Wu 



momentum (pVA)u at the tail; w is equal to the shedding of the average kinetic 

 energy (pAvy2) at the rate u. Finally, from (6) it follows that the average 

 thrust is 



Obviously, w > 0, as should be expected from the general argument stated 

 before. When T is positive, the hydrodynamic efficiency of swimming is de- 

 fined, as before, by (43). 



Lighthill reasoned that high efficiency can be achieved if v << h^, but V 

 and h^mustbe positively correlated, i.e., h^v > (for otherwise, P < 0, 

 hence T is also negative). Furthermore, there are two side conditions that 

 the inertial lift and moment of the body (with known mass distribution) must 

 balance, respectively, the hydrodynamic lift and moment in order to free the 

 body from any recoil. 



Two specific examples have been given by Lighthill. When the body mo- 

 tion is a standing wave, the efficiency is always <0.5. K the body motion is a 

 progressive wave, . 1" 



h(x,t) = b(x) cos a)(t-x/c) ; - • ; -• (54) 



then 



T= ipACO 

 4 



a;2b2(l-U2b'(0^) 



P ^ - /jUA(e)a)2b2(l-U/c) 



(55) 



(56) 



An estimate shows that -q can be as high as 0.9 at c = 1.25u provided that the 

 slope of the amplitude profile is negligible at the tail. We observe that T can- 

 not be positive unless c > u , which is a general feature as expected. 



It should be noted that for this category of slender-body motion, it is es- 

 sential that the fish must have a tail edge structure so that A(P) > 0, since T, 

 P , and w are all proportional to A{l). In reality, however, typical body shapes 

 of fishes, aside from being slender, usually are rather planar and have side 

 edges that may be regarded as sharp. In such cases, the vortex sheet shed 

 from the sharp trailing edges will considerably modify the flow field, so that 

 the thrust and energy balance will no longer depend only on the flow at the tail 

 section x = ? . 



OPTIMUM SHAPE OF WAVING PLATE 



An interesting problem concerning swimming propulsion is to find the 

 optimum shape of the body motion. The special case of the two-dimensional 



1188 



