FISHERY BULLETIN: VOL 77, NO 3 



Integrating Equation 20 gives I2 



h = r f/^expf--/) 

 \ '" / 



"' = f '"' 



t/,) 



(22) 



(23) 



To nondimensionalize, we now divide all velocities 

 by Uo  Substituting Equations ( 13), (14), ( 18), (22), 

 and (23) into S (Equation (17)) and simplifying 

 results in 



S = 



(1 - Uf) + In 



1 - U, 



1 -u 



u 



1-f/,. Ur 



in — + ain — 



i-t/, 



(a-l){Ui-Ui) + In 



l-Ur 



(241 



where the line over the speeds signifies values 

 divided by U^, . The ratio S serves now as a quan- 

 titative criterion, showing which mode of swim- 

 ming is more efficient. The advantages of the 

 comparison approach are now clear, as the only 

 parameter specific to the fish is a which, as men- 

 tioned before, can vary between 1 and 2 only. All 

 other factors, such as fish mass, frontal area, and 

 the numerical coefficient of drag, have cancelled 

 out. dropping many of the uncertainties of such 

 calculations. 



Results and Discussion 



Equation (24i is now studied for various values 

 of the parameters involved. As mentioned before, 

 the numerical value of S indicates the relative 

 efficiency of intermittent and steady swimming. 

 When S"»l beat-and-glide swimming is more 

 costly. The three variables appearing in S are 

 bounded by well-defined limits, which makes the 

 parametric study of Equation (24) easier. The 

 nondimensional "final" velocity, at the end of the 

 beat phase, is limited by 0<t7/<l as Uq is the 

 highest velocity obtainable by the animal (see 

 Equation (ID) under present conditions. By 

 definition, 0<C/, <f/^. The yolk-sac larvae and 

 later stage larvae are combinations of elongated 



and round shapes so that the limits of « are 

 lss«=£2. The range of values of « is obtained from 

 the fact that for elongated (fish) bodies, the ratio of 

 drag in normal motion to resistance to tangential 

 relative motion does not exceed 2 in the viscous 

 domain. Only some sections of the fish bodies are 

 in pure normal motion (perpendicular to the local 

 flow direction), consequently the average ratio is 

 always <2. 



It is also clear that the larger the numerical 

 value of a, the larger the possible gains by means 

 of intermittent swimming, because the drag when 

 actively swimming is proportionally greater than 

 while gliding so that breaks in the swimming 

 phase will be more advantageous. Therefore, we 

 first take a = 2 as a reasonable upper limit. Figure 

 1 shows the results of such calculations. The coor- 

 dinates are the energy ratio S versus the nor- 

 malized average velocity t/, . The calculations 

 were performed for different values of the final 



05 

 Uc 



FiCURE 1. — The ratio of energy required per unit distance S for 

 intermittent .swimming and continuous swimming respectively, 

 versus the nondimensional average speed 0^, for various values 

 of speed at the end of the beating phase, Uf. All speeds are 

 normalized by the speed U,, attained at maximum sustained 

 thrust T after a long time. Dashed line shows locus of minimal 

 energy ratio at different 0|. Ratioof swimming to gliding drag a 

 = 2. 



600 



