The aim of the present study was to quantify the improvements in the economy and efficiency of surface swimming brought about by the use of fins over a range of speeds (v) that could be sustained aerobically. At comparable speeds, the energy cost (C) when swimming with fins was about 40 %lower than when swimming without them; when compared at the same metabolic power, the decrease in C allowed an increase in v of about 0.2 ms-1. Fins only slightly decrease the amplitude of the kick (by about 10 %) but cause a large reduction (about 40 %) in the kick frequency. The decrease in kick frequency leads to a parallel decrease of the internal work rate (int, about 75 %at comparable speeds) and of the power wasted to impart kinetic energy to the water (k, about 40 %). These two components of total power expenditure were calculated from video analysis (int) and from measurements of Froude efficiency(k). Froude efficiency(ηF) was calculated by computing the speed of the bending waves moving along the body in a caudal direction (as proposed for the undulating movements of slender fish); ηF was found to be 0.70 when swimming with fins and 0.61 when swimming without them. No difference in the power to overcome frictional forces(d) was observed between the two conditions at comparable speeds. Mechanical efficiency[tot/(Cv), where tot=k+int+d]was found to be about 10 % larger when swimming with fins, i.e. 0.13±0.02 with and 0.11±0.02 without fins (average for all subjects at comparable speeds).

Passive locomotory tools for land locomotion have been shown to improve the economy and the speed of progression for any given metabolic power. Like bicycles (e.g. Minetti et al.,2001), roller-skates and skies for terrestrial locomotion, fins can be defined as passive tools for aquatic locomotion, as well as rafts,racing shells or hydrofoils. As pointed out by Abbott et al.(1995), four basic forces act on a boat (or a body) in water: lift, weight, drag and thrust. Human powered watercrafts are designed to decrease weight (e.g. buoyancy devices) and drag(e.g. streamlined shells, reduction of speed oscillations) and to increase lift (e.g. hydrofoils) and thrust (e.g. paddles, oars, propellers). Fins are meant to improve the fraction of the force (thrust) that is useful to propel the body forwards. In other words, they are meant to improve the propelling efficiency of aquatic locomotion.

Data on fin swimming are scarce and refer mainly to underwater (SCUBA diving) experiments that focus on the differences in economy of swimming at different depths (e.g. Morrison,1973) or with different swim-fin designs (e.g. Pendergast et al., 1996). An analysis of the mechanical determinants of the improved economy brought about by the use of fins (in comparison to swimming without them) has never been attempted because quantification of the mechanical work performed during aquatic locomotion is not simple.

In the present study the energy cost, mechanical work and efficiency of swimming using the leg kick were measured/estimated with methodologies previously applied to human and fish locomotion. Using these data, we attempted to calculate a complete energy balance for swimming. The differences in economy and efficiency brought about by the use of fins allowed us to further investigate the relative importance of the mechanical determinants of aquatic locomotion in humans.

Approach to the problem

As is the case for human locomotion on land, the economy and efficiency of locomotion in water depend on the mechanical work (Wtot)that the muscles have to produce to sustain a given speed (see Fig. 1). This work is generally partitioned into two major components: (i) the work that has to be done to overcome external forces (the external work, Wext) and(ii) the work that has to be done in order to accelerate and decelerate the limbs with respect to the centre of mass (the internal work, Wint). The contribution of the elastic and viscous factors to total work expenditure is thought to play a minor role in swimming and is not considered here.

The external work in aquatic locomotion is generally partitioned into two components: Wd, the work that is needed to overcome drag that contributes to useful thrust; and Wk, the work that does not contribute to thrust. Both types of work give water kinetic energy but only Wd effectively contributes to propulsion.

The resistance to motion (and hence Wd) in swimming can be measured by towing the subjects (passive drag) or while the subjects are actually swimming (active drag). Active drag is higher than passive drag, due to changes in frontal surface area and fluid dynamics caused by arm/leg movements and, as such, is a better estimate of the force opposing motion. In contrast with passive drag, active drag is difficult to assess and is generally obtained indirectly from measures of energy expenditure (e.g. di Prampero et al., 1974; Toussaint et al., 1988).

The term Wk is a quantity even more difficult to measure than Wd. The contribution of this factor was estimated in swimming humans by Toussaint and coworkers(1988). They compared the difference in the energy consumed while swimming on the MAD (Measuring Active Drag) system to the energy consumed while swimming freely. With this device the subject swims by pushing off fixed pads positioned at the water surface and hence does not waste any energy to give water momentum; in these conditions his/her O2 reflects the energy expended to overcome drag only. However, only arm propulsion could be investigated with that set-up, since the legs are fixed together and supported by a small buoy.

An alternative way of obtaining an estimate of the term Wk comes from studies of animal locomotion. As discussed by Lighthill (1975), Alexander(1977) and Daniel et al.(1992), many animals (e.g. eels) proceed in water with undulatory movements. Waves of bending, produced by rhythmic muscular contractions, can be observed moving along the body in a caudal direction, giving the water backward momentum from both sides of the fish's body. At high Reynolds number (Re), thrust arises from the lateral acceleration of the body segments. The viscous forces, which dominate motion at low Re values, are negligible. Humans swim at Revalues of about 106, which are comparable to those of slender fish(e.g. 105 for eels, as estimated by Alexander, 1977).

During steady state aquatic locomotion, at high Re and for a given(forward) speed v, the efficiency of the undulatory movement of a slender fish is given by:
$\ {\eta}_{\mathrm{F}}=\frac{(c+{\nu})}{2c},$
1
where c is the wave speed and ηF is the Froude efficiency (e.g. Lighthill,1975; Daniel,1991). In order to produce effective propulsion, ηFmust lie between 0.5 and 1.0: the speed of the backward bending wave should be higher than the average forward speed of the fish.
ηF is also defined as:
$\ {\eta}_{\mathrm{F}}=\frac{W_{\mathrm{d}}}{(W_{\mathrm{d}}+W_{\mathrm{k}})},$
2
(e.g. Lighthill, 1975; Daniel, 1991). Expressed in this way, ηF reflects the ability of the swimming body to impart useful kinetic energy to the water. Thus, the term Wk can be calculated from ηF (as measured by means of Equation 1) and from active body drag (Wd) by rearranging Equation 2:
$\ W_{\mathrm{k}}=\left(\frac{W_{\mathrm{d}}}{{\eta}_{\mathrm{F}}}\right)-W_{\mathrm{d}}.$
3

Whereas arm propulsion (e.g. in the arm stroke) is more analogous to rowing(the hands are used as oars which move water backwards), other swimming styles(e.g. the butterfly or swimming with a monofin) resemble the undulating movements of slender fish. Waves of bending similar to the ones described for slender fish were reported for subjects swimming the butterfly stroke(Ungerechts, 1983; Sanders et al., 1995). The leg(flutter kick) is similar to the butterfly (dolphin) kick, but whereas in the dolphin kick the legs are moved synchronously, in the flutter kick they are moved alternatively, out of phase by half a cycle (see Fig. 2A). Thus, waves of bending can be expected also when swimming using the leg kick (with and without fins); hence, an estimate of the Froude efficiency can be attempted as well as an estimate of the contribution to propulsion by the use of fins.

Finally, the internal work (Wint) can be measured from kinematic analysis according to a method originally proposed by Cavagna and Kaneko (1977). When swimming using the leg kick, the internal work is likely to be similar to that of walking, as in both cases the legs move with a sinusoid-like pattern, almost symmetrically with respect to the centre of mass (see Fig. 2A). A model equation,derived from that proposed for walking by Minetti and Saibene(1992), is proposed in this paper for the calculation of the internal work of the leg kick, based on the values of kick frequency and kick depth.

The experimental protocol was approved by the Institutional review board and the subjects were informed about the methods and aims of the study and gave their written informed consent prior to participation.

Subjects

The experiments were performed on seven elite college swimmers who were members of a Division I University men's swimming team (State University of New York at Buffalo, NY, USA). Their average body mass was 71.6±7.2 kg,their average stature 1.79±0.69 m and their average age 19.9±1.3 years. The subjects' maximal oxygen consumption O2max was measured in a separate session by increasing velocity in 0.1 m s-1increments from 0.4 m s-1 up to a maximum of 1.3-1.4 m s-1, and ranged from 2.74 to 3.861 O2 min-1(3.15±0.381 O2 min-1, mean ± S.D.).

Fins

Apollo Bio-Fin Pro fins were used in this study. These fins were made of rubber, small in size and highly flexible. This type of fin was shown to be effective in increasing the economy of swimming compared with different types of fins in previous underwater (SCUBA diving) experiments(Pendergast et al., 1996). The subjects used fins of two different sizes; their length, mass and surface area are reported in Table 1. The size of the fins was determined by the foot size of the subjects.

Experimental protocol

The subjects swam (at the water surface) in an annular pool 2.5 m wide, 2.5 m deep and 60 m circumference above the swimmer's path and were paced by a platform moving at constant speed approximately 60 cm above the water surface. The speed of the swimmer was set by means of an impeller type flow meter (PT— 301, Mead Inst. Corp., Riverdale, NY, USA) placed 1.5 m in front of the swimmer and connected to a tachometer (F1-12 P Portable indicator, Mead Inst. Corp., Riverdale, NY, USA). Subjects were requested to swim with the arms hyper-extended over the head and the thumbs joined with the palms down. The forward propulsion was attained by kicking the legs with (LF) or without fins (L).

Active body drag was measured as described by di Prampero et al.(1974). Known masses (0.5-4 kg) were attached to the swimmer's waist by means of a rope and a safety belt that did not interfere with the swimming mechanics. The rope passed through a system of pulleys fixed to the platform in front of the swimmer, thus allowing the force to act horizontally along the direction of movement. This force,defined by di Prampero et al.(1974) as added drag'(Da), leads to a reduction of the swimmer's active body drag (Db); in our experimental conditions Da could be better defined as an added thrust', since it acts by facilitating the swimmer's progression in water by pulling the subject forward. At constant speed, the added thrust' is associated with a consequent reduction of O2and the energy required to overcome Db becomes zero when Da and Db are equal and opposite. At the beginning of the experimental session a load was applied to the pulley system (its mass depending on the speed and/or condition) and the subject was asked to attain the requested speed. After 3 min, once the steady state was attained, expired gas was collected (for approximately 60 s) into an aerostatic balloon through a waterproof inspiratory and expiratory valve-and-hose system supported by the platform. After 1 min the expired gas collection was terminated and the load on the pulley was diminished by approximately 0.5 kg. This procedure was repeated until, in the last step, the subject swam freely (without any added load). During each collection of metabolic data the kick frequency (KF, Hz) was also recorded. O2values were determined by means of the standard open circuit method: the gas volume was determined by means of a dry gas meter (Harvard dry gas meter, USA)and the O2 and CO2 fractions in the expired air were determined using a previously calibrated gas spectrometer (MGA 1100, Perkin Elmer, CA, USA). The O2 values obtained in the last step (without any added thrust') were used in the calculations of the energy cost of free swimming: net O2 (above rest,assumed to be 3.5 ml O2 min-1 kg-1) was converted to watts W, assuming that 1 ml O2 consumed by the human body yields 20.9 J (which is strictly true for a respiratory quotient of 0.96), and divided by the speed ν to yield the energy cost of swimming per unit of distance (C) in kJ m-1.

The swimmer's Db was estimated, at any given speed and condition, by extrapolating the O2versus Da relationship to resting O2. The power dissipated against drag was then calculated from the product of the active body drag times the speed(d=Dbν).

The experiments were carried out over a range of speeds (N=5) that could be accomplished aerobically. The range of speeds depended on the condition selected: 0.6-1.0 m s-1 (L), 0.7-1.1 m s-1(LF). Each subject participated in several experimental sessions, each corresponding to 1-3 swims at a given speed and/or condition; the swims were separated by at least 15-20 min of rest.

Kinematic analysis

During the experiments, video recordings were taken at a sampling rate of 50 Hz (Handy Cam Vision, Sony, Japan) while the subjects passed in front of an underwater window. Black tape markers were applied on selected anatomical landmarks in order to facilitate the following video analysis. The distance between the hip (great trochanter) and the knee (lateral epicondyle) was measured and recorded for each subject and each experimental session and was utilized as a calibration factor.

After the experiments, the data were downloaded to a PC and digitized using a commercial software package (Peak Motus, Co, USA). The 2-D coordinates of selected anatomical landmarks were utilized to calculate the trunk inclination, the kick depth, the Froude efficiency and the internal work. Only the data collected during steady state free swimming were analyzed.

Trunk inclination (TI, degrees) was measured with the leg proximal to the camera fully extended (see Fig. 2A), from the angle between the shoulder (acromion process) and the hip (great trochanter) segment and the horizontal. In the same frame, kick depth (KD, m)was measured from the vertical distance between the ankles (lateral malleolous). Finally, in this same frame the 2-D coordinates of the hip and the knee (lateral epicondyle) markers were recorded in order to obtain the calibration factor (for each subject, speed and condition). 1-3 passes of the swimmer were filmed for each condition and speed; data for TI and KD reported in the text are the mean of all the values measured in all the passes (in each subject and for each condition and speed).

The internal work (Wint)

The internal work of the leg kick, with and without fins, was computed from video analysis during free swimming (without any added drag) on two subjects according to the method originally proposed by Cavagna and Kaneko(1977). To measure Wint the locations of nine anatomical landmarks (wrist,elbow, shoulder, neck, hip, knee, ankle, heel, toe tip) were digitized over one complete swimming cycle. On the assumption that bilateral swimming movements are symmetrical, the 2-D coordinates obtained from the body side proximal to the camera were duplicated (shifted half a cycle) and the swimming cycle was reconstructed for the whole body (see Fig. 2A). The 3-D coordinates obtained and standard anthropometric tables (Dempster, 1959) allowed us to calculate the position and the linear and angular speed of each body segment,from which the position of the body centre of mass was also derived. When swimming with fins, the extra mass of the fins was taken into account in order to compute the segment mass/total mass fraction of each body segment. The sum of the increases, over the time course, of absolute rotational kinetic energy and of relative (with respect to the body centre of mass) linear kinetic energy of adjacent segments over one cycle were then computed by a custom software package (Minetti,1998) in order to calculate Wint.

As shown by Minetti and Saibene(1992) the mechanical internal work rate when walking(int, in W) could be described by the following equation:
$\ {\dot{W}}_{\mathrm{int}}=\mathrm{k}{\nu}^{2}f,$
4
where k is related to the inertia parameters of the moving body segments, νis the average progression speed (m s-1) and f is the stride frequency (Hz). When swimming using the leg kick, int is probably similar to that of walking, as the legs move with a sinusoid-like mechanism in both cases. In contrast to walking, in swimming using the leg kick the relative linear speed of the limbs is not constrained to the average progression speed and the vertical and horizontal oscillations of the body centre of mass are kept to a minimum, similar to cycling(Minetti et al., 2001). The term ν2, which expresses the dependency of int on the translational and kinetic energy of the moving segments, is therefore more correctly related(for swimming) to the speed of the vertical movement of the legs (s,m s-1) rather than to the average horizontal speed. Whereas in cycling the displacement of the lower limb (i.e. the kick depth) is constrained by the biomechanical arrangement of the subject—bicycle system, this is not the case for the leg kick (by analogy with the step length in walking) and, hence, to calculate s both the kick depth(KD, m) and the kick frequency (KF,Hz) should be measured. Hence the model equation proposed by Minetti and Saibene for walking (1992) was modified as follows:
$\ {\dot{W}}_{\mathrm{int}}=\mathrm{k}(2K_{\mathrm{D}})^{2}K_{\mathrm{F}}^{3},$
5
where (2KD)KF is the vertical speed of the legs (s) and the term 2KD takes into account the fact that KD is analogous to step length (the distance covered in half a cycle), whereas KF was calculated over one complete cycle (by analogy with the stride frequency in walking).

This model equation was utilized to estimate k (Systat 5, USA) in the two conditions (kL and kLF) by means of a multiple non-linear regression.

The Froude efficiency (ηF) and the kinetic energy(Wk)

The Froude efficiency ηF of swimming with (LF) and without fins (L) was calculated from the values of average forward speed (ν) and from the velocity of the backward wave (c) on the same two subjects. c (m s-1) was measured as indicated by Ungerechts(1983) and Sanders et al.(1995) from the 2-D coordinates of the hip, knee and ankle joints. As shown in Fig. 2B, each coordinate reaches its minimum/maximum displacement with a phase-shift represented by the time lag (Δt). The distance between the anatomical landmarksΔlT (the thigh length) and ΔlS (the shank length) divided by the corresponding time lag between the waves minima gives the velocity of the wave along the body(c=ΔlTtT and c=ΔlStS). From the values of c, the Froude efficiency was calculated according to Equation 1 and the term Wk was calculated according to Equation 3.

Statistics

The regressions between O2 and Da for each condition were calculated by the sum of the least-squares linear analysis model. The differences in the measured variables(e.g. C, Db, Wint) as determined in the L and LF conditions were compared by the paired Student's t-test at matched speeds (from 0.7 to 1.0 m s-1 only, N=28).

The results of a typical experiment are reported in Fig. 3 for the LF condition in one subject. The regression between Da and oxygen consumption(O2), obtained from 4-5 observations, was linear for all subjects at all speeds (mean r2=0.959±0.042; range=0.775-1.000, N=70). From these relationships the individual values of active body drag(Db=Da, at O2rest) and of net energy cost(C=O2net/ν,at Da=0) were calculated.

The energy cost of the swimming leg kick is reported in Fig. 4 as a function of the speed for both L and LF conditions. At paired speeds, the energy cost when swimming with fins was 42±2 % lower than when swimming without(LF—L=-4.5±2.5 J m-1 kg-1, P<0.001). When compared at the same metabolic power, the decrease of C brought about by the use of fins allowed an increase of progression speed of approximately 0.2 m s-1.

Average values of kick depth, kick frequency, active body drag and trunk inclination are reported in Table 2. Fins only slightly decrease (14 %) the kick depth (at the ankle level) but cause a large reduction (43 %) in the kick frequency. No significant differences in active body drag and trunk inclination were observed between the two conditions at comparable speed.

Froude efficiency (ηF), calculated by Equation 1 in two subjects, was found to be 0.61±0.02 (N=39) when swimming with the legs only and 0.69±0.02 (N=24) when swimming with fins. No differences were found in the time lag (Δt) between the data calculated from the hip-knee or knee-ankle coordinates within both L and LF conditions. As shown in Fig. 5, the wave speed (c, m s-1) was found to increase linearly with the kick frequency (KF, Hz). The strength of this relationship(r2=0.911) suggests that ηF can be accurately estimated from individual values of kick frequency and speed assuming a wavelength of 2.33 m. Using this simplified method the average values ofηF (N=35) were found to be 0.61±0.01 when swimming with the legs only and 0.70±0.04 when swimming with fins, i.e. a 16 %increase at comparable speeds (see Table 3).

From the individual data of d (W)(d=Dbν)and ηF, the term k (W)could be estimated using Equation 2. k increased with swimming speed in both conditions and was 32 % smaller in LF than in L at comparable speeds (see Table 4).

The internal work rate(int, W kg-1),as measured in two subjects during actual swimming, is reported in Fig. 6 as a function of the speed (ν, m s-1). int was found to increase with the speed in both conditions. In these two subjects int was found to range from 13 to 43 W in the L condition (N=10) and from 9 to 18 W in the LF condition (N=10).

The constant k, as obtained from the multiple non-linear regression, was found to be 13.93 (kL, N=10, r2=0.976)in the L condition and 25.55 (kLF, N=10, r2=0.832) in the LF condition. The good coefficients of determination suggest that int can be accurately estimated from individual values of kick frequency and depth and from the estimated values of k (e.g. by applying Equation 5). The so-calculated average values of int(N=35 in L and LF) are reported in Table 4: int increased with swimming speed in both conditions and was 74 % smaller in LF than in L at comparable speeds.

Along with the mean values of int and k the values of d and tot are also reported in Table 4 as well as the values of (in kW, i.e. the net metabolic power). tot increased with the speed in both conditions and was significantly reduced (36 %) by the use of fins at comparable speeds. The contribution of k and int to tot were found to be speed-independent within both conditions. The contribution of k to tot is approximately the same in the L and LF condition (24±1 %, mean ± S.D. at all speeds). On the contrary, whereas int is a major determinant of tot in the L condition(40±1 % of tot,average at all speeds), its contribution is reduced to a half in the LF condition (18±13 % of tot, average at all speeds).

As shown in Fig. 7, at comparable speeds fins reduce k by 32 %, int by 74 % and tot by 36 % (whereas d is unchanged). The decrease in brought about by the use of fins (42 %) is proportional to the decrease in tot (36 %), so that the mechanical efficiency of swimming (ηM: tot/) is only slightly higher when swimming with fins: it ranges from 0.08 to 0.12 in the L condition and from 0.11 to 0.17 in the LF condition. On average (all subjects at comparable speeds), fins increase the mechanical efficiency of swimming by about 10 % compared to use of legs alone (see Table 3).

Energetics of swimming

Existing data on the energetics of fin swimming refer mainly to underwater swimming (SCUBA diving) (see Pendergast et al., 1996), and show that the effects of different fins on the energy requirements of swimming depend on the type of fin used. Large, rigid fins are energetically demanding but improve the maximal attainable speed,whereas flexible (small sized) fins improve the economy of swimming at submaximal cruising' speeds (Lewis and Lorch, 1979; McMurray,1977; Pendergast et al.,1996). Whereas fish can self-adjust the stiffness of the tail according to their needs (Lewis and Lorch,1979; Alexander,1977), humans must select the fin design according to the task to be performed. In accordance with these observations, the flexible fins used in this study were found to decrease the energy cost of swimming by approximately 40 % at sub-maximal speeds of progression compared to using legs alone. As indicated in Fig. 4, at any given metabolic power the reduction in C brought about by the use of fins is associated with an increase in the speed of progression of approximately 0.2 m s-1 (the descending curves representing iso-metabolic power hyperbolae). Thus the effect of using fins on economy of water locomotion is similar to that observed for passive locomotory tools in land locomotion(Minetti et al., 2001).

Substantial variation in the energy cost of swimming using the leg kick was observed in the L condition (Fig. 4). The differences in economy of swimming in subjects of comparable skill (assumed here) and at comparable speeds are attributed to the anthropometric characteristics that determine the subject's position in water. These differences can be quantified by measuring the underwater torque(T′, a measure of the tendency of the feet to sink) by means of an underwater balance. T′ was indeed found to be a major determinant of the energy cost of swimming at low to moderate speeds (see Pendergast et al., 1977). At moderate speeds, T′ (a static measure of the subject's position in water) relates to the subject's dynamic position in water (e.g. the trunk inclination, TI). Hence TI can also be expected to be a determinant of C. When speed increases, hydrodynamic lift acts on the body, which assumes a more streamlined position in the water and the difference between static and dynamic position (T′ and TI) lessens.

C and TI were found to be linearly related (the larger TI, the larger C) in the L condition, with an average correlation coefficient (for the five speeds considered) of 0.759±0.082(range: 0.710-0.881, N=7 for each regression, P<0.05 except for 0.7 ms-1). This suggests that part of the variability of C observed in the L condition is indeed attributable to differences in anthropometric characteristics of the subjects. In contrast, even if no differences in TI were found between the L and LF conditions, the relationships between C and TI were not significant in the LF condition, which indicates that other factors more strongly influence the energy cost of locomotion when fins are used.

Work components and efficiency of swimming

As previously discussed for land locomotion(Minetti et al., 2001), an improvement in economy does not imply, per se, a parallel improvement in the efficiency of locomotion. The effects of the use of fins on the efficiency of the leg kick can be investigated only by measuring all the components of the total mechanical work.

As shown by the flow diagram in Fig. 1 (adapted from Daniel,1991) the efficiency of aquatic locomotion is determined by three factors. (1) The efficiency with which muscles use oxygen to generate work. This efficiency is usually defined as mechanical or musculo—skeletal efficiency(ηM=tot/).(2) The efficiency with which the work done by the muscles produces a useful movement (i.e. that fraction of the work that gives kinetic energy to the fluid). This component takes into consideration the fact that some of the contractile energy is lost' in accelerating the mass of the lower limbs, in deforming parts involved in thrust production (e.g. the fin's blade) and in overcoming viscous damping in the tissues; this efficiency is called hydraulic efficiency (ηH) (Alexander,1983). Whereas the contribution of the elastic and viscous factors is difficult to assess, the contribution of the inertial factors can be estimated by measuring the internal work of swimming(Wint). Hydraulic efficiency is given by:
$\ {\eta}_{\mathrm{H}}=\frac{({\dot{W}}_{\mathrm{k}}+{\dot{W}}_{\mathrm{d}})}{{\dot{W}}_{\mathrm{tot}}}.$
7
(3) The efficiency with which the work done in moving fluid goes to useful thrust (Wd); this is the Froude efficiency (ηF) and is given by:
$\ {\eta}_{\mathrm{F}}=\frac{{\dot{W}}_{\mathrm{d}}}{({\dot{W}}_{\mathrm{k}}+{\dot{W}}_{\mathrm{d}})}.$
8
The product of hydraulic efficiency (ηH) and Froude efficiency (ηF)gives the propelling efficiency(ηP=d/tot);the product of mechanical efficiency (ηM) and propelling efficiency(ηP) gives what can be called performance efficiency(η=d/)and can be checked by dimensional analysis, as reported by Daniel(1991) and Webb(1971b).

In the computation of tot=k+d+int,the elastic and viscous terms were considered negligible and the terms d and int were directly measured. The term k was obtained from the values of Froude efficiency.

In the following paragraphs the measured/calculated values of the above mentioned parameters will be discussed separately. The average values of all the efficiencies indicated in this figure are reported in Table 3.

The power needed to overcome drag(Ẇd)

The term d was calculated as proposed by di Prampero et al.(1974) and was found to be unaffected by the use of fins, at comparable speeds.

It can be debated whether the decrease of O2 observed as a consequence of adding masses to the pulley system (Da, see Fig. 2) is attributed to changes in Wd only. We found that the added thrust(Da) not only reduced the swimmer's active body drag (and hence d) but also affected the frequency of the kick: the higher Da, the lower KF. The observed reduction of O2 for any given Da has therefore to be attributed to a decrease in int and k (both proportional to KF). Since the contribution of these factors to total O2 is large at Da=0 (during free swimming) but negligible at the highest Da, it can be shown that these factors affect only the slope of the relationship between Da and O2 and not the point at which the regression crosses the Da axis, thus they do not affect the determination of Db.

Thus, even if the procedure to determine active body drag is based on several assumptions, this is still, in our opinion, the best method proposed so far to estimate resistive forces to aquatic locomotion in humans.

The internal work rate(Ẇint)

To our knowledge, no data for int have been reported before (nor even considered as a source of energy expenditure) for swimming humans. The internal work rate when swimming using the leg kick accounts for approximately 40% of tot in the L condition and for about 20% of tot in the LF condition(its contribution to totbeing almost independent of the speed). Hence, fins halve the contribution of inertial factors to energy expenditure in aquatic locomotion. In absolute terms, int increases more than threefold over the selected speed range in both conditions and can account for up to 50 W at the higher KF observed. Not considering this parameter could therefore lead to a severe underestimation of the overall swimming efficiency.

In this paper the model equation proposed Minetti and Saibene(1992) for walking was adapted to describe the relationship between int, kick frequency and kick depth when swimming. The value of k, a constant which takes into account the inertial parameters of the moving segments, turned out to be comparable to that calculated by Minetti and Saibene(1992) for walking (k=21.64). In the L condition, k was found to be half of that calculated for walking(kL=13.93), in agreement with the fact that only the lower limbs are moving and the oscillation amplitude is smaller. The larger value of k in the LF condition (kLF=25.55) reflects the effect of the added fin mass on the inertial parameters of the lower limbs.

KF was found to be the main determinant of int in swimming. Indeed, as shown in Table 2, values of KD increase only slightly, whereas large variations in KF are observed with speed in both conditions. Moreover,fins only slightly decrease the amplitude of the kick but cause a large reduction in the kick frequency. Hence, as a first approximation, the term(2KD)2 in Equation 5 can be considered as constant and int can be calculated from int=kKF3. When the int data as directly measured in the two subjects in both conditions were pooled, the equation became int=6.9KF3(N=20, r2=0.789; where int is in W and KF in Hz). The coefficient of determination is only slightly lower than the one obtained with the model equation(r2=0.841, see above) suggesting that this approximate version of the equation can be safely utilized to estimate int when data for KD are not available.

There is some debate about whether external and internal work have to be separately calculated and summed in walking (assuming no energy transfer between the two) or jointly computed, allowing for any possible transfer, but in cycling (Minetti et al.,2001) and swimming the lower limb movement minimally affects the movement of the centre of mass. Moreover, in swimming, limb movement occurs in an orthogonal axis with respect to progression and the vertical and horizontal oscillations of the centre of mass are kept to a minimum.

The Froude efficiency (ηF)

In this paper the Froude efficiency was estimated by applying the equation proposed by Lighthill (1975)and Alexander (1977) for animal locomotion in water (Equation 1) to human swimming. This method is based on the calculation of the speed of the bending waves that move along the body (of the swimmer or the fish) in a caudal direction.

As shown in Fig. 5, wave speed (c, ms-1) increased linearly with the kick frequency(KF, Hz). Since kick frequency is the reciprocal of the wave period (T, s), the slope of the above linear relationship is the wavelength (λ, m). As shown by the high determination coefficient, the wavelength was found to be quite constant for different subjects, speeds and conditions (λ=2.34±0.18 m, N=20) and similar to the values measured by Sanders et al.(1995) in male swimmers using the butterfly stroke (2.24±0.25 m). The good agreement between the average values of wavelength measured in the different studies, the small standard deviation and the high determination coefficient(r2), suggest that the linear relationship between wave speed and KF described above can be utilized to predict the Froude efficiency of swimming using the leg kick from measurements of kick frequency (KF) and average speed (ν).

That c depends essentially on KF while the wavelength of the propulsive wave (λ) is almost constant at different swimming speeds was also observed and reported by Webb for the rainbow trout Salmo gairdneri(1971a). Fish with similar body forms are characterized by similar specific wavelengths(λ/L, where L is the fish length). Depending on the fish swimming type, λ is larger (carangiform mode) or smaller(angulliform mode) than L (Webb,1971a). The data reported above (λ=2.3 m and henceλ>body length) are compatible with the morphological observation that humans are relatively thicker for their length and resemble, if we can risk such a comparison, a trout rather than an eel.

The values of Froude efficiency measured in this study are not far from those measured in the rainbow trout by Webb (according to the method proposed by Lighthill), who reported a range of values for ηF of 0.61-0.81 at speeds between 0.1 and 0.52 ms-1(Webb, 1971b).

The kinetic work rate not useful for thrust production(Ẇk)

The data for k were obtained according to Equation 3 and using the measured/calculated values of d and ηF. k data for swimming humans are scarce; the only other values reported in the literature were obtained by means of the MAD system. With this method k is calculated as k=O2eqFREEO2eqMAD)/ηwhere O2eqFREEis the energy expended when swimming freely (expressed in W), O2MAD(W) is the energy expended when swimming on the MAD system and η is the efficiency of swimming (as obtained by graphical analysis). In those conditions (arm stroke only with the legs floating), k was found to range from 16 to 64 W at speeds of 1.0-1.3 ms-1(Toussaint et al., 1988). Since the values of kincrease with speed, it is reasonable that the values found in the present study (9-31 W in both conditions) are lower than those reported at higher speeds using the arm stroke. The two sets of data are not, however, directly comparable. In fact, the term k as defined by Toussaint et al. is given by the sum k+intsince it was obtained from values of tot and d only (indeed, they correctly defined their efficiency as propelling').

For values of Froude efficiency ranging from 0.50 to 0.75 (a reasonable range for human locomotion in water), the power wasted when imparting kinetic energy to the water (k) is bound to range from k=dF=0.5) to k=1/3 dF=0.75). Not taking into account this parameter can therefore lead to an underestimation of the overall swimming efficiency, as discussed above for int and previously emphasized by Toussaint et al.(1988). Indeed k accounts for approximately 25% of tot(for both conditions and at all speeds).

Hydraulic efficiency (ηH) and the propelling efficiency (ηP)

From the data reported in Table 4, the hydraulic efficiency can be calculated as:η H=(k+d)/tot);ηHwas found to be 0.59 in L and 0.82 in LF conditions, corresponding to a 40%difference at comparable speeds (see Table 3). As indicated by Alexander(1983) the efficiency of a propeller is given by the product of ηH×ηF(or, in other terms,η P=d/tot). The propelling efficiency turned out to be 0.36 in the L and 0.58 in the LF condition, i.e. 62% larger in LF than L at comparable speeds (see Table 3). In both cases the LF—L difference was found to be independent of the swimming speed.

Propelling efficiency has also been estimated, in competitive swimmers, by means of the MAD system when swimming using the arm stroke, and it was found to be comparable to that calculated in this study: 0.53(Toussaint et al., 1988) and 0.56 (Berger et al., 1997) at speeds between 0.9 and 1.35 ms-1.

All these values can be compared to the values of propelling efficiency of other locomotory devices for aquatic locomotion. As reported by Abbott et al.(1995), human-powered vehicles with drag-device propulsion (such as boats propelled by poles, oars and paddles) are characterized by propelling efficiencies of about 0.65-0.75. Even though these values are larger than those reported for swimming, about one third of the subject's power output is bound to be wasted using these locomotory devices. Human-powered propeller-driven boats, which can reach greater propelling efficiencies, have been developed since the 1890s for practical transportation purposes (Abbott et al., 1995). Their development almost completely ceased when gasoline-driven outboard motors were introduced; propellers with efficiencies exceeding 90% are currently in use on human-powered watercrafts of recent development (e.g. the flying fish; Abbott et al., 1995).

The efficiency of a propeller is higher if a large mass of fluid is accelerated to a low velocity than if a small mass is accelerated to a high velocity (Alexander, 1977). Since fins increase the propelling surface they would be expected also to increase propelling efficiency (as experimentally determined). The increase inη P observed in this study (62%) can be compared to the increase of propelling efficiency (7%) obtained by the use of hand paddles when swimming the arm stroke (Toussaint et al.,1991). The increase in ηP with fins compared to without fins or between fins and hand paddles may be partially explained by the higher propelling surface of fins compared to feet (fin/foot surface area:3.5; see Table 1) and hand paddles (hand paddles/hand surface area: 0.026/0.018 m2=1.45).

Mechanical efficiency (ηM)

The mechanical efficiency of swimming with and without fins, at a given speed, was calculated from the ratio of total mechanical power(tot=k+d+int)to total metabolic power (). Mechanical efficiency ranged from 0.08 to 0.17 in both conditions and at all speeds (0.11 and 0.13 for L and LF,respectively; mean for all subjects at comparable speeds). These values are higher than those reported for swimming humans. In those studies, however, any contributions of internal and/or kinetic work were neglected. When internal and kinetic work are not accounted for, efficiency values(η=d/)range from 0.03 to 0.05 (in the L condition), which is comparable to that reported by others for front crawl swimming 0.05-0.08(Toussaint et al., 1988),0.03-0.09 (Pendergast et al.,1977) and 0.04-0.08 (Holmer,1972) and compatible with the fact that the leg kick is a less effective way of moving in water than the arm stroke (e.g. Adrian et al., 1966).

Locomotory (mechanical) efficiency is generally investigated to get insight into how muscles (the actuators) work in situ. The challenge is to compute all the components of the external and internal work (as well as taking into account the contribution of elastic energy storage and release,viscous damping in the tissues and so on...) in order to obtain the best possible estimate of muscle efficiency.

By taking k and int into consideration in the computation of

$${\dot{W}}_{\mathrm{tot}}$$
⁠, a better estimate of ηM in swimming humans was obtained than in previous studies. The values are still lower than those expected from the thermodynamics of muscle contraction (0.25-0.35 at optimal contraction speeds; Woledge et al., 1985),however. The gap' between the measured values of ηM and the optimal values of muscle efficiency could arise from an underestimation of
$${\dot{W}}_{\mathrm{tot}}$$
, from an overestimation of
$${\dot{E}}$$
and (obviously enough) from muscle inefficiency itself.

Among the factors that might contribute to an overestimation of

$${\dot{E}}$$
(the metabolic power above resting conditions) is an underestimation of `basal metabolic rate' (which was not measured in this study but assumed to be 3.5 ml min-1kg-1), e.g. because a larger fraction of
$${\dot{E}}$$
is utilized for thermoregulation in water in respect to land locomotion. Moreover, as previously pointed out by Gaesser and Brooks (1975) for humans and Stevens and Dizon(1982) for warm-bodied fish,basal metabolic rate increases with progression speed/work rate. If this can be accounted for, higher values of ηM would result mainly at the higher investigated speeds.

Among the factors that can contribute to an underestimation of

$${\dot{W}}_{\mathrm{tot}}$$
(particularly in the case of LF) is the work done in deforming parts involved in thrust production. These are expected to be higher in LF (the fins' blade) than in L(the foot's sole).

As far as muscle efficiency itself is concerned, inefficiency should arise when the muscles are not working in the optimal range of either their force/length and/or force/speed relationships.

The leg kick is quite an ineffective way of using the lower limb muscles. The range of motion of the hip and knee joints is more restricted than in walking and cycling (an appreciable bending is observed only in the recovery,almost passive, phase of the cycle, see Fig. 2A) so that the leg extensors probably do not contract at their optimal length (maximal force of the knee extensors occurs at a knee angle of about 110°; e.g. Narici et al., 1988).

The other factor that is known to affect muscle efficiency (and hence mechanical efficiency) is the contraction speed at which the muscles are working. From studies of muscle physiology, it is known that only at a given contraction velocity is the maximum efficiency reached(Woledge et al., 1985). While we did not directly measure the contraction speed of the lower limb extensor muscles, it is reasonable to assume that the kick frequency is strongly associated to it. Frequencies of about 1.0 Hz have been suggested as the optimum one for Type I fibres (in cycling; Sargeant and Jones, 1995). Fins decrease the kick frequency from 1.59±0.25 Hz in L to 1.02±0.25 Hz in LF conditions (average at all speeds and for all subjects) and hence are expected to increase ηM by allowing the muscles to work more efficiently (as found in this study and for other locomotory tools on land; see Minetti et al.,2001).

Performance efficiency (η)

The rate of useful work production divided by total rate of energy expenditure has been used as a measure of performance for many biological systems (Daniel, 1991). In the step diagram of Fig. 1,performance efficiency is defined as the ratio of useful power (necessary to generate thrust) to total energy expenditure

$$\left(\frac{{\dot{W}}_{\mathrm{d}}}{{\dot{E}}}\right)$$
⁠. This concept is useful for briefly summarizing the results found in this study.

Since fins do not affect

$${\dot{W}}_{\mathrm{d}}$$
(at comparable speeds), the increase in economy (42%) observed when fins are used must produce an increase in performance; the ratio
$$\frac{{\dot{W}}_{\mathrm{d}}}{{\dot{E}}}$$
is equal to 0.03-0.05 in L and 0.07-0.09 in LF and corresponds to a 77%difference at comparable speeds. This increase is almost completely explicable on the basis of the observed increase of propelling efficiency (66%). Of the increase in ηP, one third has to be attributed to an increase in ηF(13%) and two thirds to an increase of hydraulic efficiency (40%). The increase in ηP is obtained essentially through a 43% decrease in KF (which leads to a 74% decrease of
$${\dot{W}}_{\mathrm{int}}$$
) and through an increase in the propelling surface (3.5 times higher with fins), which allows for the acceleration of larger masses of fluid at lower speeds(Alexander, 1977).

Conclusions

A complete energy balance during swimming using the leg kick, with and without fins, was attempted by combining methodologies previously applied to human and fish swimming. From the combination of these techniques, the economy(C), total mechanical work (Wtot), propelling efficiency(ηF) and mechanical efficiency (ηM) of swimming were computed. While the breakdown of the individual components of performance efficiency helps in understanding why swimming with fins represents an advancement in human powered locomotion in water, the overall gain in propulsion is far from being commensurate with what muscles are expected to produce based on their performance in land locomotion. Despite the lowering of energy expenditure and an increase of 0.2 ms-1 in swimming speed at equivalent metabolic power, other solutions with different locomotory devices should be pursued to increase Froude efficiency (e.g. to decrease

$${\dot{W}}_{\mathrm{k}}$$
⁠), hydraulic efficiency (e.g. to decrease
$${\dot{W}}_{\mathrm{int}}$$
) and muscle efficiency in order to further improve swimming performance in humans.

List of symbols

• c

velocity of the backward wave

•
• C

energy cost of swimming

•
• Da

•
• Db

active body drag

•
• E

total metabolic power

•
• f

stride frequency

•
• KD

kick depth

•
• KF

kick frequency

•
• lS

shank length

•
• lT

thigh length

•
• L

swimming without fins

•
• L

fish length

•
• LF

swimming with fins

•
• Re

Reynolds number

•
• s

speed of vertical movement of legs

•
• t

time

•
• T

wave period

•
• TI

trunk inclination

•
• T′

underwater torque

•
• v

average forward speed

•
• $${\dot{V}}_{\mathrm{O}_{2}}$$

oxygen consumption

•
• Wd

work needed to overcome drag that contributes to useful thrust

•
• Wext

external work

•
• Wint

internal work

•
• Wk

work needed to overcome drag that does not contribute to thrust

•
• Wtot

total work

•
• η

performance efficiency

•
• ηF

Froude efficiency

•
• ηH

hydraulic efficiency

•
• ηM

mechanical efficiency

•
• ηP

propelling efficiency

•
• λ

wavelength

We acknowledge the conceptual assistance given by Sawson Samimy and by Prof. Joseph Mollendorf. The technical assistance of Chris Eisenhardt, Dean Marky and Frank Moldich is gratefully acknowledged as well as the patience and kind cooperation of the swimmers. We also wish to thank Theo Bampouras for helping in the video data analysis. This research has been partially supported by the United States Navy, NAVSEA, Navy Experimental Diving Unit, contract N61 33199C0028.

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