Scaling the tail beat frequency and swimming speed in underwater undulatory swimming

Undulatory swimming is the predominant form of locomotion in aquatic vertebrates. A myriad of animals of different species and sizes oscillate their bodies to propel themselves in aquatic environments with swimming speed scaling as the product of the animal length by the oscillation frequency. Although frequency tuning is the primary means by which a swimmer selects its speed, there is no consensus on the mechanisms involved. In this article, we propose scaling laws for undulatory swimmers that relate oscillation frequency to length by taking into account both the biological characteristics of the muscles and the interaction of the moving swimmer with its environment. Results are supported by an extensive literature review including approximately 1200 individuals of different species, sizes and swimming environments. We highlight a crossover in size around 0.5–1 m. Below this value, the frequency can be tuned between 2–20 Hz due to biological constraints and the interplay between slow and fast muscles. Above this value, the fluid-swimmer interaction must be taken into account and the frequency is inversely proportional to the length of the animal. This approach predicts a maximum swimming speed around 5–10 m.s−1 for large swimmers, consistent with the threshold to prevent bubble cavitation.


Introduction
Beyond a few centimeters in length, most aquatic vertebrates propel themselves through the water by deforming their spines and propagating deformation waves through the body [1].Fish, cetaceans, reptiles, amphibians, and birds oscillate the head, body, tail, and/or fins, as appropriate, with a variety of gaits described by a specific classification [2].Despite the complexity of treating each case separately, the kinematics of underwater undulatory swimmers can be captured to first order with a few parameters such as the wavelength of the deformation λ, the tail beat amplitude A and the tail beat frequency f .There is now considerable evidence that λ and A are strongly related to animal length L, regardless of the size and shape of the animal, or the swimming conditions.In the example of fish, the wavelength scales as the animal length, with a factor of the order of unity [3,4].The same applies to the tail beat amplitude that follows A 0.2L [5][6][7][8] from tadpoles of a few centimeters to whales of 20 meters in length (Methods 4.1).These simple allometric scaling relations reveal general physical laws that are valid over several orders of magnitude in size.Momentum balance and minimal energy expenditure associated with the hydrodynamic interaction between the moving body and the surrounding water appear to drive the selection of the amplitude, as well as the determination of the swimming speed, U 3.3Af [5,[8][9][10].Therefore, the swimming speed is proportional to the oscillation frequency for a given swimmer and scales as Lf , with a proportionality factor between 0.4 and 1 for fish and cetaceans [3,[11][12][13][14].Taking the average factor, the relationship U 0.7Lf is therefore a very good approximation of the swimming speed to within a factor of 2 at most.The swimming speed is thus intrinsically linked to the oscillation frequency, but unlike the aforementioned scaling laws that follow clear and widely documented trends over several orders of magnitude in length, no consensus has been reached on the law that sets the tail beat frequency.Most studies agree that the frequency decreases with the length [5] and scaling laws f ∼ L −n with an exponent n ranging between 0.5 and 1 are often reported together with models referring to biological constraints, to the hydrodynamic interactions of the swimmer with its environment or even to the effect of gravity [15][16][17][18][19].The difficulty of establishing a clear law and identifying the mechanisms at play is related to the fact there is only a factor of 100 between the highest frequency recorded in the smallest fish and the lowest frequency recorded in the largest cetaceans, typically 20 and 0.2 Hz respectively (Fig. 1), while measurements show a large dispersion for a given length.Firstly, several parameters such as the previous training or the condition of the animal, its age or sex, and the water or body temperature influence the final performance of the swimmer [20][21][22].Secondly, the frequency is not fixed for a given swimmer but is the parameter that is adjusted to determine the swimming speed, and a factor of 10 in frequency can easily be observed for a given specimen [5].
Obtaining experimental results that maintain homogeneity in all these features for a wide range of aquatic animal sizes is simply impossible and it is thus not surprising to find diverse experimental laws in the literature.In particular, comparing animals with the same level of activity, such as sustained, prolonged or burst, would be necessary [23] but difficult to implement in experiments [24,25].As a consequence, instead of focusing on a specific activity gait, we propose to gather all data available in the literature, regardless of the level of activity or any other specific characteristic, to build a database of more than one thousand entries.This approach provides a complete picture of the dependency of the frequency with the length and captures both the main trend and the dispersion associated with a given length.We thus propose a frequency selection mechanism that, on the one hand, balances the swimmer's muscle force and the reactive forces generated by the fluid when the animal is in motion and, on the other hand, considers the type of muscles, slow and fast.This works uncovers allometric relations for the swimming frequency and speed that take the form of scaling laws in the limits of very small or very large swimmers.

Tail beat frequency measurements
We collected about 1200 data points from references listed in Methods 4.1, with no discrimination on the basis of activity level or any of the other parameters mentioned above, to avoid any possible bias in the length-frequency relationship.In Fig. 1, it appears that the tail beat frequency is correlated to the length, with all measurements located within a band in the L − f plane.For most of the lengths, the upper bound is well identified as the burst activity level [26][27][28] and the frequency varies approximately by a factor of 10 for a given L. For the longest animals, typically cetaceans with L > 5 m, the magnitude of the band decreases.As we will discuss later, this decrease is most likely associated with a lack of burst frequency measurements: unlike the smaller animals, these animals were only observed in their natural environment and were not forced to swim at peak activity.If we define the upper and lower bounds of the band as fast and slow, we observe that both follow the same behavior: the frequency is constant and maximum at small lengths, typically [f ] fast 20 Hz and [f ] slow 2 Hz respectively, before decreasing at larger L. The change in tendency occurs around L = 0.5 − 1 m.Since the span in f observed for a given L is of the same order of magnitude as the span in f of a given specimen to adjust its speed, we conclude that the frequency intervals are primarily associated with variations in the level of activity.In the next section, we interpret the data set by taking into account both the muscle activity and the mechanical interaction of the swimmer with his environment.Building on previous models, we account for limitations of muscle in terms of both maximal force and frequency to explain the transition around L = 0.5 − 1 m between a regime characterized by a frequency that is constant and a regime where the frequency decreases with length.Finally, we consider the nature of the muscles, slow versus fast, to predict the slow and fast bounds of the band in the L − f plane.

Model and scaling laws
Locomotion occurs as a result of undulating movements produced by the contraction of blocks of muscle segments [29].While the muscles contract on one side of the swimmer, those on the opposite side relax, alternately flexing the entire body from side to side.Vertebrate muscles share a large number of structural and functional features [3] that can be reasonably well described by a limited number of parameters.This is the case for the relationship between the force in the muscle, F , and the muscle contraction velocity, v, as described by Hill's muscle model [30]: where F 0 is the maximum isometric force generated in the muscle, obtained as v tends to zero, and v 0 is the maximum contraction velocity over which no force can be produced.In its dimensionless form, the force F/F 0 is a decreasing and convex function of the velocity v/v 0 , whose degree of curvature is quantified by the parameter κ (Methods 4.2).Since the muscle fibers work in parallel, the maximum force scales as the cross area of the muscle: F 0 ∼ L 2 σ 0 where σ 0 is the maximum isometric force per unit cross-sectional area [31].The contraction velocity v drives the tail velocity and we can expect the scaling v ∼ Af , which leads to v/v 0 = f /f 0 , where f 0 is the maximum tail beat frequency expected from the physiological limit of the muscle.f 0 can be inferred by measuring the twitch contraction time T of the muscle [32], that is, the period of a single contraction and relaxation cycle produced by an action potential within the muscle fibers, in order to deduce the maximum frequency in the form f 0 = 1 2T since one period consists of two antagonistic contractions.
The swimmer's movements set the surrounding fluid in motion to propel it along.Given that the only force exerted by the animals on the fluid is provided by the muscles, there must exist a balance between the forces produced by the muscles and the reaction force of the fluid.Above a few centimeters in length, aquatic organisms have a mode of locomotion based on inertia [10]: while the body is oscillating, boluses of water of mass ρL 3 are set in motion with an acceleration Af 2 normal to the tail, resulting in a lateral force that scales as ρL 3 Af 2 , where ρ is the density of water.As discussed in the Introduction and in the Method 4.1, A is proportional to L and thus the lateral force that pushes the fluid scales as ρL 4 f 2 .By balancing the latter with the force exerted by the muscle (Eq.( 1)), we obtain: where we have introduced the length L c that marks the crossover between two regimes with different scaling laws: Eq. ( 2) predicts nicely the two bounds of the frequency band.In Unlike f 0 and L c , κ has large standard deviations from the fitted values; in fact κ weakly shapes the fit since it only plays a role in the transition between the two limit regimes.These fits are also predictive: on the example of humans and underwater undulatory swimming, also called dolphin kick, the fit of the fast bound predicts a maximum kicking frequency around 3-4 Hz for a swimmer of about 2 meters, in good agreement with data recorded in elite swimmers [33].
In addition, this model reconciles the two approaches and validates each in its own length range.For L L c , the frequency is fixed by a biological constraint, f 0 , in the spirit of the approach initiated by Wardle [32] who proposed that the maximum tail beat frequency corresponds to the maximum frequency expected from the muscles.
For L L c , the frequency decreases with length as a consequence of the interplay between another biological constraint, here σ 0 , and the interaction of the swimmer with its environment [16,18,31].In this limit, the model predicts that the frequency is given by f = cL −1 , where we have introduced the speed c = L c f 0 = σ 0 /ρ, that we estimate to be [c] slow 2.0 m.s −1 and [c] fast 10 m.s −1 for the slow and fast bounds, respectively.The ratio of the wavelength of deformation λ to the body length ranges from 0.74 to 1.14 for anguillorm and thunniform swimmers, respectively [3,4], and c λf can also be interpreted as the speed of the wave propagating along the swimmer's body.Still in the perspective of a mechanical approach, we remark here that the scaling f ∼ L −1 from Eq. ( 5) is also consistent with an approximation of the swimmer by an elastic beam of length L, radius R and Young Modulus E. In such a case, the natural frequency of bending waves scales as f e ∼ R L 2 E ρ [34].By assuming geometrical similarity, R ∼ L, and body elasticity compatible with stresses generated by muscle fibers E ∼ σ 0 , the natural bending frequency scales as the tail beat frequency f of large animals, defined in Eq. ( 5).This suggests that large animals undulate their bodies to resonate with their natural bending modes, as suggested by various studies [35][36][37].
Finally, we now understand why there are so many different exponents in the literature resulting from attempts to describe the frequency-length relationship as a scaling law (e.g., in [15][16][17][18][19]).Special attention must be paid to the analysis of a range of lengths, because 1) the scaling laws are only valid in either of the two limiting regimes and 2) the data must be measured at the same activity level for the comparison to be meaningful.

Considering the type of muscle fiber and comparison with biological data
In order to connect the biology of swimmers to the fast and slow bounds in Fig. 1, we take into account the different types of muscle fibers.Muscle fibers can be roughly characterized as fast or slow, in part because the latter type has a much lower level of ATP activity and a smaller contraction speed but increased activity of oxidative enzymes.Therefore, slow fibers are intended to produce forces over a prolonged period of activity [38], while fast fibers use anaerobic chemical reactions and are adapted to rapid movements, while producing higher forces.As a result, fast and slow muscles are primarily solicited in burst and sustained activity levels, respectively [39].In what follows, we hypothesize that the presence of the lower frequency boundary of the f − L graph reflects the use of slow fibers, while to reach high frequencies, fast fibers are exploited.The fast and slow frequency bounds in Fig. 1 will therefore be modeled with the same Eq.( 2), but with parameters values that account for fast and slow muscles, respectively.
The parameters obtained from the fits match the biological data very well.Wardle et al. suggests inferring f 0 from measurements of the twitch contraction time [32], resulting in values between 5 and 25 Hz for 4 cm to 2.3 m fish [11].These measurements are in agreement with the value [f 0 ] fast = 21 Hz that fits the fast bound.Given that σ 0 200 kN.m −2 for fast muscles is rather constant among species [40], the estimate L c ∼ 0.7 m is in excellent agreement with the fit [L c ] fast = 0.5 m.For slow muscles, the maximum frequency is smaller, and we employ the same approach to study the slow bound, although measurements are rarer in this case.In the example of 10 cm salmon and 1 m sharks, f 0 ranges from 0.5-2 Hz, again in good agreement with the [f 0 ] slow = 2.0 Hz obtained from the fit of the slow bound.Measurements of σ 0 for slow muscles are found between 20 and 80 kN.m −2 [33,[40][41][42].If we take 50 kN.m−2 as the typical value, we find L c ∼ 3.5 m, whose order of magnitude is coherent with the value [L c ] slow = 1 m found from the fit.Finally, the values of κ adjusted for the fast and slow bounds, [κ] fast = 4 and [κ] slow = 9, are in the range of values recorded in vertebrates, typically between 2.5 and 10 [43,44].
This framework with six parameters, three for each bound, appears coherent because the filament arrangement in striated muscles is very similar along all vertebrates [45].
In addition to the activity level, temperature has an effect on the twitch contraction time, and therefore on f 0 [32,46].Cold water generally depresses locomotory muscle function: the lower the temperature, the lower f 0 is and a factor of 5 difference can be easily found between 2 and 30 • C.This can lead to deviations from the main trend at extreme temperatures.This is probably the main explanation of the surprising low tail beat frequency of Greenland sharks in Arctic waters (L 3 m and f 0.15 Hz in Fig. 1), a factor of 2-3 below the fit of the slow bound [17].Conversely, there are examples of thermal acclimation of fish in warm waters that could lead to f 0 values as high as 50 Hz [47].

Scaling the swimming speed
The relationship U 0.7Lf intrinsically relates the tail beat frequency to the swimming speed to a very good approximation, with a factor of 2 at most for fish and cetaceans for U .This allows us to infer scaling laws for a given activity level: For small animals, we therefore expect swimming speed to increase with length, whereas it should saturate at a constant value, 0.7 σ 0 /ρ from Eqs. ( 3) and ( 6), for large animals.This is consistent with studies that found a tail beat frequency scaling as L −1 for large swimmers, and also found a nearly constant swimming speed in this case [18,48].Still for large swimmers, the swimming speed should range approximately between 1 and 8 m.s −1 depending on the activity level, given the values of f 0 and L c found from the fits of the slow and fast bounds, respectively.In order to properly describe the mechanisms at play, it is essential to compare data at the same activity level.We therefore rely on the study by Hirt et al., who reported data on maximum swimming The black and gray thick lines represent the fast and slow boundaries predicted by the model and fitted parameters in Fig. 1, respectively.Thin lines are the scaling laws in the limit of very small and very large swimmers.
speed [49].In Fig. 2, we superimpose these data on the expected swimming speeds inferred from all frequency measurements.Various points need to be discussed.First for L 0.5 − 1 m, we observe that both data sets exhibit the same upper limit.For most lengths, the match is perfect and only small differences are observed, but they are at most a factor of 2. This regime is consistent with the scaling law expected in the limit L < L c : quantitatively, Hirt et al. found that the maximum swimming speed scales as U ∝ M 0.36 , on average, or equivalently U ∝ L, in agreement with the scaling law proposed in Eq. ( 4).
Second, for L 0.5 − 1 m, we observe a significant difference.While our scaling law inferred from frequency measurements predicts a constant speed, around 5-10 m.s −1 , data gathered by Hirt et al. suggest a humped shape with a maximum around 30-40 m.s −1 obtained for L 1 − 3 m followed by smaller speeds for larger animals.In fact, we propose that the two data sets differ in the two regions for two different reasons.
1.While Hirt et al. did an enormous amount of work gathering data from the literature, we suggest that the maximum is artificial if we apply relevant filters.Most of the highest maximum speed data collected by Hirt et al., for fish ranging from about 1 to 3 m in length, are estimates or predictions based on in vitro physiological measurements.These measurements have been shown to significantly overestimate the expected maximum speed of what were thought to be the fastest swimmers, like billfish.First, these fish have lengths L > L c and consequently their tail beat frequencies are significantly smaller than the maximum frequency f 0 expected from the muscles: we predict that a 2 m-long swimmer with f 0 = 20 Hz would swim with a maximum tail beat frequency four times smaller, below 5 Hz (Fig. 1), and thus would have a maximum speed reduced by a factor of 4 in comparison to predictions based on the twitch contraction time only.Second, recent estimates based on measurements of twitch contraction times of anaerobic muscles also provide upper bounds lower than 10 m.s −1 for billfish and other large marine predatory fish [14].
Actually, reported values of maximum speeds agree with this argument and refute some incredibly huge values that had been estimated for animals of this size.For marlins, direct measurements using speedometers showed that the observed maximum swimming speed was around 2.25 m.s −1 [50], considerably lower than the estimates of 30 m.s −1 [51].Burst speeds of sailfish also show values around 8 m.s −1 measured with high-speed video and accelerometry [28], a value much smaller than 30 m.s −1 [51].In addition, there is some theoretical evidence that the maximum speed should be smaller than 15 m.s −1 , because cavitation should appear at greater speeds, which should damage the flesh of the swimmer [52].In Fig. 2, we have used open translucent squares to represent data that were not actual measurements but estimates, or that were taken from non-peer-reviewed studies (Methods 4.4).If we remove these points from the analysis, we find that the two upper bounds of the data sets match very well, with the exception of four data points obtained for tuna and barracuda that are still significantly faster than our fit of the fast bound (open opaque squares in Fig. 2).Note that all of these points were measured using rod-mounted devices that measure the speed at which the line is pulled from the reel when a fish is hooked and pulling on the line ( [5,53,54] and Methods 4.4).Given the large fluctuations in the measurements made with this method [53], it is likely that it overestimates the maximum speed, which is also supported by the fact that barracuda and tuna do not show particularly high maximum frequencies [14].
2. For L 5 m, we attribute the discrepancy to a lack of tail beat frequency measurements for very long swimmers, typically cetaceans, at a burst level of activity.This would explain the jump in frequencies around L ∼ 5 m in Fig. 1 for the fast bound.In this figure, the measured frequencies above L = 5 m correspond to swimming speeds between 1 and 4 m.s −1 [16,19,55].Swimming speeds up to 10 m.s −1 were recorded in sperm whales [56], for which tail beat frequencies were unfortunately not measured, but which should be consistently higher than those plotted in Fig. 1 for the same length.Unlike sperm whales, killer whales and some other large marine predators, most cetaceans are filter feeders and do not have predators due to their size.Therefore, they do not often need to move at maximum speeds, which would favor data closer to the slow bound than the fast bound.
Following these considerations, it is reasonable to consider that the maximum speed is constant for L 0.5 − 1 m, with a typical value around 5-10 m.s −1 , in very good agreement with the prediction 0.7f 0 L c inferred from the fit of the fast bound.From the definition of L c , it means that U ∼ σ 0 /ρ for L > L c and that swimming speed is directly associated with the maximum stress generated by the muscles to push the surrounding water.In their model, Hirt et al. state that heavier (and consequently longer) animals need more time to accelerate to achieve maximum speed and this fact would prevent the heaviest animals from being the fastest.Here we suggest that the effect of a finite acceleration time would be a second-order effect, unlike the other locomotion modes running and flying [49].

Discussion
From the results, we conclude that length and activity level determine swimming frequency and speed at the leading order.The data collected on natural swimmers are explained using a simple model that accounts for biological characteristics, through Hill's muscle model, as well as the interaction of the undulating swimmer with its environment.This model requires only a few parameters that could be refined in the future to account for specific characteristics of each swimmer (body temperature, swimming gait, etc.).This work broadens our understanding of animal locomotion but should also help in designing biomimetic and autonomous swimming robots [57][58][59][60][61], with the constraint that artificial swimmers have their own internal characteristics that replace the biological ones discussed in the present study.
Our study highlights a crossover at a length L c ∼ 0.5 − 1 m that separates two limits: while small swimmers are constrained by biology only, large swimmers are constrained by their environment as well.For a given activity level, different scaling laws are found for swimming frequency and speed in the two limits.This should also be the case for other quantities, such as the muscle power for locomotion.For this quantity, our model predicts scaling laws in L 5 and L 2 for very small and very large swimmers, respectively (Methods 4.2).Measurements of oxygen consumption [24,25] over a wide range of lengths and activity levels might be a way to test these predictions.In the framework of the model, L c also marks a significant change in the way muscles are used.Small swimmers use muscles at their maximum speed but negligible force in comparison to their maximal capabilities (Methods 4.2).Large swimmers exhibit the opposite behavior.Given that muscle power for locomotion scales as the product of force, frequency and length, we expect muscle power for locomotion to be negligible in these two limits in comparison to the maximum power available.Remarkably, only intermediate fish, with lengths around L c , would use the full capacity of muscle power to undulate and move through water.In light of this comment, we can question in the future whether intermediate fish are more likely to use economic locomotion strategies (e.g., intermittent swimming [62], schooling [63], etc.) compared to very small and large swimmers.

Data and allometry plots
We retrieved the length-frequency data available in the literature for a total of 1202 animals, with a range of different species, morphologies and sizes.We regrouped the data according to the classical division of vertebrates: amphibians, fish, birds, reptiles and mammals.In figures, these data points are displayed in yellow, blue, red, green and purple, respectively.In the cases where the length data were not reported [16] but instead the mass of the animals, the length was deduced through the allometric relation deduced in Fig. 3b L = 0.44M 0.33 (L in meters and M in kilograms), assuming geometric similarity.Additionally, we recovered length-amplitude and length-mass data.The references we used are provided in Tab. 1.

References
Frequency vs length or mass Amplitude vs length or mass Mass vs length Amphibians [64] [64] -Fish [5,7,17,26 Our approach is based on two relationships verified through the data in Tab. 1. First, we plotted A versus L in Fig. 3a and verified that A is proportional to L, as originally shown by Bainbridge [5], but extended here to 359 different specimens and four orders of magnitude in length.In fact the best fit with a power law gives A = (0.183±0.002)L 0.981±0.005, which shows that both quantities are proportional.The best proportionality relation gives A = (0.187 ± 0.001)L.The geometric similarity for aquatic animals is shown in Fig. 3b for 432 different individuals.The best fit of the data gives M = (12.14± 0.33) L 3.04±0.02, with M in kilograms and L in meters, which is consistent with the geometric similarity characterized by an exponent of 3.This law extends Economos's relation (M = 11.27L2.95 [86]) over four orders of magnitude in length or ten in mass.Forcing the exponent of the relation to be exactly 3 gives M = (12.09± 0.33)L 3 or its dimensional homogeneous form M = (0.0121 ± 0.0003)ρL 3

Hill's muscle model
equations for tetanized muscle contraction (Eq.( 1)) can be rewritten to express the force in the muscle F as a function of the swimming frequency f : F 0 is the maximum isometric force generated in the muscle and f 0 is the maximum tail beat frequency.In its dimensionless form, the force F/F 0 is a decreasing and convex function of the frequency f /f 0 , whose degree of curvature is quantified by the parameter κ (Fig. 4a).From our analysis, very small animals (L L c ) swim at maximum frequency and negligible force (f = f 0 and F F 0 ), while very large animals (L L c ) swim at negligible frequency and maximum force (f f 0 and F = F 0 ).Intermediate sized animals (L ∼ L c ) swim at intermediate frequency and force (f f 0 and F F 0 ).An estimate of muscle power for locomotion P is obtained by multiplying the muscle force by Lf , an estimate of the speed of muscle contraction.P ≈ F Lf can be expressed as a function of the tail beat frequency following Hill's equation: The power is drawn in Fig. 4b and F 0 Lf 0 gives an estimate of the maximum power that can be delivered by the muscle.This graph highlights the fact that very small and large swimmers do not use the full capacity of muscle power, unlike intermediate sized swimmers (L ∼ L c ).
It is also possible to derive the expected scaling laws for muscle power as a function of length.To do this, we take F ∼ ρL 4 f 2 from the interaction of the undulating swimmer with its environment and f = f 0 or σ 0 /ρ/L for very small or large swimmers, respectively.This gives the following scaling laws in these two limits:

Characterization of the burst and sustained activity levels
All measurements are located within a band in the L − f plane.The fast and slow bounds corresponding to the burst and sustained activity levels, respectively, are defined as follows.First, the length axis is divided into N intervals equally distributed in logarithmic scale.Second, in each interval, the maximum and minimum frequencies are identified, as well as all data points that are found between 90% and 100% of the maximum frequency value and between 100% and 110% of the minimum frequency value.Finally, these two sets of points are averaged independently to give one point for each bound inside each interval.
For each bound, the three parameters of the model (L c , f 0 and κ) are obtained by fitting the N averaged points (one per interval) with Eq. ( 2) using the least absolute deviations (LAD) method.The robustness and precision of the fit is probed by varying N and taking the mean and standard deviation to characterize each parameter.The number of intervals, N , was varied between 10 and 50.This range ensured a statistically significant number of points while at the same time avoiding empty intervals.In practice, we selected nine values uniformly distributed between these two extreme values.

Filtering of maximum speed data
We investigated the origin of the data gathered by Hirt et al. [49] to establish objective criteria on data selection.Of all the measures reported in the study, we found three classes of data that were not as reliable as the others.First, we identified data coming from non-peer reviewed papers.Second, some of the data are only estimates, not actual measurements.Third, we found that all data obtained with rod-mounted devices are above the main trend, which could be artificial due to the high fluctuations in these measurements [53].All these three classes of data points are summarized in Tab.[49] that are either provided by non-peer-reviewed articles, given as estimates, or obtained using rod-mounted devices.We list the type of animal together with the corresponding mass and speed, the length using Methods 4.1 if not provided, and references.

1 )Figure 2 :
Figure2: Swimming speed U as a function of length L. Following the law U = 0.7Lf , we show our estimates of swimming speed from the tail beat frequency measurements displayed in Fig.1(closed circles): amphibians (yellow), fish (blue), reptiles (green), birds (red), and mammals (purple).Brown squares correspond to the data gathered by Hirt et al. for maximum swimming speeds[49] using the mass-length relationship (Methods 4.1).Open translucent squares represent either non-peer reviewed papers or data coming from estimations and not actual measurements.Open opaque squares represent data obtained using rod-mounted devices.The other data are represented by closed opaque squares.The black and gray thick lines represent the fast and slow boundaries predicted by the model and fitted parameters in Fig.1, respectively.Thin lines are the scaling laws in the limit of very small and very large swimmers.

Figure 3 :
Figure 3: a) Tail beat amplitude and b) animal mass as a function of animal length.The solid lines represent the best power-law fits of the data.

Figure 4 :
Figure 4: a) Muscle force and b) muscle power for locomotion as a function of tail beat frequency, as predicted by Hill's muscle model.The curves are drawn with κ = 5 and the quantities are represented in their dimensionless form.

Table 1 :
References reporting relations between length, frequency, amplitude or mass of swimmers.

Table 2 :
2. Data used in Hirt et al.