FISHERY BULLETIN: VOL, 72, NO. 4 



directly measured using a set of dial calipers 

 read to 0.01 cm. 



Many of the initial sequences of an excursion 

 when viewed with respect to the x -axis as defined 

 above showed the appearance of a wave along the 

 proximal portion of the fish while the rest of the 

 body coincided closely to thex-axis. This indicated 

 strong X -dependence of the amplitude in the ini- 

 tial portion of the excursion. However, after three 

 frames an almost symmetrical amplitude wave 

 was observed. Thus the amplitude in the first sev- 

 eral frames was taken as the maximum length of 

 the wave above the x-axis (Figure 3). 



The wave length was taken as that length be- 

 tween two successive crossings of thex-axis by the 

 displacement wave form. During the later part of 

 the excursions no crossings from positive to nega- 

 tive were observed and at this point the 

 wavelength was taken as twice the value from one 

 tangent of the body on the line of motion to the 

 other (Figure 3). 



The position of the midpoint between the eyes 

 after each frame was monitored to yield x(0. Each 

 successive movement of that reference point was 

 recorded in the manner outlined above and the 

 distance moved during each frame noted. 



The projected length xp(t) was taken as the 

 length between the two points representing the 

 projection of the tail and snout tip position on the 

 X-axis and was used in a manner to be described 

 later. 



The wave position Xuit) was taken as the pro- 

 jected length of the body from the point where A (^ 

 is measured to the snout tip (Figure 3). 



The points representing the functions described 

 above at each unit of time, i.e., one frame, were 

 collected for 18 excursions which were randomly 

 selected from the larval anchovy feeding films. 

 The functions were then nondimensionalized by 



division by body length and plotted. The geometric 

 form of the resulting function was then used as a 

 guide in selecting an appropriate descriptive func- 

 tion. The parameters of these functional forms 

 were then fitted by computer in the least squares 

 sense using a nonlinear steepest descent approach 

 (Conway, Glass, Wilcox, 1970). The graphical rep- 

 resentation of the proposed body displacement 

 function with the internal functions fitted in this 

 manner was found to coincide very closely with 

 the actual body displacements seen in the films. 



In the derivations for total excursion work, 

 W^ , the integral for tangential viscous reactive 

 work contains s, the distance along the fish 

 body, explicitly. The function satisfying F{x, t) 

 = s is extremely complicated for the complete 

 wave-form displacement function using all the 

 fitted internal functions and is almost impossible 

 to calculate explicitly. The alternative used 

 here is to extrapolate back from the measured 

 X;,(^) to yield s (x, t). 



We know the function F(x, t) = s satisfies 



F(Xp, t) = L 



where L is the length of the fish body. Since the 

 maximum amplitude ever encountered in this 

 study was around 0.2 L and the mean integra- 

 tion distance never greater than ttI2, we can 

 calculate, for purposes of comparison, the differ- 

 ences between the true length of a pure sine 

 wave of amplitude A and its projected length. 



The unperturbed or no sine-wave form for a 

 7r/2 interval of integration yields simply jtI2. 

 The sine-wave projected length is, for y = As\YiQ, 



I 



77/2 



Vl + A'" cos"" e d G 



Direction of 

 forward movement 



= V 



1 + A^ E \7r/2 



W: 



A 



1 + A- 



where E{<^ ,k) is the elliptic integral of the sec- 

 ond kind (in this case a complete elliptic in- 

 tegral of the second kind). Taking A ^ 2.0 cm 

 we get using A = 0.2L. 



Figure 3.— Diagram illu.strating the identification of (t)l2, 

 Xw(t) and A(t) from photographic records (see text). 



= Vl + 0.16 E{ttI2, 0.37) = 1.625. 



890 



