Fluid Mechanics of Swimming Propulsion 



and assume that its inverse function t = t(T) is unique, so that u = u(t(T)) is 

 a one- valued function of t. Regarding w and f as functions of z and t , (17) 

 becomes 



3F Bw Bw 

 Bz Br Bz 



+ — . - : .. : (24) 



where 



F(z,T) = f(z,T)/U(T) = <I)(x,y,T) + iW(x,y,T) . (25) 



Application of the Laplace transform 



F(z,s) 



j e"^"" F(z,t) dr (Res>0) (26) 







to (24), under zero initial conditions, yields 



dF / d 



- + s I w . 

 dz \ dz 



Integrating this equation from z = -«, using conditions (22), and expressing F 

 in terms of w, and vice versa, we obtain its imaginary part at y = as 



$(x,0^,s) = -v(x,0^,s) - s J v(xj,0^,s) dxj , (27) 



or 



X 



v(x,Oi,s) = -W(x,Oi,s) - s J e^^''*"''^$(Xj,Oi,s) dx^ 



(28) 



for all X. On the plate, v(x,0±,s) = v(x,s), which is the Laplace transform of 

 (18), we have 



?(x,o^,s) = $j(x,s) + AqCs) , (|x|<l) (29a) 



where 



X 



?j(x,s) = - (^+ sj J V(Xj,s) dxj (|x|<l) (29b) 



and 



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