Fluid Mechanics of Swimming Propulsion 



waving plate has been treated by Wang (1966), who adopted a discretized Fourier 

 representation of the body motion. For the simple harmonic motion given by 



h(x,t) = Re [hj(x) eJ"*] , (Ixj <1) ,,....-• ,. .^../.j :■• (57) 



let hj be represented by an (N + l)-term Fourier series 



N ^^ : 



hj(x) = - /Sq + Z] ^n ^°^ "^ • ^^ ^ *^°^ ^^ ^^^^ 



n= 1 



where /3^ is given by (41) with the time factor deleted. Then the coefficients 



Kq and \j can be expressed in terms of /S^'s. Let /3^ = /3^ + j /3^', /3^ and 

 /3^" being both real, and we define the vector 



3= (/j;,/3^,/3;,/3';, ..., /3^,/3;i) , ^^ (59) 



in which /S^' may always be set equal to zero as the reference phase. Then the 

 thrust and power coefficient can be written as ^ . ...... 



0^=^*23, Cp = cr^ty^ (60) 



where 3* is the transpose of 3, and 9 and f are (2n + 1) x (2n + 1) symmetric 

 real matrices. 2 is nonsingular and has real eigenvalues of both signs for 

 N > 1 and for all a > 0, implying that the origin 3 = is a saddle point of C^.. 

 Also, y has eigenvalues of both signs for all a > 0. ...--■...■■■ 



We consider the problem of maximizing C^p, which is required to be posi- 

 tive, under one of the two constraints 



(C-l): Cp < Pq . (61a) 



or • .-" ..' .li/'r ■" ■ . ■'•■• ■'■■ -■• " ~i'-'^ ■■■ 



(C- 2) : Cp(t) < Pq , (0<t<tj) (61b) 



where P^ and Pg are specified positive constants. This constrained optimiza- 

 tion problem is equivalent to that of maximizing a new function 



C* = Cj. - \(c73*y'3-Po) . (62) 



where \ is a Lagrange multiplier. Setting the derivatives c^ with respect to 

 all components of 3 to zero yields 



2(a) 3= \^y(a)3 . (63) 



Let 3° denote the optimum solution; then, since 2 is nonsingular, 3° satisfies 



a2-»(cT)!P(a) 1° ^ }v-'^l'' . (64) 



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