Wu 



The time average of the quadratic quantities T, P, and w, can be readily 

 obtained by substituting the solution (33), (37) into (13) giving 



w = -pu|x„ + \J {J - (?2 + §2)} , ^,.,^.. ,_^,.^,^ ,^ ..,. (39) 



-pUV Re{(X, + \j)[je(a)/3* + j(l-e(a))/3*]} , (40) 



where 



9 r h,(x) COS r\6 

 ^n = ^eJ-M-^ ^j^dx, (x=cos5. n=0,l,...)- (41) 



-1 (l-x2) 



Finally, the average thrust T follows from (6), f = (p- w)/u. The result (39) 

 shows that w > 0, since it is known from C{o) = ff + jg that J > (?^ + §^ for 

 a > and the equality holds only if cr = 0. Thus, w > in general, w = 

 only when o- = or ^q + \j =0. The first special case a = is the trivial 

 steady motion, whereas the second case corresponds to the condition that the 

 circulation around the plate remains zero for all t and hence no trailing vor- 

 tex sheet is shed from the body, since the strength of the vortex sheet at the 

 trailing edge is 



7(1, t) 3 -77 Re ^ eJ("t-^)l . ^42) 



When no vortex is shed, k^ + x.^ = 0, it is seen from (39) and (40) that w, P, 

 and hence T all vanish, even though the plate may still be waving. For any 

 other unsteady motion (a > 0) we must therefore have the inequality (14). 

 When T is positive, we may define the hydrodynamic efficiency as 



V = uf/P = 1 - W/P . (43) 



The principal features of the solution may be seen from the following spe- 

 cific example: 



h(x,t) = — (x+1) cos (kx- cot) (|x| <1) . /^^\ 



The thrust coefficient C-j. = T/[(i/4)np\j^] is plotted versus the reduced fre- 

 quency a = oj(./2V {l being the chord) for k = (1/2) k£ = 77 in Fig. 2, in which 

 the experimental results of Kelly (1961) are also shown for comparison (these 

 data include the skin-friction drag). The theoretical result shows that c^- is 

 positive for a > k , or when the wave velocity 



c = w/k = (aA)U (45) 



is greater than the swimming speed u, and C^ is negative for < c < u, or 

 a < K . This qualitative feature has already been predicted earlier. 



1184 



