A Vortex Theory for the Maneuvering Ship 

 = r = , for -00 < t ' < +co , 



(U + u)ij^ + viy + wi. 



constant , (•^) = for t' <0 , = fooCt') for t' >0 , 



(1) 



(^\ ^ =0 for t' < , = foiCt') for t' > , 

 (^) ^ = for t ' < , :^ f Q 2( t ' ) for t ' > ; 



^00 » ^0 u ^0 2 ^^® given functions of the reduced time t ' ; their squares and 

 products are negligible. 



The velocity potential of the absolute motion is: 



0(M,t') = %,m + Yl %\.(^^ fokCt') + 



^>^^(U) + J^ \(M,t') 



(2) 



In order to study the structure of the wake, we assume firstly that 



foo(t') = , fo2('t') - for t' > . 



Let us consider the bound and free vortices which generate W^CM, t '). 



At time t ' >0 , a vortex generated in a small interval (t' , r ' + dr'), with 

 0<t' <r' + dr' < t' , is for instance, made of two filament vortices: one of them 

 is of intensity dpd^, r[(F,T'), and lies on a closed contour made of an arc P'm^,P 

 on s', and on a u -shaped arc PP^,P^,,P', where P^,P^/ is deduced from the 

 segment PP' by a translation nearly equal to -i^(t ' - r'); the other one is of 

 intensity dpd^, TjCP.r'), and lies on the arc Pm^,P' on S" and on the arc P'P^,P^,P 

 defined above. 



Consequently, the total intensity on the arc PP^,P^,P' is equal to 



dpd^,rj(P,T') = dpd^,[r;(P,T') - r^(P,T')] . 



As explained on Fig. 6, the first filament vortex is equivalent to a set of 

 three vortices (a), (b), (c) of the same intensity dpd^,rj(P,T'). Vortex (a) is 

 lying on P'm^,p and on the segment PP'; vortex (b) is lying on the arc PKP' and 

 on the segment P'P; vortex (c) is lying on the arc P'KP and on the arc PP^,P^,P'. 



Vortex (a) is equivalent to a distribution of normal doublets on the part 

 S'(P,t') of s' behind the arc P'm^,P, and on the part ^^(P) behind the segment 

 PP' of the surface 1^ generated by PP' when P and P' describe the (Joj-line. 

 Vortex (b) is equivalent to a distribution of normal doublets on S^cP). Vortex 



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