Slender Body Theory for an Oscillating Ship 

 The first order term in tpg is well known, see [5]: 



(Po = -2 r {lny(77i-f)2+(^i-0' - lnx/(7^i-f)2+(^j+0' 



■F(^j,Od^. (2.20) 



■^c ( ^ , ) 



In the next sections the corresponding term in cpj and fpj will be derived 

 for the two cases (2.13) and (2.14). 



THE CASE OF LOW FROUDE NUMBER AND HIGH FREQUENCY 



First the potential ^^ fo^ ^^^^ case will be considered. From that result 

 the final form for cp^ can easily be obtained. The Greens' function G2 is trans- 

 formed into a slightly different form by separating the poles and introducing a 

 new variable. 



\ 



2i f'' de C I I 1 \ q(^i + 0+^(^l-^)^°^/^ / , rx ■ n\A 



±— ^^^^____ I - 1 q e ^ cos jq(77 - f ) sin c^ldq 



^^ \ V47COse-l \ \q-qi q - q J 



-^] ^^"tJ ^ cos{q2(77^-f)qsin6^) — 



\/l-47 



cos 



q, d^ 



2 r qdq f q4( ^i+Oq "f^C^i"^)'' =°^ ^ r , c. ■ r,\ ' 



^£ J, q-ij \'^^ 'y '^ Wi + 47cos^ 



Lj 



■y/l^ycos 5 



7 



Lf qdq r"" q3(?j + Oq-|-q3(fr^)qcos5 r iq3(7,j-f)qsin5 -iq3(77j-f)q sin^l ^ 



q,d^ 



^^ Jl '^" ■'0 yi+4ycos5 



(3.1) 

 where Lj and L^ are defined by 



' q=l 



By one time partial integration with respect to ^ after changing the order of 

 integration the contribution of the first three integrals to the potential (p^ be- 

 comes, if e tends to zero: 



^jJp Jp J "" i cos ^ V47 cos t? -1 J U-qi q - ^2 / 



s b 



q(^l + iq(fi-f) cos9 

 X e cos (qC"??! - f) sin Uj dq 



(3.2) (Cont.) 

 173 



