ue is a convenient reference velocity as it is, by definition, greater than both Uf and U1 (otherwise the animal would not be able to attain Uf with thrust Te). Thus, by dividing by Ue, non-dimensional values conveniently bounded between 0 and 1 are obtained. Using Ue only as a normalizing factor does not even require knowledge of Te.
The time required to accelerate from Ui to Uf can be written as
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where the bar indicates division by Ue.
We now have explicit expressions for all the terms required to calculate the energy expenditure for the three-phase motion, and compare it to uniform swimming. Next we need to establish pairs of values 
i and f that will fit experimental observations.
Au et al. (1988) have shown that swim lengths were, on average, about twice as long as jump lengths. We shall take this ratio of swim length to jump length k as a measurable quantity. Thus,
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and substituting eqs. (6), (16) and (21) into (25), after some manipulation
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but M1g = W(1 + m), see (12); and ß
= Te by definition, so that
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and substituting (24) into (27), we see, that if m, k and W/Te are given, we have an equation which relates i and f. Here m 0.2, and k 2. Recalling that Te is a reference thrust, we take this to be the maximal value attainable. While, direct measurement of thrust for locomoting dolphoins has not been accomplished static measurements of up to 1.5 times body weight were measured, and about 0.6 times the body weight at maximum oxygen consumption and minimum lactate production. (Fish and Rohr, 1999). Some results for the ratio W/Te were also quoted recently by Frank Fish (personal communication). These appear in Table 1.
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We can now obtain
i for any given f, from eq. (27), when m, k and W/Te are known, as the product rtb appearing in (27) is a function of i and f only (eq. 24). Returning to the basic question addressed in this paper we now compare energy requirements for the three-phase (leap-coast-burst) motion to those of the equivalent steady swimming mode.
The total distance crossed in one cycle of the three-phase mode is
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Or, from (16) and (21), applying the fact that the speed during the jump is Uf, and using the non-dimensional values for i and f
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eq. (29) can be further simplified as
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The total energy required for one cycle of the three-phase mode is Ej + Eb, as the coast phase does not require energy. From eqs. (1) and (10), recalling that the thrust during the burst phase is Te, and using (23)
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The energy cost of steady swimming near the surface at the same average speed is, from eqs. (1)–(3)
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where Uav is the average forward speed during the whole cycle. From (30), (31) and (32) the ratio of energy required for the three-phase mode, compared to steady swimming is, per unit distance crossed (as the distances are equal)
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The only unknown quantity in (33) is av, the normalized average speed. This can be calculated from
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all of which have been obtained previously. From eqs. (7), (17), (24) and (30), after some manipulation,
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We now have all the expressions required for a parametric study, which will establish ranges of speeds for which the three-phase mode is less costly in terms of energy. It is of interest also to compare the ratio of energies required R, to an equivalent ratio, comparing burst and glide swimming to steady swimming as in Weihs (1974). This ratio (eq. 18 in Weihs, 1974) is, in present notation, and again considering both modes to occur at the surface (so that cancels out),
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A comparison of R and R2 will indicate the speed ranges at which each of these two behavioral modes is advantageous, as a function of ß.
Combination of the curves requires one further step, as the reference velocity Ue taken from (23) is a function of both ß and W/Te from (23) and (12)
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To unify the curves to a single abscissa, we define a nominal maximal velocity for W/Te = 1, ß = 1, and = 4.5. Thus
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and we can use a unified abscissa Uav/Un obtained in a manner similar to Uav, (35)
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RESULTS AND DISCUSSION |
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SYNOPSISINTRODUCTIONANALYSISRESULTS AND DISCUSSIONReferences |
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We start with a straightforward comparison of the energy required by the leap-burst and coast strategy and uniform speed swimming close to the surface, as a function of normalized average swimming speed
n. As mentioned in the previous sections, accepted values for the different parameters appearing in the equations (33, 35, 36 and 39) are used. The added mass fraction in the longitudinal direction is taken to be m 0.2, the ratio of weight to thrust is 0.75 W/Te 2 and the drag penalty parameter is assumed to be 1 < ß 3. The results in the following figures should be taken as indicative of the quantitative dependency of the energy ratios R and R2. Actual values of the parameters above will be different for various species.
As preparation, the energy saving for burst and coast swimming relative to uniform swimming R2, is taken from Weihs (1974). Values of R2 that are less than unity indicate energy savings. Figure 2 is an adaptation of that paper's Figure 1, showing that the optimum gains are at low speeds and tend to the value 1 – 1/ß, i.e., for ß = 3 one can gain 66% relative to constant speed swimming. Thus, at slow swimming speeds, neglecting effects of proximity to the water surface, burst and coast swimming is highly effective. However, cetaceans needing to move at high speeds, which entail high aerobic and breathing rates will have to be close to the surface. This requires a change to the leap, burst and coast mode.
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The ratio of energy required per unit distance crossed R, for the three-phase leap, coast and burst mode (eq. 33), relative to the energy required for uniform speed swimming at the same average speed,
n, is taken from eq. (39). Here again, values of R that are less than unity indicate that using the three-stage is advantageous under these conditions. For example R = 0.8 indicates an energy saving of 20%. The calculation procedure is as follows. For a given combination of ß and W/Te, a value of f, the jump speed, is chosen. Substituting this value, in eq. (27) results in a value for i. These values f and i result in the values for the average velocities av, n and R, from eqs. (33, 35 and 39).
Figure 3 shows the energy ratio R, as a function of the nominal normalized speed n, for different values of W/Te (i.e., different animals). Each panel (a, b, and c) is calculated for a different value of the swimming drag penalty ß. The case of no penalty (ß = 1 – Fig. 2a) shows very small gains only, at relatively high speeds and high thrusts. This is expected, as for ß = 1 there is no advantage in burst and coast swimming (Weihs, 1974) and the only gains can result from jumping. This is not a realistic case, as cetaceans have a large-amplitude body oscillation type of swimming for which ß > 1, as mentioned previously. Next, we see, in Figure 3b–3c, increasing values of ß. All curves here are bucket shaped, showing that there is an optimal speed for three-phase swimming for each species. The exact value of this optimum can only be determined when good measurements of all parameters are available. Some trends are obvious from this figure. First, as the weight to thrust ratio decreases (a stronger, leaner animal) the savings are greater, and obtained at a higher average optimal speed. The range of speeds, at which three-phase swimming is advantageous grows with increasing thrust and the curve is flatter so that missing the exact optimal speed is less critical.
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Figure 4 shows a different aspect of the calculations. Here R is a function of
n, for three different values of W/Te, and a range of ß values from 1–3 (the points are for ß = 1.0, 1.5, 2.0, 2.5, 3.0 respectively). The values of W/Te chosen are those quoted for Pseudorca and Tursiops, respectively and a hypothetical animal with about twice the thrust of Tursiops for comparison. From these curves one sees that the maximal energy gains are obtained for relatively high average velocities (f is around 0.8) and high ß. The gains are bigger when for smaller W/Te, i.e., a cetacean with high thrust to weight ratio has more to gain from the three-stage mode. The highest gains are less than 20% for W/Te = 1.6, about 25% for W/Te = 1.25 and 40% for W/Te = 0.75. The curves all show a minimum for different ß. This minimum is getting out of the range that is to be expected for the higher jump speeds.
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The three-phase motion (Figs. 3 and 4) cannot be directly compared to the burst and coast calculations, as the latter (Fig. 2) do not include surface proximity effects. A different comparison of the three-phase swimming mode to the Au and Weihs (1980)
model for constant speed underwater swimming interspersed with leaps is trivial, as it is clear that this is a subset of the present model.
Figure 5 shows the gains at optimal speeds for three-phase swimming vs. constant speed swimming vs. n. Full lines are for constant ß while dashed lines stand for fixed W/Te. The optimal speeds are seen to increase with increasing maximal thrust, resulting in higher gains (lower R). These optimal speeds are somewhat reduced for high ß, i.e., larger body excursions. Finally, as the relative thrust decreases, the gains due to porpoising also decrease.
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Figure 6 shows a different comparison, of three phase swimming, versus burst and coast close to the surface. This figure shows that at low average speeds, it is more effective to use the burst and coast mode, while for every ß there is an average speed at which the three phase mode becomes more energy sparing. This is a generalization of the crossover speed found by AW. As ß increases, the crossover speed is reduced (from approximately Un = 0.48 for ß = 1.5 to Un = 0.44 for ß = 3) and the total gains possible are increased.
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A lower limit for W/Te for porpoising to be possible can be obtained from (6) and (10), by writing
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so that equating (40) and (41)
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T so that finally
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We can thus conclude that for animals with low weight to thrust ratios, significant gains in energy, of up to 40% can be obtained at high swimming speeds. Actual gains for specific animals will be dependent on their physical condition. Thus accurate predictions for individual animals are difficult.
Finally, we show a sample calculation, for Tursiops, for which the most data is available.
Maximum speeds Ue are 8.2 m/sec at the surface, and 11.2 deeper down (Rohr et al. 2002). From equation (38) we calculate Un by taking ß = 3 and 1 < W/Te < 1.5 as a good range so that 1.73 < Un < 2.18 Ue. For simplicity, we now take a midrange value, i.e., Un = 2 Ue = 16.4 m/sec. So, from equation (39) for swimming at (say) Uav = 7 m/sec, n = 7/16.4 = 0.43. Taking this value to Figure 5 we see, for W/Te = 1, R = 0.67, i.e., a 33% energy saving by porpoising. If W/Te = 1.5 (as in Table 1) R = 0.82. i.e., only 18% saving. So we expect the saving due to porpoising at 7 m/sec (14 knots) to be somewhere between these two values, i.e., around 25%. From Figure 6 we see that at this speed, saving due to burst & coast swimming can theoretically be up to 40%, but this neglects surface piercing for breathing.
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ACKNOWLEDGMENTS |
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This study was supported by the Fund for Promotion of Research at Technion. I would like to thank Daphne Weihs for assistance with the calculations and graphics, Frank Fish for useful discussions and data, and Malcolm Gordon for inviting me to present this work at the SICB 2002 annual meeting.
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FOOTNOTES |
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1 From the Symposium Dynamics and Energetics of Animal Swimming and Flying presented at the Annual Meeting of the Society for Integrative and Comparative Biology, 2–6 January 2002, at Anaheim, California.
2 E-mail: dweihs{at}tx.technion.ac.il
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References |
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TOP
SYNOPSISINTRODUCTIONANALYSISRESULTS AND DISCUSSIONReferences |
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Au, D., and W. L. Perryman. 1982. Movement and speed of dolphin schools responding to an approaching ship. Fish. Bull, 80:371-379.
Au, D., M. D. Scott, and W. L. Perryman. 1988. Leap-swim behavior of "porpoising" dolphins. Cetus, 8:7-10.
Au, D., and D. Weihs. 1980. At high speeds dolphins save energy by leaping,. Nature, 284:548-550.[CrossRef]
Azuma, A. 1992. "The biokinetics of flying and swimming.". Springer, Tokyo.
Fish, F. E., and C. A. Hui. 1991. Dolphin swimming—a review. Mamm. Rev, 21:181-195.[CrossRef]
Fish, F. E., and J. J. Rohr. 1999. Review of dolphin hydrodynamics and swimming performance. SPAWAR Tech.Rep. 1801.
Gordon, C. N. 1980. Leaping dolphins. Nature, 287:759.
Hertel, H. 1966. "Structure, form and movement.". Reinhold, New York.
Hui, C. A. 1989. Surfacing behavior and ventilation in free ranging dolphins. J Mamm, 70:833-835.[CrossRef]
Lang, T. G., and D. A. Daybell. 1963. Porpoise performance tests in a seawater tank. NOTS Tech. Rep. 3063.
Lighthill, M. J. 1971. Large amplitude elongated body theory of fish locomotion. Proc. Roy. Soc., B-179:125-138.
Rohr, J. J., F. E. Fish, and J. W. Gilpatrick Jr. 2002. Maximum swim speeds of captive and free ranging delphinids. Mar. Mammal Sci., 18:1-19.
Skrovan, R. C., T. M. Williams, P. S. Berry, P. W. Moore, and R. W. Davis. 1999. The diving physiology of Bottlenose dolphins (Tursiops truncatus). J. Exp. Biol, 202:2749-2761.[Abstract]
Webb, P. W. 1975. Hydrodynamics and energetics of fish propulsion. Bull. Fish. Res. Bd. Can, 190:1-159.
Webb, P. W., and D. Weihs.(eds.) 1983. Fish biomechanics. Praeger Press, New York.
Weihs, D. 1974. Energetic advantages of burst swimming of fish. J. Theor. Biol, 48:215-229.[CrossRef][Web of Science][Medline]
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