WEIHS ENERGETIC SIGNIFICANCE IN ENGRAULIS MORDAX 



where /, is the time spent in active swimming 

 during the cycle. In order to integrate Equation (8) 

 we have to obtain the functional dependence of (/ 

 on the time / . This is done with the aid of Equation 

 (4). Taking the thrust applied to be constant (T,, ), 

 Equation (4) can be solved in closed and general 

 form. It was previously shown (Weihs 1974) that 

 constant thrust production is the most efficient 

 procedure so that we can use this assumption here, 

 recognizing that any other behavior will probably 

 be more wasteful in terms of energy. After some 

 rearrangement. Equation (4) is 



T, 



'0 



du aK „ „ 



-77+ u =0 (9) 



at m m 



with boundary condition 



u = L(, at ^ = 



(10) 



where (/, is the initial velocity at the beginning of 

 the beat phase, which is equal to the speed at the 

 end of the gliding phase. The solution is 



(t/, - t/o) exp 



(-f') 



+ f/n 



(11) 



where U„ = T„iK is the final velocity which the 

 larva would attain if the beating phase were sus- 

 tained for a long enough period. U^ is higher than 

 the final speed actually obtained during the beat 

 phase Ui. Substituting Equation (11) into Equa- 

 tion ( 8) and integrating, recalling that the speed at 

 ty is U f, we have 



E^ = mf/o (t/o - Uf) + aKUo^ti. (12) 



To find t, . we substitute Uf in Equation (11) with I 

 = f, and obtain, after algebraic reduction 





(13) 



so that the energy required during one beat-and- 

 glide cycle is 



mUn 



The energy per unit distance crossed during one 

 cycle in intermittent swimming is 



E„ = 



'i + '2 



E^ 

 Uj 



Uidi + ^2) 



(15) 



where /, , and l.^ are the distances crossed during 

 beating and gliding phases, respectively. The sum 

 of /, and 4 is taken to be equal to the continuous 

 swimming speed U,. multiplied by the total cycle 

 time, as we compare energy required for crossing 

 the same distance /, + l.^. Thus, t/, is also the 

 average velocity during the whole intermittent 

 swimming cycle 



U, 



t^ + tn 



(16) 



To compare energy expenditure in intermittent 

 and continuous swimming, we define a ratio S 



E,. 



S = 



E„ (ti + lo) 



aKU,^(ti + /a) aA'(/i + l^f 



(17) 



Intermittent swimming is more efficient only 

 when the numerical value of S is less than unity. 

 To calculate values of S we need explicit expres- 

 sions for /, , /j, <, , and /j • The beat phase velocity 

 has already been found, Equation ( 1 1 ), so that /, is 

 easily obtained by integration, using ti . Equation 

 (13), as the upper bound 



m 

 ~aK 



U„ - Ui 



Ui) 



(18) 



The gliding phase of intermittent swimming is 

 described by 



du 



m —- + Ku 

 dt 







(19) 



which is obtained from Equation (4) when no 

 thrust is applied, and gliding drag is experienced 

 by the fish. The relevant initial condition is 



(20) 



(21) 



with the time spent gliding ^, obtained from the 

 fact that at /2 the speed is back to U, . After some 

 reshuffling 



599 



