A Vortex Theory for the Maneuvering Ship 



Substituting in Eq. (16), we get: 



dr;' 



1 C"^ r 



[ ' TE ap^P^'^P^^'p*^'^'^] "^^ 



Putting 



Vp(T?')dT7' , 



we obtain 



E V^p^ [E K(^) Kp] f;(-) =^ JE CpUp(i7^) 



p L n -I P 



or 



V^> = s 



^EV^) b„^ 



Therefore 



F*(T?',t') = E V rT,') F*(t') 



is known. 



Of course, for t ' = +co, the motion is steady, and consequently 



1 = F*(T]',+00) = ^ Vp(T,') . 

 P 



This equation must hold for any value of 77' in the range Vp' <v' "^ Vp • 



Obviously there is a contradiction unless F*(T]',t') is independent of v' 

 That leads to 



r\(v',f) = r^^(v') F*(t') 



Then, Eq. (16) gives 



839 



