Slender Body Theory for an Oscillating Ship 



After a rotation of coordinates this becomes 



i>(^,0) = B(^^,e^) + (^-^^)^ P(^^,6^) - (0-e^)^ Q(^^,e^) + . 



with 



and 



P(^o-^o) = I- 



D.. + D,, + ^(D.. - D,,)2 +4D 



'iS " ^e 



ff ^et 



Q(^o'^o) 



D^A + D.« - 7(0.. - D..)' + 



^f "6'< 



4D 



^e 



(3.7) 



The first order term in (3.5) originates from the neighbourhood of the sta- 

 tionary point and therefore if (^^,9^) is an interior point, this term becomes: 





qB(q,^o)e^''^^°''°^dq f-"4p(f-f„)2 "- ^ 



q- 1 



i./"'^'^-^"' -I 



dcf e 



Q(^-^n)' 



de 



r r B(q,5„) e' ' °'^°^ qdq 

 = ^e F(^o,OdM "' , , ifP>0, Q>0 



(q-l)v^ 



Inserting for D(^,5) the expression (3.6) the result becomes 



cos 5j, = 0(e) , D^^ = 0(e) , D^^ = Q ^b + 0(^^) - 



^0 = ^1 + 0(6), Dg^ = 0(e), Ti(^,,e,) = ±q^3e(77j-f) 



With (3.7), (3.8) and (3.9), for (3.5) the result 



±i(vi-f)iB<i dq 



r F(<fi,odc r 



3 = ^ F(<fi,OdC B(q,5o) e" 



q- 1 



is obtained. 



From (3.1) and (3.10) the first order term of cpj follows easily: 



(3.8) 



(3.9) 



(3.10) 



175 



