Wu 

 The corresponding optimum value of c^ is 



Cf = \CT(3°)*?(CT)3° ■ (65) 



Upon using the constraints 



(65) reduces to 



a(3°)t?(cr)3° = P , (66) 



Ct° = >^ Po ' (67) 



which signifies this \ as the maximum hydrodynamic efficiency under these 

 conditions. Consequently, C^ is maximized when 



^"' = >^;in[^2-l(a)?(a)] , (68) 



the minimal element of the set of all positive eigenvalues of [a2~i?] for a given 

 a > 0, whose corresponding eigenvector 3^ satisfy (3^)*?^^ > 0. The optimum 

 solution /3° is an eigenvector corresponding to ^'^■^[o'T^9\ satisfying condition 

 (67). 



Numerical calculation of the result has been carried out by Wang (1966) for 

 the simplest case of a flat plate with average power limitation constraint (61a) 

 for the motion having /3 = (/3^, f3[, /3l'), which corresponds to a rigid plate in 

 plunging and pitching oscillations. The numerical results of the optimum effi- 

 ciency for the subspaces: 



(M- 1) Sj = {3:/3; = /3'; = 0} and (M- 2) S = {^:^[ = 0} 

 are shown in Fig. 4. 



SKIN- FRICTION DRAG OF CETACEAN 



Recently a series of hydrodynamic experiments with several specimens of 

 different species of porpoises (Tursiops gilli, Stenella attenuata) have been per- 

 formed by Lang and co-workers (1963, 1966a,b,c) under more carefully con- 

 trolled conditions. The test results with a Pacific bottlenose porpoise (Tursiops 

 gilli) compare closely with highest predictions based upon rigid-body drag cal- 

 culations, the same power output per unit body weight as for athletes, and a pro- 

 pulsive efficiency of 85%. The maximum power output of Stenella attenuata, per 

 unit body weight, was, however, 50% greater than for human athletes; and the 

 measured drag coefficient was approximately the same as that of an equivalent 

 rigid body with a near -turbulent boundary layer. Thus, in general, no unusual 

 hydrodynamic or physiological performance was observed. Also, it has been 

 pointed out that Gray's paradox can be largely resolved by consideration of 

 duration; Gray's analysis was based on the power output of humans for a 15 

 minute period and this figure can be raised several times if based on a shorter 

 period, such as a few seconds. 



1190 



