1 2 

 - V 



2 --i 



(t) 



1 LA. 



2 1 3a' 



1 3 



2 Ba' 



A Vortex Theory for the Maneuvering Ship 



2 -, 



k =0 ™ ' 



2 



y (7) 



^ E ^Ok^-^e) fok(t') 



1 2 

 — V , 



2 "-' 



BCT 



7 *ooC"^e) + Vgi; 



3 

 3a' 



Since (b) is negligible, we obtain 



Pi(m^,t') 



3t 



E "^OkC^e) fok(t') 



L k = o 



2 



+ i Ve (m^,t') - ^ v'.Cm^.t'), 



1 , , , M 



- p (m' t ) = ^ r^ 



3a 



7 %o(^e) + VgCm^.t') i;(m^) 



3 



3cr' 



^ (8) 



'J'ooC^e) + E V'^e-t') 



In par. 5.4, we give the expression of the set of forces (?.) due to p. . Now 

 we consider the set of forces ?' due to p'(m^,t'). 



5.2. The Set of Absolute Forces Due to the Wake 



Let us assume, for instance, that we are in the case "bj" of par. 4, when 

 (u/U)^, , (Lq/U)^, are identically null, while (w/U)^, is, for t' >0, an arbitrarily 

 given function of t ' . 



According to a result of par. 4, the density of the normal doublets on s due 



to w,(m , t ') is: 



1^ e' ' 

 7l(m,t') = (^)^^ 7t(m,0+) + ( j)^ ^ Byl(rn,f) + j ^ (j)^^ S,.*(m, t'-r')dr' (9) 



when (w/U)^, is continuous for t'>0. 



The pressure pj(mg, t ') due to ¥^(mg, t ') = -yjCm, t ') is given by 



847 



