Current Progress in the Slender Body Theory 



Here K^ is a modified Bessel function of the third kind and gives the source be- 

 havior of 0( 1 ) , while the integral with respect to x. is the correction required 

 to satisfy the free surface condition (A3.1). 



Now the interaction term in the potential near the ship is found by investi- 

 gating the source potential near the line of sources, i.e., for y^ + z^ small, in 

 which case 



(1) 



Q* 



7T ^( 1) 



log yy2+ z2 + log 4-C|k| 



+ ^(-ic^-ikU) 



dk 



y/k^TJJ ig^/k^Tk^ + (-ia;-ikU)2) 



, (A3. 3) 



(log c = 7 = 0.577 



the second to f J J^^^^ and the third to f j ^^ ^ The last is the qviantity of interest 



here, and the \ integral involved can be integrated explicitly to give 



)77 . . .). The first term of (A3. 3) corresponds to </>J^°) in (3.13) 

 i f*''^^ and the third to f J^^\ The last is the quantity of interei 

 ;an be integrated expli( 



^Q*i) /3(k) coth/3(k) , 



(FS)* 

 ( 1) 



where /3(k) is defined by 



cosh /3(k) 



(-ioj- ikU)' 



(A3.4) 



(A3. 5) 



with |lm/3| < 77. The last condition appears to fail for co real, since then cosh /3 

 is real and negative. The correct interpretation is, however, obtained by taking 

 -ia' to have a small positive real part (corresponding to decaying transients) 

 which we may then let tend to zero, giving 



i(7T-a(k)) , 



if cos a(k) = ^^^^^ ^^ < 1 , < a(k) < ^ , 



/3(k) = 



177- a,(k) sgn (co + kU) , if cosh a(k) 



g|k| 



( oj + kU) 2 

 g|k| 



(A3. 6) 



> 1 , < a(k) < 00 . 



The inverse Fourier transform of (A3. 4) may be taken by use of the convolution 

 theorem, giving 



.(FS) 

 C 1), 



d^ K^i^(x-0 Q(i)^^) ■ 



where 



221-249 O - 66 - 12 



161 



