FISHERY BULLETIN: VOL. 72, NO. 4 



MEASURED MEANS 

 FITTED FUNCTION 



Xp(t)/L = 0.0029lt'-0.04l2t + 1.00 



a. ao' 



0.20 



15 



! 0.12 



i§ 



; E 0.08 



'■ 'A 

 > ? 



■■^ 0.04 

 0.00, 



X(l)/L = -0.00009654l' +0.002551^+0.00061 +0.0001396 



10 II 12 



Figure 7. — Nondimensional position, X(t)IL , and projected body_ 

 length, XpitVL, as functions of time, t, in motion frame units. 

 The graphs display the fitted curves (lines) together with the 

 original data (open circles) and points of the fitted curve at 

 corresponding time units (closed circles). 



The curve for X (t)/L, deserves some discussion. 

 Since the amplitude of the propagated wave was 

 known to be zero at ^ = 0, both X = =« or X =0 would 

 be descriptive of the initial straight-line 

 configuration. However, X = implies an infinite 

 number of oscillations varying like sin t/\ with 

 neither the function nor the first derivative exist- 

 ing as X ^ 0. Since at the end points of an excur- 

 sion a slightly perturbed wave form was observed, 

 i.e., a finite wavelength, the nondimensional 

 wavelength of the t = excursion wave form was 

 adjusted to be equal to the last. A perfect relation 



2.4|- 



2.0, 



X 



I- 

 o 



1.6 



I 1.2 



2 0.8 

 to 



Q 0.4 



z 



o 



0.0 



o MEASURED MEANS 

 • FITTED FUNCTION 



A(t)/L= 11.16 



/ 0.947 y-° / 0.947 \ 

 \l+l.02/ \t + l.02/ 



I.I 



+2.29 



J I I I L 



I I I 



■0 2 4 6 8 10 12 14 



t 



Figure 8. — Nondimensional wavelength, X {t)IL, as a function of 

 time, t, in motion frame units. The graph displays the fitted 

 function (line) together with the original data (open circles) and 

 points of the fitted curve at corresponding time units (closed 

 circles). The dotted portion of the fitted curve is discussed in the 

 text. 



would have X = »= at both end points. This, I 

 believe, does not drastically affect the results 

 since the only modulatory component at the end 

 points is the amplitude which is zero at these 

 points. This accounts for the Lennard-Jones type 

 of function which was chosen as a functional rep- 

 resentation of X(t)/L and is shown in Figure 8 

 along with the function itself. The values at other 

 than the end points together with the fact that 

 A{t) ^ at these points is sufficiently descriptive 

 of the contour wavelength to vitiate any physical 

 inconsistencies or mathematical problems that 

 may arise from the end point modification of 

 X it)/L discussed. 



The integrals representing the work per excur- 

 sion namely W^g ^ , Wg, VT^ were subdivided 

 further into smaller iterated integrals and, using 

 the mean excursion time of 12.9 frame time units 

 (-O.lOs) integrated by the method already out- 

 ^lined. The values obtained were taken to repre- 

 sent the work/excursion of an anchovy larvae of 

 length equal to the mean of the animals used in 

 the study or 1.4 cm. 



The values of the work are divided into five 

 categories as follows: 1) head energy representing 

 the value of the integral in Equation (3), 2) normal 

 energy representing the value of the 1st integral 

 of H^B ^, 3) tangential energy representing 2nd 

 integral of W q"^ , 4) body inertial energy rep- 

 resenting the 1st integral ofW^ , and 5) inertial 

 energy representing the 2nd integral of W^ . The 

 value of these five integrals in ergs/excursion 

 and their fraction of the total excursion energy 

 is given in Table 1. It is observed from the table 

 that accelerative terms such as body inertial and 

 inertial energies account for more than three- 

 fourths of all the energy used in swimming. It 

 is worthwhile noting that although this is an ex- 

 pected outcome of the peculiar behavior of the 

 anchovy larvae, it is possibly true that neglect of 

 such terms in many analyses of fish energetics 

 is cause for errors. Attention to these matters has 

 been given thorough theoretical discussion in 

 Lighthill (1970, 1971). The analysis in this paper. 



Table 1. — Excursion energy components in ergs for the 1.4-cm 

 anchovy larva. 



894 



