A Vortex Theory for the Maneuvering Ship 



Finally, we obtain 



1 



2W 



s 



...LI 



Z^ COS wt = 2 ^-'^'^ ] AL ^'^ '^ ^3^^ '*' ^ 6)L 



^hj ^°^ (^^ + 2 



^ ^B ^iB ( u) I /f^'L\^ ^0 



1-^(0) - f(^ 



1 -^ 



A "L, 



1 - f 1'^'^ 



^ ^IB ' 1bI"u" 



where 



> (11) 



WL / ^ , 77 \ 



-- COS ..t+- , 



00 



^jgCr') cos {^^ r') dr', giB(^) = J ^IB^-^') ^i" (^ ^') ^^ ' (12) 



are the cosine and sine Fourier transforms of the derivative (p^^(t'). 

 A quite similar reasoning leads to: 



C I /<^L 



1 .,,2J2W , ^ 1 \ U / I (c^L\ ^0 



X^^ cos cvt = 2^^" lAL ^13 



Xj,^ cos (a;t + ^) = lpAU2 ^- a 



T I 1 II ) -T- cos a;t , 



wL \ U / L 



> (13) 



^(0) + f; (f 



wL / ^ , 7T 



_ _ COS a,t + - 



where 



CO CO 



;(f) =1 ^(x')cos(^r')d.', g;(f)=J 0Cr') s.n(f r')dr'. (14) 



Similarly, we obtain 



\ cos a;t = ^pALUM^ f^as + Z^G^ " ^ 



2W 



a;L/U 



^ -rf?"T 



giB(c.'L/U)l /^,L\- ^0 



A l^B^Lg ^mgj "IB cJL/U J\U/ L ^"^ "^- (Cont.) 



(15) 



