VLYMEN: SWIMMING ENERGETICS OF THE LARVAL ANCHOVY 



where ^■ is the kinematic viscosity of seawater 



U.S. Navy Hydrographic 



2-c-l 



0.0119 cm2-s 

 Office, 1956). 



A computer program by Stroud (1971) using 

 16-point Gauss-Legendre integration, and the 

 above outlined integration scheme was used to 

 compute the integrals comprising W.^. The pro-> 

 gram was translated into Algol and executed 

 on a Burroughs 6700 at the University of Cali- 

 fornia, San Diego Computer Center. Accuracy 

 of the program was checked by evaluation of the 

 iterated integrals 



/ e c/v/ 



Jo "Jo 



e^' dx for various w and 



The results showed the integration scheme to be 

 accurate to the eighth decimal place in the former 

 integral when compared with tables in Rosser 

 (1948) and accurate to the fourth decimal place in 

 the latter integral using standard tables. Details 

 of the mathematical scheme are found in the ap- 

 pendix. 



In the integrations of Wj a relative convergence 

 was computed by first doing the integration over 

 the whole interval, that is, 



/o 



" J fit) 



F(x, t) dxdt. 



Then the value corresponding to one subdivi- 

 sion is computed, namely, 



Hi r Hit I 



lx= I I Fix, t) dxdt 



11 



Jo J fit 



i I 



J tl-i J n, 



t fgiti 



+ 1 / F(x. t) dxdt. 



The relative convergence is then computed as 



C 



h -h 



If this value is less than 0.05, the value 4 is taken 

 as the value of the integral. If it is greater, the 

 intervals comprising I^ are further subdivided 



and the process continued until convergence is 

 reached. Thus, if /„, corresponding to 2 " subdivi- 

 sions, and /„ +1, corresponding to 2„ + i subdivi- 

 sions, are of such values that 



- 1 



In 



<0.05, 



n + 2 



then /y is assigned the value /„ ^ i. 



The convergence is set higher than one might 

 expect because computation of the complex in- 

 tegrals of the type used in this study is manifested 

 by slow and oscillatory relative convergences 

 necessitating a great deal of computer time. How- 

 ever, when the convergence criteria was set at 

 0.05 in the integrations performed, convergences 

 were better than the critical value. The effect of 

 the higher convergence criteria is thus seen as 

 being an economic and computational conveni- 

 ence. 



RESULTS 



The plotted values of the nondimensional 

 amplitude, A{t)IL, wave position, Xw(t)IL, and 

 projected length, Xp{t)IL, along with the de- 

 scriptive functions fitted by the methods discus- 

 sed are shown in Figures 6 and 7. The points 

 comprising the curves of each represent the 

 mean value of the particular parameter in 

 question at successive units of time where one 

 time unit is Vr28 s. 



MEASURED MEANS 

 FITTED FUNCTION 



A|t)/L! 0.206 cxp [-0.044(1 -7.19)^] 



9 10 II 12 13 14 15 



Figure 6. — Nondimensional amplitude, A (H/L and wave posi- 

 tion, Xu, (t)IL, of body displacement function as functions of 

 time, t, in motion frame units. The graphs display the fitted 

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

 points of the fitted curve at corresponding time units (closed 

 circles). 



893 



