Current Progress in the Slender Body Theory 



where the hull velocity transfer functions gj(oj) are given by 



igi + ig2 + HSa = ~^^?3 - 3- Y'P^o) 



(A2.4) 



+ Jg5 + kg6 = rx(igj+j^g2+!5g3)~D><'^< 



(0) 



No slenderness assumptions have been made up to this point and the boundary 

 condition (A2.3) is still valid for an arbitrary rigid body making arbitrary small 

 motions in an arbitrary steady field 0/ ) • 



Now if the ship is slender n lies nearly in cross-sectional planes (or, more 

 accurately, from (A1.9) we see that n^ = 0(e) 02 = 0(e) 03) so that the gj may 

 be consistently approximated in the form 



3 ^ B \ ,(2D) 



'2 3^+ "3 B^j'^CO), 



3 B \ ,( 2D) 



'2 37+ "3 Tz) ^(0), 



= yg,- zg,-(n2^+ 03 — j 0^0) 



(A2.5) 



gs = -xgg- Un3 

 gg = xg2 + Un2 



all with error a factor 1 + ( e log e) which is mainly due to the replacement of 

 g!)(o) by its slender body approximation Ux + cp^^^^ + f^*^^^'' + f^o)"* from (3.6) 

 and (3.7); of these terms only 0[o)^ contributes to the gj . Notice mat in the 

 expression for gj we have used the slender body approximation to the boundary 

 condition for 0( ) 5 namely 



'2 37+ "3 Blj^(O) = -Uni- (A2.6) 



Notice also that the surge and roll velocity transfer functions gj and g4 are 

 smaller by a factor 0(e) than the other gj , since a slender body is an inefficient 

 exciter of motion in these modes. 



*Recent investigation has shown that Eq. (AZ.6) cannot legitimately be used to 

 simplify gj. The resulting values for the fluxes Qi, O4 in these modes are un- 

 changed, although the given derivation for Oj is no longer valid. 



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