FISHERY BULLETIN; VOL. 77, NO. 3 



stem from calculations based on the analysis in 

 Weihs ( 1974 ). The ratio of energy per unit distance 

 traversed in intermittent swimming to that of con- 

 tinuous swimming at the same average speed R, 

 can be shown to be, in present notation, 



intermittent swimming relatively more efficient 

 than carangiform swimmers as a is greater for 

 anguilliform swimming in which most of the body 

 is oscillated. Another result is that for greatest 

 gains, the average speed during the whole beat- 



tanh 



R = 



Uf-Uj 



tanh 



In { cosh 



tanh 



Ur - U, 



[/,.[/, 



+ t/, sinh tanh ^ 



U, - U. 



- - 1^ + a In —~ 



(251 



The computed values of /?, for a = 2, appear in 

 Figure 3. Each full line describes the values of i? 

 for a gi\er\Uf as a function of the average velocity 

 C/j.. Each of these curves ends at U ^ = [/^and, in a 

 similar manner to Figure 1, has a minimum for a 

 lower value oiU ^ . Here, however, all curves have a 

 large section in the range R' \, i.e., intermittent 

 swimming is more efficient. In fact, the slower the 

 average velocity, the higher the possible gains, as 

 shown by the dashed line which is the locus of 

 lowest values of i? as a function oitj^ . As already 

 mentioned in Weihs (1974)^ this curve goes 

 monotonously from unity at t/^ = 1 (continuous 

 swimming by definition) to 1/aat t/,, -►O. One can 

 therefore predict that fish species using the an- 

 guilliform swimming mode (Breder 1926) will find 



0.0 25 5 75 10 



Uc 



Figure 3.— The ratio of energy required per unit distance R. for 

 intermittent and continuous swimming at high Reynolds num- 

 bers, respectively, versus nondimensional average speed Uc 

 Dashed lines shows locus of minimum values of /? attainable as a 

 function of Uc. See Figure 1 for definitions 



602 



and-glide should be as low aspossible, with small 

 differences between U, and Uf. 



Anchovy, which swim in the anguilliform mode, 

 fulfill both these predictions as adults and more 

 mature larvae usually swim by means of a single 

 beat followed by a long glide, so that 1 ) t7, and Of 

 are not too different and 2) [/,. is rather low. 



Having examined the low and high Re domains, 

 where the drag coefficient is porportional to the 

 reciprocal of Re, and constant, respectively, we 

 now look at the transition regime between them. 

 Based on average swimming speeds of 0.8 body 

 length/s, larvae will be in this regime when they 

 are from 5 to 15 mm in length. Analysis of the 

 forces and energy is much more complicated here 

 because the hydrodynamic drag is 



C[) a Re" 



(26) 



where fi is not constant, but itself is a function of 

 both Re and body shape. This results in Equation 

 (4) taking the form 



T = 



du_ 

 dl 



+ aKu' 



(27) 



a differential equation that has to be solved nu- 

 merically when /3 is not zero or one (the two cases 

 discussed previously). While this in itself is a rela- 

 tively straightforward task, the generality and 

 accuracy of the previous solutions is immediately 

 lost as numerical values for the mass and K have 

 to be included. K especially is known very inaccu- 

 rately as it includes the numerical drag coefficient 

 and the frontal area (which varies at different 

 speeds and times). The setting of /3 is even more 

 problematical as it depends on the instantaneous 

 swimming speed in an empirical manner (which 



