Wu 



as the tail moves downward, to a positive maximum (in clockwise sense) when 

 the tail is at the lowest position. The velocity field due to this vortex system 

 is clearly in the form of a jet moving in the positive x direction, as depicted 

 in Fig. 3. By the principle of action and reaction, the plate therefore experi- 

 ences a positive thrust. For the same reason, the thrust is negative if c < u, 

 since the shed vorticity is reversed in sense. In the case of a self-propelling 

 body, however, the backward momentum due to inertial forces and the forward 

 momentum due to the skin friction exactly balance in steady swimming. 



Fig. 3 - Jet moving in the positive x direction 



The effect of body thickness in sinusoidal motion has been discussed by 

 Uldrick and Siekmann (1964). 



Starting Stage of a Forward Swim 



A typical starting motion has been considered by Wu (1962), with the plate 

 starting with a constant acceleration from at rest, 



U(t) = at , (a > 0) 



and with h(x,t) assuming a polynomial of degree 3 in x. The small time be- 

 havior of the solution has been evaluated with the assumption of small lift and 

 moment, in order to minimize the body recoil in lateral and spinning motions. 

 The result shows that the thrust is generated at the time of order t^, whereas 

 the power is already required at the time of 0(t), the initial power being posi- 

 tive definite for arbitrary transverse motion h(x,t). When a high efficiency is 

 required in addition, the body profile appears in an S shape, with a maximum 

 and minimum of h at x = -0.564 and x = 0.295 approximately. 



SWIMMING OF SLENDER FISH 



Lighthill (1960) treated the problem of swimming of slender fish, at suf- 

 ficiently large Reynolds number, by applying an inviscid slender -body 



1186 



