Linearized Potential Flow Theory for ACVs in a Seaway 



We may now use (V. 18) in (V. 14) to derive an integral representation 

 for Af , but before doing so the integrand in the integral over S Q 

 in (V. 14) will be re-written by using the expansion for p 



ifft 



P S U*)= <*V<M ) + /?ae (x 001 + VoOl* P o 



i. e. 



Vp - iap = /3Vp +0a(x +h ) (Vp - icrp 



c s t s o> ^ 001 G 001 ott Ow 



so that the integral over S Q becomes 



// 



/3Vp G +/3«(x + h 6 ) I 2Vp G+p (Vd_ - iff) g( d^dl 



Ov 001 G 001 J o>t °t ^7 | 



We therefore have finally, 



Sh 



tf(x,y, z) = 



rj- 



1/ 



ar 



ah 



G(x, y, z; £', b, T ) + G(x, y, z; £,' -b, T ) 



■±Mfc\ 



i-lr""* v lr" c b* -b>! 



)] 



b a r,' 



+a< r 

 .b i 00 





a G 



dtf'|b o^'oV 



+ r Am ( 



S 2 G 



a 2 G 



001 SV3f L s i' s f 



'b 



— ) 



-b 



| de'dr' 



dG(x, y, z; ?',ij',o ) dG(x, y, z; I','?', o ) 



--fcjfa.. [*] [- 



L 77 = b i?:b 



1 m r ' ro 



+ ^^Jjr VP o. G + ^>001 +h G/001 ) 



av 



d*' 



17= -b 



'2Vp G + p (V-°— 

 I o {{ o^ a{ 



- iff ) G> 



d^dT? + 



239 



